exchange rate models for india - an appraisal of forecasting performance
TRANSCRIPT
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EXCHANGE RATE MODELS tr'OR INDIAAII APPRAISAL OF FORECASTING PERTORMANCE
Samrat Bhattacharya
Dissertatlon subfiitted to the University of Dethi in partial fullillmentof the requirements for the award sf the degree of
MASTEN OT PHILOSOTTTV
Department of EconomicsDelhi School of Economics
University of DelhiDelhi - I'10 (m7
India
JuIy, 2000
Dr. Paut llut(Supenristx)
OEC 1TOH
Ilr; nL $r*darmflead, D€paffismt of Ssorro,raics
Delhi School of Eoonomics ' .
UnivemityofDelki', .,,:,
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This is to ce,rtiff that this otudy *Exchangs Rate Modets for India : An appaisal offorecasting'Se ance" is based on my original'resemb {ront. My indeb,ted e to
other works/publicatiirns has been duly ackriowldgd lffiin. This study has not bffisuUmiited in part or fuil for ey omrreipU*ra ordegree of,anlr other tmiversity.
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Ackuowledgement
I am grateful to my supervisor Dr.Pami Dua for the enormous patie,lrce aria irrtsest
that she has shown towtrds my dissertatioa. It is only with her hClpfirl guidauce *rat I
could successfully complete thr prese,lrt study.
I arn also &tailkful to my co-sup€rvi$or Dr.Par&a Ssn who helpod rus to bdtei
undsrstand the thesretical foundations of my resesroh wor,k
rout whose help it was not
possible to- conslete the present st$dy.
I axn t}rankful to'Ivk.Amit, Mr.Vinayu,r, ard bfr.Sqis€y for their assisffiroe in ttre
computer lab.
SAMRAT BHATTACHARYA
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TABLE OF CONTENTS
CHAPTER I: Introduction .....1
CHAPTERII: Survey of Literature ..... .........6
Section A. Review of the Theoretical Foundations of the Models of the
Exchange Rate Determination . .. . .:. . .. . ....6
Section B Empirical Survey of Asset Markets Models. ...,:.......:.......16
Section B (I). EstimationResults ..:..... .........16
Section B(II) Out-of-Sample Forecasting Performance.. .....29
CHAPTER III: Econometric Methodology ......42
Section A. Unit Root Tests . ... .. . . ..........42
Section B. Box-Jenkins Methodology.......::.. ......48
Section C. Cointegration Methodology ......50
Section D. Vector Autoregression Methodglogy .....52
CHAPTER IV: Measures of Forecast Evaluation ... .... .. ..61
CHAPTER V : Data Source & Definitions. .. ....73
CHAPTERYI: EmpiricalResults ........:.::..::... ......74
Section A. UnitRootTests. ....:... :..:....:.....:.... ..........75
Section B. Cointegrationlong-Run equilibrium & Vector error Correction
Model ......... ......:...86
Section C. BayesianvAR... ......,......... ........:.....,95
Section. D. ARIMA . .. . :... .....9:9
CHAPTERVII: Conclusion Result. ......124
Bibliography
Appendix
Chapter I
INTRODACTION
The notion of exchange is central to economics. The analysis of
exchange and exchange ratios suggests that there are broadly three kinds of prices .
relative prices, which reflect the exchange of goods for other goods and which exist in
both barter and monetary economies ; money prices, which reflect the exchange of
money for other goods, and exists only in a monetary economy; and thirdly, the
general level of prices, which reflects the average price of all commodities and exists
only in an economy with money. However, there is a fourth kind of price which is of
our interest. This is the price of one money (or medium of exchange) in terms of
another money (or medium of exchange). Hence, rather than exchanging money for
goods or services, money can be exchanged for another money. This price is called
the exchange rate. The exchange rate may be defined as the domestic price of foreign
currency, or as its reciprocal, the foreign price of domestic curency. In this paper, we
employ the former definition.
India followed a pegged exchange rate regime till 1991. The onset of
the economic crisis in l99l brought about a change in the government policy with the
aim of bringing India in line with the world economy. There was a seachange in the
Indian foreign exchange market after the economic liberalisation of 1991. In the pre-
liberalisation era, the basketJinked exchange rate policy regime, with the RBI
performing a market clearing role, provided very limited freedom to the market. The
post-1991 period, however, saw a movement towards a market-determined exchange
rate regime following the recommendations of the High Level Committee on Balance
of Payments (Chairman : Dr.C.Rangarajan,1991). The Report of the Expert Group on
Foreign Exchange Markets in India (Chairman : O.P.Sodhani, 1995) aimed at
integration of domestic foreign exchange market with foreign exchange markets,
more operational freedom to dealing banks and widening and deepening of the
markets. The principle underlying the conduct of the exchange rate policy under the
market based regime is to allow the market forces determine the exchange rate with
the monetary authority ensuring that the exchange rate reflects the fundamentals of
the economy.
In the background of the above discussion, it becomes evident that spot
exchange rate in India is fast becoming a very important market-determined variable.
One needs to better understand the behaviour of the spot exchange rate in the new
open economic environment in India. There is now more need to produce forecasts of
the exchange rate as it affects the economic agents in a far greater way than it used to
do a decade ago. Business houses, in this new environment, need to have 'good'
forecast of the exchange rate so that they can take adequate measures to minimise the
exchange rate related risks. Government also needs a 'good' forecast of exchange rate.
This is more so for an underdeveloped country like India where imports are a major
component of trade. Any major fluctuation in the exchange rate could affect import
(as well as exports) adversely leading to a deteriorating trade balance. This is more so
for the commodities like crude oil. So it will be advantageous for the government to
have 'reliable' and 'good' forecast of the exchange rate so that they can hedge to avoid
any adverse implications. Moreover, given the poor performance of the exchange rate
models, it becomes challenging to an academician to model the exchange rate
dynamics such a way so as to generate 'useful' forecasts.
The most challenging question that a forecaster faces is that " Is
Random Walk the best forecasting model ?". This question has haunted forecasters
since the seminal work of Meese and Rogoff(1983). Lot of research work has got into
this but without much sucsess. The present study is another attempt in this direction.
Here an attempt has been made to model the exchange rate dynamics iir a way so as to
generate accurate, rational and efficient forecast. The present study is the culmination
of the two earlier research work by the same author. Bhattacharya (199S) attempted to
model exchange rate by Box-Jenkins methodology, while another study (1999) tested
various competing models of exchange rate determination. In both these papers,
random walk turned out to be a better performer over other competing models, barring
the univariate ARIMA models, in terms of out-of-sample forecast performance. The
present study attempts to extend the earlier work and attempt to model the spot
exchange rate to generate reliable forecasts.
OBJECTIVES :
We lay out some specific objectives of the proposed study.
Most important and the root of all the heated discussion lies the argument that the
exchange rate follows a random walk. So it becomes utmost important to check
whether exchange rate follows a random walk or not. We work with the US-India
spot exchange rate. We propose to use the traditional unit root tests like
(Augmented Dickey-Fuller) ADF and Phillips-Perron. However, given the
drawback of these traditional and most popular test we intend to employ other
tests for unit root, like KPSS (Kwiatkowski, Phillips, Schmidtand Shin (Tgg2))
test, Bayesian unit root test. For a detailed discussion of these tests the readers are
requested to refer to chapter IfD.
To test the various competing models of exchange rate determination based on
economic fundamental namely moneta"ry models of exchange rate determination
which includes flexible-price monetary model, Dornbusch's sticky price monetary
model, Frankel's real interest differential model and Hooper-Morton model. These
models are the most frequently tested models of exchange rate determination
based on fundamentals of the economy. (For a detailed discussion on these
competing models refer chapter II).
The structural models impose ad-hoc restrictions on the coefficients of the
estimation models. To avoid any arbitrary restriction on the data generating
process, one moves into the realm of atheoretical modelling like Vector
Autoregressive Regression (VAR). The concept of Cointegration helps us to test
the monetary models in a cointegrating framework and leads to the estimation of
vector Error Correction Models (VECM) based on the long-run cointegrating
relationship for generating out-of- sample forecasts.
Economic forecasting involves not only data and statistical/econometric models,
but also the forecaster's personal belief about how the economy behaves and
where it is heading at any moment. So a task of forecaster, in practice, is to blend
data and personal belief according to a subjective procedure. Bayesian Vector
Autoregressions (BVAR) model attempts to blend forecaster's subjective belief
and data in a scientific way (refer chapter III).
All the competing models have been ranked on the basis of their forecasting
performance. A battery of forecast evaluation measures has been carried out to
assess the quality of the forecasts generated from the competing models. We use
the time series properties of the actual and predicted series to evaluate the
forecasting performance of the competing models.(All these tests are discussed in
Chapter IV).
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The present study is organised as follorvs : Chapter II reviews the
literatufe regading the theoretical foundations of the monetary models, their
ernpirical validation and out-of-sample forecasting performance ; Chapter III outlines
the econometric methodology used in the present work; Chapter IV describes the
various forecast Evaluation measures employed in tho pr6$ent study; Chapter V grves
the data source and definition of the variables used in this study; Chapter VI provides
a detailed aualysis of the empirical recults obtained and finally Chaptor YII coastudes
the study, outlining few limitations of the present study.
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Chapter II
SURWY OF LITERATURE
This chapter is organised as follows. Section I gives a brief theoretical
exposition to the various theories of the exchange rate determination. We discuss the
flexible-price monetary model, Dornbusch's sticky price formulation, Frankel's real
interest differential model, and Hooper-Morton model. We have also briefly described
the basics of the portfolio balance model of exchange rate determination. In Section II
the empirical results pertaining to the asset market models of exchange rate
determination have been outlined with Section II.A. giving an account of the
estimation results of the asset market models of exchange rate determination with a
focus on the monetary models. Section II.B briefly outlines the out-of-sample
forecasting performance of the asset market models of exchange rate determination.
A. Asset Market Models : A Theoreticul Exposition
Prior to 1970s the models of exchange rate determination were based
on relative price levels and trade flows. Trade elasticities were thought to underlie the
supply and demand curves in the foreign exchange market. Since the exchange rate
began to float in the 70s, their fluctuations have resembled those of asset market
prices. Rather than following the movement of relative price levels, exchange rate
movements seem to be dominated by monetary conditions. The theoretical literature
has correspondingly turned to the asset market models of exchange rate
determination. The theoretical assumption that all asset-market models share is the
absence of substantial transaction costs, capital control, or other impediments to the
flow of capital between countries. However, beyond this common point, the asset
market models diverge in a number of different and complex routes.
In one class of asset-market models, domestic and foreign bonds are
perfect substitutes. Portfolio shares are infinitely sensitive to expected rates of return.
The uncovered interest parity must hold. However, then bond supplies become
irrelevant, and the exchange rate is determined in the money market. Such models
belong to the 'monetary approach' of exchange rate determination, which focuses on
the demand and supply of money. Within the monetary approach, however, there are
two different models.
In the first type of models I Frenkel (1976),Bilson (1978) ] it is
assumed that prices are perfectly flexible. Consequently, changes in the nominal
interest rate reflect changes in the expected inflation rate. The relative increase in the
domestic interest rate compared to the foreign interest rate implies that the domestic
currency is expected to loose value through inflation and depreciation. Demand for
the domestic currency falls relative to the foreign currency, which causes it to
depreciate instantly. So, we get a positive relationship between exchange rate and
nominal interest differential.
The second strand of models I Dornbusch (1976) ] assumes that prices
are sticky, at least in the short-run. Consequently, changes in the nominal interest rate
reflect changes in the tightness of monetary policy. When the domestic interest rate
rises relative to the foreign rate it is because there has been a contraction in the
domestic money supply relative to the domestic money demand without a matching
fall in prices. The higher interest rate at home attracts a capital inflow, which causes
the domestic currency to appreciate instantly. So, we gd a negative relationship
between the exchange rate and the nominal interest differential. This model is also
coined as the 'overshooting model' as the domestic currency appreciate
instantaneously more than it will in the long-run.
In the other class of the asset-market models, domestic and foreign
bonds are imperfect substitutes. This is the 'Portfolio-Balance approach' of exchange
rate determination, in which asset holders wish to allocate their portfolios in shares
that are well-defined functions of expected rates of return. According to the portfolio-
balance model, the relative quantities of the various assets and of the rate of
accumulation ofthese assets exert profound first-order effects on the exchange rate.
(A) THE FLDilBLE-PNCE MONETARY APPROACH :
Apart from the assumption that uncovered interest parity holds
continuously, the flexible-price monetary model relies on the twin assumption of
continuous purchasing power parity (PPP) and the existence of stable money demand
functions for the domestic and foreign economies.
Monetary equilibrium in the domestic and foreign country, respectively, are given by
and
7n= p+Q y-)"i
m* =p*+0y*-)"i*
(1)
(2)
where m: log of the money supply
p : log of the price level.
y: log of the domestic real income
/ : money demand elasticity with respect to income.
)" : money demand semi-elasticity with respect to interest rate.
( x denotes the foreign variable)
explicitly as being fundamentally relevant for understanding the evolution of the
exchange rate. Rather, the relevant concepts relate to three groups ofvariables : first
are those which are determined by the monetary authorities, second are those which
affect demands for domestic and foreign monies, and third are those which affect the
relative price structures.
The flexible price model has been criticised for its assumptions of
continuous PPP. Under continuous PPP, the real exchange rate,i.e., the exchange rate
adjusted for differences in price levels cannot vary. But, one of the prime
characteristics of the floating exchange rate regime has been the wide gyrations in the
real exchange rates between many of the major currencies. This led to the second
generation of monetary models pioneered by Dornbusch (1976)'
(B) Dornbusch's' Overshoofins' Monetarv Model :
In this model PPP does hold in the long-run, so that a given increase tn
the money supply raises the exchange rate proportionately as in the monetarist model,
but only in the long-run. However, in the short run, because of sticky prices, a
monetary expansion leads to a fall in the interest rate. This leads to capital outflow,
causing the exchange rate to depreciate instantaneously to give rise to the anticipation
of appreciation at just suflicient a rate to offset the reduced domestic interest rate, so
that the uncovered interest rate parity hold. This model thus explains the paradox that
countries with relative high interest rates tend to have currencies whose exchange rate
is expected to depreciate. However, the above analysis is done under the assumption
of full employment so that the real output is fixed. If output, on the contrary, responds
to aggregate demand, the exchange rate and interest rate changes will be dampened.
11
From (l) and (2),we get
The PPP condition is
(m -m *) = (p - p *)+0 0 - y x1 * l(i -i*) (3)
s=p-p*
Using the PPP condition we get the formulation of the monetary model
which has been extensively used in the empirical literature :
s = (m - m*) - Q0-yx) +)"(i-i*) (s)
This says that an increase in the domestic money stock, relative to the foreign money
stock, will lead to a rise in s, that is, in a fall in the value of the domestic currency in
terms of foreign currency (depreciation). An increase in domestic output appreciates
the domestic currency, i.e. a fall in s (/>0). This is because an increase in domestic
real income creates an excess demand for the domestic money stock. As agents try to
increase their (real) money balances, they reduce expenditure and prices fall until
money market equilibrium is achieved. As prices fall, PPP ensures an appreciation of
the domestic currency in terms of foreign culrency. Similarly, a rise in the domestic
interest rate reduces the demand for money and prices increase to maintain the money
market equilibrium and via PPP leads to a depreciation of the domestic culrency
(2>0).
There is another alternative but equivalent way of formulating the flexible
price model by imposing the uncovered interest parity condition on equation (5).
Uncovered interest rate parity implies that
(4)
(6)i-ix = E(As)
Z(As) : the expected depreciation of domestic currency'
s: log ofthe spot exchange rate.
where,
Combining (5) and (6), we get
s - (m - m*)- 0 0 - y*)+zE(As)
If the expectations are assumed to be rational, then by iterating forward, we can obtain
the following' forward looking' solution
s, = (t+2)'Zr|}7'l(*-*.)" ,*,td!",*i*{* y,**" I (8)
The superscript " e " stands for the expectations which are conditioned on information
set at time t. From equation (8) it gets clear that the monetary model, with rational
expectations, involves solving for the expected future path of the " driving variables "
- that is, relative money supply and income.
The assumption that the prices relevant for money market equilibrium
are the same as those relevant for the PPP can be relaxed by allowing for the price
level to be a weighted average of the prices of non-tradable goods and internationally
traded goods I Frenkel and Mussa (1985)]:
p=o px+(1-o) pr
p*=on p** +(1-o*)p*,
where , p, and pt denote the logarithm of the prices of non-tradable and tradable
goods, and o denotes the weight of non-tradable goods in the price index.
PPP holds only for tradable goods, so that
pr=S*p*r
This gives us the monetary model equation of the form
s=(m-m*)+00 - y*)+ )"(i-i *)+of(pr - pr>(P* , * p* *))
[assuming o=o*)
The monetary approach differs from the elasticities approach to
exchange rate determination in that concepts like exports, imports do not appear
(7)
10
Although the exchange rate will still depreciate, it may no longer overshoot, and the
interest rate may actuallY rise.
The overshooting model retains the money demand function and
uncovered interest pality condition of the flexible-price monetary models' Thus,
m= p+g y- )"i (l)
m* =p*+0 y*-Ai* Q)
i-i4= E(As) (3)
However, it replaces the instantaneous PPP condition with a'long run version .
s=p-P
In the short ruq when the exchange rate deviates from its equilibrium path, it is
expected to close the gap with a speed of adjustment 0 :
E(As) = -0(s-s)
The equation of exchange rate determination is given by
s - (m *m *)- Q 0 - Y*)*(1/AX i -i *)
s-(m-m*)-\0-y*)+y (i-i*) lr= -]la
The sign of the coefiicient of the relative income term is same as that of the flexible
price monetary model, Since the prices are perfectly flexible in the long run,
proportionality between money supply and prices holds in the long run which by PPP
(which also holds in the long run) imply a coefftcient of one of the relative money
supply term. However, the difference with the flexible price monetary model arises
when one considers the sign of the interest differential. In case of the flexible price
monetary model the sign of the coeflicient of the interest differential is positive,
whereas in case of the Dornbusch model it is negative (f <0)'
t2
Frankel (1979) argued that a drawback of the Dornbusch (1976)
formulation of the sticky-price monetary model was that it did not allow a role for
differences in secular rates of inflation. He develops a model in that he emphasizes
the role of expectation and rapid adjustment in capital markets. The innovation is that
it combines the Keynesian assumption of sticky prices with the 'flexible-price'
assumption that there are secular rates of inflation.
Equation (4) of the Dornbusch model is replaced by
E(As) = -d(s-F)+(n-r*)
where, fr ,fr * are the current rates of expected long run inflation at home and abroad.
This says that the expected rate of depreciation is a function of the gap between the
current spot rate and an equilibrium rate, and of the expected long run inflation
differential between the domestic and foreign countries. The theory yields an equation
of exchange rate determination in which the spot rate is expressed as a function of the
relative money supply, relative income level, the nominal interest differential (with
sign hypothesized negative), and the expected long run inflation differential (with sign
hypothesized positive).
It will be very fruitful to consider an equation with alternative testable hypotheses:
s - (m - m *) + 0 Q - y *) +a (r - r *)+ B (n - r*)
The alternative testable hypotheses are as follows :
Flexible-price Model
Sticky-price Model
Real Interest Differential Model
'. d>0,0<0,F=0
: a<0,5<0,F=0
: a<0,Q<0,p> 0
13
(c) THE PORTFOLTO BAL-ANCE MOpEL :
The literature on the monetary models of the exchange rate
determination focuses the role of the exchange rate in maintaining continuous
portfolio balance among existing stocks of financial assets. It assigned the exchange
rate no role in balancing the flow of demands and supplies of foreign exchange arising
from trade in goods and capital.
As in the flexible-price and sticky-price monetary models, the level of
the exchange rate in the Portfolio Balance model (PBM) is determined, at least in the
short run, by supply and demand in the markets for financial assets. The exchange
rate, however, is a principal determinant of the current account of the balance of
payments. The PBM is inherently dynamic model of exchange rate adjustment, which
combines the asset market, the current account, the price level, and the rate of asset
accumulation. One of the main feature of the PBM is that it assumes imperfect
substitution between domestic and non-money assets. In addition, the PBM is stock-
flow consistent, in that it allows for current account imbalances to have a feedback
effect on wealth and hence, on long run equilibrium.
We consider a small open economy model due to Branson et.al (1977),
where domestic residents hold domestic money stock, M, which are dominated in
home currency; domestically issued non money assets B ( i.e., domestic bonds ) ; and
foreign-issued non money asset F, which are denominated in foreign exchange. The
current account in the BoP gives the rate of accumulation of F overtime. The total
supplies of the three assets M, B, and F, to domestic holders are given at each point of
time. The rate of return on F is given by ,*, fixed in world capital market, plus the
t4
expected rate of increase in the exchange rate, s . The rate of return on B is the
domestic interest rate r, which is to be determined in the domestic financial market.
The asset market equilibrium conditions are given by
(l) M = m(r,r* + s)W [Money market] (m, <0,m, <0)
(2) B = b(r ,r* + s)W [Home asset market] (0, > 0,b 2 < 0)
(3) sF * -f (, ,r* + s)W [Foreignasset market] (.f , < O, .f , > O)
G) W = M +B+sF I Wealth constraint ]
The assets are assumed to be gross substitutes, so that I r,l ,
I tl *a I frlrln ,l
Thg case where the assets are perfect substitutes is given bV "f ,=b r-)o, in which
case equations (2) and (3) collapses to the uncovered interest rate parity condition :
r=r"+i
and the financial sector of the model collapses to the money market equilibrium
condition. The main implication of the above equations is that the exchange rate is
determined not just by money market conditions, as in the monetary model, but also
by conditions in bond markets.
The novel feature of the portfolio balance model is that it allows for
the dynamic stock-flow interaction between the exchang e rate, the current account
and the level of wealth. For instance, an increase in money supply would be expected
to lead eventually to a rise in domestic prices, but a change in prices will affect the net
exports and hence will have implications for the .current
acmunt of the balance of
payments. This in turn affects the level of wealth which, in adjustment to long-run
equilibrium, feeds back into asset market and hence exchange rate behaviour.
Therefore, the reduced form equation used for estimation purpose is of the form
s = g(m,m" ,b,b* ,CA,CA.)
15
where, b denotes domestic (non-traded) bonds, m denotes the domestic money supply,
and CA denotes the cumulated domestic current account balance ( * stands for the
foreign variables).
However, it is worth to point out that the introduction of the flow
component in the determination of the exchange rate is not unique to the portfolio
balance model. Hooper and Morton (1982), for instance, attempted to incorporate the
flow dynamics in the Dornbusch-Frankel sticky price formulation by allowing
changes in the long-run real exchange rate. These long-run real exchange rate changes
are assumed to be correlated with unanticipated shocks to the trade balances. This
enabled them to introduce the trade balance in the exchange rate determination
equation.
B. Empirical Survev of Asset Market Modek
B.L Estimation Results
Frenkel (1976) tested the flexible price version of the monetary model
for the Deutsche mark - U.S. Dollar exchange rate over the period 1920-23. This
period corresponds to the German hyper-inflation. Frankel argued that during the
period of hyperinflation, domestic monetary impulses will overwhelmingly dominate
the monetary equation, and thus the domestic income and foreign variables can be
dropped. Frankel reported results supportive of the flexible-price model during this
period. His estimated regression equation is
logs = - 5.135 + 0.9751og M + 0.59llogn
(0.731) (0.050) (0.073)
*: 0.994, D.W. : 1.91. (standard effors are given in the brackets)
l6
The elasticity of the exchange rate with respect to the money stock does not differ
significantly from unity (at 95Yo confidence level) while the elasticity of the spot
exchange rate with respect to forward premium(capturing the expected inflation and
hence, expected depreciation) is positive and significant.
Bilson (1978) tested for the Deutsche Mark - Pound Sterling exchange
rate (with forward premium substituted for the expected change in the exchange rate
and without any restrictions on the coefficients on domestic and foreign money) over
the period January 1972 throttgh April 1979. His results were in accordance with the
monetary approach. Putnam and Woodbttry (1979) estimated the monetary model for
the Sterling - Dollar exchange rate over the period 1972-74, and reported that most of
the estimated coeflicients were significantly different from zero at 5Yo significance
level.
Frankel (1979) considered the real interest differential model for the
mark - dollar exchange rate over the period luly 1974 - Feb.1978. He combined both
the features of the flexible price monetary model and sticky price monetary model to
obtain a real interest differential (RD) model. Frankel used long-term bond interest
differential as an instrument for the expected inflation term, on the assumption that
long-term real rates of interest are equalized. He was able to reject both the flexible -
and sticky - price versions of the monetary models in favour of the real interest
differential model. Frankel also tested the possibility that the adjustment in capital
markets to changes in the interest differential is not instantaneous by including lagged
interest differential as a regressor. However, the coeffrcient of the lagged interest
differential is insignificantly less than zero suggesting of the idea that capital is
perfectly mobile.
t7
Driskill (1981) presented an estimate of an equation representative of
the Dornbusch overshooting model for the Swiss franc - U.S. dollar rate for the period
1973-77 (quarterly data) and reported results largely favourable to the sticky-price
model. The novel feature of this p4per is the incorporation of trade balance responses
to relative price changes in the exchange rate equation. The major findings are as
follows : (a) the exchange rate overshoots in the quarter in which a monetary change
takes place by a factor of 2; (b) the exchange rate adjustment path to full equilibrium
is not monotonic but rather exhibits periods of appreciation and depreciation.
However, price adjusts monotonically; and (c) PPP holds in the long-run.
Although monetary models performed reasonably well up to 1978, the
euphoria was short-lived once the sample period is extended. Dornbusch (1980),
HaSmes and Stone (1981), Frankel (1982) and Backus (198a) cast serious doubts
about the ability of monetary models to track the exchange rate in-sample : few
coefficients were correctly signed; the equations had poor explanatory power as
measured by the coefficient of determination; and residual autocorrelation was a
problem.
Dornbusch (1980) estimated the flexible price monetary model for the
dollar-mark exchange rate using a quarterly data for the period 1973:2 to 1979:4. The
explanatory variables are relative nominal money supplies (logarithm of Ml,
seasonally adjusted), relative real income (logarithm of gross national product at 1975
prices, seasonally adjusted), nominal short-term interest differential (yield on money
market instruments) and nominal long-term interest differential (yield on domestic
government bonds). The estimated regression equation for the 1973:2 - 1979:4 period
is
18
s = - 0.03 (m - m. ) -1.05 (y- y. )+0.01 (i-i )+0.04 (i-i )
(1.e0) (2.07)
R2: 0.33 , D.W. : 1.83. (t-statistics are given in the brackets)
The above equation offers little support for the monetary model with
most of the coefficients being insignificant, and the overall explanatory power of the
equation being very low. The model also suffers from a very high serial correlation
problem. Dornbusch points out that this poor performance of the flexible-price
monetary model can be attributed to the breakdown of the PPP in the short run.
Haynes and Stone (1981) estimated it" Frankel's real interest
differential (RID) model for the period luly 1974 to February 1978 (Frankel's original
sarnple) and from July 1974 to April 1980 based on Cochrane-Orcutt procedure.
While for the shorter period, all the coefficients have signs supporting RID model,
the situation changes dramatically for the longer sample period. Not only the
coefficient of determination falls drastically from 0.61 to 0.38, the signs of relative
money supply and relative income are inconsistent with all monetary models.
Especially disturbing to the monetary approach is the fact that the sign on the relative
money is significantly negative rather than positive. This evidence from the longer
sample is similar to estimates of Dornbusch (1980) and Frankel (1982). Frankel called
this phenomenon - the price of mark rising as its supply is increased - the "mystery of
the multiplying mark ".
In response to this apparent collapse of monetary models, Dornbusch
and Frankel each offers modifications. Dornbusch (1980) specifies the current
account as a significant determinant of the exchange rate within both rational
expectations and portfolio balance models. Frankel (1982) extends the money demand
equation underlying the real interest differential model to include a wealth proxy.
(-0.07) (-0.e7)
t9
Empirical evidence based upon data extended beyond February 1978 supports the
modifications. Haynes and Stone (1981) have an alternative explanation for the poor
performance of the monetary models. They point out that all the above equation
specifies each explanatory variable in relative form which restricts that a given
increase in each of the domestic variables to have the same effect on the spot
exchange rate as an equivalent decrease in the foreign variable. Such subtractive
linear constraints are dangerous because linear restrictions, in general, not only yield
biased parameter estimates, but also it can lead to sign reversals when the variables
are positively correlated. Unconstrained estimates show that the model explains the
mark-dollar exchange rate equally well before and after February 1978. Furthermore,
evidence in their study tends to support the Chicago variant, which stresses the
significance of secular rates of inflation, over the RD and the Keynesian special case.
Driskill and Shefkin (1981) argued that the poor performance of the
monetary model could be attributed to the simultaneity bias introduced by having the
expected change in the exchange rate(implicitly) on the right hand side of the
monetary equations. One potential way of correcting this problem is to use the
rational expectations solution of the monetary model. Hoffrnan and Schlagenhauf
(1983), Woo (1984), Finn (1986) implemented a version of the "forward solution"
flexible-price model formulation, and found support for the rational expectations
model.
Hoffrnan and Schlagenhauf (1983) considered the flexible price
monetary model where the exchange rate is considered as the relative price of two
monies, implying that the exchange rate is determined by the relative demands and
supplies of those monies. Assuming a Cagan relative money demand function, the
spot exchange rate equation (in logarithm form) can be written as
20
s t =k -fm, -fr*, *T /*, -q y,- € (i", -i,)
By assuming that uncovered interest parity and rational expectations hypothesis holds,
the spot exchange equation can be written as
,, = +;k * fi*, - fi*., - fi ,, *fi f ,*, * rlr, r,.,The above equation has one unobservable variable E, s ,*, .
By applying a mathematical expectation, they arrive at the following form
,, = k+]-it=l-l't E,lm *i- nt*,*j-!,*j+y*,*;)l+e fi'l+e'
This equation illustrates the point that exchange rates depend upon
current and expected future values of exogenous variables specified by the monetary
model. Thus, changes in the expected value of these variables can result in abrupt
changes in the spot exchange rates. However, the appearance of expected future
values of exogenous variables which are unobservable requires the specification of
the process generating the exogenous variables. Hoffman and Schlagenhauf assumed
a differenced AR(l) specification for all the exogeneous variables :
Lm, = p *Lm ,*t* Fi, Qa)
L** , = p* ^L,m* ,_r+pr, (2b)
Ly , -- p ,Ly,-t*/4, Qc)
Ly"t = p* ,Ly* ,_r*Fo, Qd)
The j-period forecasts from the above AR(l) model was used to replace the
unobservable expected values in the exchange rate determination equation. The
appropriate way of estimating the rational expectations monetary model is to estimate
equations (1) and (2) as a system to account for the implicit cross-equation parameter
21
restrictions. A likelihood ratio test is performed to check the validity of the
restrictions. Hoffinan and Schlagenhauf applied the model to dollar/deutschemark,
dollar/franc and dollar/pound exchange rate. Thre results are for the monthly data
covering the period 1974'.06 to 1979'.12. The likelihood ratio test indicated that the
restrictions implied by REH could not be rejected for any of the currencies
considered.
Woo (1984) reformulates the monetary approach by ascertaining that a
money demand function with a partial adjustment mechanism had more empirical
support than a money demand function mechanism had morye empirical support than a
money demand function which assumed instantaneous stock adjustment. His study
covered the time period 1974.3 to 1981:10 for the dollar mark exchange rate. A
rational expectation hypothesis model was estimated and the restrictions implied by it
could not be rejected. Finn (1986) also considered the simple flexible-price monetary
model and its rational expectations extensions. The US-UK exchange rate over the
period 1974:5-1982.12. l{rrs result confirms to the rational expectations version of the
flexible price monetary model. The test of coeffrcient restrictions 'could not be
rejected (at 5Yo significance level) for the REH version, but was strongly rejected for
the simple version for the monetary model.
Backus (1984), on the other hand, didn't find many statistically
significant coeffrcients for the Dornbusch model. Papell (19S8) argued that the price
and exchange rate dynamics underlying the Dornbusch sticky-price model cannot be
captured by single-equation estimation methods. He reduced the structural model to a
reduced form, vector-autoregressive, moving-average model with nonJinear
constraints. He found support for the Dornbusch model for the period 1973:l to
19844. Barr (1989) empirically implemented a version of the sticky-price model
22
formulated by Buiter and Miller (1981). The model performed satisfactorily, in-
sample.
With the advent of the cointegration methodology there has been a new
fillip to the research in the asset market models of exchange rate determination. The
asset market models are considered as theories of long-run equilibrium. McNown and
Wallace (1989) tested the monetary model of exchange rate determination as a theory
of long run equilibrium. They used the Engle-Granger (1987) two-step cointegration
methodology to test for the presence of co-integration among the variables of the
monetary model. This hypothesis of cointegration has been tested for five
industrialised countries - U.k., Japan, Germany, France, and Canada - the U.S. Test
and estimations employed monthly data covering the period of floating exchange rates
- beginning in April 1973 for all countries except Canada and U.K., which
commenced floating in July 1970, and June T972, respectively, and upto the end of
1986. Their results were generally unfavourable to the monetary approach both in the
case of restricted model (which assumes equality of foreign and base country) and the
unrestricted model. Only the restricted model for France with U.S. as base country
supports the hypothesis of cointegration.
Given the problems with the Engle-Granger test, the above result was
not surprising. MacDonald and Taylor (1991) used the multivariate cointegration
technique proposed by Johansen and Juselius (1990) to test for the long run
relationship between monetary variables and the exchange rate. They also considered
four of the five countries in the McNown and Wallace study, namely, Germany, U.S.,
Japan, and U.K. Their study covers the time period January 1976 to December 1990.
They took Ml as the money supply, income is measured by IIP and interest is long
term rate given in IMF's International Financial Statistics. The money supply and
23
industrial production series are seasonally adjusted. Two interesting results stemmed
from their work. First, the presence of cointegrating vectors provided a valid
explanation of the long-run nominal exchange rate. Two, for the German Mark - US
Dollar rate, a number of popular monetary restrictions cannot be rejected.
MacDonald and Taylor (1993) used the data for the deutsche mark -
U.S. dollar exchange rate over the period January 1976 to December 1990. Their
major empirical findings are as follows. First, the static monetary approach to the
exchange rate determination has got some validity when considered as a long-run
equilibriurh condition. Secondly, when the exchange rate fundamentals suggested by
the monetary model are assumed, the speculative bubble hypothesis is rejected and
thirdly, the full set of rational expectations restrictions imposed by forward looking
monetary model are rejected. However, their testing procedure of the rational
expectation version of the monetary model is different from the earlier tests of the
forward looking models of exchange rate determination (Hoffman and Schlagenhauf
(1980) Woo (1985) and Finn (1986)). MacDonald and Taylor's method of obtaining a
forward looking solution relies on the multivariate cointegration methodology and its
application to present value models. The forward looking solution of the flexible-price
monetary model can be written as
s, = (t +A)'Zrhr' E(*,*,t1,)
where, x,=lm' *f y'f,.
This is the basic equation of the forwardJooking monetary approach to the exchange
rate @MAER). An implication of the present value model of the exchange rate is that
the exchange rate should be cointegrated with forcing variables contained i, 4 .
24
This implies that
Lt= s, -ffi,+m* r+r7y,-0!', u /(0)
Previous researchers (Hoffman and Schlagenhauf (1980) Woo (1985) and
Finn (1986)) implemented the present value exchange rte model in first-difference
form. However, as Engle and Granger (1987) pointed out that if a vector of variables
are cointegrated, then an empirical formulation in first difference misspecifies the data
generating the process. Thus, if the variables are cointegrated, one should follow
Campbell and Shiller (1987) to test the forward restrictions.
We first need to estimate a VAR of lag length p for the vector
g = lA^xr,....,...,A x t_ p+t, L r,.....,L,_ o*rl
Define g' and h' as selection vectors with unity in the (p+l)th and first elements
respectively, so that,
Lr=g/2,
and Lx,=h/r,
(2)
(3)
The multiperiod forecasting formula is given by
Elr,*rlH ,)= Ak z, (4)
where, .F/ , is the restricted information set consisting of current and lagged values of
L, and Ar,.
st - xt = ,i t*l,E(Lx,,tlIt)
Projecting both sides onto .F/, and using (2), (3), and (a) we obtain,
8t z, - ,!, t#l' htAiz,
- ht VIAQ - WA)-' ,,
25
FMAER imposes on the VAR for (L, , M,)' the following 2p linear restrictions :
Ho : gl (I - WA) - h' ryA - A
We can also define the " theoretical spread " as
Li = h' VA(I * ryA)-t zt
Thus, testing the restrictions is tantamount to testing H , . L,=t ,. Manifest
differences in the behaviour of the time series of I, and lr would be indicative of
economically important deviations from the null hypothesis.
Hendrik and Jayanetti (1993) used a black market exchange rate to
allow them to consider a longer time period which may enable cointegration to better
capture the true long-run relationship between exchange and the explanatory variables
as suggested by the monetary approach. They limit their analysis to India, Pakistan
and Sri Lanka and worked with the annual data. They estimated an equation of the
form
so = ao * or,so + az(m - m*) + at(y - y-) + aa(r - r*)where, s, is the black market exchange rate, s o is the offrcial spot exchange rate.
The oflicial exchange rate is included because the black market exchange rate is
dependent on the official exchange rate and the administration of the official market.
If the set of policies and institutions that govern the legal exchange market is stable,
statistical analysis should find that the black market is related to both official
exchange rate (the spillover effect) and the monetary variables (the underlying shifts
in supply and demand) in a stable manner. They found strong support for the
monetary model using Johansen cointegration methodology. For each country the
26
model with no trend was rejected. For India the cointegrating vector, when
normalised on s, is obtained as [ 1, -0.596, -0.967,0.528, -0.054].
Benjamin and Mo (1995) undertook a study based on monthly data for
US-German exchange rate. They tested for the cointegration for all the three versions
of the monetary model : Frenkel-Bilson, Dornbusch-Frankel, and Hooper-Morton. In
all the three models only one cointegrating vector is detected which suggest that the
long-run relationship exists for the US dollar / Deutschmark exchange rate and the
economic fundamentals. Engsted (1996) reexamines the performance of the monetary
models of the exchange rate (MMER) during the German hyperinflation period of
1920s. The purpose of this paper is to derive and test the cointegration implications of
Frenkel's (1976) model. Also, based on the cointegration result, he derives and test
the exact restrictions that the rational expectations imposes on a bivariate monetary
model for exchange rate and money supply. They used the concept of multi-
cointegration to test for the restrictions imposed by the rational expectations version
of the monetary model. Despite the very strong assumptions inherent in the model,
viz, cagan-type money demand function, instantaneous Purchasing Power Parity,
rational expectations, the exact version of the MMER gives a very accurate
description of the deustchemark-sterling exchange rate during the German
hyperinflation period.
Choudhry and Lawler (1997) applies the Johansen cointegration
technique to examine the validity of the monetary model of exchange rate
determination as an explanation of the Canadian dollar - US dollar relationship over
the period of the Canadian float (October 1950 - May 1962). All data are monthly and
seasonally unadjusted. The money stock variable used for both countries is Ml, the
long term interest rate is represented by the long-term government bond rate, and the
27
industrial production is used as a proxy for income. The exchange rate is expressed as
Canadian dollar per US dollar. The ADF tests are applied with monthly seasonal
dummies turning out to be I(1). A single cointegrating vector is identified whose
coefficients conform to the restrictions imposed by the monetary model (only the
proportionality relationship between money supply and exchange rate is getting
rejected) which lends support to the interpretation of the model as describing a long-
run equilibrium relationship. This support is reinforced by the results derived from the
associated error-colrection model, which identify a short-run tendency for the
exchange rate to revert to the equilibrium value defined by the estimated long-run
model.
Diamandis, Georgoutsos and Kouretas (1998) re-examines the
monetary model of exchange rate determination from a long-run perspective using the
monthly data from January 1976 toMay 1994 for the Deutschemark - Dollar, Dollar-
Pound and Yen-Dollar exchange rates. A novel feature of the analysis is the
implementation of the testing procedure suggested by Paruolo (1996) to examine for
the presence of I(2) and I(1) components in a multivariate context. Two cointegrating
vectors are identified for all cases by using the maximum eignvalue test statistic. To
identify the two cointegrating vectors, independent linear restrictions on each vector
was imposed. Some commonly imposed restrictions on the monetary model were used
for one vector while the other was restricted to provide the Uncovered Interest Parity
relationship. However, the Likelihood Ratio test rejected all the jointly imposed
restriction and they were unable to associate the forward-looking version of monetary
model with either vector, conditional upon the other describing the Uncovered Interest
Parrty condition. This outcome may be attributed to the failure of the UIP condition to
hold in the long run. However, the unconditional version of the monetary model to the
28
exchange rate may still be a valid framework for interpreting the long-run movements
of the Deutschmark-dollar, pound-dollar and yen-dollar exchange rate.
B.II. Out - of - Ssmole Forecasting Performance
A model is often judged by its out-of-sample forecasting performance.
A model may have very good in-sample properties like high adjusted squared
correlation coeffrcient, good in-sample forecasts, no serial correlation problem, but it
may perform very poorly in terms of out-of-sample forecasting performance. A model
is finally used to generate out-of -sample forecasts and if it performs poorly in this
respect, then one may have to reduce his/her confidence on the model.
The monetary models - flexible price monetary model, sticky price
monetary model, real interest differential model - performed well in-sample till 1978-
79. These models, when tested on the extended time period covering upto early 80s,
also performed well in-sample, albeit after some modifications (see Dornbusch
(1980), Frankel (1982). However, the picture is totally different when one considers
the out-of-sample forecasting performance of these models.
Meese and Rogoff (1983) compares time series and structural models
of exchange rates on the basis of their out-of-sample forecasting accuracy. Each
competing model is used to generate forecasts at one to twelve month horizons for the
dollar/mark, dollar/yen, and trade-weighted dollar exchange rates. A rolling
regression methodology is adopted to generate dynamic out-of-sample forecasts. In
this methodology, the parameters of each model are estimaed on the basis of the most
up-to-date information available at the time of a given forecasts. The competing
structural monetary models are flexible-price monetary (Frenkel-Bilson) model, the
29
sticky-price monetary (Dornbusch-Frankel) model, and the Hooper-Morton model. A
variety of univariate time series techniques are also applied to the data. They also
considered an unconstrained vector autoregressive (WAR), composed of variables in
equation (I). A convenient normalization for the estimation of the VAR is one in
which the contemporaneous values of each variable is regressed against lagged values
of itself and all other variables. For example, the exchange rate equation is given by
s t : ai1 s .-t+ ........... + a in s t-n * B' t X - r * .......... + B' i, X - n * u'tit
where, Xit = {* rm* rlr! * rfr rT* , rfr' ,fr" ,TBTTB *) .
To reduce overparameteizatron of the VAR, they constrain the domestic and
cumulated trade balances to have same coefficients. The VAR model is important
because it does not restrict any vaiables to be exogenous a priori, and is therefore
robust to estimation problems like simultaneous equation bias, which plagues the
above discussed structural models.
Each model is initially estimated for each exchange rate using
data up through the first forecasting period, November 1976. As mentioned already,
forecasts are generated at one, three, six and twelve month horizons. The purpose of
considering multiple forecast horizons is to see whether the structural models do
better than time series models in the long run, when adjustment due to lags and / or
serially correlated error term has taken place. It is expected that when lags and serial
correlation are fully incorporated into the structural models, a consistently estimated
true structural model will outpredict a time series model at all horizons in large
sample. Out-of-sample forecast accuracy is measured by three statistics - Mean Error
(ME), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE). Table
below gives the RMSE for the US dollar / pound sterling exchange rate at one, six and
30
twelve month horizons over the November 1976 through June 1981 forecasting period
for exchange rates for representative versions of each model.
Root Mean Square Forecast Errors
(in percentage terms)
(Source : Meese and Rogoff (1983))
The above table is a representative of the general results obtained by Meese and
Rogoff. None of the models achieves lower, much less significantly lower, RMSE
than the random walk model at any forecasting horizon. The structural models, in
particular, failed to improve over the random walk model in spite of the fact their
forecasts are based on realised values of the explanatory variables. Allowing for
separate coefficients on domestic and foreign real incomes and money supplies
yielded no gain in out-of-sample forecasting accuracy. They cited several possible
reasons to explain the poor performance of the structural models. They concluded that
there could be problems with respect to the building blocks of the structural exchange
rate models : uncovered interest parity, proxies for inflationary expectations, goods
market specifications, and the common money demand specification.
A lot of research has gone into to refute the conclusions of Meese and
Rogoff. As discussed earlier, Woo (1985) and Finn (1986) estimated the rational
expectations version of the flexible-price model. Woo (1985) argued that a money
demand function with a partial adjustment mechanism had a more empirical support
than a money demand function which assumed instantaneous stock adjustment.
Model Random
Walk
Forward
Rate
Univariate
Autoregression
VAR Frenkel
-Bilson
Dornbusch-
Frankel
Hooper-
Morton
$/pound I Month 2.56 2.67 2.79 s.56 2.82 2.90 3.03
2 Month 6.4s 7.23 7.27 t2.97 8.90 8.88 9.08
3 Month 9.66 11.62 13.35 21.28 t4.62 t3.66 L4.J I
3l
Following Goldfeld (1973), they included a lagged money term in the money demand
specification to capture the partial adjustment in money holdings. The study focused
on mark-dollar monthly exchange rate covering the period T974:3 to 1981:10. He
used the last twenty observations in the sample, that is, 1980:3 to 1981:10, for the out-
of-sample forecast comparison with the random walk model. The result showed that
the rational expectation version of the monetary model outperform the random walk
model at every forecasting horizon under the mean-absolute-error criteria. In terms of
root- mean-square-effor also, the structural model outperform the random walk
model.
Finn (1986) evaluates the forecasting performance of monetary and
random walk models of the exchange rate. Instrumental-variable estimates of the
simple monetary model are not supported by the data, while the full-information-
maximum-likelihood estimates of its rational expectations counte rpart are. Monthly
data is used over the period 1974:5 to 1982:12 for explaining the dollar-sterling
exchange rate. The forecasting period embraces 1980:1 through 1982:12 and the
forecasting performance is evaluated for the rational expectations version of the
monetary and the random walk models. In terms of root-mean-square-effor and mean-
absolute-error, the two models are very closely ranked for the one- and six-month
forecasts - the random walk model performing marginally better at one month
horizon and rational expectations monetary model marginally better at the six-month
horizon. For the twelve month horizon forecasts, both models are closely ranked - the
random walk model performing marginally better on the mean-absolute-error criterion
and better by approximately 1.1 percentage point on the root-mean-square-elror
criterion. In light of the above results, Finn concluded that the rational expectation
monetary models forecasts as well as the random walk model.
32
Schinasi and Swamy (1989) reexamines the forecasting performance of
models reported by the Meese and Rogoff (MR) without imposing the restriction that
the regression slopes are fixed over time. Major result of their study is that when all
coefficients are allowed to vary, the conventional models of exchange rates employed
by MR yielded more accurate forecasts than their fixed coefficient counterparts and
more accurate than the random walk models. The study, however, supported most of
the MR conclusions regarding fixed coefficient models, but contrary to the MR study,
Schinasi and Swamy found significant improvements in the forecasts of fixed
coefficient models that include lagged adjustment. The structural models estimated by
MR and used for forecasting nominal spot exchange rates (do11ar-mark, dollar-yen,
dollar-pound) is given below:
s, = Bo+ Br(mt- m.t)+ Bz(yt-y.)* Bz(r,-r.,)+ p+(tTt" -v,". )+ Bs(TBt-T?.,)+u,
Schinasi and Swamy argued that the sequential estimation of fixed
coefficient regressions (that is,'rolling regression') is not the appropriate technique for
capturing the variations in coefficients overtime. At the high level of aggregation of
exchange markets, there is little reason to believe that behavioural parameters are
fixed. There is a wide diversity of participants in foreign exchange rate markets with
relatively small and highly variable market shares. Even if each participant reacted to
macroeconomic developments according to a stable fixed coefficient reaction
function, it is difficult to argue that macroeconomic variables would be related to
exchange rates by a simple fixed coefficient relationship, without also assuming that
individual reaction functions are identical.
We briefly present the stochastic coefficient representation of the
exchange rate models :
yt=X'tB
p,= B+a
&=Q tt*vt
E(vt) = Q
E(vtv") = Ao if Fs and 0, otherwise.
where, xt,Bt,B,tt,vt are all (kxl) vectors, (D and A,a a.rg (kxk)mafrcos, .rr
represents the vector of the explanatory variables in (1), and yt is the natural
logarithm of the spot exchange rate. In (3), each coefficient in each period, B i, , has
two components : a time-independent fixed coefficient, B, , and a time-dependent
stochastic component, 6;r.
Combining (2), (3) and (4a), we can view the stochastic coefficient representation as a
fixed coefficient model with effors that are both serially correlated and
heteroscedastic, where the form of serial correlation and heteroscedasticity is very
general :
yt = X't*I,lt
l,lt=X'ttt
tt=@tt-t*lt
They estimated all the competing models from March 1973 through March 1980 and
generated the multistep-ahead forecasts for the period April 1980 through June 1981.
The competing models include, apart from the stochastic coefficient model, fixed
coefficient model, and random walk model. The stochastic coefficient model turn out
34
to produce superior forecasts than the fixed coefficient model and also outperformed
the random walk model. They also generated multistep-ahead forecasts of the Box-
Jenkins (1970) type time series models, ARIMA(I,1,0), ARIMA(O,1,1) and
ARIMA(1,1,1) and found them to be inferior to the multi-step ahead forecasts of the
random walk model with or without drift.
As already discussed, with the advent of the technique of cointegration
and VAR there has been a new lease of life to this topic. Many of these papers
exploited the long-run and short-run properties of the monetary models to generate
out-of-sample forecast that outperformed the random walk in terms of RMSE and
MAE. MacDonald and Taylor (1993) used a vector effor correction model to generate
out-of-sample forecasts that are superior to those generated by a random walk
forecasting model. They found for the deutschemark-dollar exchange rate existence of
a cointegration relation that corresponds to the static monetary approach exchange
rate equation. Thus, the monetary model can be interpreted as having at least long-run
validity. According to the Granger representation theorem, if a cointegration
relationship exists among a set of I(1) series then a dynamic error-conection of the
data also exists. So they estimated the error-correction model for the initial period
1976:l to 1988:12 and reserved the last 24 datapoints, corresponding to the period
1989:01 through 1990:12 for post-sample forecasting performance. They performed a
d5mamic forecasting exercise for a number of forecasting horizons. The dynamic
elror-correction model outperforms the random walk model at every forecast horizon
as shown in the table below :
35
Dynamic Forecast Statistics
approx y equa
percentage differences divided by 100.
(Source : MacDonald and Taylor (1993)).
This shows that imposing the monetary model as a long-run equilibrium condition on
a dynamic, error-colrection model led to dynamic exchange rate forecasts" at every
forecast horizon considered.
Hoque and Latif (1993) compared the forecasting performance of
unrestricted VAR model, a Bayesian VAR model, a structural model and an error
correction version of the structural model. The purpose was to obtain the best
forecasts for the Australian dollar vis-d-vis the US dollar in terms of root mean square
error (RMSE) of forecasts. For estimation quarterly data from 1976Q1 to 1990Q1 has
been used. The period 1990Q2 to 1991Q1 has been used for ex post prediction. Five
variables chosen for VAR / BVAR system included exchange rate, current account
balance, three month forward rate, relative long-term interest rate, and relative price
level. The BVAR model was estimated with several degrees of tightness (2), decay
(d) andweights (w) as follows: )":0.I,0.25,0.3; d:1,2;w:0.01,0.15. A
structural model (due to Wallis(1989)) was also constructed to compare its forecasting
Horizon(Months)
RMSE fromError- Correction Model
RMSE from RandomWalkModel
t2 0.131 0.148
9 0.103 O.TI2
6 0.081 0.088
J 0.043 0.0s3
2 0.032 0.040
1 0.028 0.030
Note : Figures are logarithmic differences and are therefo re approximatelv equal to
36
perfoflnance with the multivariate time series model. It was found that the structural
model performs best in terms of RMSE than either of the two time-series models.
BVAR model performs better than the unreskicted VAR, but not as well as the
structural model. An attempt was also made to improve the forecasting performance
of the structural model by considering the time-series properties of the variables
involved in the exchange rate equation. The cointegration property among the
variables have been exploited to generate forecasts from an eror-coffection model.
The error-correction version of the structural model displayed a better forecasting
performance compared to the simple structural model.
Liu, Gerlow and Irwin (1994) analyses the forecasting accuracy of fuIl
vector autoregressive (FVAR), mixed vector autoregressive (MVAR) and Bayesian
vector autoregressive (BVAR) models of the US dollar lYen, US dollar I Canadiart
dollar, and US dollar / Deutsche mark exchange rates. The VAR models are based on
the theoretical model of monetary / asset exchange rate determination developed by
Driskill et.al (1992). The models are estimated over the in-sample period L973:3 -
1982:12. For the out-of-sample period covering 1983:1 through 1989:12, l-, 3-, 6-,
and l2-month forecasts are generated. Performance criteria include bias tests,
informational content tests, and market timing ability tests. The variables included in
the model are logarithm of the exchange rate, logarithm of the relative real income,
the logarithm of relative price levels, the interest rate differential and the trade balance
between the two countries. They employed the Litterman's prior to estimate the
Bayesian VAR.
In terms of bias tests, BVAR model's performance is better than FVAR
and MVAR. To determine if the forecasts generated from the alternative VAR models
contain additional information beyond a random walk process informational content
37
test developed by Fair and Shiller (1989, 1990) was employed. In the case of the US
dollar I Yen exchange rate, at forecast horizons of 1- through 6-months, forecasts
generated by the FVAR, MVAR, and BVAR models did not contain additional
information beyond that produced by the random walk forecasts. However, the BVAR
model dominated the FVAR and MVAR at the l2-month forecast horizon, and it
contained additional information not generally not found in a random walk model. In
terms of market timing ability test, the forecasts generated from the FVAR models,
for all three exchange rates, have no significant market timing value. In other words,
the FVAR model is not capable of significant predictions of the directional movement
of the exchange rates. MVAR forecasts exhibit significant market timing ability
across all forecast horizons for the US dollar I Canadian dollar exchange rate.
Forecasts generated from a BVAR model also have significant market timing value
for the US dollar I Catadian dollar rates and US dollar lYen across 3-to 12-months
forecasts horizons. Thus, out-of-sample forecasting performance indicate that the
forecasting performance of restricted VARs (MVAR and BVAR) is substantially
better than that of the unrestricted VARs (FVAR).
Benjamin and Mo (1995) used the multivariate cointegrating
methodology to generate long-run forecasts of the US dollar / Deutschmark exchange
rate.' They used three competing structural models - Frenkel-Bilson, Dombusch-
Frankel and Hooper-Morton - and in all the three models, only one cointegrating
vector was detected, suggesting that the long-run cointegrating relationship exists
between the exchange rate and economic fundamentals. They initially estimated the
models for the period 1973:04 - 1988:07 and out-of-sample forecasts were obtained
by using the rolling regression for the period 1988:8 - I993:l (60 months). The
forecasts were evaluated in terms of RMSE and MAE statistics. The forecasts were
38
generated using the error-correction versions of the three models. These forecasts
from the structural model clearly outperformed the random walk model at every step.
This finding was especially significant in that the multistep-ahead forecasts of the
structural models outperformed even the one-step-ahead forecasts of the random walk
model.
Chinn and Meese (1995) examined the predictive performance of the
standard structural exchange rate models using both parametric and nonparametric
techniques. They examined four bilateral rates (Canada, Germany, Japan, and the
U.K.) relative to the US dollar, using monthly data for the period 1973:03 - 1990:11.
They argued that the post-Bretton Woods era was too short to extract reliable
estimates of he long-run elasticities by direct estimation (either by Engle-Granger
methodology or by Johansen and Juselius). So they impose a set of coefficient
restrictions for each of the candidate models and used them to generate the error-
correction term.
Chinn and Meese estimated the monetary models by OLS and
instrumental variables (IV) procedures in unconstrained first differences, and error
correction models. At one month forecast horizon, the structural models procedure
discouraging result compared to the naiVe random walk model (with or without drift)
in terms of RMSE statistic. The predictive performance of the structural models
shows some improvements at horizons beyond one-month period. Error correction
models with an elror correction term lagged once do not, in general, produce the best
RMSE at the yearly predictive horizons.
Bhawani and Kadiyala (1997) employed the black market data for
exchange rates in developing countries to investigate the forecast performance of
several exchange rate models. Unlike other studies which used the ' actual ' realized
39
values of the exogenous variables, this study employed expected future values of the
exogenous variables (predicted outside the model). As representatives of the structural
monetary models they estimated the reduced form equation for Bilson-Frenkel
flexible price model and sticky price Dornbusch model. They also estimated an effor-
correction version of the structural models.
The models have been used to generate forecasts at one, three, six and
twelve month forecast horizon for the Indian rupee / US dollar, Mexican peso / US
dollar, and Pakistan rupee / US dollar bilateral spot exchange rates. Forecasts for all
models are based on rolling regressions. Initial sample period for India covers the
period 1913-89, for Mexico it covers 1982-89 and for Pakistan it covers 1978-88. The
numberof 1-, 3-,6-,12-monthaheadforecastsequal 36, 34,31 and25respectively.
Three criterion are used for forecast evaluation - ME, MAE, RMSE. The error-
correction version of the Bilson-Frenkel model outperformed the simple random walk
model at all forecast horizon for the Indian rupee / US dollar exchange rate. While in
case of Pakistan the effor-coffection model outperforms the random walk model at all
forecast horizon, except the one-month forecast horizon, for Mexico peso / US dollar
the simple random walk model exhibits the least forecast error at all horizons,
followed by the effor-coffection model.
Choudhry and Lawler (1997) estimated an eror-correction version of
the monetary model for the Canadian dollar - US dollar exchange rate over the period
of the Canadian float 1950-62. To test the adequacy of the monetary error-correction
model a forecasting exercise was carried out. Forecasts are generated for three, six,
nine, and twelve month forecasting horizon over the period June 1961 to May 1962.
For the comparison purpose forecasts were also made with two alternative models - a
simple random walk model and a random walk model with drift. RMSE statistic has
40
been used for evaluating the forecasting performance. The error-corection model
outperforms both the random walk models across the range of forecast horizons.
Reyiew of the Survey of Literature
soon after the breakdown of the Bretton Woods Agreement. More specifically, the
monetary models performed well in the years 1975-T980. (Frankel (1976), Bilson
(1 978), Frankel (197 9).
coefficients, persistent serial correlation problem in the 1980s, following which
few modifications to the original structure were put forward. (Dornbusch (1980),
Frenkel (1982), Haynes and Stone (1981), Driskill (1981).
Meese and Rogoff (1983). Given the general poor performance of the monetary
models vis-a-vis the random walk forecasts, a lot of research has gone into
refuting the negative conclusions reached by Meese and Rogoff (1983).
a long-run equilibrium phenomenon. (MacDonald & Taylor (1991, 1993, 1994),
Choudhry and Lawler (1997), Diamandis, Geougoutsos & Kouretas (1993)).
autoregression formulation of the monetary models of the exchange rate
determination produced forecasts which beats the random walk forecasts, mostly
in the developed country context. (MacDonald & Taylor (1993, 1994), Liu,
Gerlow and Irwin (1994), Bhawani & Kadiyala (1997), chaudhry & Lawler
(1ee7)).
41
Chapter III
E CONOME TRIC METHOD OLOGY
The present chapter discusses the econometric methodology used to
generate forecasts of the exchange rate, while the next chapter takes up the discussion
of various forecast evaluation measures. Starting point of any time series analysis
involves checking of the stationarity properties of the series under study. This is
achieved by the traditional unit root tests of (Augmented) Dickey-Fuller test (ADF),
Phillips-Perron (PP) test as well as two other test of unit root - KPSS Test, which is
essentially a non-parametric tests and Bayesian unit root test. Various unit root tests
are performed to get a clear picture about the presence of stochastic trend in the data.
These tests are described in more detail in section A. Section B discusses the
univariate modeling motivated by the Box-Jenkins three step methodology. Section C
takes up the discussion of the concept of cointegration among the variables which
helps us to impose economic structure on the variables under consideration. Section D
discusses the atheoretical modeling involving vector autoregression (VAR) models as
proposed by Sims (1980). This includes full VAR (FVAR), vector error correction
model (VECM) and Bayesian VAR (BVAR).
Section A : Unit Root Tests
Unit root hypothesis has received much attention in the economic and
econometric literature since the seminal work of Nelson and Plossar (1982). A non-
stationary series has the following properties :
42
(a) There is no long-run mean to which the series returns.
(b) The variance is time-dependent and goes to infinity as time approaches infinity.
Unit root becomes important in the context of spurious regression involving several
non-stationary varalbles as proposed by Granger and Newbold (1974). A spurious
regression has high R', t-statistics that appear to be signifrcarfi, but the results are
without any economic meaning. So it is very important to find out the stochastic
properties of . the variables under study so that we do not ran into such spurious
regression problems.
In the first modern attempt to test for the unit roots, Nelson and Plossar
(1982) tested 14 historical macroeconomic time series for the US by the Augmented
Dickey-Fuller (ADF) test. They analyzed the logarithms of all of these series (except
for the interest rates, which was treated in levels) and found empirical evidence to
support a unit root for 13 of them (exception being the unemployment rate). Meese
and Singleton (1982) studied various exchange rate time series and could not reject
the null hypothesis of a unit root. In the present thesis, we work out the sequential
testing procedure of the (augmented) Dickey-Fuller test as opposed to the simple
Dickey-Fuller test performed by the above mentioned studies.
(i) (Auementedt Dickey-Fuller Test :
Dickey-Fuller consider three different equations that can be used to test
the null of unit root :
Ly,
Ly, *Et
L)tri+t I Et
(1)
p
= aU*r + \fiLyni+t * tt
p
= ao * atlt-t + l_, BiLyn+r
= ao + arlrt + azt + f,^ fri
(2)
(3)Ly,
43
Here the nullof the unit root is given by
i}t :0.
The first equation is a pure random walk model, the second add an intercept or drift
term, and the third includes both a drift and linear time trend.
Enders (1995) suggests a sequential testing procedure to test for the
presence of a unit root when the form of the true data generating process (DGP) is
unknown. The motivation of doing this sequential procedure can be traced to Cambell
and Perron (1991). They pointed out that the major problem with the Dickey-Fuller
(DF) test is that tests for a unit root is conditional on the presence of the deterministic
regressors and tests for the presence of the deterministic regressors are conditional on
the presence of a unit root. This follows from the fact that if the estimated regression
includes deterministic regressors that are not in the actual DGP, the power of the unit
root test against a stationary alternative decreases as additional deterministic
regressors are included. Furthermore, if the estimated regression omits a deterministic
trending variable present in the true DGP, such as azt, the power of the t-test goes to
zero as the sample size increases. This necessitates the following sequential testing
procedure ofunit root.
Step 1. Estimate the most general model (eq.(3)) and test the null of unit root
(ar : 0) by € s statistic. If the null is rejected, conclude that unit root is not present in
the series and stop the sequential procedure. If the null hypothesis is not rejected we
proceed to step 2.
Step 2. At this stage we determine whether trend term is needed to be included
in the eq.(3). This is achieved by testing the significance of the trend term under the
null of unit root by using e B, statistic or Q. statistic. If the trend is not significant
we proceed to step 3. On the other hand, if the trend is significant we go back to step
44
1 and retest for the presence of unit root using the standardized normal distribution. If
the null of unit root is rejected stop the sequential procedure and conclude that the
series is stationary. If not, then conclude that unit root is present and ptoceed to
step 3.
Step 3. Estimate equation (2), that is, one without a trend but a drift (constant)
term and test the presence of unit root by using the g ustatistic. If the null of unit root
is rejected conclude that the series does not contain a unit root and stop the sequential
procedure. Ifnot, then proceed to step 4.
Step 4. Here we determine whether a drift term is needed to be included in our
regression model. This is done by using 6 ,, statisti c or Q , statistic. If the drift term
is not significant proceed to step 5. If, on the other hand, it is significant, we return to
step 3 and retest the null of unit root by using the standard normal distribution.
Rejection of null of unit root will lead us to abandon the sequential procedure and
conclude that the series does not contain a unit root. Non-rejection of the null will
lead us to step 5.
Step 5. Finally, we estimate the simple model, that is, one without a drift or
trend term (eq.(l)).We use the g statistic to test for the presence of unit root. If the
null is rejected, we conclude that the series does not contain a unit root. Otherwise, we
conclude that it contains a unit root.
(ii) Phillips-Peruon Test :
The distribution theory supporting the DF test assumes that the errors
are statistically independent and have a constant variance. Phillips-Perron developed a
test that allows that disturbances to be weakly dependent and heterogeneously
distributed.
45
They considered the following regression :
!r=a o*a r!,q*ll,
lt = do * drla + d2(t - Tl2) + 1t1
where T : No. of observations.
The most useful test-statistics are as follows :
Z (t a. ),: Use to test the null hypothesis a r =l
Z(td): Use to test the null hypothesis dr = 1.
Between the Dickey-Fuller and Phillips-Perron test, the later is
preferred because it has better power. This implies that if the null of unit root is not
rejected by the DF test but rejected by the PP test, then we rely on the PP test and
conclude that the series does not contain a unit root.
(iii) KPSS Test:
A feature of the Dickey-Fuller and Phillips-Penon tests is that they
make the unit root the null hypothesis and given the low power of the former test it is
very difficult to reject the null of unit root. KPSS, therefore, argue that in trying to
distinguish between stochastic and deterministic trends, it is natural to consider both
the null of trend stationary and difference stationary. They developed a test of unit
root where null hypothesis is taken to be the absence of unit root. This is essentially a
non-parametric test.
Let Y1 be a sample of T observations. KPSS assumes that the series
can be decomposed into the sum of a deterministic trend, a random walk, and a
stationary elTor :
Yt = (.t *rt*tt
Here 11 follows a random walk.
(1)
(2)
46
Let e1 , t:1,2,....T, be the residuals from regression of !, on an intercept and time
trend. Let 62 " be the estimate of the error variance from this regression (the sum of
squared residuals divided by T). Define the partial sum process of the residuals :
ts' = ,E,
u' ,r: lrzr.....rT.
Then, the LM statistic is given by
LM =\5, ,I 6, ,i=1
KPSS uses an estimator of sample error variance (6', ) of the form
TIT
^s21/1 = T-'Zu' ,+27 -1!w(s,/) +\e,e,-,
Y" w(s,l) is an optional lag window that corresponds to the choice of a spectral
window. KPSS uses the Bartlett window, w(s,l)=t-t/*1 , which guarantees the
non-negativity of the estimated sample variance. The lag parameter / is set to correct
for residual serial correlation.
In event of testing the null hypothesis of level stationarity instead of
trend stationarity, we define et as the residual from the regression of y on intercept
only (i.e. € t = !,-y), the rest of the test statistic is unaltered. The test is an upper tail
test. The critical values are given in KPSS (1992).
(iv) The Sims-Bayesian Unit Root Test :
Sims (1988) argues that conventional tests for the presence of unit
roots, such as DF tests, are fundamentally flawed. The relevant question should be
how probable is null of unit root relative to the other competing hypotheses. The
classical econometricians cannot give the probability that a hypothesis holds. What
47
they can tell us is whether a hypothesis is rejected or not rejected (Koops, 1992).
Further, while the classical inference is sharply affected by the presence of a trend and
drift term, the Bayesian flat prior theory is not.
Consider the following autoregression model :
lt=Pit-t*8t
The test statistic is the square of the conventional rstatistics for P = t. This is
compared to the Schwarz criterion which has an asymptotic Bayesian justification.
This is approximately given by
r = 2log (#) - bg (oil + 2tog(t - 2 tts1
where, o' o= , o' is the variance of a , and for monthly data s: 12.Z v,G *r)'
" Alpha" gives the prior probability on the stationary part of the prior; the remaining
probability is concentrated on p = 1 If t2 > t we reject the null hypothesis of unit
root.
Section B : Box-lenkins Methodology
Box and Jenkins (1976) popularized a three-stage method aimed at
selecting an appropriate model for the purpose of estimating and forecasting a
univariate time series. This is found to generate forecast which is as good as, if not
better than the multivariate models. The advantage of autoregressive integrated
moving average (ARIMA) models is the relatively little information set used in
estimation and forecasting - only the .lagged values of the dependent variable and
emor term are required. The ARIMA (p,d,q), where p is the order of the
autoregressive process, q is the order of the moving average process and d is the
degree of integration
o'
48
o@)0-B)" , = O(B)u ,+6 ,
where, the polSmomials in the backward operator (B) are given as follows :
Q(B) = r-Q, B -.......-Q o B'
e@) =l-0 tB--.-...-o ,B'
In the identification stage, we visually examine the time plot of the
series, autocorrelation function, and partial autocorrelation function. We use the tests
of stationarity - (Augmented) Dickey-Fuller test and Phillips-Perron test - to check
formally whether the series contain a unit root or not. If, from the above tests, we
conclude that the level is non-stationary we then work with the difference of the
series. The number of differencing to be done depends on whether the differenced
series is stationary or not.
In the estimation stage, we fit various plausible models and
significance of their coefficients are examined. Our aim is to choose the most
parsimonious model with no serial correlation in the eror term. The two most
commonly used model selection criteria are the Akaike Information Criterion (AIC)
and Schwartz Bayesian Criterion (SBC) :
AIC = T ln(residual sum of squares) + 2n
SBC : Tln(residual sum of squares)+nln(T)
where, n: number of parameters estimated (p+q+possible constant term)
T: number of usable observations.
Ideally, the AIC and SBC should be as small possible. Of the two criteria, the SBC
has superior large sample properties. The serial correlation test is performed by using
the Lung-Box-Pierce Q statistic
LQ=n(n+rZ*
49
This follows a chi-square distribution under the null hypothesis of no serial
correlation in the error term.
' The third state in the Box-Jenkins methodology involves diagnostic
checking. Here, we primarily see whether the residuals from an estimated model are
serially uncorrelated. Any evidence of serial correlation implies a systematic
movement in the series that is not accounted for by the ARMA coefficients included
in the model. The Ljung-Box Q statistics of the residuals are used to test the presence
ofserial autocorrelation in the residuals.
Finally, we can use the selected model to obtain the forecast of the
series.
Section C : Cointesrstion Methodolow
Granger and Newbold (1974) warned of a serious empirical
consequence of estimating models with non-stationary variables. When both the
dependent variable and the explanatory variables in a time-series regression are non-
stationary, spurious correlations are likely to occur: variables appear to be significant
when in fact they are not. The symptoms of this spurious correlation include a high R2
combined with a low Durbin-Watson statistic. These synrptoms are familiar features
of estimated exchange rate models [Boothe and Glassman (1987)]. This led to the
concept of co-integration as proposed by Engle-Granger (1987) to test for the
presence of spurious regressions. The test of co-integration allows us to find out
whether the non-stationary variables have any meaningful relation among them.
50
(i) Engle-Graneer Two Steo Methodolosy (EG) :
Suppose there are two variables Y1 and21, both being I(r). The long-
run equilibrium relationship is estimated by
Yt=00+8121+e1
The estimated residuals are saved. If the deviations from long-run equilibrium are
found to be stationary, the Yl and 21 series are cointegrated of the order (1,1). So in the
second stage, we perform the unit root tests of DF/ADF and PP to check the
stationarity of the estimated residuals.
There are quite a few problems with Engle-Granger procedure of
testing the presence of Cointegration among the variable. One major problem with the
two-step procedure is that the estimation of the long-run equilibrium regression
requires that the researcher place one variable on the left-hand side and use the others
as regressors. So, in practice, it is possible to find that one regression indicated the
variables are cointegrated, whereas reversing the order indicates no cointegration.
Moreover, it is not possible to find the presence of multiple cointegrating vector
among the variables. For all these drawbacks, it is suggested to test for the presence of
cointegration among the variables in a multi-variate framework by using Johansen-
Juselius Maximum likelihood method (JD (1990).
(ii) Johanseru-luselius Cointesration Test :
Consider the p-dimensional vector autoregressive model with
Gaussian errors
! , = At! ,_t*......+A *! rtc+Y.D+e ,
where .y, is a pxl vector of stochastic variables, D is a vector of nonstochastic
variables, such as seasonal dummies or time trend. Johansen test assumes that the
51
variables in y , are I(l). For testing the hypothesis of cointegration the model is
reformulated in the error-corection form
Ly , = f , A.y,-, *.......tf o-, L! ,-o*r+lIy ,-r+ p+y .D*, ,
The hypothesis of cointegration is formulated as a reduced rank of the lI -matrix
H ,:fI - o0'
where a and p are (pxr) matrices of full rank. The null hypothesis implies that the
process Ay, is stationary, "y, is nonstationary, but B'y, is stationary relations
among nonstationary variables. This is basically an eror-correction formulation
which allows for the inclusion of both differences and levels in the same model
thereby allowing one to investigate both short-run and long-run effects in the data.
The number of distinct cointegrating vectors can be obtained by
checking the significance of the characteristic roots of II matrix. The tests for the
number of characteristic roots that are significantly different from unity can be
conducted by using the following two test statistics:
ltrnr" = -, ,Jr ln (1 - ;*l)
).max - -T In (1-,tr,+l)
where, 7 ,', ar" the estimated values of the characteristics roots (also called
eignvalues) obtained from the estimated II matrix; T is the number of usable
observations.
Section D : Vector Autoreeression (VAR) Methodology
Vector autoregression (VAR)
critique to the structural macroeconometric
approach has been developed as a
modeling where arbitrary coefficient
52
restrictions and lag structures are imposed on the data-generating process. VAR
allows the data to speak for itself by allowing data to determine the lag structure and
doing away with the arbitrary exogeneity assumption that the structural econometric
models often make. Moreover, estimation of VAR becomes very important in the
context of multivariate cointegration tests proposed by Johansen-Juselius.
Consider a pth order VAR of n variables
! t = Ao+At!,_tl .......+A p!,_p+e t
where y , is an (nxl)vector containing each of the n variables included in the VAR,
A o is (nxl) vector of intercept terms, A ris (nxn) matrices of coefficients and
e ,is an (nx1) vector of error terms. VAR modeling has been popularrzed due to
Sims (1980). The variables included in the VAR are selected according to the relevant
economic model. VAR model scores over the traditional structural modeling by
abstaining from assuming exogeneity / endogeneity of the variables under
consideration. Rather, it treats all the variables symmetrically. When all the variables
included in the VAR have the same lag length in each of the equation of the VAR
system then it is known as a FUIMR (FYAR). Each separate equation can then
been efficiently estimated by simple OLS. However, it is possible to employ different
lag lengths for each variable in each equation. Such a system is called Mixed VAR
(MVAR). If some of the VAR equations have regressors not included in the others,
seemingly unrelated regressions (SUR) provide efficient estimates of the VAR
coefficients.
The choice of the lag length in VAR is an important issue, as the
inclusion of unnecessary lags will lead to severe overparameteization problem as
inclusion of lags quickly consumes the degrees of freedom. . The two commonly used
measures are the multivariate generahzation of the AIC and SBC :
53
AIC=Tlogl2l+2
SBC = rlog lll + Nlog (7)
where, l>l is tfre determinant of the variance / covariance matrix of the residuals; N is
the total number of parameters estimated in all equations. The model based on the
lowest AIC and SBC is chosen.
Another test that is often employed for selecting the lag length is the
likelihood ratio test. Let X, and X, be the variance / covariance matrices of the
unrestricted and restricted system respectively. Asymptotically, the test statistic is
given as
LR = e-c)( tog lX,l _ log l>,1)
has a chi-square distribution with degrees of freedom equal to the number of
restrictions in the system. Here, logll,l is the natural logarithm of the determinant
of I, ; T is the number of usable observation; and c is the number of parameters
estimated in each equation of unrestricted system. If the calculated value of the test
statistics exceeds the tabulated critical value, we reject null hypothesis, i.e., we reject
the restriction.
A block exogeneity test is useful for detecting whether to incorporate
a variable into a VAR. Consider a three variable system x, y and z. To test whether
variable z should be there in the system is similar to testing whether lags of z in x and
y equations are zero. This cross equation restriction is tested by performing a
likelihood ratio test. For this we need to estimate the x and y equations using p lagged
values of x, y and z and calculate the (unrestricted) variance I covaiance matrix X , .
Reestimate the two equations excluding the lagged values of z and calculate the
54
(restricted) variance / covariance matrix X, . Next, we calculate the likelihood ratio
statistic :
LR = Q-c)( log lX.l - log l>,1)
This statistic has a X' distnbution with degrees of freedom equal to the p lagged
values of z.
(i) Vector.Enor Coruection Model (VECM) :
Given that the variables are integrated of order 1, that is, I(1), and that
the variables are cointegrated I as concluded from the Engle-Granger and Johansen-
Juselius test for cointegration ], we can construct an error-correction model captures
both the short-run and long-run dynamics which may significantly improve the
model's forecasting performance. Consider the two variable system with y and zberng
the two variables. The two variables are integrated of order I and furthermore they are
cointegrated, that is, there exists a linear combination among them which is an I(0)
variable. Given this one can estimate the error-correction model of the following form
Ly , = f ,+ F ,a,_r+28,,(i)Ay ,_,*ZF ,r(i)Lz t_i*t yti=l
Lz , = F ,+ 0 ,a,_r+\F ,, Q)Ly ,_,*ZF ,r(i)a,z t-i* € ,ti=1 i=1
where O,-r= !,-r-fr 12,-, is the error-correction term, p, is the parameter of the
cointegrating vector, and e y,,€ ,, are the white noise disturbances.
Multivariate geteralization of the VECM is given as below
Ly,= f ,A.y,_r*.......*Io_, L/,_o*r+I-I.y Fr+p+Y.D+e ,
OLS is an efficient estimation strategy if each equation contains the same set of
regressors.
55
However, it will be appropriate to mention that the above formulation
of the eror-coffection model is only one of the approach to the problem and is
certainly does not exhaust the literature. The early development of the error-correction
model was very much the work of the London School of Economics by Phillips,
Sargan and Hendry. Phillips (1954,1957) introduced. the terminology of error
correction to economics in his analysis of feedback control mechanisms for
stabilization policy. The current popularity of error-correction model could largely be
attributed to the David Hendry, whose influential article with Davidson, Srba and Yeo
(1978) on aggregate time series relationship between consumer's expenditure and
income , proved to be a very important cornerstone. Work by Engle and Granger
(1987) on the effor-corection model is different because they take into consideration
the cointegration property of the time series data.
(ii) Bavesian Vector Autoreeression (BVAR) :
It is often rightly pointed out that economic forecasting is an art,
perhaps because it involves not only data and groups of equations, or statistical
models, but also the forecaster's personal beliefs about how the economy behaves.
The Bayesian approach to statistics, a general method for combining beliefs and data
in economic forecasting models. Bayesian Vector Autoregression (BVAR) modei has
been developed explicitly along Bayesian lines, provide modelers more flexibility in
expressing their beliefs as well as an objective way to combine those beliefs with
historical data.
BVAR models strikes a balance between Unrestricted Vector
Autoregression (UVAR) and structural econometric modeling. UVAR model relates
56
future values of a vector of variables to past values of that vector. From Bayesian
point of view, UVAR models allows the data speak for themselves. This implies
complete ignorance on the part of the modeler regarding the value of the coefficients
included in the UBVAR. The forecasting problems of large UVAR models stem from
the fact that economists often have too little data to isolate in a model's coefficients
only the stable and dependable relationships among its variables which leads to poor
forecasting performance.
On the other hand, in the structural econometric models which are
widely used for economic forecasting, overfitting problem of the UVAR model is
tackled by including in each equation of the model only a few variables (or lags of
variables) that economic theory suggests are most directly related to the variable that
the equation forecasts. Such exclusion of variables from an equation amounts to
certainty that their coefficients are zero. Certainty is an absolute belief, not subject to
revision by any amount of historical evidence. Although these restrictions can prevent
overfitting in a structural model, they are often too rigid to accurately express the
modeler's true beliefs and tend to cause useful information in the historical data to be
ignored.
A BVAR model have striking similarity with a UVAR as in both types
of models each variable is allowed to depend on the current and past values of all the
variables included in the model. But at the same time it also differs from a WAR
model and resemble a structural econometric model by using prior beliefs to reduce
overfitting. However, the sources of the prior beliefs and the ways they are used are
generally different in a BVAR model than in a structural model. Whereas economic
theory is the main source of prior beliefs, a BVAR to guess which values of all the
57
coefficients will lead to the best forecasts and to specifr an extensive system of
confidences in each coefficient.
Steps in Building s BVAR:
The first step is to construct an unrestricted VAR model. A n-variable
unrestricted VAR can be written as
Y (t) = A(L)Y (t)+ x(t)+u(t), t : 1,2,........, T.
Y(t) is an (nxl) vector of variables observed at time t, A(L) is an (nxn) matrix of
polynomials in the lag operator L, X(t) is an (nxnk) block diagonal matrix of
observations on k deterministic variables,p is the (nkxl) vector of coefficients on
the deterministic variables, u(t) is the (n xl) vector of stochastic disturbances, andX
is an (nxn) contemporaneous covariance matrix. The coefficient on Z o is zero fot
all the elements of A(L), i.e., only lagged values of elements of y appear on the
right-hand side.
The i-th equation of the model is given by
y(i,t) =\laU, j,c) y(j,t - j) + x' 1t1B1i) + u(i,t)j=t r=t
where, y(i,t) and u(i,t) are the ith elements of y(t) and u(t) respectively, BQ) is
the (ftx1) subvector of B corresponding to the ith equation and a(i,j,r) is the
coefficient on the c thlagof the jth variable of the ith equation.
The second step in the BVAR modeling is to speciff
the priors for the coefficients of the variables. BVAR model assumes an independent
normal prior distributions for each of the nz m YARcoefficients, such that a(i, j,r)
is assumed to have mean 6(i, j,c) and variance s'(i, j,r). The value chosen for
58
any particular 6 (i,7, r) would represent the ' best guess ' for the value of a (i, j,r) ,
and the value chosen for the corresponding ^s 2
Q, j,c) would reflect the degree of
confidence in that guess (smaller values reflecting greater confidence). The Minnesota
prior is the most often used prior which takes into account the fact that many
economic time series follows a random walk.
The Minnesota prior means are given by
5(i,i,r1=1 ,f i= j and r=l-0 otherwise
The Minnesota prior simplifies the choice of values for the s(i, j,r)
by specifying the following standard deviation function :
s(i, j,r) =ly sG)f(i, j)l(.{i) (4)sj
where s,is the standard error of a univariate autoregression for y(i,t), y is the
'overall tightness'parameter, g(r) is a function which describes the tightness on the
rthlag relative to first lag, and f (i,j) is a function giving the tightness on the jth
variable in the ith equation relative to the ith variable. Since the variables in the model
are likely to be of different magnitudes, the ratio of the standard errors, j, tt
included as a rescaling factor to make units comparable. Equation (4) reduces the
number of prior variandes fuom nzm to n2+2 parameters. Each f (i,j)can be
thought of as the i, jth element of an (nxn) matrix which must be specified. We
assume a symmetric prior for f (i, j)
f (i,j)=t if i=i=w otherwise
59
where w is a constant. The syrnmetric prior reduces the problem of choosing n2
parameters to the problem of choosing a single hyperparameter, w. The value of w
gives the relative tightness on the coefficients of the ' other ' variables in the ith
equation.
The choice of the lag tightness function, g(r), should be such that it
reflects the increasing confidence that coefficients are close to zero as the length of
the lag increases. Two possible functions are possible
Harmonic Function : SG) = r-o
Geometric Function : SG) = dt-l
In both cases, the single decay parameter, d, must be chosen. For the harmonic
function, the choice of a larger d implies more rapidly increasing tightness and thus a
more rapidly decreasing s(i, j ,c) as lag length increases. For the geometric function,
the choice of a smaller value of d implies more rapidly increasing tightness. The
overall parameter, y, gives an overall measure of confidence in the prior, with smaller
values coresponding to greater confidence.
60
Meusures of Forecast Evaluution
The present chapter discusses alternative forecast evaluation measures
for judging the forecast performance of the competing models. This is essential to
determine the qualitative performance of the forecasts. Moreover, it has been noticed
from the various empirical works that the ranking of the forecasts change quite
substantially under altemative forecast evaluation criterion. Given this fact it may be
upto the forecaster to give subjective weights to the alternative forecast evaluation
criterion to na:row down to one particular forecasting model. Again, the alternative
criterion may give very rankings to the forecasts from the various models at different
forecast horizons. So one may, in practice, find different models producing the best
forecast at different forecasts horizon.
I. Measures of Forecast Accuracy :
The crucial object in measuring forecast accuracy is the loss fturction
L(Y ,*tr,t ,*,,,,), often restricted to L(e,*0,,), which charts the "loss", "cost" or
"disutility" associated with various pairs of forecasts and realizations. Here f ,*0., is
the k-period ahead forecast errors and e *k,t = Y ,*o -t,**,, is the k-period ahead
forecast effors. In addition to the shape of the loss function, the forecast horizon (k) is
also crucial importance. Rankings of forecast accuracy may be very different across
different loss functions and / or different horizons. This result has led some to argue
the virtues of various " universally applicable " accuracy measures. Clements and
6t
Hendry (1993), for example, argue for an accuracy measure under which forecast
rankings are invariant to certain transformations.
Nevertheless, let us discuss a few stylized statistical'loss functions,
because they are widely used and serve as popular benchmarks. Accuracy measures
are defined on the forecast errors, € r*k,t=Y ,**-Y ,**,, , or percent errors,
p t+t.t = (Y ,*r, -t ,*0,,) lY ,*r,. The most common overall accuracy measures are the
Root Mean Squared Error (RMSE) and Root Mean Squared Percent Error (RMSPE)
defined below :
RMSPE =
Two other forecast accuracy statistics that have appeared in the literature are
RMSE=ff*,,
MAE = +*l ,.r)(1) Mean Absolute Error :
(2) Mean Absotute Percentage Error : MA\E = lLl, ,.r,,1
Anyway, the above measures do not provide much direct information
about whether something better might be achieved in terms of forecasting. It is
common to compare the performance with a set of naive forecasts, given by
\. _t/t t+k,t - I t'
Theil's inequality statistics helps in this comparison :
I r r ,*r -t ,**,,)'t=1
T
Itr ,*o-Y ,)'t=1
**o
U2=
62
U2:0 implies Perfect Forecast.
U2 : 1 implies forecast is as accurate as the naTve forecast
model, that is, the model should not be used for forecasting.
IJ2 > I implies forecast inaccurate relative to naive forecast.
U2<I implies forecast accurate relative to naTve forecast.
II. Forecast Rationalilv :
Forecast accuracy measures discussed above gives us an indication of
the relative performance of alternative forecasting models. However, it is essential to
check the properties of .the forecasts generated from the competing models
individually. There are a number of notions of rationality of the forecast which
includes those of unbiasedness and efficiency. One of the properties of the optimal
forecast is that forecast "rro..
have a zero meafl.In other words, optimal forecasts are
unbiased estimates of the actual series. In the present context it means that bias test
determine if the model forecasts are systematically higher or lower than actual
exchange rate. This is.also often referred as a requirement of weak rationality which
implies that forecasts are consistent in the sense that forecasters are not systematically
mistaken their forecast.
Tests ofunbiasedness are based on a regression ofthe form :
A,*n:d+BP *h*€ t+h (1)
The joint hypothesis a=0 and B =1 entails unbiasedness. An F-test is a valid test
statistic for the joint hypotheses if the error term, p, is i.i.d. Howeyer, a problem
associated with the above regression is that serial correlation is introduced into the
error term for equations corresponding to 3 -, 6-, 9-, and 12-month ahead forecasts.
For h > 1 (where h is the forecast horizon), the forecast horizon will exceed the
63
sampling frequency (assumed to be 1), so that forecasts overlap in the sense that they
are made before knowing the error in the previous forecast. Thus this test of
rationality does not rule out serial correlation in the error process of a moving average
of order (h-1). It also seems likely that conditional heteroscedasticity exists in the
error term. To correct for these problems, a heteroscedastic, autocorrelation consistent
covariance matrix (Newey and West (1987) is used to estimate the standard errors of
coefficients in the above equation. With the use of the Newey-West estimator, the
joint test statistic is distributed as a Chi-squared.
Although the joint hypothesis a = 0 and B= 1 is popularly described
as a test of unbiasedness, it can also be viewed as a test of efficiency, in the sense of
checking that forecasts and their errors are uncorrelated. If there is a systematic
relationship between the two, then this could be exploited to help predict future errors,
and could be used to adjust the forecast-generating mechanism accordingly. Mincer
and Zarnowitz (1969) used the concept of efficiency in this sense. They define
forecast efficiency as the condition that p =l in equation (1), so that the residual
variance in the regression is equal to the variance of the forecast eror.
When the data are non-stationary, integrated series, then a natural
further requirement of the relationship between the actual and forecasts is that they
are cointegrated. Infact, Cheung and Chinn (1998) proposed a test of rationality
based on the time series properties of the actual and the predicted series : the forecast
and the actual series (a) have the same order of integration, (b) are cointegrated and
(3) have a cointegrating vector consistent with long run unitary elasticity of
expectations. This means that cointegrating vector involving actual and the predicted
series should be (1 -1). We extend the unbiasedness test in this cointegrating
64
framework by testing the restriction (1 -1 0) on the cointegrating relationship
involving Actual, forecasts and a constant.
Hendry and Clements (1998) argued that the above tests of unbiasednsess or
rationality.may be too slack in that they are satisfied by more than one predictor, or
conversely, too stringent given the typical non-optimality of most forecasts, stemming
from the complexity of economic relationships, and the open-ended number of
variables that could conceivably affect the variable(s) of interest. Such criterion may,
therefore, be of limited value as means of forecast selection leading to a plethora of
forecasts, or non at all, ifonly these criteria are used.
It has been pointed out that the regression based test of unbiasedness
rest on strong statistical assumptions. Unsystematic forecast errors need not have
fixed variance through the sample period, nor need they be normally distributed. Such
deviations from the classical assumptions may compromise the efficiency of the
regression statistic. Moreover, standard testing procedures associated in (1) are only
valid as5rmptotically when the disturbances are correlated with future values of the
regressors. To deal with these problems, Campbell and Ghysets (1995) introduced a
nonparametric testing methodology to assess the unbiasedness of forecasts. The focus
of this nonparametric approach is on the median of the forecast effors rather than the
mean. However, for symmetric distributions with finite mean, median-unbiasedness
and mean-unbiasedness are equivalent. Nonparametric tests may be more reliable than
the regression-based procedures, particularly in small samples.
We consider the tests of the unbiasedness of one-step ahead forecast
effors. Let the one-period ahead forecast effors be written as
E ir=Sr-s" r-,
65
Define a function
u(z)=1if z>0
= 0 , otherwise
The role of the u (z) function is simply to indicate whether the forecast error is
positive or negative.
Consider the signed test statistic :
st =
where t:|,2,......,T is the number of forecast errors.
Under the null hypothesis that the forecast errors are independent with zero median,
the sign statistic is distributed with Bi(T,llz),that is, as the binomial distribution
with the number of trials T and probability of success 0.5. In large samples, the
studentized version of the statistic is standard normal,
s ,-T l2 - N(0,1)
lr /4Thus, significance may be assessed using standard tables of binomial or normal
distributions. It should be noted that the sign test does not require distributional
symmetry.
Consider another statistic given by
T
w , =2"@,, )'R* ,,t=l
where R* ,, is the rank of lE ,,1 with lE ,,1,. lz' ,.1 teing placed in ascending
order.
Under the null hypothesis that forecast errors are independent and symmetric about
mean zero (and hence about azero median), W , is distributed as Wilcoxon signed
rank test. The intuition of the test is that if the underlying distribution is symmetric
T
Z"@ u)t=l
66
about zero, a "very large" (or "very small") sum of the ranks of the absolute values of
the positive observations is "very unlikely" to be high. The exact finite-sample
distribution of the signed-rank statistic is free from nuisance parameterS and invariant
to the true underlying distribution. Moreover, in large samples, the studentized
version of the statistic is standard normal,
*,-ryP- N(0,1)
The results from the bias test will enable us to tell whether forecasts are unbiased
predeictors of the actual exchange rate. A forecast may be beating the random walk
forecasts, one of the main focus of the present study, but it may be actually biased.
Then it may be difficult to tell unambiguously whether we should use those forecasts
even when they are beating the random walk.
III. Testing the equalilv of orediction mean squared erroys :
The comparisons of mean squared error (MSE) or root mean square
effors (RMSE) are merely descriptive indicating one set of forecasts has made
relatively small errors than another. This does not any way tell us whether the
difference in the MSE or RMSE between the competing the forecasting models is
significant or not. So one may get a lower MSE or RMSE compared to the simple
random walk model but that difference may not be statistically significant. So it is
very essential to find out whether the difference in the MSE or RMSE arising are
statistically significant or not.
Diebald and Mariano (1995) proposed a test of the null hypothesis of
no difference in the accuracy of two competing forecasts which allow for forecast
r (T +DQr +D
67
effors that are potentially non-Gaussian, non-zero mean, serially correlated, and
contemporaneously correlated. Suppose that a pair of h-steps ahead forecasts have
produced errors (e ,, ,e 2,) , t=1,2,......n. The quality of a forecast is to be judged on
some specified function g(e) of the forecast error, e. Then, the null hypothesis of
equality of expected forecast performance is
Els@u)-s(er,)l=o
Define d , = g(e u) - g(e r,)
Then, the test statistic based on observed sample mean is given by
7 =n-'la ,t=1
One problem is that the series d , is likely to be autocorrelated. It can be shown that
the variance of d is, asymptotically,
v G) * n-t ll ,*zfr o J
k=l
where f o is the kth autocovariance of d , . This autocovariance can be estimated by
f r=n-'\{a,-a)@,-o-d)t=k+l
The Diebold-Mariano test statistic is then
s, = [ tr @)]''' .a
Under the null hypothesis, this statistic has an asymptotic normal diskibution.
Diebold and Mariano considered mean squared eror as the standard of forecast
quality, that is, g(e) = e' .
68
V. Information Content Test :
Forecast encompassing tests enable one to determine whether a certain
forecast incorporates (or encompasses) all the relevant informatioh in competing
forecasts. The idea of forecast encompassing was formalized and extended by Chong
and Hendry (1986). Suppose we have two forecasts, f' ,*0,, artd t' ,*r,,,. Consider the
regression
Y ,*k = F o+ F ,t' ,*0,,+ F ,t' ,*k,,*t ,*k,,
If (B * F, F, ) = (0,1,0), one says that model 1 forecast encompasses model 2, and
if (P o,B,F r):(0,0,1), then model 2 forecast encompasses model 1. For any
other (B o ,0 , , P , ) values, neither model encompasses the other, and both forecasts
contain useful information.
Fair and Shiller (1989, 1990) take a different but related approach
based on methodology. Their test is popularly known as the Information Content
Test. They argued that many econometric models are used to forecast economic
activity which differ in structure and in the data used. So their forecasts are not
perfectly correlated with each other. This necessitates the developing of some test
which enables to find out whether each model have a strength of its own, so that each
forecast represents useful information unique to it, or does one model dominate in the
sense of incorporating all the information in the other models plus some. They
developed the following regression based test
(Y,*o-Y,) = d+ P(Yt,*k,,-Y,)+y(Y' t+r,z-Y,)*€,*r.,
If neither model I nor model 2 corrtun any information useful for k-period-ahead
forecasting of Y ,, then estimates of B an d 7 should be zero.In this case the estimate
of the constant term a would be the average of the k-period-change in Y. If both
69
models contain independent information of the k-period-ahead forecasting, then p
and y should both be non-zero. If both models contain information, but the
information in, say, model 2 is completely contained in model I and model 1 contains
further relevant information as well, then B but not 7 should be non-zero. So one
estimates the above equation for different model's forecasts and test the hypothesis
H ri? =0 and the hypothesis ,E/ z:T=0. lf , is the hypothesis that model I's
forecasts contain no information relevant to forecasting k period ahead not in the
constant term and in model 2, and H , is the hypothesis that model 2's forecasts
contain no information not in the constant term and in model 1.
Fair and Shiller's test bears some relation to encompassing tests but is
not exactly identical to it. For instance, Fair and Shiller does not constrain B arf y to
sum to one, as usually the case for encompassing tests. However, it is not difficult to
perform the forecast encompassing test in the Fair-Shiller framework. We can test the
null hypothesis (a, f ,y): (0,1,0) or (0,0,1). Under the null of forecast
encompassing, the Chong-Hendry and Fair-Shiller regressions are identical. When the
variable being forecasted is integrated, however, the Fair-Shiller framework may
prove more convenient, because the specification in terms of changes facilitates the
use of Gaussian asymptotic distribution theory.
YI. Market Timing Test:
Studies by Gerlow and Irwin (1991) show that statistical evaluation
measures may not yield results consistent with the actual trading profits generated by
exchange rate models. Hence, it is useful to consider a measure of economic value of
a model. Henriksson and Merton (1981) developed a test which is essentially a test of
70
the directional forecasting accuracy of a model. The directional accuracy has been
shown to be highly correlated with actual trading profits.
From a sample of N actuals and forecasts and their probabilities, form
the following contingency table, and test the independence of actuals and forecasts
Forecasts I
Actuals
<0 >0
<0
>0
P ,, (O ,,) P ,, (O ri) P,.(O,)
P ,, (O .i,) P ,, (O ii) I-p t (O i)
P,(o.,) l-p r.(o i) 1(o)
Where p ,i is the joint probability that an observation will belong to the ith row and
jth column, p.i end pi. tre the marginal probabilities, with . denoting summation
over either columns or rows. In parentheses we denote the number of observations in
each cell.
Thus,
Pr=b and Pr=P.t
Pzzl- p,
The null hypothesis that a direction-of-change forecast has no value is that the
forecasts and actuals are independent, in which case p i = p r. p .i, for all ij. (for a
brief discussion on directional analysis of forecasts refer Ash, Smyth and Heravi
(1ee8)).
The formal H-M test relies on a theorem in Merton (1981) that shows,
without recourse to a model of equilibrium returns,that a necessary and sufficient
condition for a rational investor to modiff his prior beliefs is that the sum of the two
7t
conditional probabilities of a correct
implies that the forecast has no value
forecasts, pttpz, exceeds one. This also
when actual and forecasts of a series
distributed independently and pttpz=\. Consequently, the test proposed
Henriksson and Merton tests whether actual and predicted series are independent.
The uniformly most powerful unbiased test for independence is
R.A.Fisher's Exact Test (Fisher (19a1)) which is identical to H-M's test for predictive
values. Thus the nonparametric test proposed by H-M is asymptotically equal to the
simple X2 testof independence in a 2x 2 contingency table. The true cell probabilities
are
by
are unknown, so one uses a consistent estimates i, , =! ^ao
one consistently estimates the expected cell counts under the null,
^ o,o,E ,, = +. One, therefore, can construct the statistict,o
Under the null, C - Z' ,.
b -o ''. Theno
E r=p i. P .i ,bY
-E u)'(o,ic=zi,i=l E,j
72
Chupter V
Data Soarce and Deftnitions
We use monthly data in the present study. The time period covered in
our study is from January 1993 to JuJy 1999. The exchange rate used in this study is
Indian Rupee-US Dollar bilateral spot exchange rate defined as the number of Rupees
per unit of Dollar. Money supply is measured by Ml for both India and US. Output
is proxied by the General Index of Industrial Production (IIP). For India, IIP figures
for the period January 1993 to March 1998 are given for the base year 1980-81:100,
while for the period April 1998 to July 1999, IIP figures correspond to the base year
1993-94:100. The two series are spliced to obtain the whole series at 1980-81 prices.
For US, the IIP figures correspond to the base year 1992:100.Interest rate figures for
both countries coffespond to 3-month Treasury Bill rates. Prices used in this study are
Consumer Price Index (CPf numbers. For India, we use the CPI figures for the
Industrial Workers (base : 1982:100); while those for US corresponds to the CPI for
all commodities (base : 1982:100). All the above mentioned data for India are
obtained from Handbook of Statistics on Indian Economy, Reserve Bank of India,
1999 ; while for US they are obtained from the Federal Reserve Bank, Minneopolis
website hnp://www.stls.frb.org/fred/. Monthly data for imports and exports are
obtained from the Monthly Abstracts of Statistics, Government of India, and given in
terms of rupees. Monthly output, prices, money supply and trade balance data are
seasonally adjusted.
73
Chapter VI
Empirical Results
This chapter discusses the empirical findings of the present study.
First, we present the unit root test results on various time series variables. We carried
out a host of unit root tests - (Augmented) Dickey-Fuller (DF), Phillips-Perron (PP),
KPSS and Bayesian unit root test. Results of the unit root tests are reported in
Section A. A11 the variables under the present study tums out to be an integrated
process of order 1. Given that the series are I(1) it is natural to test for the presnece of
cointegration among the economic variables of interest. We employ the Engle-
Granger (EG) and Johansen-Juselius (JJ) tests of cointegration to estimate the long-
run equilibrium relationships. Section B describes the cointegration test results and
the existence of the monetary model as a long-run equilibrium relationship. We also
test for the presence any particular version of the monetary model in the Indian
economy. This section also reports the estimation result of the vector error correction
model for the purpose of generating forecasts of the exchange rate. Section C
describes the estimation results of the various formulations of Bayesian vector
autoregression based on the Minnesota prior and thereafter, forecasts are generated
from these models. We also estimate an univariate ARIMA model (Section D) by
employing Box-Jenkins methodology for generating forecasts of the exchange rate.
This often serves as the benchmark model for forecast comparison. A forecasting
exercise has been carried out whereby forecasts are generated for one to twelve
month forecast horizon. Finally, we run abattery of forecast evaluation tests to assess
the quality of the forecasts, reported in Section E.
74
A11 variables, except three-month treasury bill rates for both countries,
are given in logarithm. Our initial estimation period is from January 1993 to
December 1996. A rolling regression technique has been adopted whereby one
sample point is added and estimation is carried out. This is continued till the last
sample point, viz. July 1999, is reached. Unit root tests have been carried out by this
rolling regression technique. This will enable us to find out whether the stochastic
trend is present in the variables under consideration for whole of the forecasting
period, that is, from January 1997 to July 1999. Unit root tests have been performed
on both the individual variables, that is on the spot exchange rate, IIP of both
countries, money supply (M1) of both countries, and on the three-month treasury bill
rates, as well as on the relative value of the variables, that is, on the relative money
supply, relative interest rate and relative IIP.
Section A : Unit Root Tests
We first carry out the traditional Dickey-Fuller (DF) and Phillips-
Perron (PP) test to find out the presence of stochastic trend in the series. Logarithm
of the bilateral spot exchange rate is a natural candidate to begin the analysis as this
is the central variable of the present study. Tablesl.l (A) and (B) gives the DF and
PP test results for the levels of the variables. We have presented here the full sample
result for all the variables, that is, covering the period January 1993 to July 1997.
The conclusion remains unchanged for any of the sub-periods starting from the
period I 993 :01 -1996:12.
DF test requires a sequential testing procedure, starting from a general
model which allows for a deterministic trend in the data to a simple model which
75
tests for the presence of a pure random walk. In the general model the statistic that
tests for the null of the unit root.is given by e ,. The calculated value of the test
statistic is -2.3064 whereas the critical value is -3.41 (at 5% significance level). Since
it is a left-tail test, this leads to the inference that a unit root is present in the spot
exchange rate. Cambell and Perron (1991) argued that absence / presence of a
deterministic trend affects the stochastic trend inference. So it is natural to test for a
joint hypothesis of the presence of a stochastic trend but absence of a deterministic
trend. This is given by the / , statistic and it is a F-test. The calculated value of the
/ rstatistic is3.2314 which is less than the critical value of 6.25 (5% significance
level) implying the presence of a stochastic trend but no deterministic trend. We then
move to the next model - one with a constant and no trend. Here the null of unit root
is evaluated by the f , statistic. As indicated in table 1.1.(A), the null hypothesis gets
accepted at 5oh significance level. We use this regression to carry out another F-test
to find out whether the DGP (Data Generating Process) is a random walk with drift.
This is carried out by using / , statistic. The constant term is turning out to be
insignificant thereby rejecting the possiblility of a random walk with drift. We
therefore proceeded to find out whether the data generating process (DGP) is a pure
random walk. This has been caried out by estimating the simplest specification - one
with no constant and no trend. Here the relevant.statistic is q . The calculated value
of this statistic is 2.2078. Given this is a left-tail test, the null hypothesis of unit root
does not get rejected atany of the 1, 5 and 10% significance level. So from the DF
test one can conclude that the series is integrated of order 1, i.e.,I(1).
Given the low power of the DF test which has the implication that it
tends to accept the null hypothesis of unit root 'too' often one needs to substantiate
76
the result of the DF tests with some other tests. Phillips-Perron (PP) test is a natural
candidate since it also has the same null of the presence of a unit root and has a better
power than the DF test. Result of the PP test for the levels is given in Table 1.1(B).
We are concerned with the statistics Z(td, )and Z(ta- , ) whose critical values
corresponds to those of e ,and 6 , statistics. The calculated value of the
Z (t d , ) statistic is -2.2456 which is less than the critical value at lYo, 5o/o, and l}Yo
significance level, indicating the presence of a unit root. This conclusion is
reinforced by the Z (t a. , ) statistics. Thus both the DF and PP tests leads to the
same conclusion of the presence of a stochastic trend in the logarithm of the spot
exchange rate series, that is, it is I(1).
To find out whether the exchange rate series is integrated of higher
order, unit root tests have been carried out for the first difference of the series.
Results of the DF tests are given in Table 1.2 (A). The calculated e s statistic
(-4.01 86) falls in the critical region at arry of the 106, 50 and, l0o/o significance level.
Given the sequential nature of the DF test, we stop where the null of unit root gets
rejected. The results of the PP test, given in Table 1.2 (B) also supports the DF
results. So, from the DF and PP tests we can finally concluded that the spot exchange
rate is integrated of order 1, i.e., it is I(1).
We now look into the time series properties of other variables which
may help to explain the movements of the exchange rate - output (proxied by the
index of industrial production (IIP)), money supply (as measured by Ml) and interest
rates (measured by the three-month treasury bill rates). This choice of the variables is
motivated by the monetary model. The results of the Dickey-Fuller and Phillips-
Perron tests are given in Table 1. We refrain from giving a detailed description of the
test procedure here. However, the table below gives the final conclusion from these
tests.
Summary of the Dickey-Fuller and Phillips-Perron test
Time Span : January 1993 - July 1999
presence
is I(1).
Except for the Indian money supply, as given by Ml, there seems to be
unanimous conclusion by both the tests that the series under consideration are I(1).
Only in Indian Ml there seems to be different conclusions produced by the two
different tests. So we may need to look into some other unit root tests to come into
any clear cut conclusion regarding this variable. In fact, latter on we perform both the
KPSS and Bayesian unit root tests to cross-check our conclusion from the DF and PP
tests.
We are, however, more interested to cany out a detailed unit root
analysis of the relative value of the variables - relative IIP, relative money supply and
relative interest rates - as these variables will be later used to explain the exchange
IS,
Exchange Rate Y Y
Indian IIP Y Y
Indian Treasury Bill Rate Y Y
Indian M1 N Y
US IIP Y Y
US Treasury Bill Rate Y Y
US M1 Y Y
Y o Unit root in the series under eratlon
18
rate behaviour. The results are given in Table 2. Consider the results for the
logarithm of the relative index of industrial production. Dickey-Fuller test results for
the levels are given in Table 2.I(A). The calculated values of the C E ,e , and E
statistics are -1.6628, -2.117 and 0.1319 respectively. All of these falls in the non-
rejection region at l, 5 and l0o/o significance level. The PP test results (given in
Table 2.1(B) also supports the conclusion of the DF test. So a stochastic kend is
present in the relative IIP. To find out whether the series is integrated of higher order,
we undertake DF and PP test on the first differences of the relative IIP. Results are
reported in Table 2.2 (A) and (B). The DF test on first differences fails to reject the
null of unit root at l%o and 5%o significance level. However, the PP test strongly
reject the null of unit root in the relative IIP series. So we can conclude that the
relative IIP series is I(1).
We now look at the logarithm of the relative money supply as given by
M1. DF tests reported in Table 2.1(A) indicate that the series is nonstationary in
level. The calculated values of the e , ,C , and g statistics are -1.1841, -1.2796 and
-2.4556 respectively. The series is tuming out to be nonstationary at loh significance
level. However, at 5%o and LlY, significance level it is coming out to be stationary.
The PP test on the levels (Table 2.1(B)) strongly supports the null of unit root at any
of the three significance level under consideration here. The DF and PP tests on first
difference of the series strongly rejects the null of unit root. In fact, the calculated
value of the f , statistic is -5.0838 (Table 2.2 (A)) which leads to the rejection of the
null of unit root in the first step only. Thus, we can conclude that the logarithm of the
relative money supply is I(1).
Finally, we consider the relative interest rates measured by the
difference between the two country's three-month treasury bill rates. This variable is
79
not in logarithm term. The calculated values of the e e ,e o and g statistics are
2.007, -2.1418 and -1.770 respectively. This clearly indicates the presence of unit
root in the relative interest rate series at lo/o and 5o/o significance level. The PP test
results on the levels strongly supports the DF conclusion. The calculated value of
Z (t d , ) and Z (t a. , ) are -1.5205 and -1.5685 respectively (Table 2.1 (B)), which
falls in the acceptance region. Thus both the tests clearly points out towards the
presence of a stochastic trend in the level of the series. To infer on the degree of
integration we carry out the DF and PP tests on the first differences of the series.
Results are given in Table 2.2 (B). Both DF and PP test concludes that the first
difference ofthe relative interest rate is stationary. So the relative interest rate series
is I(t).
Summary of the Dickey-Fuller and Phillips-Perron test
Time Span : January 1993 - July 1999
presence
is I(1).
It is a well established fact that standard unit root tests fail to reject the
null hypothesis of a unit root for many economic time series. In these standard unit
root tests (DF and PP) the unit root is the null hypothesis to be tested. An explanation
for the common failure to reject a unit root is simply that most economic time series
are not very informative about whether or not there is a unit root.
Relative IIP Y Y
Relative Ml Y Y
Relative 3-Months TB Rate Y Y
the o Unit root in the series on ls
80
DeJong et. al. (1991) provide evidence that the DF tests have low power against
stable autoregressive alternatives with roots near unity, and Diebold and Rudebusch
(1990) show that they also have low power against fractionally integrated
alternatives. Kwiakowshi et. al. (1992) developed a unit root test, popularly know as
theIKP,SS test, which tests the null hypothesis of stationarity against the alternative of
a unit root. We intend to apply this test to crosscheck our inferences from the
standard DF and PP tests.
KPSS test consists of two test statistics - ry r, where the null of level
stationarity is tested, and rl , , where the null of trend stationarity is tested, against
the alternative of a unit root. In the calculation of the KPSS test statistic, the choice
of lag truncation parameter / is very important. In the presence of large and
persistent positive serial correlation, the long-run variance S'1f typically increases
monotonically in l, so that KPSS statistic decreases as / increases. KPSS
recommended calculation of ,S2 (/) out to a value of / such that the long-run
variance estimate and hence the KPSS test statistic have " settled " down. However,
setting / too high can result in significant loss of power. Keeping this trade-off in
mind we have chosen the truncation lag parameter / to be equal to five.
Table 3 and 4 reports the KPSS test for the levels of the variables
under this study. At the outset it should be remembered that all the variables except
the interest rates are in logarithm. Table 3.1. presents the result of the KpSS test
when the null hypothesis is that the series is level stationary. For the exchange rate
series, the null of level stationarity is getting rejected even at 1% significance level at
all values of / as the calculated value of the test statistic is greater than the critical
value. Table 3.2 gives the KPSS test result where the null hypothesis is trend
81
stationarity. In the case of exchange rate, the calculated value of rl , statistic exceeds
the critical value at lo/o significance level at all /. Thus the KPSS test supports the
conclusion of the DF and PP tests that the bilateral India-US spot exchange rate
indeed contain a unit root.
KPSS test has been applied individually to the levels of all the series
under the present study. As far as the null of level stationary is considered, except for
the Indian and US three-month treasury bill rate and the US Ml, the null of level
stationary is getting rejected atlYo significance level at all I for all the variables.
The US TB rate is getting rejected at 5% significance level atl>3, while that for US
Ml the rejection level is taking place at 10% significance level for all /. In case of
the Indian treasury bill rate, for l>3,the calculated value of the ? rstatistic is less
than the critical value (at any of the 1, 5 and 10% significance level), implying that
the null of level stationary is not getting rejected. However, given that the power of
the KPSS test declines as / increases, we conclude broadly that the null of level
stationary is getting rejected for the Indian three-month treasury bill rate. In the case
of null of trend stationary, the calculated value of the ry "
exceeds the critical value at
10% significance level at all 1, except for the US IIP. For US IIP, at l>3 the null of
trend stationary is not getting rejected even at 10% significance level. Invoking the
same argument as in the case of the Indian treasury bill rate we can broadly conclude
that US IIP is indeed trend stationary. Thus, KPSS test on the level of the variables
under the present study reinforces the DF and PP tests result that all the series indeed
contain a unit root. Table below summarizes the result of the KPSS test.
82
(Level)
implies that unit root is present in the relevant series.
We also test for the level and trend stationarity for the relative value of
the variables - relative money supply, relative IIP and relative treasury bill rate. A11
the variables, except the relative interest rate, are given in logarithm term. The results
of the KPSS test are given in Tables 4.1 and 4.2.Table 4.1 give the results for the
KPSS test when the null hypothesis of level stationary. For all I , the null hypothesis
gets rejected at 5o/o significance level as the calculated value of the ? , statistic
exceeds the critical value. Table 4.2 reports the calculated value of the 7 , statistic
for testing the null hypothesis of trend stationary. For relative IIP and relative money
supply series, the calculated value of the test statistic exceeds the critical value at lo/o
significance level for all /. For the relative interest rate series, the calculated value
Exchange Rate N N
Indian IIP N N
Indian Treasury Bill Rate N N
Indian Ml N N
US IIP N N
US Treasury Bill Rate N N
US Ml N N
N stands rejection of the respective null In other words, it
83
of the r7 , statistic exceeds the critical at l0%o significance level for all / except at
/:5. So we can conclude that all the series of relative terms indeed contain a unit
root, reaffirming the resulting obtained from the DF and PP tests.
Summary of the KPSS Test
Time Span : January 1993 - JuIy 1999 \
(Levels)
rr N rr stands for the rejection o respectlve nu is. In other words, it
implies that unit root is present in the relevant series.
As mentioned earlier, most economic time series are not very
informative about whether or not there is a unit root, which attributes towards the
poor performance of the standard unit root tests. Bayesian unit root analysis offers an
alternative means of evaluating how informative the data are regarding the presence
of a unit root, by providing direct posterior evidence in support of stationarity and
nonstationarity. In Bayesian analysis the choice of the prior distribution occupies a
very important position. When there is no a priori belief regarding the distribution of
the parameter, a diffuse or non-informative prior is used. Often, a uniform prior is
used to represent the ignorance over the parameter space. This is known as the flat
prior. We employ this flat prior in our analysis.
Sims (1988) notes that it would be appropriate to put some probability
a uniformly on the interval (0,1) and some probability (1- o ) on p=1, where p is
ative Interest Rate
Relative Money Supply
84
the autoregressive parameter. A lower limit for the stationary part of the prior is also
specified such that prior for p is flat on the interval (lower limit, 1). Following Sims
we take a : 0.8 since for this level the odds between stationarity and the presence of a
unit root are approximately even. The results of the Bayesian unit root test are
reported in Table 5. As usual, all the variables except the interest rates are taken in
logarithm term.
Table 5.1 reports the Bayesian unit root result for the level of the
exchange rate series is presented. Here the squared t (0.046) is less than the Schwarz
limit (8.195) thereby strongly supporting the presence of a unit root in the the
exchange rate series. The'marginal alpha'is also above 0.90. 'Marginal alpha' is the
value for alpha at which the posterior odds for and against unit roots are even. A
higher value of ' marginal alpha ' favours the presence of unit root. A high ' marginal
alpha ' of 0.9215 supports the presence of unit root in the exchange rate series. For all
the series reported in the Table 5.1. the squared t is less than Schwartzlimit indicating
the presence of a unit root in the level of these series. However, it should be noted that
the ' marginal value ' is very low for two series - Indian Ml and US Ml. Also,
although for these two series the squared t is less than Schwarz Limit, they are not
very less than the latter value. So one need to cautiously interpret the result of unit
root for these two series and need to be substantiated with other unit root tests.
Table 5.2. reports the Bayesian unit root test for the relative variables -
relative IIIP, relative money supply and relative interest rate. For all the series the
squared t is less than the Schwarz limit indicating the presence of unit root in all
these series. However, the ' marginal alpha' for all the three variables are not very
high. So it would be appropriate to substantiate the Bayesian test with other unit root
test to conclude about the nature of these time series.
85
Finally, we present a summary of the results obtained from the various
unit root test regarding the presence of a unit root in the series under consideration.
Summary of the Unit Root Tests
Time Span : January 1993 - July 1999
(Levet)
presence
Section B : Cointegration, Long-ran Equilibrium Relationship and
Vector Ewor Coruection Model
(i) Cointegration:
From the above discussion on unit root tests we can conclude that all
the series under consideration are non-stationary. Granger and Newbold (1974)
pointed out the possibility of spurious regression in the event of running an OLS
Exchange Rate Y Y Y Y
Indian IIP Y Y Y Y
Indian Treasury Bill Rate Y Y Y Y
Indian Ml N Y Y Y
US IIP Y Y Y Y
US Treasury Bill Rate Y Y Y Y
US M1 Y Y Y Y
Relative IIP Y Y Y Y
Relative Treasury Bill Rate Y Y Y Y
Relative Money Supply Y Y Y Y
" y " stands for the presence o unit root in the series ton.
86
involving non-stationary variables. A direct fallout of such an event is the presence
of very high R2 (orR') among the " economically unrelated " variables. A common
indicator of spurious regression is R2 > d, where d is the Durbin-Watson statistic.
However, two I(1) series could infact have some economic relationship between
them and hence are " co*integrated ". Engle and Granger (1987) developed the
concept of cointegration based on the time series properties of the variables where
they talked about the possibility of a linear combination the I(1) variables which is
(0). This is a rather special condition, because it msans that all the series
individually have extremely important long-run components but that in forming a
linear combination these long-run components cancel out and vanish.
We present the result of the Engle-Granger (EG) cointegration test in
Table 6 for the full sample period January 1993 to July 1999. We work with the
relative value of the variables - relative IIP, relative three-month treasury bill rates,
relative money supply as given by Ml, and exchange rate. A11 variables except the
relative three-month treasury bill rate are in logarithm. Our choice of the variables is
motivated by the monetary models of exchange rate determination (discussed in
Chapter II). To recapitulate, monetary models of exchange rate determination
postulates existence of an equilibrium relationship among exchange rate, money
supply, interest rate,output, prices and expected inflation, depending on the variant of
the monetary model we are estimating. We first test for the Flexible-price and
Dombusch's sticky price version of the monetary model which postulates an
equilibrium relationship between the exchange rate, money supply, interest rate and
output.
The calculated value of the DF statistic (constant, no trend regression)
is -3.3001 which is greater than critical value of -3.81 (10% significance level) and
87
hence lies on the non-rejection zone. This implies that the residuals from the
regression are non-stationary and hence the variables are not co-integrated. Even
when we allow for a trend in the regression equation the residuals are turning out to
be non-stationary. The PP statistics also reinforces the DF results.
It is well established, both theoretically and empirically, that EG test
has several problems. First, it is very sensitive to the normalisation in the sense that
with one variable as dependent variable , it may not reject the null of no-cointegration
but with some other variable as dependent variable it may show cointegration.
Furthermore, it presumes that only one co-integrating vector exists among the
variables, thus ruling out the possibility of multiple cointegrating relationships
among the variables, which may be justifred by the economic theory.
Johansen (1988) and Johansen and Juselius (JJ) (1990) developed a
cointegration test among the variables which takes into account the drawbacks of the
EG test. It uses the maximum likelihood estimates to find out the number of co-
integrating vectors among the variables. JJ test treats all the variables symmetrically
and thus it prerequires an estimation of vector autoregression (VAR) where apriori
all I(1) variables are assumed to be endogenous. Optimal lag length selection is an
important problem for the estimation of VAR. There is apparantly a trade-off
involved in the lag length selection problem. A smaller lag length will preserve
degrees of freedom but could induce serial correlation problem in the system,
whereas a longer lag length would exhaust the degrees of freedom very quickly but
remove the serial correlation problem.
We are interested in choosing a lag length which gives us a VAR with
no serial correlation problem in the individual equation of the system since presence
of serial correlation will lead to forecasts which consistently overpredicts or
88
underpredicts the variables concerned. Also, it Cheung and Lai (1993) pointed out
that serial correlation is a serious problem for the Johansen approach and that the
usual lag length selection criteria (Akaike Information Criterion and Schwarz
Bayesian Criterion) may be inadequate, particularly in the presence of moving arerage
errors. In the present study we employ four tests to select the optimal lag length of
VAR on which we would be performing the JJ test - Akaike Information Criterion
(AIC), Schwarz Bayesian Criterion (SBC), Likelihood ratio test (LR), and LM tests
for serial correlation. The result of these tests are given in Table 7. The criterion for
choosing lag length on the basis of AIC or SBC is to choose that lag length that gives
us the minimum of these two statistics. Both these statistics point towards a lag length
of 1. The LR test, which under null hypothesis, follows a chi-square test, also gives us
a lag length of 1. We therefore re-estimated the VAR system with one lag of all the
variables and tested for the presence of serial correlation in the system. We found
serial correlation problem in the system. This leads us to move to a higher lag length
at which there are no serial correlation problem. We choose a lag length of three.
It is worth to point out that we are working with the relative value of
the variables. One prime reason for restricting the coefficients to be equal for both
countries is to preserve degrees of freedom given relatively small post-liberalisation
period in India. Ideally we would like to allow the coefficients to be unequal for both
countries. Whether this affects the forecasting performance is not very clear from the
literature, for instance, Meese and Rogoff (1983) found that even after allowing the
coefficients to vary across the countries there was no significant improvement in the
forecasting performance of the monetary models.
We employed the block-exogeniety test to determine whether to
incorporate a variable into a VAR. This will enable us to determine whether lags of
89
one variable granger cause any of the variables in the system. This is again a
likelihood ratio test and follows a chi-square distribution under the null hypothesis.
The results of the block exogeneity test is given in Table 8. The results clearly
indicate that null hypothesis of non-inclusion of a particular variable is getting
rejected for all the variables. For instance, if we test the null hypothesis that the lags
of the exchange rate is not present in any of the equation of the system, the LR test
statistic value is given by 25.3496 with a p-value of 0.003 indicating that we can
reject the null hypothesis even at 1% significance level. Similar results hold for other
variables indicating that none of the variables under consideration can be excluded
from our system of VAR.
With the lag length of three in VAR, we proceed to calculate the two
statistics - I ^u* and )" oo, statistics - used in the JJ test. 2 *u* has got a much
sharper alternative than )" ,,," since the later tests the null hypothesis that the number
of distinct cointegrating vectors is less than equal to r against a general alternative
whereas the former tests the null hypothesis that number of cointegrating vectors is r
against the alternative of (r+1) cointegrating vectors. So we rely on the )" ,u* statistic
to pin down the number of cointegrating vectors. The results of the cointegration test
is given in Table 9. Trace statistic rejects the null of no-cointegration at 5oh and lo/o
significance leve. 2 nu* statistic indicates that there are two cointegrating vectors as
the calculated values of the statistic exceeds the critical values at 5Yo and ljoh
significance level.
variables motivated by the monetary models of exchang e rutedetermination.
90
among the variables suggested by the monetary models. While 2 oo" statistrcs
gives us one cointegrating vector, the ), -u* statistic indicates the presence of two
cointegrating vectors.
vectors between the exchangerate, money supplies, ou@uts and interest rates.
(ii) Long-run Equilibrium Relationship :
We present the estimation result of the cointegrating vectors in Table
10. As noted earlier, we have two cointegrating vectors. The signs of the second co-
integrating vector confirms to those postulated by the monetary models of exchange
rate determination. More specifically, it confirms to the flexible-price version of the
monetary models of the exchange rate determination.
For convenience we report below the long-run economic relationship
that makes economic sense
s = 7.8110 + 1.3669 (m-m.)-2.5361(y-y-)+0.3902(r-r.)(2t.57)** (21.07)x* (31.00)*x (9.98)**
where, s is the bilateral spot exchange rate (Rupee-Dollar exchange rate), y is output
as proxied by the Index of Industrial Production (IIP), r is the interest rate as proxied
by the three month treasury bill rate, and m is the money supply as measured by the
M1. (* indicates a corresponding variable for the foreign country, here it is USA). All
the variables are in logarithm, except the interest-rate. The figures given in the
brackets are the calculated value of the likelihood ratio of testing the null of the
significant presence of the relevant variable in the cointegration relati.onships. Under
91
the null of no significant presence, the statistic follows a chi-square distribution with
one degree of freedom. Given that the critical value of chi-square with one degree of
freedom is 5.84 at l0o/o significance level and 3.24 at 5o/o significance level, we can
conclude that the variables are significantly present in the cointegrating vector.
The long-run relationship confirms to the flexible-price version of the
monetary model. As postulated by the flexible-price version of the monetary models,
exchange rate is negatively related to the relative income differential. The mechanism
is that as domestic income goes up relative to the foreign income, there is an excess
demand for money in the domestic economy. To clear the money market, prices
decrease which in tum, leads to an appreciation of the domestic currency via the
Purchasing Power Parity (assuming to be holding continuously in this class of
models). This then ensures the negative relationship between the exchange rate and
the output. Moving on to the interest differential term, our results confirm to the
theoretical signs of the flexible-price monetary model. Here, the exchange rate is
positively related to interest differential. As domestic interest goes up relative to the
foreign interest rate, which basically reflects higher domestic expected inflation,
people hold less domestic money, thereby creating an excess supply in the domestic
money market. To clear the money market, prices increase and as Purchasing Power
Parity condition is assumed to hold continuously, this leads to a depreciation of the
domestic curency (an increase in s). Finally, relative money supply and the spot
exchange rate are positively related, again confirming to the theoretical conclusions of
the flexible-price version of the monetary models. An increase in the domestic money
supply vis-a-vis the foreign money supply creates an excess supply in the domestic
money market. Domestic prices increase to clear the money market which, via the
purchasing power parity condition, leads to a depreciation of the domestic currency.
92
LOne of the restrictions, implied by the monetary models, that is
frequently tested is the long-run proportionality betrveen money supply and exchange
rate. We have tested this restriction in our cointegrating framework. This test imposes
a value of one to the money supply coefficient and carry out a likelihood ratio test.
Under the null hypothesis of the restriction being valid, the test statistic follows a chi-
square statistic with one degrees of freedom. The calculated value of the statistic is
given as X2 (1)=19.848 withap-valueof 0.000. Thisimpliesthattherestrictionof
long-run proportionality between the exchange rate and money supply gets rejected at
the conventional l' , 5o/o and l0% significance level. However, as noted by
MacDonld and Taylor (1993, 1994) and Choudhry and Lawler (1997), the rejection of
this proportionality restriction does not invalidate the monetary model as a long-run
equilibrium condition given that signs of the cointegrating vector corresponds to those
postulated by the monetary model. We, therefore, conclude that the flexible-price
monetary model is a valid framework for analysing exchange rate behaviour in the
Indian Economy.)
We now focus on the second cointegrating vector which does not
confirm to the economic theory of the monetary models of exchange rute
determination. We present the second long-run equilibrium relationship given as
s =3.7770 + -0.1158(m-m.)+2.0145 (y-y" )-0.5409 (r-r.)(3.ee43) (0.0877)(0.046) (0.767)
(1.3018)(0.2s4)
(1.3851)(0.23e)
First parenthesis gives us the likelihood ratio value of testing the null
hypothesis of no significance of the relevant variable in the cointegrating
relationships. The second parenthesis contains the corresponding p-values. Only the
constant term was turning out to be significant at the SYosignificance level. All other
93
variables are not only incorrectly signed but also statistically not significant as
indicated by a very high p-value. Plots of the two cointegrating vectors clearly bring
out the mean -reverting tendency of the economically meaningful cointegrating
vector. The other co-integrating vector, one not confirming to the monetary model,
does not show the same mean-reverting tendency.
ERROR CORRECTION TERMS
Fotu
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
T
\.aA r^ /\nn.A-^ Ar, ,- rrv vv
^\A^ .J\\^^\- J"w
YEAR
ECMl is the plot of the error correction term which confirm to the Monetary model
ECM2 is the plot of the error correction term which does not confirm to the Monetary model
(iii) Vector Error Correction Model :
We use the economically meaningful cointegrating vector to develop a
vector effor colrection model (VECM) to generate out-of-sample forecasts. Engle and
Yoo (1987) pointed out that forecasts taken from cointegrated systems are 'tied
together'because the cointegrating relations must 'hold exactly in the long-run'. They
demonstrate in a series of Monte Carlo experiments that incorporating cointegration
into the forecasting model, can reduce mearl squared forecast erors by up to 40%o at
94
medium to long forecasts. Lin and Tsay (1996) found that for simulated data imposing
the'correct'unit-root constraints implied by cointegration does improve the accuracy
of the forecasts. However, they obtained a mixed results for the real data sets.
Result of the full sample estimation of the vector error correction
model (VECM) is given in the Table 11. The estimation results are not very
encouraging, except for the error-correction term which is negative (-0.08032) and
significant (t-statistic is -3.1284). This shows that the any deviations from the long-
run relationship gets corrected by 8 percentage points each period. The individual
t-statistics of other regressors are mostly not significant, except the first lag of the
relative interest term. But this is not totally unexpected because of the possible
presence of the multicollinearity problem among the lagged regressors. But what is
really important for the effor-corection model is that the error correction term should
be negatively signed and statistically significant, thereby justifying the estimation of
the error-correction model, depicting the short-run adjustments to the long-run
equilibrium.
Section C : Bayesian Vector Autoregressions
Continuing with our multivariate analysis, we turn to the bayesian
estimation of the monetary models. Selection of priors is the most important task
involved in the bayesian estimation of the vector autoregressive models. Bayesian
model assumes an independent normal prior distribution for each of the coefficients.
"""iV" use the most commonly used Minnesota prior which involves specification of
three "hyper-parameters" - )", overall tightness parameter ; w, symmetric weights
given to the lags of other variables in the equation; and d, decay parameter controlling
95
for the decreasing importance of the lags of the variables. Since the ultimate motive of
developing a bayesian VAR is to generate forecasts which beats the random walk, we
choose that prior which minimises the average of the one to twelve-step mean squared
errors and Theil's U statistics for the out-of-sample forecasts. That is, we estimate the
bayesian model initially with each prior over the period Jan. 1993 - Dec.1996, and
then used a rolling regression method to generate a sequence of one to twelve-step
ahead forecasts. This gives us, for example, 31 one-step ahead forecasts, 29 tfuee-
steps ahead forecasts and so on. We compute the root mean square erors and Theil's
U statistics for each of the forecast horizons and take an average of these statistics.
We choose that prior which gives us the minimum of the ayerage root mean square
erors and Theil's inequality. However, this is also not a very easy proposition as
there could be an infinite number of such combinations. We relied on the past
empirical studies on the bayesian analysis to restrict the prior choice space among the
combinations of )" :0.2,0.1, w :0.4,0.5, 0.6, and d: 1.0, 2.0 (refer Doan (1990),
spencer (1993), Todd (198a)). We use a harmonic decay function to tighten up the
prior with the increasing lags.
The result of this analysis is given in Table 12. We start with the
parameters of the prior recommended by Doan (1990). The overall tightness, I , and
the harmonic lag decay, d, are set at 0.2 and 1 respectively. A symmetric interaction
function - f(ij) - is assumed with w:0.5. The average of the root mean square errors
(RMSEs) and Theil's U statistic is given by 0.05417 and 0.9097 respectively. We then
did a grid search for the optimal values of the prior. We kept the overall tightness
prior equal to 0.2, but reduced the value of w to 0.4. It should be noted that reducing
the value of w, that is, decreasing the interaction, tightens the prior. We have marginal
improvement in terms of average RMSE (0.05376) and Theil's U (0.9035). Keeping
96
the harmonic lag decay fixed at 1.0 and varying the values of )"and w, we find that
), : 0.1 and w:0.4 produces the minimum of the average RMSE and Theil's U. We
then varied the harmonic lag decay, d, parameter to 2. So searching over all the
parameter values we conclude that the combination of ),:0.1, w : 0.5, and d:2.0 is
optimal as this gives us the minimum average RMSEs and Theil's U. We use this
prior to generate the forecasts from the bayesian VAR.
We attempted to estimate the Frankel real-interest differential and
Hooper-Morton model of exchange rate determination. As akeady discussed in the
chapter II, Frankel's model incotporates an expected inflation term in the exchange
rate determination equation, while the Hooper-Morton model additionally introduces
a cumulative trade balance to capture the role of stock-flow interaction in exchange
rate determination. Various proxies have been tried out in capturing the expected
inflation differential between countries, most popular being the long-run government
bond rate differential and past twelve-month inflation differential. However, because
of the lack of time series data on long term government bond rate for India and
inherent backward nature of the other proxies, we decided to fit an ARIMA model to
the inflation series of both the countries and generate twelve-month ahead forecasts as
the proxy for the expected inflation differential. For the cumulative trade balance, we
restricted ourselves to the bilateral cumulative trade balance between the two
countries to remove third counky effect that often get introduced if we take overall
trade balance for both the countries.
Dickey-Fuller and Phillips-Perron test results for the inflation series for
both countries, generated from the consumer price index numbers (CPD, and the log
of the cumulative bilateral trade balance (in logarithm term) are given in the Table 13.
The results indicate that we can reject the null of no-unit root for both the series. So
97
we went on to fit an ARMA model for both the inflation series. Our total sample
period spans 1980:01 to 1999:07, but we adopt a rolling regression method to
generate twelve-month ahead forecasts where our initial estimation period is from
1980:01 to 1993:01. Forecasted inflation series for both the countries turned out to be
stationary.
We face a stumbling block due to the fact that both the expected
inflation differential and the bilateral cumulative trade balance series tumed out to be
stationary. This prevented us from testing the theory in the cointegration framework
in the sense that we cannot include them directly in the cointegration relationship and
check whether their signs confirm to the theory. But still to find out whether the
presence of these variables does play any role in forecasting of exchange rate, we use
these variables in developing a bayesian vector autoregression. We use a non-
informative or flat prior on these I(0) variables in our Minnesota prior selection
framework. We employ the same prior as we did earlier in the bayesian estimation of
the monetary model. We call this model Bayesian VAR with deterministic variable
(BVARD).
As discussed earlier, there is a debate whether it is appropriate to
restrict the coefficients to be equal for the home and foreign countries in the monetary
model. We argued that because of relatively small post-liberalisation period we
refrain from allowing different coefficeints for the two countries. However, the
bayesian approach helps us to avoid the degrees of freedom problem. We therefore
estimate a bayeseian version of the monetary model where we allow the coefficients
of the regressors to vary between the two countries. This we term as an
o' LJnrestricted Bayesian Vector Autoregression (UBVAR) ". One clarification
should be done at this stage. Our usage of the word ounrestricted' have got a special
98
meaning of allowing the coefficients of the regressors to vary across the countries in
the exchange rute determination equation. One should not be counfused by
considering bayesian VAR as unrestricted model as in the bayesian VAR we always
impose restrictions in the form of the prior values of the parameters. We employ the
same grid search method as in the restricted bayesian model selection. The results are
reported in Table 14. The combination of )":0.1, w - 0.4, and d:1.0 is turning out to
be the optimal prior based on the average of the RMSE and Theil's U statistics. We
use this prior specification to generate the forecasts from the unrestricted bayesain
vector autoregression (UBVAR).
Section D : Autoregressive Integrated Moving Average Models (ARIMA)
So far we have discussed multivariate models which require a
relatively large information set for estimation and updating the forecasts from these
models may be a costly procedure. There is one class of univariate models,
popularized by Box-Jenkins, which uses very little information set for estimation and
forecasting and updating forecasts from these models are very easy compared to the
multivariate models. These are autoregressive moving average (ARIMA) models.
These models are popularly used for generating short-term forecasts and they performI
quite well in terms of out-of-sample forecasting performance compared to the
multivariate forecasting models.
First step in building an ARIMA model is to look at the autocorelation
function (ACF) and partial autocorrelation (PACF) functions to have an idea about the
nature of the data generating process. The plots of ACF and PACF are given in the
appendix. Plots of ACF and PACF on the levels of the log of the exchange rate shows
99
that ACF dies down very slowly while PACF has one spike dt lag 1 after which it dies
down very rapidly. Problem with this visual inspection is that this kind of ACF and
PACF behaviour is generated by both an AR(l) process and a simple random walk
process. So it is not possible to conclude from the naked eye whether the series is
stationary or non-stationary. This information is very crucial as ARIMA modeling
presupposes the series to be stationary. ACF and PACF plots of the first difference of
the series shows that both of them dies down very rapidly. This increases the
suspicision that the log of the spot exchange rate series could be non-stationary.
Infact, when we perform the formal tests of unit root on the spot of the exchange rate,
discussed at the beginning of the chapter, it indeed turned out that the log of the
exchange rate series is first-difference stationary.
We proceeded to fit an ARIMA model on the first difference of the log
of the exchange rate series. We give the estimation results for various specification of
the ARIMA models spanning the time period 1993:01 - 1996:12 (our first estimation
period) and 1993:01 - 1999:07 (full sample period) in Table 15. We choose that
ARIMA specification that passes through a battery of model adequacy measures -absence of serial correlation, significant coefficient estimates and satisff the principle
of parsimony. Apart from the above mentioned battery of tests one also needs to keep
in mind that the chosen model satisff the two important requirements of inevitability
and stationarity. Based on all these criterion, we choose the ARIMA (2,1)
specification. This model has been used to generate the forecasts of the exchange rate
and compared with the multivariate forecasts from the VAR models.
100
Section E : FORECAST EVALUATION
Monthly exchange rate forecasts were generated for the simple random
walk (RW), ARIMA, vector error corection model (VECM), bayesian vector
autoregression (BVAR), bayesian vector autoregression with deterministic variables
(BVARD), and unrestricted bayesian vector autoregression (UBVAR) models across
I-,3-, 6-,9-,and l2-month forecast horizons. It should be noted that we work with
the logarithm value of the spot exchange rate and forecasts are also generated for the
logarithm value of the spot exchange rate. For compariosn for forecast performance
we have also developed two multivariate models - a level VAR (LVAR), which is
justifiable given that the variables are cointegrated, and a bayesian vector error
correction model (BVECM), which is nothing but the usual bayesian BVAR
augmented by the effor coffection term. A flat prior is used on the coefficient of the
elror colTection term.
Evaluation tests were carried out over the out-of-sample period
1997:01-1999:07- We adopt a rolling regression estimation methodology for
generating the out-of-sample forecasts. In this method we initially estimate all the
models over the period 1993:01-1996:12 and forecasts are generated for the period
1997 01-1999:07. Next, we increase our estimation period by adding one sample
point and reestimate all the models for the period 1993:01-tr997:01and based on this
we generate our out-of-sample forecasts. This forecasting strategy gives us 31 one-
month ahead ,29 three-month ahead, 26 six-month ahead, 23 nine-month ahead, and
20 twelve-month ahead forecasts. For the error-coffection models, however, the
cointegrating vector was obtained from the full sample estimation and was fixed at
their long-run values while estimating the models for the various sub-samples.
101
(i) Descriptive Statistics :
Descriptive statistics on forecast accuracy are reported in Table 16.
Forecast error is calculated as the spot rate minus the forecast rate. Table reports two
most frequently used descriptive statistics - root mean square error (RMSE) and
Theil's U statistic (U). It also reports three other statistics that also sometime appear in
the forecasting literature - mean absolute error (MAE), root mean square percentage
error (RMSPE) and mean absolute percentage error (MAPE).
One Month Forecast Horizon :
We first look at the one-month ahead forecast performance of the alternative models.
One of the most challenging task that a forecaster faces is to generate forecasts that
beats the naiVe forecasts charucteized by simple random walk forecast, especially at
the short forecast horizon. The most popular statistic to find out these is Theil's
inequality statistic. The model which beats the random walkforecast has a value of U
less than.l. In case of one-month ahead forecasts, only BVECM model has U>1.
VECM produces the best forecasting performance with U:0.86629, followed by
UBVARI, BVARD and LVAR. In terms of RMSE, again, VECM produces the best
forecast as it possesses the minimum RMSE (0.01287). This is less than the RMSE of
the RW forecast (0.01486). ARIMA occupies the second position with a RMSE value
of 0.013943 followed by UBVAR with RMSE of 0.013971. LVAR, BVAR, and
BVARD very marginally outperforms the RW model in terms of RMSE. In terms of
other three reported statistics, VECM and ARIMA consistently outperforms the RW
model in the one-month horizon. For the other models, the picture is not that clear.
For instance, in terms of RMSPE, LVAR, BVAR, BVARD, BVECM and UBVAR
102
BVECM and UBVAR outperforms the random walk, while all of them show a poor
performance in terms of MAE and MAPE. We, however, rely on the Theil's U and
RMSE to conclude that most of the models beat the random walk forecasts at the one-
month horizon with VECM model posting an all-round impressive performance
followed by ARIMA, UBVAR, BVAR, BVARD and LVAR.
Summary of One Month Ahead Descriptive Statistics
Noie : The numbers in the cells are the rankings of the various forecasting models
across various forecast evaluation measures at this forecast horizon.
Three Month Forecast Horizon :
We now move on to the three-month ahead forecasting performance
analysis of the competing models. Here the rankings of the various models changes
substantially compared to the one-month ahead forecasts. VECM, which was tuming
out to be the best model in terms of one-month ahead forecasting performance, posted
a very poor perfofinance in the three-month horizon. Indeed, it failed to beat the
random walk forecasts in terms of RMSE and also had a Theil's U value greater than
one. ARIIMA model continued with its impressive performance by beating the
I I t is worth to point out again that the term Unrestricted Bayesian VAR has been used in this paper toindicate that the coefficients of the regressors in the exchange rate determination model are allowed to
Descriptive
StatisticsRW ARIMA VECM LVAR BVAR BVECM BVARD UBVAR
RMSE
THEIL'S U
MAE
RMSPE
MAPE
r03
random walk model in terms of all the descriptive statistics. But it was the IIBVAR
which produced the best forecast in this horizon by posting the lowest RMSE
(0.029712) and Theil's U (0.90191) among the competing models. BVAR, BVARD
and LVAR also outperformed the RW model in terms of RMSE and Theil's U. For
other statistics, the situation is the same as the one-month ahead forecasts with the
models out-performing the random walk model in terms of RMSPE, while
performing worse in terms of MAE and MAPE. Thus we can conclude that in the
three-month horizon, it is the UBVAR which occupies the top rank followed by the
ARIMA, BVAR, BVARD, and LVAR. BVECM continued to show a poor
performance in the three-month ahead forecast horizon also while VECM joined this
group.
Summary of Three Month Ahead Descriptive Statistics
Note : The numbers in the cells are the rankings of the various forecasting
across various forecast evaluation measures at this forecast horizon.
Descriptive
Statistics
ffiTHEIL'S U
MAE
RMSPE
MAPE
RW
)
5
2
5
2
ARIMA VECM LVAR BVAR BVECM BVARD UBVAR
2
2
1
I
I
8
8
7
8
6
6
6
6
7
7
aJ
J
4
4
5
7
7
8
6
8
4
4
5
J
4
1
1
J
2
J
vary across the countiries.vary across the countrres.
104
Six Month Forecast Horizon :
At the six-month ahead forecast horizon, it is the UBVAR model
which gave the best forecasting perfonnance in terms of the descriptive statistics.
VECM, which posted a very poor perfofinance at the three-month horizon, bounced
back to register an impressive performance at the six-month horizon by beating the
random walk forecast both in terms of RMSE and Theil's U statistics. ARIMA
continued with its impressive performance by occupying a close second position after
the UBVAR model. BVAR also outperformed the random walk forecasts at this
horizon. BVECM continued with its dismal performance at the six-month horizon
while the LVAR forecasts were unable to beat the random walk forecasts. In terms of
RMSPE, all the models which have U<1, beats the random walk forecasts. Contrary
to earlier forecast horizon, ARIMA and UBVAR beats the random walk forecasts in
terms of MAE and MAPE criterion also. So we can conclude that at the six-month
forecast horizon AR[MA, UBVAR, BVAR, and BVARD outperformed the random
walk forecasts at the six-month ahead forecasts.
Summary of Six Month Ahead Descriptive Statistics
Note : The numbers in the cells are the rankings vanous
across various forecast evaluation measures at this forecast horizon.
DescriptiveRW ARIMA VECM LVAR BVAR BVECM BVARD UBVAR
Statistics
RMSE
THEIL'S U
MAE
RMSPE
MAPE
6
6
aJ
6
J
2
2
1
1
1
5
5
6
J
4
7
7
7
7
7
J
aJ
4
5
6
8
8
8
8
8
4
4
5
4
5
1
I
2
2
2
105
asting models
At the nine-month ahead forecast horizon, ARIMA, VECM,
BVAR, BVARD and UBVAR models beats the random walk forecasts by the virtue
of having a Theil's U statistic value of less than one and RMSE less than the
corresponding figure for the random walk model. UBVAR model again outperformed
all other models in terms of RMSE (0.06398) and Theils U statistics (0.785805).
VECM comes a close second with a RMSE value of 0.06615 and U:0.812410.
BVAR model occupies the third position followed by the ARIMA and BVARD.
LVAR and BVECM model performed very poorly and was unable to beat the random
walk model as they have a Theil's U value of greater than one. In terms of RMSPE
and MAE the models with U<l beats the random walk model, while in terms of
MAPE, ARIMA, VECM and UBVAR model beats the random walk forecast. So at
the nine-month ahead forecast horizon I-IBVAR model outperforms all other models
in terms of all the statistics followed by the VECM.
Summary of Nine Month Ahead Descriptive Statistics
Note : The numbers in the cells are the rankings of the various forecasting models
across various forecast evaluation measures at this forecast horizon.
Descriptive
Statistics
mTHEIL'S U
MAE
RMSPE
MAPE
RW ARIMA VECM LVAR BVAR BVECM BVARD UBVAR
6
6
6
6
4
4
4
2
J
1
2
2
J
1
2
7
7
7
7
7
J
aJ
5
5
5
8
8
8
8
8
5
5
4
4
6
1
1
1
2
J
106
Twevle Month Forecast Horizon :
At the twelve-month forecast horizon, the rankings of the models
remains unchanged compared to the nine-month ahead forecasts.'VECM model
outperformed all other models including the random walk model, followed by the
UBVAR, BVAR, BVARD, ARIMA and LVAR. A11 of these models also beats the
random walk forecasts.
Summary of Twelve Month Ahead Descriptive Statistics
Note : The numbers in the cells are of the varibus forecasting models
across various forecast evaluation measures at this forecast horizon.
Major ftndings From Descriptive Statistics :
considered.
better than the univariate benchmark model across the forecasting horizon.
However, no one multivariate model consistently out-predicts the forecasts from
the ARIMA model. At one month forecast horizon it is the VECM, at three and
six month it is UBVAR, at nine month they are IIBVAR, VECM and BVAR,
DescriptiveRV/ ARIMA VECM LVAR BVAR BVECM BVARD TIBVAR
Statistics
RMSE
THEIL'S U
MAE
RMSPE
MAPE
7
7
7
6
5
5
5
4
6
6
6
7
7
1
3
2
aJ
aJ
8
8
8
8
8
4
4
J
5
6
2
2
5
2
2
4
4
107
while in the twelve month they are VECM, IIBVAR, BVAR and BVARD. A11
these rankings are based on RMSE and Theil's U statistic.
The performance of the ARIMA model deteriorates over the longer forecast
horizon. This is expected as ARIMA model is specifically used to generate
forecasts over the short and medium forecast horizon. Infact, it consistently
occupies the second position at one, three, and six month ahead forecast horizon.
VECM vs Bayesian Vector Autoregression : No clear cut conclusion emerges
from the present analysis - it depends on the forecast horizon. At one month and
twelve month forecast horizon, it is the VECM which outperforms all the bayesian
models. However, at other forecast horizon bayesian models, specifically,
UBVAR, outperforms the VECM. Again, this conclusion is based on the RMSE
and Theil's U statistics.
Among the bayesian models, it is the UBVAR which is turning out to be the best
performer across the forecast horizon followed by the BVAR.
) Forecast performance of all the models deteriorate as we move to the future which
is evident from the increasing RMSE, RMSPE, MAE and MAPE. This confirms
to the theory that forecast errors increases as we try to generate forecast in the
distant future.
From the above discussion on the descriptive statistics, we conclude
that BVECM and LVAR are not performing as well compared to the other competing
models. So we drop these two models from our further analysis. We concentrate on
the following 5 models - VECM, BVAR, BVARD, UBVAR and ARIMA - in our
further analysis of the quality of the forecasts.
108
getting accepted at 10 , 5o/o and 10% signifrcance level. For the other models, the null
of unbiasedness is getting accepted at only 1% significance level and not at 5o/o or
t0% significance level. The efficiency test results arc far more encouraging compared
to the one-month forecast horizon. For all the models null of efficient forecasts are
getting overwhelmingly accepted. Thus, we can conclude that at three-month forecast
horizon forecasts from all the competing models are both unbiased and efficient.
We move on to the nine-month forecast horizon (Table 19.3). The
results are not very encouraging at this forecast horizon. Except the forecasts from
the ARIMA model, none of the model produces unbiased forecasts. Even for the
ARIMA model, the null of unbiasedness of forecasts get accepted at 1o/o significance
level. Moving on to the efficiency test, the results are agun turning out to be negative.
None of the forecasts are tuming out to be efficient.
Summary of Unbiasedness and Efficiency Tests :
unit root. This holds across all the forecast horizon.
Actual and Predicted series are cointegrated, at one, three and nine month forecast
horizon. While they are cointegrated by both the Engle-Granger and Johansen-
Juselius methodology at one and nine month ahead forecast horizons, they are
cointegrated only by the Johansen-Juselius method at six month forecast horizon.
At one month ahead forecast horizon, none of the forecasts are unbiased. Only
ARIMA and VECM turned out to be efficient (Mincer-Zarnowitz) at this forecast
horizon.
111
A t*h:d+BP,*o*€,*r,
where A ,*o is the actual series, P ,*o is the predicted series and h is the forecast
horizon.
The null of unbiasedness is given by the joint hypothesis of
a,=0, and F =l
while, the null of efficiency is given by
f=lGiven the existence of cointegration between the actual and predicted
series, it is natural to test for the above hypothesis as restrictions on the co-integrating
vector. The restrictions on the co-integrating vector could be thought of as the
following restrictions : (1 -1 0) as the unbiasedness restrictions and (1 -1) as the
efficiency restrictions, where the elements in the bracket corresponds to the actual
series, predicted series and a constant respectively. Results are given in Table 19.
Table 19.1 reports the result for the one-month ahead forecast horizon. As far as the
bias test is considered, none of the model's forecasts are turning out to be unbiased as
the calculated value of the chi-square statistic exceeds the critical value at the
conventional significance levels. In case of the efficiency test, ARIMA and the
VECM's forecasts are turning out to be efficient as the null hypothesis cannot be
rejected at lo/o,5o/o and 10% significance level for ARIMA and lo/o for the VECM.
All other tests fails to pass this test. Thus, at one-month forecast horizon, only
AzuMA and VECM qualiff among the competing models as far as the efficiency
criterion is considered. All other models fails to pass both the tests.
At the three-month forecast horizon, the unbiasedness and efficiency
criterion is satisfied by all the competing forecasts (Table 19.2). We first focus on the
unbiasedness test results. For the ARIMA and VECM the null of unbiasedness is
110
(ii) Forecast Rationality : Unbiasedness and Efficiency Tests
(a) Parametric Tests :
It is important to consider the time series properties of the actual and
the predicted series with the advent of the unit root and cointegration literature
(Cheung and Chinn, 1998 ; Hendry and Clements, 1993). Two of the prime
requirements for ' consistent ' forecasts are (1) actual and predicted series should be
integrated of the same order, and (2) they should be co-integrated. Otherwise the
forecast effors will have unbounded variance. The summary results of the unit root
tests, across the forecast horizon, are reported in Table 17. We employ the traditional
Dickey-Fuller and Phillips-Perron test of unit root to assess the order of integration of
the actual and predicted series. The results indicate that the actual and predicted series
are integrated of order 1, that is, they contain a stochastic trend. Thus, the first
requirement of consistent forecast, viz., the actual and the predicted series are
integrated of the same order gets satisfied. We employ the Engle-Granger and
Johansen-Juselius cointegration test to find out the existence of cointegration between
the actual and predicted series. The summary results are given in Tables 18. Results
indicate that the actual and the predicted series are co-integrated by both EG and JJ
methodology at one- and nine-month forecast horizon, while it is co-integrated by the
JJ methodology at the three-month forecast horizon but not by the EG method.
However, the null of no-cointegration could not be rejected by EG and JJ method at
six- and twelve-month forecast horizon. For all the cases where co-integaratiotn
exists, there is only one co-integrating vector.
Two of the desirable properties underlying a rational forecasts are
unbiasedness and efficiency properties of the predicted series (refer chapter IV). To
test for the unbiasedness and efficiency we consider the following regression
109
unbiased or efficient.
efficiency tests could be attributed to the relatively small out-of-sample forecast
horizon. Given alarge out-of-sample forecast period, one may able to overturn the
negative results obtained above.
(b) Non-Parumetric Tests :
So far we have discussed the parametric test of unbiasedness
and effrciency and the results have been, in general, very discouraging. However, as
discussed in the chapter IV, parametric unbiasedness test does not perform well in a
small sample size. Our relatively small sample period of 3l forecasted values is a
good justification for carrying out non-parametric tests of unbiasedness which has
better small sample properties. We employ two non-parametric tests - sign-test and
wilcoxon rank-sum test. While the former tests for median unbiasedness rather than
mean unbiasedness, the later one, under the assumption of symmetric distribution of
forecast elrors, tests for mean unbiasedness. The results of these tests are reported in
Table 20. Sign-test is a two tail test where we reject the null hypothesis if S <r or if
S>n-f, where n is the number sf *'ve and -'ve errors and t is the critical value. In
case of one-month ahead forecast errors, t:9.993. So reject the null of median
unbiasedness at 5oZ significance level if S is less than equal to 9.993 or if S is greater
than equal to 21.007 .In case of ARIMA forecasts at one-month forecast horizon the
calculated value of S is 15 which falls in the no-rejection zone. We, therefore,
conclude that ARIMA forecasts are unbiased based on the sign test.
As mentioned earlier, we also calculate the Wilcoxon rank sum test to
check for median unbiasedness which under the assumption of s5rmmetric distribution
tt2
also implies mean unbiasedness. We reject the null of mean (: median) : 0 at the
significance level a if thecalculated uul'rr" of the statistic $fRS) exceeds W ,-o,, or
if it is less than W orz. If the calculated value of WRS is between W orz arrd W ,_o,,
or equal to either quantile, accept the null hypothesis. In case of one-month ahead
forecasts, we reject the null hypothesis if WRS > 348 or if WRS < 148, otherwise we
accept the null hypothesis. For ARIMA forecasts, the calculated value of WRS is 273
which does not fall in the critical region. Thus we can conclude from the Wilcoxon
rank-sum test that the ARIMA forecasts are mean unbiased. So both the non-
parametic test of biasedness gives us the same result that ARIMA forecasts are
unbiased at one-month ahead forecasts. This is in contrary to the parametric test of
unbiasedness where the nullof unbiasedness was strongly getting rejected. Given the
fact that these tests have good finite-sample power and are insensitive to deviations
from the standard assumptions of normality and homoscedasticity that are very
critical for carrying out the parametric tests, we conclude that the ARIMA forecasts
are indeed unbiased.
We carried out non-parametric unbiasedness test, both sign and
Wilcoxon rank-sum tests, for the forecasts generated from the VECM, BVAR,
BVARD and UBVAR. The results are reported in the Table 20. The result clearly
indicates that for none of these models the forecasts were biased at the one-month
forecast horizon, except the VECM forecasts. In case of the forecasts from the
VECM, the null of unbiasedness is getting rejected by the Wilcoxon rank-sum test.
However, the sign test accepts the null of median unbiasedness. So, in general, the
nonparametric test overturns the conclusions of the parametric tests of unbiasedness
discussed earlier where none of the forecasts are turning out to be unbiased. We
refrain from performing the non-parametric bias test as either the test procedure could
ll3
be very conservative, even asymptotically, or it could have avery low power (Diebold
andLopez (1996)).
Summary of the Rationality Tests :
for all the models at one month ahead forecast horizon. At three month forecasts
horizon all the models produce unbiased forecasts. Again, at the nine month
ahead horizon, none of the model's forecasts are tuming out to be unbiased.
of forecasts from all the models, except the VECM. For forecasts from the
VECM, while the sign test does not reject the null of unbiasedness, the Wilcoxon-
rank sum test does reject the null. Given that the Wilcoxon-rank sum test assumes
that forecasts errors are symmetrically distributed, one may fall back on the sign
test to conclude that forecasts are unbiased.
form ARIMA and VECM are efficient. None of the other model's forecasts are
efficient. In case of three month ahead forecast, all the forecasts are turning out
to be efficient. However, at the nine month ahead forecast horizon, the results are
again tuming out to be negative.
(tii) Equality of Forecast Eruors Test :
Diebold and Mariano (1995) pointed out that mere looking at the
various forecast accuracy statistics and concluding that one model is outperforming
the other is not correct. We employ a test developed by them which tests for the null
hl.pothesis of no difference in the accuracy of the two competing models. We use the
tt4
loss-function of the form g (e) = e' , which allows one to test whether there is any
significant difference between the root-mean-square effors of the competing models.
This is important because it may so happen that in terms of root-mean-square emors
some model may be outperforming the random walk model but there may not be any
statistically significant difference between the two root-mean-square elrors.
We present the results of the Diebold-Mariano (DMS) test in Table2l.
Let us first consider the one-month ahead forecasts. The calculated value of the DMS
for the pair of simple random walk (SRW) model and the ARIMA model is 1.2535,
which is insignificarrt at 10% significance level. (This test statistic, under the null
hypothesis, asymptotically follows a standard normal distribution, and it is a two-tail
test). This implies that there is no significant difference between the forecast error of
the SRW and ARIMA models. The DMS between the SRW and VECM is -3.7199
which is statistically significant at 10% significance level, implying that there is
significant difference between the forecast errors of these two models. This in turn
implies that the RMSE for the VECM is different from that of the SRW model. Given
that the RMSE of the VECM is lower than that of the SRW model, we can conclude
from the Diebold-Mariano test that the RMSE of the VECM is significantly lower
than that of the SRW model at one-month ahead forecast horizon. However, for other
models - LVAR, BVAR, BVARD and tiBVAR- the DMS is not statistically
significant implying that the forecast effors of these models are not significantly
different from that of the SRW model.
We proceed to test for the equality of the forecast errors at other
forecast horizon between simple random walk forecasts and other competing models.
The results are again not very encouraging. At the three-month horizon, forecast
erors of none of the competing model is significantly different from that of the
115
simple random walk forecast. At the six-month ahead forecast horizon, agun, none of
the model's forecast error is different from that of the simple random walk model. At
the nine-month ahead forecast hoizon, only the ARIMA model's forecast errors is
statistically different from that of the simple random walk model. Given that it has a
lower RMSE (infact, lowest among the competing models) we can conclude that
ARIMA model is significantly out-performing the simple random walk model. At the
twelve-month ahead forecast horizon, it is again the ARIMA model that has got
statistically significant different RMSE from the simple random walk model.
Summary of the Diebold-Mariano Test :
significantly different from the SRV/ model. Forecast effors are also significantly
different (as a pairwise) between the ARIMA and VECM, VECM and BVAR,
VECM and BVAR, and between VECM and UBVAR. Difference of forecast
effors also imply that there is statistically significant difference between the root
mean square elTors from these models.
At three month ahead forecast horizon, test fails to provide evidence of
significantly different forecast effors among the competing models.
At six month ahead forecast horizon, none of the model's forecast errors are
statistically different from that of the random walk model. Only cases where
forecast effors were significantly different from each other are those of BVAR and
UBVAR, and BVARD and LIBVAR.
At the nine month ahead forecast horizon, it is only the ARIMA model which
produces forecast erors that are significantly different from that of the SRW.
ll6
At the twelve month ahead forecast hoizon, again, none of the model's forecasts
are significantly different from those of the random walk model.
(iv) Information Conterut Test :
Fair and Shiller (1989, 1990) noted that the superiority of a particular
model in terms of forecast accuracy does not necessarily imply that the forecasts from
other models contain no additional information. Moreover, when the RMSEs are close
for two forecasts (this is particularly true in our study), little can be concluded about
the relative merits of the two. This led Fair and Shiller develop an Information
Content Test to find out whether one set of forecasts has more 'information' than the
competing models. This will help us to conclude whether the forecasts from the
competing model has more information relative to the simple random walk model.
The results of the information content test is given inTable22.
At the one month forecast horizon, only the forecasts from the VECM
have more information than the simple random walk forecasts. Considering the
ARIMA and VECM as a pair, we found that the coefficient of the ARIMA term is
insignificant whereas the VECM term is statistically significant at 5Yo significance
level. This implies two things. One, VECM has information beyond that provided by
the simple random walk model. Second, information of ARIMA model is completely
contained in VECM and VECM contains further relevant information than the
ARIMA model. Other pairs with the ARIMA model do not have coefficients
significant implying that none of the model contains any information useful for the
one-month ahead forecast of the spot exchangerate. Considering other combinations
with the VECM, viz., (VECM,BVAR), (VECM,BVARD) and (VECM,UBVAR), we
tt7
find that in all these cases the coefficient of the VECM term not only to be positively
signed but also statistically significant at 5%o significance level. However, the
coefficient of the other variable with the VECM is always turning out to be
statistically insignificant. For the BVARD and UBVAR pair, the coefficient of the
BVARD term was significant but negatively signed which " is a perverse result in
economic terms as this implies that the information in the BVARD models is
negatively correlated with the actual exchange rate changes " (Liu, Gerlow and
Irwin (1994)).
Beyond one-month ahead, the results of the information content test is
not very encouraging. At all other forecast horizon, viz. tltree to twelve months, either
the coefficients are statistically insignificant or they are significant but negatively
signed implying a perverse relation in economic terms. This implies that none of the
models at more than one-month ahead forecast horizon contains information which is
useful to forecasting the spot exchange rate beyond that provided by the simple
random walk model. It should be noted that forecast errors are autocorrelated of order
MA(k-l), where k is the forecast horizon, for which we have used the Newy-West
heteroscedastic-autocorrelation consistent estimator for carrying out information
content test beyond one month horizon.
Summary of the Information Content Test :
(VECM) forecasts which contains 'information' beyond that contained by that of
a random walk model. 'Information' from all other model are contained in the
VECM forecasts.
ll8
encouraging. None of the model contained any 'information' beyond that of the
simple random walk model.
(v) Direction-of-change Forecast Analysis :
Direction-of-change forecasts are often used in financial and economic
decision making (e.g. Leitch and Tanner, 1991). The question as to whether a
direction-of-change forecast has value involves comparison to a naive benchmark -
the direction of change forecast is compared to a ' naive ' coin flip. Cumby and
Modest (1987) developed a test based on Merton's (1981) work for evaluating the
direction-of-change forecasts. They tested the null hypothesis that a direction-of-
change forecast has no value by testing the null of independence between the actual
changes and forecasted changes. The test statistic follows a chi-square distribution,
under the null of independence, with one degree of freedom.
Results are reported in Table 23. At one-month forecast horizon, none
of the models predicted changes in the spot exchange rate accurately. This is because
none of the calculated value of the chi-square statistic are statistically significant. The
picture improves somewhat at the three month ahead forecast horizon. Here forecasts
from the VECM and UBVAR models predicted changes in the exchange rate that has
some value to the consumers as both of them has calculated chi-square statistic which
is very marginally significant at l0o/o significance level. It should be mentioned again
that predicted changes has 'value' if the actual and predicted changes are not
independent. At the six-month ahead forecast horizon, it is only the VECM which
predicted changes in the exchange rate that has got some value in the sense that the
119
actual changes and predicted changes of the exchange rate are statistically not
independent. At nine- and twelve-month forecast horizons, the direction-of-change
forecasts results are again not positive since all of the models produces forecasts that
is statistically independent from the actual changes.
Summary of the Direction-of-Change Analysis :
competing models are statistically independent from the actual changes in the
exchange rate.
depreciation and appreciation of the spot exchange rate that is not statistically
correlated with the actual changes.
has some 'value' in the sense that the predicted changes in the exchange rate is not
statistically independent from the actual changes.
Major Findings of the Forecusting Exercise :
Across the forecast horizon, that is, from one to twelve month ahead forecasts,
simple random walk forecasts are beaten by most of the models as shown by the
descriptive statistics such as root mean square effors (RMSE), Theil's U
statistic (U).
The benchmark univariate ARIMA(2,1,1) model produced better short and
medium period forecasts, viz. one, three and six month ahead forecasts, as
indicated by RMSE and U ; however, its performance deteriorates in the long-run
forecast horizon of nine and twelve months.
120
The multivaiate models able to outperform the ARIMA model across the forecast
horizon. At one month horizon, it is the vector effor correction model (VECM), at
three and six month horizon it is the Unrestricted Bayesian VAR (IJBVAR)2, at
nine month horizon they are VECM, UBVAR, and BVAR, while at the twelve
month honzon they are VECM, BVAR, BVARD and UBVAR.
Among the multivanate models, the performance is mixed in terms of out-
performing the random walk model. While BVAR, BVARD and TIBVAR
consisitently outperforms the random walk across all the forecast horizon in terms
of RMSE and U, BVECM always performs worse than the random walk forecasts
in terms of these descriptive statistics.
VECM and LVAR have shown mixed performance. VECM ou@erforms the
random walk model at all forecast horizon, except at the three month horizon.
LVAR outperforms the random walk forecasts only at one and twelve month
forecast horizon.
From the analysis of the descriptive statistics, it can be concluded that none of the
model performs best across all the forecast horizon. While VECM forecasts
occupies the first position in one and twelve month forecast horizon, while
UBVAR occupies the top slot in three, six and nine month forecast horizon.
VECM forecasts are turning out to be efficient at one month ahead forecast
horizon, although there is no clearcut evidence on the unbiasedness of these
lorecasts at this horizon.
At one month ahead forecast horizon, other models which includes ARIMA,
BVAR, BVARD and UBVAR, turning out to be unbiased in terms of non-
parametric tests of unbiasedness.
2 It is worth to mention again that we use " Unrestricted Bayesian VAR " to point out the fact that in
121
All the forecasts satisfy the condition of unbiasedness and efficiency at three
month ahead forecast horizon. Although VECM performs very poorly in terms of
the descriptive statistics at this horrzon, it is also turning out to be unbiased and
efficient.
At one month ahead forecast horizon, VECM forecast erors are statistically
different from all other model's forecast elrors including that of the random walk.
This evidence, combined with the fact that VECM has the minimum RMSE,
implying that it is significantly lower than the competing models.
In terms of the information content test, also, it is the VECM which gives the best
performance at the one month forecast horizon. VECM forecasts contained
'information' beyond those contained in other competing models including the
random walk.
a As regards the performance of the models in correctly predicting the appreciation
and depreciation of the exchange rute, it is the VECM and ARIMA models which
significantly predicts the same at the three month forecast horizon. This
corroborates our earlier hypothesis that VECM forecasts being unbiased and
efficient, inspite of having very poor descriptive statistics, could only imply that it
is correctly predicting the increase/decrease of the exchange rate.
From the above discussion of the various tests of forecast evaluation,
one can conclude that we can show some confidence in the forecasts at one and three
month forecast horizon. At one month ahead forecast horizon, VECM forecasts has an
edge over its competitors in terms of RMSE, Theil's U, Efficiency and Unbiased test,
Diebold-Mariano statistics and Information Content Test. At three month forecast
this Bayesian VAR formulation we allow for the coefficients to vary across the country.
122
horizon, has an edge over its competitors in terms of RMSE, Thsil's u,
Efficiency and Unbiased testg and direction-of-change analysis. Beyond three month
forecast horizon, one has to check the forecast output very carefully as there is no
clearcut evidence on the quality of the forecasts. This is not wholly rmexpected.
Forecasts of a furancial variable l1ke the exchange rate over the long frrecast horizonI
is, in all probability, could go haywire
123
Chupter VII
Conclusion
The forecasting perfornance of exchange rate models has received
considerable attention since the breakdown of the Bretton Woods, when exchange rate
began to float. The debate opened up with the work of Meese and Rogoff (1983),
whose work on exchange rate forecasting of various theoretical and atheoretical
models of exchange rate determination brought out the poor out-of-sample forecasting
performance of the asset market models. In fact, the most serious result was that the
naiVe forecasts of the simple random walk model outperformed the various asset
market models in out-of-sample forecasting performance. Lot of research input has
gone into overtuming the conclusions reached by the Meese and Rogoff. The present
study is an attempt in that direction and can be thought as a culmination of the earlier
works by the author (Bhattacharya1998,1999).
In this dissertation we develop forecasting models involving both
univariate and multivariate time series techniques with the aim of outperforming the
random walk model in terms of out-of-sample forecasting performance. We consider
the monetary models of exchange rate of determination - flexible price monetary
model, sticky price monetary model, real interest differential model and Hooper-
Morton model. A11 these models are essentially'monetary' in nature - they assume
that exchangerate is determined by the relative demand and supply of the two monies
as exchange rate is thought to be as nothing but the relative price of two monies.
With the advent of the cointegration technique there is a new fillip to
the on-going empirical analysis of the asset market models. Exchange rate is now
considered to be determined , in the long-run, by the economic fundamentals. It is
hypothesized that if the asset market models hold, then there must be a cointegrating
t24
relationship between the exchange rate and the variables determining it. We
,therefore, expected to find a long-run equilibrium relationship between exchange
rate, money supplies, interest rates and outputs as postulated by the monetary models
of exchange rate determinhtion. The presence of cointegration among these variables
confirmed the monetary models as a valid framework for evaluating exchange rate
beltaviour.
The next question that needs to be addressed is which version of the
monetary model is supported by the empirical study. This could be determined by
signs of the variables in the cointegrating vector. Our empirical study supports the
flexible-price version of the monetary model of exchange rate determination. Given
the existence of an economically meaningful cointegrating vector, we proceed to
develop an effor correction model which would capture the short-run adjustments
towards the long-run equilibrium. Finally, forecasts are generated from this vector
error coffection model (VECM).
Alternative time series models are also developed for generating
competing forecasts. A commonly used univariate time series model is developed by
using the Box-Jenkins methodology. These class of models have empirically shown to
produce good short and medium-run forecasts. Other multivariate models developed
in the present study are bayesian in nature where a modeler's prior beliefs are
incorporated in the model estimation. In the present study we developed various
bayesian vector autoregressive models by using the Minnesota prior developed by
Litterman (1979).
Four different bayesian VAR model have been developed. First is the
bayesian VAR (BVAR) model which is simply the bayesian counterpart of the
monetary model used in the cointegration analysis. Second one incorporates the
t2s
expected inflation and cumulative bilateral trade balance between India and USA as
deterministic variables by using a flat prior on them in an otherwise BVAR model.
This we name as bayesian VAR with deterministic variable (BVARD). This is
basically a Hooper-Morton model developed in the bayesian framework. Third model
in the bayesian class is what we call a bayesian vector elTor correction model
(BVECM) which incorporates the error correction term (with a flat prior on it) in the
BVAR model, where the error correction term is obtained from the cointegrating
relationship. Finally, we develop the unrestricted bayesian vector error correction
(UBVAR) model. The term ' unrestricted ' has been used to denote the fact that we
have allowed the coefficients to vary across the countries in the bayesian framework.
That is, instead of assuming Minnesota prior on the coefficient of the, say, relative
money supply, we now assume the same prior on separate coefficients of the money
supply term of each country.
We employ a rolling regression method of estimation and generated
the out-of-sample forecasts from the competing models. Forecasts are generated for
one, three, six, nine and twelve month ahead forecasts. We carried out a battery of
tests to assess the quality of the forecasts from the competing models. Among the
tests we used to track the performance of forecasts includes various descriptive
statistics like root mean square errors, Theil's U statistic, unbiasedness and efficiency
tests, Diebold-Mariano equality of forecasts eror test, information content test and
direction-of-change analysis. We use the time-series properties of the actual and
predicted series to evaluate the forecasting performance of the competing models.
On the basis of various forecast evaluation criterion we conclude that
at one month ahead forecast horizon, it is the vector effor corection model (VECM)
which outperforms all the models including the simple random walk model, while at
126
the three month horizon, it is the unrestricted bayesian VAR (UBVAR) which
occupies the first position. Although beyond three month horizon, multivariate models
based on economic fundamentals as well as the univariate ARIMA model
outperforms the random walk forecasts in terms of the various descriptive statistics,
these forecasts do not seem to satisff the desirable properties ofa good forecast. This
lead us to conclude that beyond the three month horizon one should interpret the
forecast results very carefully.
We conclude here by noting down few limitations of the present study.
implicitly imposed the restriction of equal coefficienls between the domestic and
foreign country by working with relative money supplies, relative outputs and
relative interest rates. This may be a very restrictive assumption. Because of our
relatively small sample size, given by the post-liberalisation period, we are forced
to take this route to avoid degrees of freedom problem.
dependent on the definition of the money supply employed. With M3 as the
measure of the money supply, we are unable to find any long-run equilibrium
relationship that confirms to the monetary models of the exchange rate
determination.
because of the absence of time series data on the three month treasury bill rate for
India. Regular auction of the Indian three month treasury bill rate takes place from
January 1993 only.
t27
We also tried out call money rate in the case of India as the measure of the short-
term interest rate which gave us alarger sample size, that is, from August 1991 to
July 1999. However, we are unable to find any long-run equilibrium relationship
by using either M3 or Ml as the measure of the rironey supply.
Thus, it seems that the existence of the monetary model as a long-run equilibrium
is not very robust to the choice of the variables included in the determinants of the
fundamentals.
exchange rate determination. One would like to test for the portfolio balance
model of exchange rate determination which allows for imperfect substitutability
of assets. However, we refrain from doing this analysis as data on the breakdown
of asset holding by economic agents were not available.
To sum up, in this dissertation we had undertaken an exercise to find
out the forecasting performance of the monetary models of the exchange rate
determination. Our prime motive was to obtain forecasts from the monetary models
of the exchange rate determination which beats the simple random walk forecasts.
We have been successful in generating forecasts from the flexible-price monetary
model which beats the simple random walk forecasts. However, our analysis is
restricted by the relatively small post-liberalisation period in India. In future, we
would like to evaluate the forecasting performance of the portfolio balance models of
the exchange rute determination. Given the on-going research on the market
microstructure approach, we would like to model the agent's behaviour in the foreign
exchange markets which would help us to better forecast the exchange rate
movements than the traditional macroeconometric approach.
r28
Table: 1.1@)
PHILLPS-PERRON TEST
Models:
(l)!t = d.o * d.rta * d.z(t - Tl2) + p1
(2) y, = a*o *a*, !,-t*lt,
Time Period : Janua.ry 1993 - July 1999
Levels
ModelNull HypothesisTest Statistic
Exchange Rate
Indian IIP
Indian CPI
I,ndian Treasury Bill Rate
Indian Ml
US IIP
US CPI
US Treasury Bill Rate
US M1
(1)
H o:d, --lZ (tdr)
-2.2456
-1.7275
-2.1986
-t.4379
-2.0032
-2.t257
-1.4693
-t.0721
-3.6085
(2)
H o:a't =lZ(t a.)
0.2886
-1.7648
-0.9084
-1.3888
-2.6941
-0.4101
-1.5310
-r.8529
-2.3792
Table : 1.2(A)
DICKEY-FULLER TEST
Time Period : February 1993 - Juty 1999
First DifferencesModelNull HypothesisTest Statistic
Exchange Rate
Indian IIP
Indian CPI
Indian Treasury Bill Rate
Indian Ml
US IIP
US CPI
US Treasury Bill Rate
US M1
(3)
Ho:a, =at =00,
4.6993
2.6307
2.84s0
s.3971
3.0426
3.7011
(3)
H o:a, =0tr1
-4.0186
-3.0609
-2.0997
-2.3831
-5.9088
-4.0596
-3.2828
-2.4647
-2.7023
a)H o:a, =0
tr$
-2.5604
-2.1545
-2.3972
-2.t721
-2.6333
Q)
4,q*r=0,
3.2826
2.4504
2.8761
2.3637
3.5581
(1)
Ho:ar=0T
-t.6432
-1.1070
-2.4155
-2.1105
-2.6835
fable: 1.2(B)
PHILLPS.PERRON TEST
Time Period : February 1993 - Juty 1999
First Differences
Critical Values
Perron Test
ModelNull HypothesisTest Statistic
Exchange Rate
Indian IIP
Indian CPI
Indian Treasury Bill Rate
Indian Ml
US IIP
US CPI
US Treasury Bill Rate
US Ml
(1)
H o:d, =1
Z (td,)
-t0.3200
-12.2440
-6.8433
-7.7047
-10.6940
-9.8343
-7.4110
-7.7271
-4.3868
(2)
H o'.a* t =l
4o.r)-10.0100
-12.1010
-6.8330
-7.7566
-9.9658
-9.8943
-7.3344
-7.3726
-4.0959
Dickey-Fuller Test
SignificanceLevel
aa
-3.96-3.41
-3.13
0,
ffi6.255.34
xu 0'
3.654.59
3.78
x
T%
5%t0%
-3.43
-2.86-2.s7
-2.58-1.95-1.62
Significance Z (t dr) Z(t a. )Level
t%5%10%
-3.96 -3.43
-3.41 -2.86-3.13 -2.57
TABLE :2.1(A)
DICKEY.FULLER TEST
Time Period : January 1993 - July 1999
Levels
TABLE :2.1(B)
PHILLPS - PERRON TEST
ModelNull HypothesisTest Statistic
(3)
H o:a, =0xr
(3)
H o:a, =a,
0,
(2)
Ho"a,=0xu
(2)
4:q-4-0,
(l)H o:a, =0
T
Relative IIP
Relative CPI
Relative InterestRateRelative Money
-1.6628
-2.7484
-2.0067
-1.1841
2.2043
3.9054
2.3198
t.t29t
-2.ttt7
-0.7423
-2.1418
-1.2796
2.6783
4.6t31
2.3766
3.6534
0.1319
2.6064
-1.1770
-2.4556
Levels
ModelNull HypothesisTest Statistic
R"lrt"- IP
Relative CPI
(1)
H o:d, =lZ (td,)
(2)
H o:a* t =l
fg,)-2.2603
-0.7545
-1.568s
-r.7250
Relative Interest Rate
Relative Money Supply
-1.9434
-2.2526
-1.5205
-1.4638
TABLE .2.2(A)
DICKEY.FULLER TEST
Time Period : January 1993 - July 1999First Differences
TABLE | 2.2(B\
PHILLPS. PERRON TEST
Notes : All the variables are in logarithm term, except the relative interest rate.Exchange rate is the bilateral Indian Rupees / US Dollar exchange rate.Relative IIP is given by the logarithm difference of two country's Industrial
Production Index.Relative CPI is given by the logarithm difference of two country's Consumer Price
Index for industrial workers.Relative interest is given by the 3-Month Treasury Bill rate differential.Relative money supply is given by the logarithm difference of two country's money
supply where Ml has been used as the measure of the money supply.
ModelNull HypothesisTest Statistic
Relative IIP
Relative CPI
Relative InterestRateRelative MoneySupply
(3)
H o:a, =0
_::_-3.0418
-2.0403
-3.3488
-5.0838
(3)
H o:a, = a,
_0,4.6279
2.s354
5.6113
(2)
H o:a, =01u
-2.7660
-2.1861
-3.3130
(2)
4:q-'q=0,
3.82s8
2.4977
s.4948
(t)H r:a, =0
__:_2.6678
-1.2858
-0
First Differences
ModelNull HypothesisTest Statistic
R.trt"- m
Relative CPI
(1)
H o:d, =l
3!:!--12.3050
-6.6488
-7.9321
-11.0020
(2)
H o'.a* t =lZ(t a.)
Relative Interest Rate
Relative Money Supply
-12.1940
-6.666r
-7.93t8
-10.6130
APPENDIX
Table : 1.1(A)
DICKEY.F'ULLER TEST
Models:p
(1) A !t = at !trtZO,t!y_l+ii=2
P
(2) L ! t = ao * at !,-t +ZbiL y y_1*ii=2
p
(3) A ! t = ao + azt + at !rt +Z4L ! y_t*i,'- t
Time Period : January 1993 - July 1999
Levels
Notes : IIP stands for the Index of Industrial ProductionCPI stands for the Consumer Price Index NumberTreasury Bill rate is of the maturity period of 3-Months
All the variables, except the treasury bill rate, are in logarithms.
ModelNuliHypothesisTest Statistic
Exchange Rate
Indian IIP
Indian CPI
Indian Treasury Bill Rate
Indian Ml
US IIP
US CPI
US Treasury Bill Rate
US Ml
(3)
Ho"a,=g
_::_-2.3064
-t.448t
-2.8842
-t.8278
-2.0472
-2.1925
-t.3499
-1.0191
-2.3681
(3)
Hr"a, =s,
_d,3.2314
1.9739
4.5t26
t.7200
3.8109
2.436t
2.0188
2.3850
2.8041
-0(2)
H o:a, =g
__t,0.2139
-1.7890
-0.9665
-1.8553
-2.5350
-0.4081
-t.6176
-1.8638
-1.8328
(2)
4:q:4=
--!r-2.4t66
6.2192
6.0261
1.7284
8.5396
33.t250
65.5110
2.3370
1.72s7
(1)
H o'.a, =Q
c
2.2078
2.9411
3.2585
-0.4789
3.0340
8.1562
1 1.1750
"0.73t6
-0.3056
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Bayesian Unit Root Test(limit : 0.5, alpha : 0.8)
Time Period : January 1993 - July 1999(Levels)
Table: 5.2
Bayesian Unit Root Test(limit:0.5, alpha = 0.8)
Time Period : January 1993 - JuIy 1999
Variables
Exchange Rate
Indian IIP
Indian CPI
Indian Treasury BillRate
Indian Ml
US IIP
US CPI
US Treasury BillRate
US Ml
Squared t Schwarz Limit Marginal Alpha
8.195 0.9215
8.038 0.7316
10.333 0.9568
6.787 0.7170
8.494 0.3593
10.s41 0.9727
11.109 0.9330
7.263 0.6219
8.049 0.2761
0.046
2.808
0.914
1.703
6.426
0.t67
2.617
3.747
6.753
Variables
Relative IIP
Relative CPI
Relative Interest Rate
Relative Monev S ly
Squared t Schwarz Limit
6.327
9.7t4
7.014
8.514
Marginal
5.644
0.591
3.t47
2.737
0.2t91
0.9s02
0.5797
0.7818
, Tahle r 6
. EnglrGrmgorTe*t'of Cofutegrafioni'..
'::. Sarnple Psrtod : Jirriurry 1993 - July 1999
Test Type Constant, No Trend eonstant, Tremd
Dickey-Fuller
Phillips-Perron
-3.3001
-3.2383
-2.3381.
-3.1413
No Coirrtegration
No Cointqgration
Table : 7
Lag-length Selection for Unrestricted Vector Autoregression
Sample Period : January 1993 - JuIy 1999
Notes: (1) AIC stands for Akaike Information Criterion(2) SBC stands for Schwarz Bayesian Criterion(3) LR stands for the Likelihood Ratio Test Statistic(4) LM stands for the Langrange Multiplier test for Serial Correlation(5) In column (a) the value corresponding to 3 lags is the LR test statistic valuefor testing the null of 3 lags versus 4 lags. (p-values are given in the bracketsbelow)
No. of Lags
(1)
AIC
(2)
SBC
(3)
LR
(4)
LM
(s)
4lags -19.4383 -17.3371
3 lags -19.5818 -t7.974721.2656(0.168s)
18.557(0.2e23)
t9.3664(0.2s01)
No serialCorrelation
SerialCorrelation
SerialCorrelation
2 lags
1 lag
-19.7607 -18.6483
-19.9292 -19.3112
Table 8
Block-Exogeneity Test
Sample Period : January 1993 - JuIy 1999
Number of Lags in the VAR: 3
Notes : (1) p-values are given in the parenthesis.
(2) under the null hypothesis, LR statistic follows a chi-square distribution with 9degrees of freedom.
Null Hypothesis : Variable is not present in Likelihood Ratio (LR) Statistic
Spot Exchange Rate
Relative IIP
Relative Interest Rate
Relative Money Supply
25.3496(0.003)
20.$a6(0.01s)
18.7968(0.027\
30.3714(0.0000)
Johansen-Juseliustl*;iation Test Resuu
Sample Period : 1993:01 - 1999:07
Number of Lags in VAR: 3
Cointegration test based on the Trace Statistic
(Constant in the cointegrating Vector)
Cointegration test based on the Maximum Eignvalue Statistic
(Constant in the cointegrating Vector)
Eignvalues
0.37339
0.25635
0.06584
0.027t0
Nul1Hypothesis
AlthemativeHvpgqE,t
r)1
::-
6s.299s
29.7746
7.2642
2.0883
s%c.Y. t0% c.Y.
49.9500
31.9300
17.8800
7.s300
r-0
r<l
r<2
r<3
r>2
r>3
r=4
s3.4800
24.8700
20.1800
9.1600
Y""".0.37339
0.25635
0.06584
0.02710
Nul1 Althemative
35.5249
22.5t04
5.1759
2.0883
5% C.V. t0% c.Y.
25.8000
19.8600
13.8100
7.5300
Hvoothesis
r=0
r<l
r<2
r<3
r=7 28.2700
22.0400
15.8700
9.1600
r=2
r=3
r=4
Table : 10
. Cointegrating Vectsrs
Sample Period : January 1993 - July 1999
Number of lags in VAR: 3
Note:Sisthebilateralspotexchangeratg (y-y' ) istherelativellP, (i-i*)istherelative interest differential, (*-*' ) is the relative money supplies and C is theconstant. * indicates a foreign variable.
S (v- v. ) (i-i ) (m-m') C
1.0000
t.0000
-2.4145 0.5409 0.0058 -r.2363
2.5361 -0.3902 -t.3669 -7.8110
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Table: 12
Bayesian Vector Autoregression Parameter Selection
Simple Flexible Price Monetary Model with Log of exchanle rate,log of Relative' UP, Relative 3-Months Treasury Bill Rates, and Log of Relative Money Supplygiven by Ml.
THEIL'S INEQUALITY STATISTICS
Harmonic Lag : d: 1.0
Ilarmonic Lag : d:2.0
ROOT MEAN SQUARE ERRORS
Harmonic Lag : d: 1.0
Priors l" = 0.1 )'"=0.2
ffi0.9097920.916s19
w:0.4w:0.5w:0.6
0.8756060.8792030.882678
Priors 7" = 0.1 )u=0.2ffi0.8916140.896867
w:0.4w:0.5w:0.6
0.8756060.8651120.867559
Priors l" = 0.1
ffi0.0520290.052225
)"=0.20.0537680.0s41740.054s89
w:0.4w:0.5w:0.6
Harmonic Lag: d:2.0
Priors l. = 0.1
ffiffi?v=0.2
ffi0.0s27470.053062
w:0.4w:0.5w:0.6
0.0509910.051 130
TABLE : 13
DICKEY.F'ULLER TEST
Models:p
(1) A !t = ar !t_rtZO,tly_1+ii=2
p
(2) A !t = ao * at !,-r +ZbiL y y_t*ii=2
p
(3) A !t =ao+azt*at !,_t+Ib,A !y_r+i!-a
Time Period : January 1993 - July 1999
Levels
ModelNull HypothesisTest Statistic
Expected Inflationfor IndiaExpected Inflationfor USABilateral CumulativeTrade Balance
(3)
Ho:a, =a,
0,
12834
2.5423
(2)
4,%n=
---ir-3.23s1
1.6114
(1)
Ho:ar=0a
ffi-1.0499
(3)
H ,:a, =0T1
(2)
H o:a, =0xu
-0
-2.5606
-2.1992
-3.6565
-2.5436
-t.7923
PHILLPS. PERRON TEST
LevelsModelNull HypothesisTest Statistic
Expected Inflation for India
Expected Inflation for USA
Bilateral Cumulative TradeBalance
(1)
H o:d, =lz (tdl)
(2)
Ho"a*
Z(t a,^0069
-5.5164
-3.5262
r=1-r)
-6.8640
-5.9s24
-3.9806
Table: 14
BAYESIAN VAR PARAMETER SELECTION
Simple Flexible Price Monetary Model with Log of exchange rate, log of rrps, 3-Months Treasury Bill Rates, and Log of Money Supplies given by Ml.
THEIL'S INEQUALITY STATISTICSHarmonic Lag z d: 1.0
IlarmonicLag:d:2.0
ROOT MEAN SQUARE ERRORS
Harmonic Lag : d: 1.0
llarmonic Lag : d:2.0
Priors t!!_0.8324120.8436270.856376
]:!2-0.8452380.8661760.885023
w:0.4w:0.5w:0.6
Priors l" :0.1 ]:!2-0.842t040.8587040.873395
w:0.4w:0.5w:0.6
0.8375280.8473570.858362
Priors l. = 0.1 _?u:0.2
0.049t490.0504780.051632
w:0.4w:0.5w:0.6
0.0488350.0495810.050431
Priors l, = 0.1 )v=0.2
w:0.4w:0.5w:0.6
0.0492980.0499530.050687
0.049t020.0s01590.051063
ACF AND PACF OF LOG OF THE SPOT EXCHANGE RATE
AGF : LEVELS
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u1o
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NLAGS
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ILoo-
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1
0.5
0
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tOO)(Y)I-F
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AGF : FIRST DIFFERENCE
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Trble : 17
Summary Results of Unit Root Tests on Actuals and Predicted Series
i.li,!'
Notes : "Y' stands for the non-rejeetion of the null of unit root.We employed Dickey-Fuller and Phillips-Perron unit root tests.
Models/Forecast ACTUALS ARIMA VECM BVAR BVARD UBVARHorizon
lStep Y Y Y Y Y Y
3 Steps
6 Steps
9 Steps
12 Steps
Y
Y
Y
Y Y Y Y
Table: 18
Summary Results of Cointegration between Actual and Forecasted Series
Notes : "EG' stands for the presence of cointegration by the Engle-Granger procedure."JJ" stands for the presence ofcointegration by the Johansen-Juselius procedure.'\IOC" stands for no cointegration by either Engle-Granger or Johansen-Juseliusprocedure.
Models/ForecastHorizon
ACTUALS BVAR BVARD UBVAR
I Step
3 Steps
6 Steps
9 Steps
12 Steps
EG, JJ
JJ
NOC
EG, JJ
NOC
EG, JJ
JJ
NOC
EG, JJ
NOC
EG, JJ
JJ
NOC
EG, JJ
NOC
EG, JJ
JJ
NOC
EG, JJ
NOC
EG, JJ
JJ
NOC
EG, JJ
NOC
EG, JJ
JJ
NOC
EG, JJ
NOC
Table:19
Bias and Efficiency Tests of the Forecasts
Table 19.1
One-Month Ahead Forecasts
Models Bias Test
l1i1-E10.0831(0.006)
25.7335(0.0000)
84.8132(0.0000)
86.0046(0.0000)
88.1885(0.0000)
Efficiency Test(1 -t
ARIMA r.7984(0.180)
3.8779(0.04e)
40.1989(0.0000)
37.7419(0.0000)
27.507t(0.0000)
VECM
BVAR
BVARD
UBVAR
Table 19.2
Three-Month Ahead Forecasts ,?
Models Bias Test
l1-rg_3.562s(0.168)
3.0312(0.220)
8.5920(0.014)
8.1199(0.017)
8.5609(0.013)
Efficiency Test(1 -1)
ARIMA 0.0603(0.806)
0.1002(0.7s2)
0.3904(0.s32)
0.3104(0.s77)
0.1599(0.68e)
VECM
BVAR
BVARD
UBVAR
- _1.3Fq!!:]iw:tr
Teble 19.3
Niue-Month Ahead tr'orecasts
Bias Test Test(1-l 0 -r)
ARIMA
VECM
8.7265(0.013)
14.4646(0,001)
12.1932(0.002)
12.2484(0.002)
12.4788(0.002)
7.0984(0.00s)
17.!,47g(0.000)
15.6964(0.000)
r6.0102(0.mm)
16.0?48(CI.0m)
BVAR
BVARD
UBVAR
Table:20
Nonfarametric Test of Unbfusodness
Models
-ARIMA
VECM
BVAR
BVARD
TJBV'AR
:'i1"':t2
20
11
11
11
Wilcoxon Rank-Sum Test
202
413
209
197
213
Notes : (1) Reject the null of median urfiiasednese,at 5yo significsnce level if th€calculated value of the sign statistic is less than equal to 9.993 or if S is greater thar equalto 21.007.
(2) Reject the null of median (= mean) unbiase&ess if the calculated valuo of theWilcoxon rank sum test statistic exceds 348 or if it is less than 148.
Table :21
Diebold-Mariano Test for Equality of Mean Square Errror
One-Month Ahead Forecasts
( * indicates significant at 5% and 10% significance level)
Three-Month Ahead Forecasts
( * indicates significant at 5% and 10% significance level)
Six-Month Ahead Forecasts
Models ARIMA
mVECM BVAR
0.9002-0.32313.9532.
BVARD
@,0.20933.9681 .0.7808
UBVAR
@0.1433
4.0129 *
1.87561.5758
SRWARIMAVECMBVARBVARD
-3.7199.-3.8118 .
Models ARIMA
@VECM
-a8642-1.0256
BVAR
m-0.76810.4857
BVARD UBVAR
ffi-0.28291.04681.71291.8107
SRWARIMAVECMBVARBVARD
0.5057-0.84840.9790-0.2936
Models
ffi:ARIMAVECMBVARBVARD
ARIMA
mVECM
ffi-0.3045
BVAR BVARD
03884-1.0748-0.19080.1338
UBVAR
0.3612-1.0364-0.2207
0.8069-0.10060.4903
4.1578 *
3.5814 *
( * indicates significant at 5% and 10% significance level)
Table 21 (Gontd..)
Nine-Month Ahead Forecasts
( * indicates significant at 5% and 10% significance level)
Twelve-Month Ahead Forecasts
Models
SRW:ARIMAVECMBVARBVARD
ARIMA VECM
037690.1815
BVAR
ffi-1.0851-1.6305
BVARD
ffi-0.9493-1.4588-0.6234
UBV
3.6993. 1.35670.3083-0.0132-0.1485-0.7423
Models ARIMA
Tssz
VECM
ffi0.5192
BVAR BVARD
ffi-0.2598-1.62670.5474
UBVAR
0.4234-0.3079-1.4459
1 .0159ARIMAVECMBVARBVARD
0.5495-0.43121.49711.7156
( * indicates significant at 5% and 10% significance level)
Table:22
lnformation Content Test
One-Month Forecast HorizonDependent Variable : Actual
Independent Variables
Constant
0.0014(0.6888)(0.4e66)
0.0089(1.4e43)(0.1463)0.0094
(1.s200)(0.13e7)0.0093
(1.3406)(0.1908)0.0069
(r.66e0)(0.1083)0.0077
(1.71e8)(0.0e65)0.0086
(1.s482)(0.1328)0.0102
(1.ss21)(0.131e)0.0087
(1.2248)(0.230e)0.0082
(1.1646)(0.2s40)
ARIMA VECM
0.7783(2.0128)(0.0538)
BVAR BVARD UBVAR
0.1495(0.33s8)(0.73es)0.4873
(1.zete)(0.2069)0.5006
(1.3468)(0.1 888)0.5076
(1.24s4)(0.2233)
-0.97s6(-1.1414)(0.2634)
-1.1792(-t.ts73)(o.2s6e)
-t.1273(-0.e106)(0.3702)
0.9232(2.7s42)(0.0102)
0.9499(2.8183)(0.0088)0.9s94
(2_e020)
(0.0071)
-1.2573(-1.se13)(0.1,228)
-1.5857(-1.50e6)(0.t424)
-1.7207(-r.4003)(0.1724)
-0.463r(-0.662e)(0.s128)-1.1323
(-1.ss76)(0.1306)
-0.486s(-0.4813)(0.6340)
-1.9372(-1.e002)(0.067s)
0.5661(0.53ee)(0.5e35)t.3637
(1.00s7)(0.3 r31)
Table 22 (Contd..)
Three-Month Forecast Horizon
Independent Variables
Constant ARIMA
ffi(-0.5018)(0.6200)0.5857
(0.ee82)(0.3273)0.5990
(0.e4t4)(o.3ss2)0.5183
(0.es8s)(0.3467)
VECM BVAR BVARD UBVAR
0.0244(1.8706)(0.0727)0.0368
(2.0108)(0.0s48)0.0382
(2.r448)(0.041s)0.0455
(2.0628)(0.04e3)
0.0392(2.1 880)(0.0378)0.0408
(2.3s08)(0.0266)0.0464
(2.137T)(0.0422)0.0424
(3.37es)(0.0023)0.040s
(2.1064)(0.04s0)
0.0361(1.8258)(0.07e5)
-0.1061(-t.6643)(0.1081)
-1.7209(-2.3706)(0.0255)
-1.9t07(-2.6737)(0.0128)
-2.2970(-2.32t7)(0.02383)
-0.0385(-0.60e3)(0.s476)-0.0388
(-0.0388)(0.s644)-0.02s1
(-0.410e)(0.684s)
-1.4653(-1.8s06)(0.07s6)
-1.6509(-2.11,66)(0.0440)
-1.9920(-1.8828)(0.0705)
21.7534(1.e724)(0.0se3)-t.3042
(-0.8462)(0.4s01)
-24.4491(-2.ts78)(0,0404)
-2.4150(-2.33e8)(0.0272)
-0.3179(-0.1837)(0.40s1)
1.0069(0.7022)(0.4888)
Table 22 (Contd..)
Six-Month Forecast Horizon
Independent Yariables
Constant
0^0654(2.6se)
(0.0140)0.0638
(2.e64s)(0.006e)0.0658
(3.0e8s)(0.00s1)0.0905
(4.26t0)(0.0003)0.0726
(4.0468)(0.000s)0.074t
(4.2e83)(0.0003)
0.0970(s. l 63s)(0.0000)0.0798
(s.6582)(0.0ooo)0.1078
(3.68s3)(0.0012)0.0888
(3.47ss)(0.0020)
ARIMA VECM BVAR BVARD UBVAR
-1.1424(-1.312s)(0.2023)r.6287
(1.5s80)(0.132e)1.5493
(1.440s)(0.1632)1.8545
(1.7726)(0.08es)
-0.1914(-0.4604)(0.8481)
-1.8766(-6.e4s4)(0.0000)
-1.9740(-6.710s)(0.0000)
-2.8901(-6.s024)(0.0000)
0.3421(0.8661)(0.3e80)0.34t7
(0.8736)(0.3e13)0.3716
(0.8e36)(0.3808)
-1.6658(-4.515)(0.0002)
-1.787s(-4.3368)(0.0002)
-2.5138(-4.tt78)(0.ooo4)
10.311l(1.se63)(0.r24r)0.9375
(0.4123)(0.683e)
-12.2369(-1.8702)(0.0742)
-0.58s2(-0.3es0)(0.6e6s)
-3.3427(-1.1028)(0.281s)-t.2186
(-0.6218)(0.5401)
Table 22 (Contd..)
Nine-Month Forecast Horizon
Constant
0.1205(5.53e1)(0.oooo)0.1199
(6.8388)(0.0000)0.1091
(7.721e)(0.0000)0.1456
(e.1 s6s)(0.0000)0.0989
(6.41s6)(0.0000)0.1031
(8.3285)(0.0000)0.1513
(12.1683)(0.0000)0.1162
(1 1.8887)(0.0000)0.1496
(e.22e2)(0.0000)0.1398
(s.6826)(0.0000)
ARIMA
+8477(-2.003)(0.0s8e)-1.t253
(-1.0721)(0.2e64)0.6362
(t.4s7L)(0.1606)1.0816
(2.t6tt)(0.0430)
Independent Variables
VECM BVAR BVARD UBVAR
-0.2358(-0.4777)(0.6380)
-0.8068(-2.07ee)(0.0506)
-1.5469(-6.st76)(0.0000)
-2.5035(-6.8618)(0.0000)
0.1 555(0.237e)(0.8144)0.7341
(1.67e2)(0.1087)0.7916
(1.8232)(0.0833)
-1.t476(-1.6074)(0.t236)
-1.9782(-4.7r37)(0.0001)
-2.9969(-s.0342)(0.0001)
0.0893(t.6672)(0.1111)0.0275
(0.367e)(0.7168)
-1.4346(-1r.5s36)(0.0000)
-0.3918(-0.2716)(0.7887)
-2.0333(-10.4634)(0.0000)-1.4262
(-0.73re)(0.4727)
Table 22 (Contd..)
Twelve-Month Forecast Horizon
Constant
0.173 t(s.e773)(0.0000)0.1469
(s.e083)(0.0000)0.ts29
(6.0033)(0.0000)0.1903
(8.51e2)(0.0000)0.1459
(6.6686)(0.0000)0.1517
(6.3306)(0.0000)0.1901
(8.2e84)(0.0000)0.1358
(11.8763)(0.0ooo)0.2884
(6.0e5e)(0.0000)
0.2595(6.8064)
ARIMA
-L7541(-2.se{e)(0.018e)-0.3219
(-0.4062)(0.68e7)-0.7243
(-0.8827)(0.38e7)0.4023
(0.s142)(0.6137)
lndependent Variables
VECM BVAR BVARD UBVAR
-0.4574(-1.337s)(0.1e87)
-0.9909(-4.43ss)(0.0004)
-0.9809(-e.0422)(0.0010)
-2.1099(-4.16s6)(0.0006)
-0.1015(-0.387e)(-0.3878)-0.2547
(-0.8378)(0.4138)0.0218
(0.105s)(0.10
-1.0301(-s.034s)(-s.034s)
-0.6921(-4.6921)(0.0002)
-1.9096(-s.304e)(0.0001)
-5.1222(-2.3273)(0.0326)2.4743
(2.s62t)(0.0202)
4.2983(r.e6o2)(0.0666)
t.9664(2.71e8)(0.0146)
-5.9470(-3.s180)(0.0026)-4.8945
(-3.8887)(0.0000 0.0012
Table : 23
Direction - of - Change Analysis
Notes : (1) Entries in the cell are calculated chi-square statistic with 1 degrees of freedom.(2) Critical value of the chi-square statistic with I degrees of freedom atl}Yo
significance level is 2.7 l.(3) * indicates that the statistic is significant at lloh significance level.(4) uindicates that the statistic is significant at 10.08o/o significance level, bindicates
that the statistic is significant at 10.06% significance level.
ARIMA
VECM
BVAR
BVARD
UBVAR
One-Month
0.6628
0.7484
0.0000
0.0000
1.6155
Three-Month
alg44
2.581t',
0.0000
0.0000
2.6966b
Six-Month Nine-Month0.1406
0.5312
0.1406
0.0000
0.0000
Twelve-Month1.2468
4.3681 *
0.1889
0.1639
0.6288
0.0553
0.1169
0.0000
0.0000
0.0000
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