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Financial Econometrics, fall 2011-2012
Professor Paulo Rodrigues; Grader Vladimir Otrachshenko
FORECASTING VOLATILITY: AN ANALYSIS
OF FTSE NAREIT US REAL ESTATE INDEX #420 Raquel Alexandra Dias; #427 Maria Helena Magro; #438 João Ramiro Santos
1. Introduction
Forecasting volatility accurately is very
important in the financial markets to investors,
traders, risk managers and researchers. For that
reason, since Engle (1982) first developed the
Autoregressive Conditional Heteroscedasticity
(ARCH), the published and working papers that
study forecasting performance of various volatility
models have increased exponentially, reaching a
number of 6921, until now.
Volatility is a statistical measure of the
dispersion of returns for a given security or
market index.2 Volatility is a proxy for risk, not the
risk itself.
The volatility is used in many financial and
economic areas (for instance, for pricing
derivative securities: it shows the extent to which
the returns of the underlying asset in options will
fluctuate between a given day and the options’
expiration); used in financial risk management
(mainly since Basle Accord in 1996); used to
calculate the reserve capital in banks and trading
houses (generally, it is at least three times the
Value at Risk – VaR); important for policy makers
1 Federal Reserve – www.federalreserve.gov
2 Investopedia.com
since there are evidences of a clear evidence of
the link between financial market uncertainty and
public confidence; and, at last, Federal Reserve
(USA) and the Bank of England (UK), among
others, take into account the volatility of stocks,
bonds, currencies, and commodities in
establishing its monetary policy. 3
This paper aims to not only model the volatility
of FTSE NAREIT US Real Estate Index but also
provide a good instrument for forecasting
purposes. In order to test the volatility model, we
believe that the best approach is to do a out-of-
sample forecasting.
Our purpose is to analyse this particular Index,
specifically its volatility. In our opinion, this is an
important feature to be analysed, since volatility
in the property commercial was believed to be
very low before the 2007/2008 subprime crisis;
and its nature and magnitude has been the
subject of much debate in the literature.
Through our empirical envidences performed
to the purposes of this analysis, we were able to
create a volatility model regression that is able to
3 Forecasting Volatility in Financial Markets: A Review;
Ser-Huang Poon and Clive W. J. Granger (2003)
forecast, and it is based on a EGARCH regression
model.
In the Section 2, we will analyse some of the
papers related to the forecastiong volatility topic,
and present the main conclusion we can draw
from them. Section 3 presents the data we have
used to run our analysis, taken form the
Bloomberg database. Section 4 introduces the
base model we have constructed in order to
analyse the volatility of this particular index, in
Section 5 we present our empirical results, this is,
the models we used in making volatility forecast
for our market indexes, namely the ARCH, GARCH
and GARCH derivatives approaches. Following, in
Section 6 we will analyse our previous models and
see which forecasts better the volatility in study
by making a out-of-sample forecast. We will finish
in Section 7 with the conclusion of our work.
In appendix, it can be found some specific
calculations and tabels that will be refered during
the paper, as well as a table with the research
papers we have consulted to improve this
research.
2. Literature Review
The econometrics and financial literature is
repleted with studies comparing the various time-
- series models and their capability to forecast
volatility. The existing models are mostly
compared in terms of four attributes: 1) the
relative weighting of recent versus older
observations, 2) the estimation criterion, 3) the
trade-off in terms of out-of-sample forecasting
error between simple and complex models, and 4)
the emphasis placed on large shocks.
The main debate has been about which of the
known models have the best performance. In one
of the earliest papers, Akigray (1989)
demonstrates that Generalized ARCH (GARCH)
outperformes the Exponentially Weighted Moving
Average (EWMA) and Historical Volatility (HIS)
models. Later, Cao and Tsay (1993) as well as
Heynen and Kat (1994) conclude that Exponential
GARCH (EGARCH) performs better for volatility
forecasting for small stocks and exchange rates.
In 2001, Bond and Wang studied the nature
and measure of the volatility in the commercial
property market in the UK, using a stochastic
volatility model.
We also based our study in the paper written
by Poon and Granger (2003), that cover 93 paper
researches and review their methodology and
empirical findings. They have concluded that
GARCH model and alternative versions (mainly
EGARCH and GJR-GARCH) are the most used in
these papers and perform better.
In adition, we also took into analysis the
Yolama and Sevil (2008) paper, whose purpose
was to employ seven different GARCH models (E-
GARCH, PARCH, TARCH, IGARCH, C-GARCH,
GARCH and GARCH-M) to forecast in-sample daily
stock market volatility in 10 different countries.
They found that asymetric volatility models
performed better in forecasting stock market
volatility than the historical model.
At last, we also took in count the 2010 paper,
written by Patton, that makes an analytical
analyses of how “less sensitive” forecast loss
functions can lead to incorrect inferences and
selection of inferior forecasts over better
forecasts, focusing on volatility forecasting.
In Table 2 (found in Appendix), we summarize
all the papers we based on.
3. Data
The study covers 36 years (Jan 1972 – Dec
2008) of monthly data of the FTSE US Real Estate
Inde’s prices4. FTSE NAREIT US Real Estate Index is
designed to present investors with a
comprehensive family of REIT performance
indexes that spans the commercial real estate
space across the US economy. The National
Association of Real Estate Investment Trusts®
(NAREIT) is the worldwide representative voice
for REITs and publicly traded real estate
companies with an interest in U.S. real estate and
capital markets. NAREIT's members are REITs and
other businesses throughout the world that own,
operate, and finance income-producing real
estate, as well as those firms and individuals who
advise, study, and service those businesses.5
According to some financial empirical results,
as the ones from Fama and French Model (1988),
short-term predictions are too noisy when
compared to medium/long-term ones. Then, our
concern at the initial stage was about the
introduction of long-term explanatory variables
that would allow us to reduce this noisy effect.
For this purpose, we have decided to include
the 10-Year US Government Bond yields, the US
Gross Domestic Product Growth Rate and the
4 Listed companies in Table 1 - appendix
5 http://www.ftse.com/Indices/FTSE_EPRA_NAREIT_Glo
bal_Real_Estate_Index_Series/index.jsp
Unemployment Growth Rate as regressors. The
reasoning to do so is pretty intuitive: first, we
have considered important to include the T-Bond
yields because since they are a proxy for the
minimum return the marginal investor is willing to
receive for engaging in an adjusted/controlled risk
investment or to pay for a limited and safe loan,
we thought it could reasonably work as a “Bottom
Line Return”. Second, we wanted to include pro-
cyclical US GDP Growth variable but, in order to
obtain consistency along the model, US GDP
Growth statistics (which is reported quarterly)
needed to be given in monthly frequency as well,
and so we have used a proxy – the Industrial
Production Index, which is commonly used in
empirical works with this same function6. The
economic intuition behind it is that the US FTSE
Index (abbreviation used for the FTSE NAREIT US
Real Estate Index) is positively correlated with the
economy, so it is expected that some of the US
GDP growth will affect the returns of our index.
Third, the Unemployment Growth Rate is also a
directly related and cyclical series, which we
wanted to test if it has any explanatory power on
US FTSE returns patterns.
All these data were taken from the platform
Bloomberg, and treated using the Eviews 7.0
software packaging.
Both prices of the FTSE US Index and
Industrial Production Index, we turned into
returns, using the below formula:
Rt = log(Pt) - log(Pt-1) , where the Pt = close
price at time t, and Pt-1 = close price at the t-1
6 As example: Efficiency of Financial Intermediaries and
Economic Growth in CEEC, Andrus Oks (2001)
period.
4. Methodology
As a first step in our research we will analyse
each of the previously explained variables, doing
stationary, autocorrelation and normalization
tests. After, we will construct a base model with
the significant variables and test it, recurring to
ARMA,correlation, etc., tests so we can have the
best model of the returns of the US FTSE Index.
4.1. Dependent Variable – US FTSE Index
As this paper aims to study the volatility
existing among the US FTSE Returns, we took US
FTSE Index as the raw input. For the purposes of
this study, we have considered important to test
each transformation of this dependent variable.
This way we could know more or less our
limitations, the corrections needed to be applied
and about what could we infer.
Once the prices themselves generally come
with struggling difficulties associated (as its non-
linearity), the level variable needed to be the
transformation of US FTSE Index into price
logarithms.
Its descriptive statistic is present in Figure I,
from which we could state that it has an expected
and made linear with an upward slopping (until
the last moment, when it drops almost vertically).
Essential tests (the same performed for ARMA
Model) are present in Figures II and III, from which
we could conclude that the level variable is
robustly non-stationary for a 10% significant level
considered (as KPSS conclusions confer
robustness to the ADF conclusions for being in
accordance) and that it has Serial Autocorrelation,
meaning it has persistent memory, mainly in its
first lag.
The same was developed for its first
differences (which is the variable in focus for this
paper). Regarding US FTSE Returns (the first
differences of the US FTSE Prices, the variable
characterized above), we perceive no trend in its
graph and a great change in its distribution (but
still not being normally distributed). From the two
stationarity tests, we could conclude that we have
statistical evidence enough to reject the null
hypothesis for the existence of a unit root and
that the series is stationary or follows an I(0)
(which is in accordance with what was expected
from first differences of a non stationary initial
series). Also, there is no Serial Autocorrelation
(despite the first lag being at the upper
boundary).
Why did we perform these tests for
stationarity and Serial Autocorrelation?
Stationarity is important to understand the effects
of events occurring in previous years in the
current period. And, if there is Serial
Autocorrelation, inferences taken from the model
might be biased.
The last test incurred was to confirm that
normality still not being applied to the distribution
of the dependent variable – the US FTSE Returns.
With the dependent variable analyzed and
properly corrected, we have proceeded with the
construction and analysis of a base model – that
means, with a regression that most accurately
models the volatility in this specific index – which
will allows afterwards building the best
forecasting model.
4.2. Base Model
Before we construct this base model, it is
needed to assess the quality of data for each of
the explanatory variables, as we were essentially
concerned with the order of integration of each
variable, meaning, the stationarity within the
series. The apprehension about this issue had to
do with the fact that not accounting for the
presence of a unit root could lead to a Spurious
Regression (regression in which independent
variable has reasonable explanatory power,
despite not being related with the dependent
variable), which then could mislead all the
conclusions derived from the model, even though
the model seems correct and well specified.
The stationarity test was developed in three
steps: first, we have computed some standard
descriptive statistics in order to get some intuition
about the behaviour of the series; second, we
have looked to the graph of each variable in time
and the correspondent correlogram (more
specifically, the autocorrelation function, with the
aim to get sense about the memory of the series);
and third, we have applied two formal stationarity
tests – the Augmented Dickey-Fuller (ADF) and
the Kwiatkowski-Phillips-Schmidt-Shin (KPSS).
These two tests confer robustness to the
conclusions taken on stationarity. More
concretely, ADF test has as its null hypothesis the
existence of a unit root, though the series is non-
stationary, and contrarily the KPPS test the null is
that the variable is indeed stationary. As results of
their performance, four outcomes appeared: in
two of them, the tests agreed the (in) existence of
stationarity; and, in the remaining two, their
conclusions conflicted (remaining inconclusive).
Linearity issue was corrected through the
transformation of the series in logarithmic
variables, as we did for the Industrial Production
Index and US FTSE Index prices. According to the
Table 3 - Appendix, whenever previous tests
proved the existence of non-stationarity in the
non-transformed level series, they actually were
stationary in their first differences. Curiously, the
logarithm of the US FTSE prices was already
stationary in level as well as Unemployment Rate.
Regarding this last one, we used the
unemployment growth rate since we are trying to
explain the first differences of prices of the US
FTSE Index. (For full description see Appendix
Figures VIII to XXIII)
Summarizing the results, given that both the
ADF and KPSS tests confirmed the existence of
non-sationarity in our original series, we have
transformed non-linear and price denominated
series into logarithms and then we have taken the
first differences (either for the logarithmic and for
the rates denominated series) in order to
eliminate the unit root, which resulted in robust
stationary variables. Nevertheless, the use of the
differences implicitly eliminates any possible long-
rung relationship, since we were expecting that
the variable would stabilize and its first
differences would collapse to zero in the future:
level variables are generally I(1), with the
exception of Unemployment Rate and US FTSE
Index Returns, and first differences are all I(0) ().
To assess the performance of the explanatory
variables, we have developed the following
contemporaneous equation:
Results presented in Table 4 - Appendix
showed statistical evidences against our initial
intuition: the parameters of the Industrial
Production Index and the Unemployment Growth
Rates first differences revealed to be insignificant
to explain US FTSE returns, meaning that they
have lower explanatory power and so they should
be disregarded for the next phases. So, we were
left with the significant first differences of the T-
Bond yields. Financially interpreting the significant
parameter of yields in the returns regression is
that an increase of 100 b.p. in the 10-year T-Bond
yields’ differential is expected to decrease the US
FTSE Returns by approximately 0.03%, on average,
ceteris paribus.
Once identified and excluded the non-
significant variables, we have regressed again the
US FTSE Returns in order to the remaining
significant variable, the 10-years T-Bond Yields’
differential, with the intention of understanding
what is not considered separately, instead in
accumulatively included in the residuals.
Observing the regression’s correlogram:
There are two important conclusions to be
highlighted: first, according to the Auto-
Correlogram Function (ACF) and the Partial Auto-
Correlation Function (PACF) (Figure XXIII -
Appendix), there is the need to introduce at least
three or four lags of the dependent variable and
three lags of the error term. Therefore, it seemed
that introducing an ARMA structure in the
regression it might help to explain even more the
returns, capturing explainable residuals from the
error term.
In order to choose the best ARMA model, we
have relied on four comparison criterions: (1) the
adjusted R-squared, (2) the Akaike Information
Criterion (AIC), (3) the Schwarz Information
Criterion (SIC) and (4) the number of
insignificance variables. The regression applied
was then:
The results from this kind of trial and error
process are expressed in Table 5 – Appendix.
In accordance with the information presented
on the previous table, the ARMA(3,3) Model is the
more consensual one when considering all the
four criteria applied: it is the one with the lowest
AIC (of approximately -3.196), it has the third and
the fourth best Adjusted R-squared and SIC,
respectively, and it is also the single one without
insignificant ARMA terms (just the constant,
which is not extracted from the model once it
ensures that once it ensures a more consistent fit
through the reduction of the slope’s sensitivity
(since in the graph there is clearly an intercept).
The number of lags economically understandable
since a reasonable number of companies
composing the US FTSE Index that report their
earnings in a quarterly frequency (so, the results
announcement of the last three months can
actually impact the following period). Going even
further in the past influence, a year of lags could
be significant; however, it would be difficult to
find economic reasons for it.
In conclusion, our Base Model is as follows:
In Table 6 – Appendix, it can be seen the
output resulted from the Base Model. From it, we
concluded that all the variables are significant and
the model as a whole has statistical and economic
logic behind: the impact of the yields is negative
and small and the magnitude of Auto Regressive
and Moving Average components are quite high.
4.3. Analysis of the Base-Model
Once tested and selected the Base Model we
are finally able to focus on the objective of this
paper: modelling the volatility of US FTSE Index
Returns. As a first step, it was necessary to ensure
the inexistence of Serial Correlation in the Base
Model residuals through the performance of the
Breusch-Godfrey Test with three lags of residuals.
Table7 - Appendix exhibits the results of this test:
the LM-statistic is reasonably small (assuming a
value of 0.823), and so it lies in the null hypothesis
region, we do not reject the absence of serial
correlation.
Furthermore, it was also necessary to test for
non-linearity of the previous regression to
reinforce the correct specification of the model,
for what we have applied the Ramsey RESET Test
through the following auxiliary equation:
, b = 3
Running the equation above, we have got the
results summarized in the Table 8 - Appendix,
from which we concluded that the Ramsey RESET
Test rejected the null hypothesis for the absence
of linearity, since the log likelihood ratio is higher
than the Chi-square Critical Value for three
degrees of freedom (corresponding to the number
of restrictions in the model).
Finally, it was necessary to infer about
Normality of the model’s residuals. The
Autoregressive Conditional Heteroscedasticity
(ARCH) family requires errors to be well behaved,
which means that errors must follow a Gaussian
White Noise. In order to perform a Jarque-Bera
Test for Normality, it is required to check the
errors histogram. Figure XXIV - Appendix shows
statistical evidence to reject the null hypothesis
for normally distributed errors: the Jarque-Bera p-
value close to zero rejects the null hypothesis for
the joint test stating that the skewness and the
kurtosis excess equal zero. Nonetheless, this
particular problem does not affect in any
circumstances our research and results, and
therefore it was not corrected.
Summing up, there was reasonable statistical
evidence supporting the inexistence of serial
correlation in the specified model. The application
of Autoregressive Conditional Heteroscedasticity
frameworks were due to two main reasons: first,
the Adjusted R-squared was relatively small (only
8.5%), and second, attempting to the Figure XXV -
Appendix, we have comprehended that much of
residuals’ variability was still left to be explained.
5. Empirical Results
5.1. Modelling Volatility
The Unconditional Variance is the expected
variance for the US FTSE Returns, without any
restrictions. Conditional Variance defines the
expected variance for the same variable but
restricted to the information disclosed in the prior
periods [so, we can state Conditional Variance is
denoted by E (σ2t | Information made available
on the market until the current period t)].
In order to infer about US FTSE Index volatility
forecasts, we used five models – the
Autoregressive Conditional Heteroscedasticity
(ARCH), Generalized Autoregressive Conditional
Heteroscedasticity (GARCH), the Exponential
Generalized Autoregressive Conditional
Heteroscedasticity (EGARCH), the Threshold
Generalized Autoregressive Conditional
Heteroscedasticity (TGARCH) and the Generalized
Autoregressive Conditional Heteroscedasticity in
Mean (GARCH-M). Intuitively, there would be no
need to apply the two simplest models – ARCH
and GARCH – once it does not have into account
the differential impacts of negative and positive
shocks, which, for this specific case of US FTSE
Index, is important to consider because of the
Real Estate Bubble in 2008’s that changed
Conditional volatility for this sector onwards –
seen in the Figure XXVI Appendix. Moreover,
ARCH and GARCH Models face statistical
problems, the ones that boosted the development
of GARCH’s extensions (nevertheless, still not
being able to correct all of the drawbacks of their
base models. For instance, ARCH Model’s order is
decided with basis on trial and error methodology
(even if achieved a good model, we cannot never
know if it is the best fitted to the data); also, the
number of lags (order) needed to capture all
conditional variance might be too large (and so
the model becomes not parsimonious, existing
better alternatives to it); finally, negativity
constrains might be violated (as the number of
parameters increase, the more likely there would
be parameters with negative estimated values).
ARCH Model is falling into disuse as GARCH Model
already takes into account all of its terms
[checking GARCH (q; p) equation, it can be
explained in other words as the ARCH (q) plus the
sum of p lags of the dependent variable]. GARCH
Model only improved some ARCH Model’s
limitations (for instance, it decreased the
probability of estimating negative parameters and
it needed a lower number of lags to capture the
same, becoming the model more parsimonious), it
still having some drawbacks: the violation of non-
negativity conditions by the estimated model still
occurring; GARCH Model also does not account
for leverage effects (despite not being a relevant
modelling restriction for the Index studied in this
paper), although they can account for volatility
clustering (concept noted by Mandelbrot to
describe large changes tend to be followed by
large changes, of either sign, and small changes
tend to be followed by small changes”) and
leptokurtosis (the property of having a similar to a
“bell” distribution but with fatter tails) in series;
moreover, it does not allow direct feedbacks
between Conditional Volatility and Conditional
Mean.
All these models assume to have well behaved
and normally distributed errors. If this is not true,
it must be taken into account, for instance,
through Maximum Likelihood Estimation (MLE)
with robust variance/covariance estimators,
known as Quasi-Maximum Likelihood Estimation
(QMLE). As stated previously, for this specific case
residuals are well behaved, basing on the
Correlogram.
5.1.1. ARCH(q)
The ARCH Model is the simplest model of the
Autoregressive Conditional on Heteroscedasticity
family. The idea behind it is to model the variance
of the error term using an autoregressive
framework of a given order. The biggest
contribution of the model is that some time series
variable could be explained, not by changes in
exogenous variables but rather by the volatility
embodied in the series. This means that, there is a
non-linear relation between the original times
series and the i.i.d. shocks underlying it. Yet, these
models allow for the presence of linearity in
modelling the mean.
The truth is that ARCH has not been used
by practitioners or academics, essentially because
applying this framework brings a set of difficulties.
At the top, there is the non-negativity constraints
that might violated specially when including many
lags and also, the complexity in choosing the right
number of lags. Still, the most important is that
when modelling stock returns volatility we must
have in mind that this is a persistent
phenomenon. Meaning, that if the volatility is
high today it is expected to remain high
tomorrow, though to explain the volatility we
must include the volatility itself what does not
happen with ARCH.
This is the simplest model on Conditional
Heteroscedasticity, the easiest to handle and to
give us a first insight about the presence of ARCH
Effects within residuals. It takes care as well as of
clustered errors and nonlinearities (which was
kind of “forced” through the resource to
logarithms, whose important and widely known
function is the one of make variables linear and
progressing smoother); jointly contributing to
better forecasts. These factors were the motives
why we insisted in apply ARCH Model, despite
knowing ahead that it would be almost
immediately denied by realized evidence for the
reasons stated above.
Assuming that the Base Model residuals are
well behaved but still non-homoscedastic, we can
set up a model with the aim of modelling the
variance of the residuals. More specifically it can
be expressed by following expression:
ARCH (q)
In Table 9 - Attachments are presented
the results for some ARCH models. In view of that,
we are able to conclude that the best model is
ARCH (2) essentially because is the one o makes
all regressors of the base model statistically
significant. This means that the unexplained
variation of US FTSE Returns from two months
prior have explanatory power over the
contemporaneous volatility. Over again, this
emphasizes the persistence character of the
volatility.
Finally, in the next sections we will discuss
more complex models that will enable us to
overcome some of the mentioned problems, at
the same time that we perform an out-of-sample
analysis.
5.1.2 GARCH(q,p)
GARCH model is the simplest extension to the
previously presented ARCH Model. This is true as
its expression is the ARCH equation added of a
term:
GARCH (q, p)
With ut being the residuals from the Base
Model and the conditional variance (computed
from relevant past information). The difference to
the previous model lies on the last term which is
the p optimal number of lags of the dependent
variable – Base Model’s Residuals.
This is the simplest model from GARCH
frameworks, correcting then some of ARCH’s
problems – its parsimony, its over-fit and its lower
frequency of non-negativity constraints violation.
Still, remaining common flaws demanded simple
GARCH Model to be tuned through the
reinvention/ extension of GARCH.
Bearing the theoretical background above in
mind, we have run GARCH Model covering all the
lags’ combinations until the 6. We considered half
of a year for any peculiar reason, it was just
because some models improved until reach the
fifth lag, but then all of them became worse. In
order to select the best orders’ combination of
GARCH Model, we have applied Likelihood Ratio
Test (the highest the best, given the null and
alternative hypothesis, presented in Table 10 and
Figure XXVII – Attachments) and Akaike
Information Criterion (the lowest the best,
immediately extracted from Eviews outputs). So,
through the collection of these two determinants
for each combination, given by the two tables on
the same attachment, we could conclude that
GARCH (4; 3) was the best one within GARCH’s
Models:
GARCH (4; 3)
This means that variance is kind of persistent
for four months lagged; in other words, volatility
remains smooth for the period during what new
information from the companies is not publically
disclosed.
Even though, it turned some explanatory
variables insignificant (four ARMA variables are
now not significant), which might implicitly mean
that it would not be the model with the best fit to
the data we had. We were already expecting this
occurrence as returns are unpredictably volatile
and reacting more hysterically to negative rather
than positive effects; together with GARCH’s
boundaries, intuitively we thought we would need
to use further extensions to ARCH Model, once
this Index had a special and negative impact
recently.
For the following GARCH Model extensions, we
assumed Normal Error Distribution. This might not
be true as it accounts for volatility clustering but
kind of ignores their leptokurtosis inherent
characteristic. Nevertheless, it develops an
irrelevant role for our subject and conclusions.
5.1.3 EGARCH(q,p)
This extension/ asymmetry of the GARCH
Model takes into account the different impact of
negative and positive shocks on the sector, given
by:
EGARCH (q , p)
Since it is modelling the logarithm of the
variance, the variance will be always positive,
there is no need to impose non-negativity
constrains on parameters, and asymmetries/
leverage effects are allowed and accounted in the
term (whenever it is positive, it reflects that
positive shocks on the market generate less
volatility than negative shocks of similar
magnitude). In financial theory, this last
phenomenon is designated by overreaction of
common investors, and it is related to their level
of risk aversion and their deviations from
characteristics assumed for rational and efficient
investors (for instance, the preference for national
assets rather than the best ones). Although, if
numbers obtained were too small, we could have
some problems with values tending to infinite.
Again, applying the same procedures
sequentially explained for GARCH, we concluded
that EGARCH (5; 5) Model was the best fitted
against all the other options tested:
EGARCH (5, 5)
Financial intuition behind its orders may be
related to the volatility adjustments to analysts’
expectations or other more direct signals given to
investors, generally disclosed previously to the
firms’ announcements and so returns capture the
prior five months.
Observing more carefully the Table 11 and
Figure XXVIII in Attachments, we could notice that
four of coefficients came negative (moreover, the
first one was strongly and significantly negative),
which might be perceived against the non-
negativity constrains for the parameters’
property. However, there is no misunderstanding:
EGARCH’s parameters can be negative, meaning
EGARCH does not impose any non-negativity
constrains, once it is a logarithmic variance and so
it is always positive.
5.1.4 TGARCH(q,p)/ GJR Model(q,p)
This second extension of GARCH Model also
accounts for asymmetries, it was also designed in
a way to capture leverage effects between returns
and volatility; although, it performs differently
through a dummy variable, being specified by:
TGARCH (q, p)
Basically, according to a Journal of Finance and
Economics about the EGARCH and TGARCH
Models “The leverage coefficients of the EGARCH
model are directly applied to the actual
innovations while the leverage coefficients of the
GJR model can connect to the model through an
indicator variable. For this case, if the asymmetric
effect occurs, the leverage coefficients should be
negative for the EGARCH model and positive for
the GJR model”, conferring robustness to each
other conclusions.
In which is the dummy variable:
equals the unit if the shock occurred in the market
was negative (if < 0) and equals 0 (so,
annulling the last term of the equation above)
otherwise (if the shock occurred in the market
was positive, inferred by a > 0). This way, a
negative shock will weigh more, which is in
accordance with financial theory: negative shocks
have higher impact on the economy than a
positive one, even if they have the same
magnitude – overreaction (concept explained in
EGARCH).
Running all over again the same procedures
from the last two models, the best one (as we
could conclude from Likelihood Ratio and the
Akaike Information Criterion in the Table 12 and
Figure XXIX was TGARCH (4,3):
TGARCH (4,3)
It has the same number of orders as GARCH, so
we could input to it the same reasons.
In order to apply the GARCH-in-Mean
framework, it was necessary to define the best
previously presented model to work as its base
model. This way, the one with the highest
Likelihood Ratio and the lowest Akaike
Information Criterion – the EGARCH (5,5) – was
applied to test for GARCH-M.
5.1.5 EGARCH-M(q,p)
The GARCH- type of model, initially proposed
by Engle et al. (1989), differs from previous by not
changing the specification model for the
conditional variance, but instead for including it as
an explanatory variable into our initial structural
model. The idea underlying the use of EGARCH-M
is that investors should be rewarded with higher
returns whenever they take additional risks. This
financial thinking resulted in the GARCH-M as risk
strongly relies on the volatility existing on the
market, which is represented by:
If δ from (assuming that
follows a normal distribution with zero mean
and variance) is positive and statistically
significant, it means that an increase in the
conditional variance will increase the risk and so
the return demanded by investors, upgrading the
returns’ Index. Although intuitively this seems a
good idea, in practice these types of models
revealed controversial findings. By doing some
literature review we found that there are several
cases in which the risk-return relationship turned
out to be negative. Are examples of this the
followings, Lie et al. (2005) and Guedhami and Sy
(2005), while others found no significant relation
at all like Shin (2005) and Baillie and DeGenarro
(1990). Consequently at that point an economic
rationing behind this particular finding must be
drawn. Some authors refer that this negative risk-
return relationship is explained by a simple
understanding of business cycle, namely when the
economy is at a peak of a business cycle, when
typically expected returns are low, the better-
than-habit consumption levels make investors
more risk tolerant and thus require a lower
reward-to-risk.
Despite the fact that the prior justification
seems reasonable, another point of view
regarding the negative trade-off of risk-return
exists. Namely, the one presented by Lanne and
Saikkonen, which states that the negative
parameter on the conditional variance (or in any
other specification) is due to the non-exclusion of
the constant term in the base model when it is
included. Accordingly, when applying the GARCH-
M framework one should impose that the
constant term is equal to zero. Theoretically
makes sense, since we do not expected that
returns is explained by a constant, meaning that,
for the next period one expected the previous
value (referring to past values) plus some
structure, in which the conditional variance can be
included (also, in our case is given by the Yields’
Differential) and a random shock.
In this sense, there are various practical
aspects to bear in mind when including the
conditional variance modelled by an
Autoregressive Conditional Heteroscedasticity
framework as a regressor. Fundamentally, the
model specification can be linear or logarithmic in
the conditional variance. Therefore three different
regressors can be introduced in the base model:
first the conditional variance itself ( ), second
the conditional standard deviation component ()
and thirdly the logarithm of the conditional
variance ( ). In addition to this, there is the
necessity of specifying the underlying model,
which in our case is given by the EGARCH that
corresponds precisely to our best one, selected
previously.
As previously stated, one have to choose the
best EGARCH specification, in this sense we run
the three presented specifications which are
presented in Tables 13, 14 and 15 - attachments.
From the previous tables we are able to exclude
the conditional variance approach at once (Table
13), because it is the only one that is not
individually significant. A possible explanation
could be that the “number” of variance per se as
no financial meaning, it is not much informative, is
if we had a problem of scale between the
variables. By performing the remaining
regressions, we found that the conditional
standard deviation and the logarithm conditional
variance are indeed statistically significant.
However the parameters on these regressors are
slightly negative even though it is strongly
statistical significance, meaning that we are back
to the scenario presented above.
Again, in order to choose the best model we
applied some criteria comparison as previously,
namely, looked to the log-likelihood ratio and to
Akaike information criterion plus the insignificant
variables in the structural model. In particularly,
given the wide range of tests performed there
was one fundamental condition that had to be
met in order to the model e accepted, that the
parameter on the conditional variance is
statistically significance. Otherwise it would not
make sense to use the model in the forecasts.
Having this in mind the conclusions are
straightforward, for the case of the inclusion of
the conditional variance the best model is the
EGARCH-M (4; 3), essentially because all base
model’ parameters are significant and it as the
best others criterions. In what relates the
logarithm of the conditional variance, the best fit
has been obtained by the EGARCH-M (3; 4), the
reasons are the best log-likelihood and AIC with
the lowest number of insignificant parameters.
Finally, the best conditional standard deviation
model is the EGARCH-M (3; 3), because it had the
best AIC information criteria and none of the
parameters were statistically insignificant.
In Table 16 - Attachment we present a
comparative analysis of the best fitted models, in
order to achieve the best in-mean specification.
Accordingly, the overall model and specification is
EGARCH-M (4; 3) fundamentally because it had
the best AIC of all the analysis as well as the log
likelihood, simultaneously with no insignificant
parameters.
6. Forecasting Volatility
6.1. Out-of-Sample Analysis
As we have previously observed,the best
models we have found that best fit the volatility
are the ARCH(2), GARCH(4;3), EGARCH(5,5),
TGARCH(4,3) and GARCH-M(4;4). In order to see
which one of them forecasts most perfectly the
volatility of the US FTSE Index, we performed an
out-of-sample analysis, concerning the period
between Jan 2009 and Nov 2011. We have
choosen this particular sample size to perform this
analysis because, since it incorporates the 2008
mortgage crisis, it accounts for the crisis effects
on volatility (as mencioned before, there are
empirical financial researches that have found
that “(…)As shown, stock market volatility displays
a strong countercyclical pattern—peaking just
before or during recessions and falling sharply late
in recessions or early in recovery periods.(…)” 7).
Since we are currently living a crisis again (the
Euro Crisis), we wanted to have a crisis effect in
our model, so it would better forecast the
volatility for the period 2009-2011, and most
importantly, so it could accurately forecast the
volatility in this index for the next periods ahead.
The out-of-sample forecast analysis are used
by forecasters to determine if a proposed leading
indicator is potentially useful for forecasting a
target variable. The steps for conducting an out-
of-sample forecasting experiment are as follows:
1) Divide the available data on the target
variable and the proposed leading indicator
(both stationary) into two parts: the in-sample
data set (roughly 80% of the data – in our
particular case, and due to the reason preciously
above explained it ascended to ≈93%) and the
out-of-sample data set (the remaining 20% of
the entire data set – in our case ≈7%). As
noticed, this first step was already made.
2) Once chosen the in-sample data set, is
should be choosen competing forecasting
models – already explained in section 5. With
these forecastng models (EGARCH, GARCH,
TGARCH, GARCH-M and ARCH) . It is these
competing models that we are going to run an
out-of-sample “horserace” with.
3) To run a horserace (i.e. forecasting
competition) between these models, we must
7 Stock Market Volatility: Reading the Meter, Hui Guo
(2002), Economic Synopses Nr. 6
“roll” each model through the out-of-sample
data set one observation at a time while each
time forecasting the target variable the chosen h
periods ahead.
4) Now to decide the winner of the
horserace between the models, we must
calculate the Average Loss associated with these
various models: the “standard” average loss
functions Mean-Squared Error (MAE), Mean
Absolute Error (MSE) and the Mean Absolute
Percentage Error (MAPE):
In statistics, the mean square error or MSE of
an estimator is one of many ways to quantify the
difference between an estimator and the true
value of the quantity being estimated. MSE is a
risk function, corresponding to the expected value
of the squared error loss or quadratic loss. MSE
measures the average of the square of the
“error.” The error is the amount by which the
estimator differs from the quantity to be
estimated. The difference occurs because of
randomness or because the estimator doesn’t
account for information that could produce a
more accurate estimate.
The MSE is the second moment (about the
origin) of the error, and thus incorporates both
the variance of the estimator and its bias. For an
unbiased estimator, the MSE is the variance. Like
the variance, MSE has the same unit of
measurement as the square of the quantity being
estimated. In an analogy to standard deviation,
taking the square root of MSE yields the root
mean squared error or RMSE, which has the same
units as the quantity being estimated; for an
unbiased estimator, the RMSE is the square root
of the variance, known as the standard error.
The MAE measures the average magnitude of
the errors in a set of forecasts, without
considering their direction. It measures accuracy
for continuous variables. The equation is given in
the library references. Expressed in words, the
MAE is the average over the verification sample of
the absolute values of the differences between
forecast and the corresponding observation. The
MAE is a linear score which means that all the
individual differences are weighted equally in the
average.
Where N is the number of observation in the
out-of-sample data, is the observed and real
volatility – calculated through the squared
differences of the returns ; and
is the volatility forecasted by the model in
analysis.
5) The forecasting method that has the
smallest MAE and MSE average losses in the
out-of-sample forecasting experiment is the
superior forecasting method. If one forecasting
method has a better MAE measure while the
other forecasting method has the better MSE
method then you have a split decision. Then
the only way you can determine a winner
between the competing forecasting models is to
break down and choose one of the average loss
functions to base your choice on, either the MAE
average loss function or the MSE average loss
function.
After we have performed the above steps
described, we were able to conclude which of
these models have the best and strongest
forecasting power (see Table 17 in Appendix). The
one with the lowest both MSE and MAE is the
EGARCH model, which goes along with our
expectations and other papers on the same
subject of forecasting volatility.
6.2. After October 2011
After we have chosen the best model that
most accuretly forecasts the volatility of the FTSE
NAREIT US Real Estate Index, this is, the E-
GARCH(5;5). Taking into account the forecast of
the future 10-year US Treasury bond yields and
the returns of the index in analysis for the next 3
months (November 2011, December 2011 and
January 2012), we were able to forecast the
volatility for the same period, as seen in the table
below.
After Oct 2011
E-GARCH (5;5) Forecast model
2011M10 0.013876
2011M11 0.014090
2011M12 0.011810
2012M01 0.010485
As we can see, the is expected to remain around
1%. This low volatility can be explained by the
interventation of the US Government on the
mortgages agencies in this last 2/3 years, and due
to the fact that real estate investments still
represent a lower risk than other investments like
sotcks or option. (Real estate investments have
strong history of total return. Over a period of 30
years from 1977 to 2007, almost 80% of the total
U.S. real estate return came from income flows.
This helps bring down volatility, as investments
which rely on income returns result in being less
volatile than the ones relying heavily on capital
value returns)8.
7. Conclusion
We have compared the forecasting ability of
several volatility models – the ARCH, GARCH,
EGARCH, T-GARCH and EGARCH-M - focusing on
four issues: (1) the proper weighting of older
versus recent observations, (2) the relevance of
the parameter estimation procedure, (3) the
proper weighting of large return surprises, and (4)
the effect of a recession in the returns and
volatility.
On the face of it, one could argue that the
empirical evidence provided in this paper suggests
that it is possible to produce a volatility
modelling model that can be used to forecast
the volatility of the FTSE NAREIT US Real Estate
Index. Moreover, we have found empirical
evidences that in fact the best model to do this
forecast is the E-GARCH (5; 5) volatility model,
since it takes into consideration the negative 8http://www.comparebroker.com/blog/2011/10/20/
is-it-wise-to-invest-in-real-estate-amid-economic-uncertainties/
shocks of such crisis as the 2008 mortgage bubble
and the current Euro crisis we are facing now –
that even though it most felt in Europe, it has
some effects on the US real estates returns and
volatility.
Our evidences are coherent with other papers
like Chang Su (2010)9, that also tested models to
forecast volatility and concluded that EGARCH
model accomodates better the leverage effect,
volatility persistence, fat tails and skewness.
9 Application of EGARCH Model to Estimate Financial
Volatility of Daily Returns: The empirical case of Chine; University of Gothenburg
APPENDIX
Company Listed In FTSE NAREIT US Real Estate
Index – TABLE 1
A Acadia Realty Trust
Alexandria Real Estate Equities, Inc.
American Campus Communities, Inc.
American Realty Capital Healthcare Trust
American Realty Capital Properties, Inc.
American Tower Corporation
Anderson-Tully Company
Apartment Investment & Management Company
Apollo Residential Mortgage Inc.
Apple REIT Nine, Inc.
Apple REIT Six, Inc.
Archstone
Ashford Hospitality Trust, Inc.
AvalonBay Communities, Inc.
Acadia Realty Trust
Alexandria Real Estate Equities, Inc.
American Campus Communities, Inc.
American Realty Capital Healthcare Trust
American Realty Capital Properties, Inc.
American Tower Corporation
Agree Realty Corporation
American Assets Trust
American Capital Agency Corp.
American Realty Capital New York Recovery REIT, Inc.
American Realty Capital Trust
Americold Realty Trust
Annaly Capital Management, Inc.
Apollo Commercial RE Finance, Inc.
Apple REIT Eight, Inc.
Apple REIT Seven, Inc.
Arbor Realty Trust, Inc.
ARMOUR Residential REIT
Associated Estates Realty Corporation
B Behringer Harvard Multifamily REIT I
Behringer Harvard Opportunity REIT II
Berkshire Income Realty
Blackstone Real Estate Advisors
Boston Properties, Inc.
BRE Properties, Inc.
Brookfield Office Properties
Behringer Harvard Opportunity REIT I
Behringer Harvard REIT I, Inc.
BioMed Realty Trust, Inc.
Boardwalk REIT
Brandywine Realty Trust
Broadstone Net Lease, Inc.
C Camden Property Trust
Capital Trust, Inc.
Capstead Mortgage Corporation
Carey Watermark Investors Incorporated
Cedar Realty Trust, Inc.
Chesapeake Lodging Trust
CNL Lifestyle Properties, Inc.
Cole Credit Property Trust II, Inc.
Cole Credit Property Trust, Inc.
Colony Financial, Inc.
CoreSite Realty Corporation
Corporate Property Associates 15
Corporate Property Associates 17 - Global, Inc.
CREXUS Investment Corp.
CYS Investments, Inc.
Campus Crest Communities
CapLease, Inc.
Care Investment Trust, Inc.
CBL & Associates Properties, Inc.
Chatham Lodging Trust
Chimera Investment Corporation
Cogdell Spencer Inc.
Cole Credit Property Trust III, Inc.
Colonial Properties Trust
CommonWealth REIT
Corporate Office Properties Trust
Corporate Property Associates 16 - Global, Inc.
Cousins Properties Incorporated
CubeSmart L.P.
D DCT Industrial Trust Inc.
Derwent London Plc
Digital Realty
Duke Realty Corporation
Dynex Capital, Inc.
DDR Corp.
DiamondRock Hospitality Company
Dividend Capital Total Realty Trust Inc.
DuPont Fabros Technology, Inc.
E EastGroup Properties, Inc.
Electric Infrastructure Alliance of America, LLC
Equity Lifestyle Properties, Inc.
Equity Residential
Excel Trust, Inc.
Education Realty Trust, Inc.
Entertainment Properties Trust
Equity One, Inc.
Essex Property Trust, Inc.
Extra Space Storage, Inc.
F Fair Value REIT-AG
Federal Realty Investment Trust
First Industrial Realty Trust, Inc.
First REIT of New Jersey
Forest City Enterprises, Inc.
Federal Capital Partners
FelCor Lodging Trust Incorporated
First Potomac Realty Trust
Forest Capital Partners LLC
Franklin Street Properties Corp.
G Gables Residential Trust
Getty Realty Corp.
Glimcher Realty Trust
Global Income Trust, Inc.
Government Properties Income Trust
General Growth Properties, Inc.
Gladstone Commercial Corporation
Global Growth Trust, Inc.
Global Logistic Properties
Gramercy Capital Corp.
H Hammerson PLC
HCP, Inc.
Hersha Hospitality Trust
Hines Global REIT, Inc
Home Properties, Inc.
Host Hotels & Resorts, Inc.
Hatteras Financial Corp
Health Care REIT, Inc.
Highwoods Properties, Inc.
Hines Real Estate Investment Trust, Inc.
Hospitality Properties Trust
Hudson Pacific Properties, Inc.
I Independence Realty Trust
Inland American Real Estate Trust, Inc.
Inland Real Estate Corporation
INREIT Real Estate Investment Trust
Investors Real Estate Trust
Industrial Income Trust, Inc.
Inland Diversified Real Estate Trust, Inc.
Inland Western Retail Real Estate Trust, Inc.
Invesco Mortgage Capital Inc.
IStar Financial Inc.
J Japan Retail Fund Investment Corporation
K KBS Legacy Partners Apartment REIT, Inc.
KBS Real Estate Investment Trust II, Inc.
Kenedix Realty Investment Corporation
Kimco Realty Corporation
KBS Real Estate Investment Trust I, Inc.
KBS Strategic Opportunity REIT, Inc.
Kilroy Realty Corporation
Kite Realty Group Trust
L Land Securities Group PLC
Lexington Realty Trust
LTC Properties, Inc.
LaSalle Hotel Properties
Liberty Property Trust
M
MAA
Mack-Cali Realty Corporation
Medical Properties Trust Inc.
MHI Hospitality Corporation
MPG Office Trust, Inc.
Macerich
MCR Development LLC
MFA Financial, Inc.
Monmouth Real Estate Investment Corporation
N National Retail Properties, Inc.
Northstar Real Estate Income Trust, Inc.
Newcastle Investment Corporation
NorthStar Realty Finance Corporation
O Omega Healthcare Investors, Inc.
One Liberty Properties, Inc.
P Parkway Properties, Inc.
Pennsylvania Real Estate Investment Trust
Piedmont Office Realty Trust, Inc.
Post Properties, Inc.
Prologis, Inc.
Public Storage
Pebblebrook Hotel Trust
Phillips Edison - ARC Shopping Center REIT
Plum Creek Timber Company, Inc.
Potlatch Corporation
PS Business Parks, Inc.
R RAIT Financial Trust
Rayonier Inc.
Regency Centers Corporation
RioCan
RREEF America REIT II, Inc.
Ramco-Gershenson Properties Trust
Realty Income Corporation
Resource Capital Corp.
RLJ Lodging Trust
RREEF America REIT III, Inc.
S Sabra Health Care REIT, Inc
SEGRO PLC
Shaftesbury PLC
SL Green Realty Corp.
Sovran Self Storage, Inc.
Stag Industrial, Inc.
Steadfast Income REIT
Summit Hotel Properties Inc.
Sunstone Hotel Investors, Inc.
Saul Centers, Inc.
Senior Housing Properties Trust
Simon Property Group, Inc.
Societe De La Tour Eiffel
Spirit Finance Corporation
Starwood Property Trust, Inc.
Strategic Hotels & Resorts, Inc.
Sun Communities, Inc.
Supertel Hospitality, Inc.
T
Tanger Factory Outlet Centers, Inc.
The Community Development Trust
Two Harbors Investmet Corp.
Taubman Centers, Inc.
Thomas Properties Group Inc.
U UDR, Inc.
Urstadt Biddle Properties, Inc.
UMH Properties, Inc.
V Ventas, Inc.
Vornado Realty Trust
Verde Realty
W W. P. Carey & Co. LLC
Watson Land Company
Wells Real Estate Investment Trust II, Inc.
Wereldhave USA, Inc.
Weyerhaeuser
Washington Real Estate Investment Trust
Weingarten Realty Investors
Wells Timberland REIT, Inc.
Westfield, LLC
Winthrop Realty Trust
Literature Review – TABLE 2
Title Author Year
Forecasting Stock
Market Volatility and
the Application of
Volatility Trading
Models
Jason Laws and
Andrew Gidman
2000
A Measure of Fundamental Volatility
in the Commercual Property Market
Shaun Bond and Soosung Hwang
2001
Forecasting Volatility
in Financial Markets: A
Review
Ser-Huang Poon
and Clive W. J.
Granger
2003
Forecasting Volatility Louis H.
Ederingtion and
Wei Guan
2004
Forecasting Volatility
in the Financial
Markets
John Knight and
Stephen Satchell
2007
Forecasting World
Stock Markets
Volatility
Abdullah Yalama
and Guven Sevil
2008
Volatility forecast comparison using
imperfect volatility proxies
Andrew Patton 2010
Critical Value
Do not reject H0
Critical Value
Reject H0
Figure I – Natural Logarithm of US FTSE Prices
(Level Dependent Variable)
Figure II. – Stationarity in Level
a) Augmented Dickey-Fuller Unit Root Test (ADF)
H0: Unit Root / Non-Stationarity
H1: ~ I (0) / Stationarity
a) Kwiatkowski-Phillips-Schmidt-Shin Unit Root
Test (KPSS)
H0: Stationarity
H1: Non-Stationarity
Both models with Trend and Intercept because
there is a clear pattern through the observation of
the correspondent graph.
Figure III. – Serial Correlation
Figure IV. – Natural Logarithm of US FTSE Returns
(First Differences Dependent Variable)
Figure V. – Stationarity in First Differences
a) ADF
b) KPSS
Figure VI. – Serial Autocorrelation
2
4
6
8
10
12
14
16
1975 1980 1985 1990 1995 2000 2005
annual_Yields_10yearUSGovernmentBonds
0
10
20
30
40
50
2 4 6 8 10 12 14 16
Series: I10
Sample 1972M01 2008M12
Observations 444
Mean 7.428336
Median 7.175500
Maximum 15.84200
Minimum 2.212300
Std. Dev. 2.631976
Skewness 0.811579
Kurtosis 3.295220
Jarque-Bera 50.35327
Probability 0.000000
TABLE 3 – Test-Statistics for Stationarity Tests
i_10 d_i10
ln_indprod d_ln_indprod
Unem Growth
ADF Test:
Constant
-18.16305
-7.092998 Constant&Trend -2.628751
-2.289439
-2.704203*
KPSS Test: Constant
0.251500
0.121693 Constant&Trend 0.341894
0.226043
0.998462
Conclusion I (1) I (0)
I (1) I (0)
I (0)
(Robust) (Robust)
(Robust) (Robust)
(Robust)
*Test valid only for 10% Significance Level ** The remaining tests are valid for 10%, 5% and 1% significance Levels
*** ”Robust” implies that both KPSS and ADF agree on the stationary test result
ANALYSIS OF THE SERIES YIELDS ON US GOVERNMENT BONDS
FIGURE VII - Graph and Descriptive Statistics of Yields on 10-Year US Government Bonds
FIGURE VIII – Correlogram: Yields on 10-Year US Government Bonds
Null Hypothesis: I10 has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 1 (Automatic based on SIC, MAXLAG=17) t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -2.628751 0.2677
Test critical values: 1% level -3.978956
5% level -3.420022
10% level -3.132657
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(I10)
Method: Least Squares
Date: 12/04/11 Time: 22:56
Sample (adjusted): 1972M03 2008M12
Included observations: 442 after adjustments Coefficient Std. Error t-Statistic Prob.
I10(-1) -0.022864 0.008698 -2.628751 0.0089
D(I10(-1)) 0.143401 0.047254 3.034718 0.0026
C 0.270099 0.096968 2.785460 0.0056
@TREND(1972M01) -0.000483 0.000178 -2.710343 0.0070
R-squared 0.038146 Mean dependent var -0.008664
Adjusted R-squared 0.031558 S.D. dependent var 0.369062
S.E. of regression 0.363192 Akaike info criterion 0.821239
Sum squared resid 57.77596 Schwarz criterion 0.858265
Log likelihood -177.4939 Hannan-Quinn criter. 0.835843
F-statistic 5.790183 Durbin-Watson stat 1.969930
Prob(F-statistic) 0.000689
Null Hypothesis: I10 is stationary
Exogenous: Constant, Linear Trend
Bandwidth: 16 (Newey-West using Bartlett kernel) LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.341894
Asymptotic critical values*: 1% level 0.216000
5% level 0.146000
10% level 0.119000
*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)
Residual variance (no correction) 3.998481
HAC corrected variance (Bartlett kernel) 58.95914
KPSS Test Equation
Dependent Variable: I10
Method: Least Squares
Date: 12/04/11 Time: 22:57
Sample: 1972M01 2008M12
Included observations: 444 Coefficient Std. Error t-Statistic Prob.
C 10.37798 0.189904 54.64865 0.0000
@TREND(1972M01) -0.013317 0.000742 -17.94527 0.0000
R-squared 0.421491 Mean dependent var 7.428336
Adjusted R-squared 0.420182 S.D. dependent var 2.631976
S.E. of regression 2.004139 Akaike info criterion 4.232801
Sum squared resid 1775.326 Schwarz criterion 4.251250
Log likelihood -937.6817 Hannan-Quinn criter. 4.240076
F-statistic 322.0326 Durbin-Watson stat 0.033841
Prob(F-statistic) 0.000000
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
1975 1980 1985 1990 1995 2000 2005
D_I10
0
10
20
30
40
50
60
70
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Series: D_I10
Sample 1972M01 2008M12
Observations 443
Mean -0.008758
Median -0.015000
Maximum 1.590000
Minimum -1.880000
Std. Dev. 0.368650
Skewness -0.266906
Kurtosis 6.103445
Jarque-Bera 183.0388
Probability 0.000000
FIGURE IX - 10-Year US Government Bonds – ADF Test and KPSS test with Constant and Trend
FIGURE X - Graph and Descriptive Statistics of Yields on 10-Year US Government Bonds’ Differential
FIGURE XI - Correlogram – on Yields on 10-Year US Government Bonds’ Differential
FIGURE XII – Yields on 10-Year US Government Bonds’ Differential - ADF Test with Constant
Null Hypothesis: D_I10 has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=17) t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -18.16305 0.0000
Test critical values: 1% level -3.444923
5% level -2.867859
10% level -2.570200
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(D_I10)
Method: Least Squares
Date: 12/04/11 Time: 22:59
Sample (adjusted): 1972M03 2008M12
Included observations: 442 after adjustments Coefficient Std. Error t-Statistic Prob.
D_I10(-1) -0.861064 0.047407 -18.16305 0.0000
C -0.007667 0.017409 -0.440435 0.6598
R-squared 0.428494 Mean dependent var -0.001488
Adjusted R-squared 0.427196 S.D. dependent var 0.483495
S.E. of regression 0.365927 Akaike info criterion 0.831750
Sum squared resid 58.91721 Schwarz criterion 0.850263
Log likelihood -181.8168 Hannan-Quinn criter. 0.839052
F-statistic 329.8962 Durbin-Watson stat 1.968687
Prob(F-statistic) 0.000000
3.6
3.8
4.0
4.2
4.4
4.6
4.8
1975 1980 1985 1990 1995 2000 2005
LN_INDPROD
0
5
10
15
20
25
30
35
3.750 3.875 4.000 4.125 4.250 4.375 4.500 4.625
Series: LN_INDPROD
Sample 1972M01 2008M12
Observations 444
Mean 4.161030
Median 4.120093
Maximum 4.612385
Minimum 3.685012
Std. Dev. 0.281439
Skewness 0.132013
Kurtosis 1.658962
Jarque-Bera 34.55972
Probability 0.000000
FIGURE XIII - Yields on 10-Year US Government Bonds’ Differential – KPSS Test with Constant and Trend
ANALYSIS OF THE SERIES INDUSTRIAL PRODUCTION (PROXY FOR GDP)
FIGURE XIV – Graph and Descriptive Statics on Industrial production Index
Null Hypothesis: D_I10 is stationary
Exogenous: Constant
Bandwidth: 0 (Newey-West using Bartlett kernel) LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.251500
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)
Residual variance (no correction) 0.135596
HAC corrected variance (Bartlett kernel) 0.135596
KPSS Test Equation
Dependent Variable: D_I10
Method: Least Squares
Date: 12/04/11 Time: 23:00
Sample (adjusted): 1972M02 2008M12
Included observations: 443 after adjustments Coefficient Std. Error t-Statistic Prob.
C -0.008758 0.017515 -0.500014 0.6173
R-squared 0.000000 Mean dependent var -0.008758
Adjusted R-squared 0.000000 S.D. dependent var 0.368650
S.E. of regression 0.368650 Akaike info criterion 0.844316
Sum squared resid 60.06899 Schwarz criterion 0.853556
Log likelihood -186.0159 Hannan-Quinn criter. 0.847960
Durbin-Watson stat 1.716230
Null Hypothesis: LN_INDPROD has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 3 (Automatic based on SIC, MAXLAG=12) t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -2.289439 0.4383
Test critical values: 1% level -3.979052
5% level -3.420068
10% level -3.132684
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(LN_INDPROD)
Method: Least Squares
Date: 12/04/11 Time: 22:21
Sample (adjusted): 1972M05 2008M12
Included observations: 440 after adjustments Coefficient Std. Error t-Statistic Prob.
LN_INDPROD(-1) -0.014756 0.006445 -2.289439 0.0225
D(LN_INDPROD(-1)) 0.237454 0.047774 4.970377 0.0000
D(LN_INDPROD(-2)) 0.181603 0.048457 3.747748 0.0002
D(LN_INDPROD(-3)) 0.186976 0.048650 3.843322 0.0001
C 0.055354 0.023697 2.335970 0.0199
@TREND(1972M01) 3.00E-05 1.42E-05 2.113942 0.0351
R-squared 0.192672 Mean dependent var 0.001768
Adjusted R-squared 0.183371 S.D. dependent var 0.007473
S.E. of regression 0.006753 Akaike info criterion -7.144160
Sum squared resid 0.019791 Schwarz criterion -7.088431
Log likelihood 1577.715 Hannan-Quinn criter. -7.122175
F-statistic 20.71512 Durbin-Watson stat 1.984700
Prob(F-statistic) 0.000000
Null Hypothesis: LN_INDPROD is stationary
Exogenous: Constant, Linear Trend
Bandwidth: 16 (Newey-West using Bartlett kernel) LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.226043
Asymptotic critical values*: 1% level 0.216000
5% level 0.146000
10% level 0.119000
*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)
Residual variance (no correction) 0.002653
HAC corrected variance (Bartlett kernel) 0.037709
KPSS Test Equation
Dependent Variable: LN_INDPROD
Method: Least Squares
Date: 12/04/11 Time: 22:22
Sample: 1972M01 2008M12
Included observations: 444 Coefficient Std. Error t-Statistic Prob.
C 3.683433 0.004892 752.9661 0.0000
@TREND(1972M01) 0.002156 1.91E-05 112.7973 0.0000
R-squared 0.966427 Mean dependent var 4.161030
Adjusted R-squared 0.966351 S.D. dependent var 0.281439
S.E. of regression 0.051626 Akaike info criterion -3.085072
Sum squared resid 1.178057 Schwarz criterion -3.066622
Log likelihood 686.8860 Hannan-Quinn criter. -3.077796
F-statistic 12723.23 Durbin-Watson stat 0.020988
Prob(F-statistic) 0.000000
FIGURE XV – Correlogram: Industrial production Index
FIGURE XVI – Industrial Production – ADF Test and KPPS Test with Constant and Trend
-.05
-.04
-.03
-.02
-.01
.00
.01
.02
.03
1975 1980 1985 1990 1995 2000 2005
D_LN_INDPROD
0
10
20
30
40
50
60
70
80
-0.0375 -0.0250 -0.0125 0.0000 0.0125
Series: D_LN_INDPROD
Sample 1972M01 2008M12
Observations 443
Mean 0.001817
Median 0.002378
Maximum 0.021455
Minimum -0.042261
Std. Dev. 0.007471
Skewness -1.286933
Kurtosis 8.517067
Jarque-Bera 684.1177
Probability 0.000000
FIGURE XVII – Graph and Descriptive Statics Industrial Production Differential
FIGURE XVIII – Correlogram: Industrial Production Diferencial
Null Hypothesis: D_LN_INDPROD has a unit root
Exogenous: Constant
Lag Length: 2 (Automatic based on SIC, MAXLAG=12) t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -7.092998 0.0000
Test critical values: 1% level -3.444991
5% level -2.867889
10% level -2.570216
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(D_LN_INDPROD)
Method: Least Squares
Date: 12/04/11 Time: 22:25
Sample (adjusted): 1972M05 2008M12
Included observations: 440 after adjustments Coefficient Std. Error t-Statistic Prob.
D_LN_INDPROD(-1) -0.419935 0.059204 -7.092998 0.0000
D(D_LN_INDPROD(-1)) -0.344922 0.058782 -5.867807 0.0000
D(D_LN_INDPROD(-2)) -0.171270 0.048262 -3.548775 0.0004
C 0.000678 0.000342 1.983102 0.0480
R-squared 0.368945 Mean dependent var -8.21E-05
Adjusted R-squared 0.364603 S.D. dependent var 0.008509
S.E. of regression 0.006783 Akaike info criterion -7.139882
Sum squared resid 0.020057 Schwarz criterion -7.102729
Log likelihood 1574.774 Hannan-Quinn criter. -7.125225
F-statistic 84.96888 Durbin-Watson stat 1.980792
Prob(F-statistic) 0.000000
Null Hypothesis: D_LN_INDPROD is stationary
Exogenous: Constant
Bandwidth: 12 (Newey-West using Bartlett kernel) LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.121693
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)
Residual variance (no correction) 5.57E-05
HAC corrected variance (Bartlett kernel) 0.000178
KPSS Test Equation
Dependent Variable: D_LN_INDPROD
Method: Least Squares
Date: 12/04/11 Time: 22:25
Sample (adjusted): 1972M02 2008M12
Included observations: 443 after adjustments Coefficient Std. Error t-Statistic Prob.
C 0.001817 0.000355 5.117352 0.0000
R-squared 0.000000 Mean dependent var 0.001817
Adjusted R-squared 0.000000 S.D. dependent var 0.007471
S.E. of regression 0.007471 Akaike info criterion -6.953199
Sum squared resid 0.024674 Schwarz criterion -6.943958
Log likelihood 1541.133 Hannan-Quinn criter. -6.949554
Durbin-Watson stat 1.288710
.03
.04
.05
.06
.07
.08
.09
.10
.11
1975 1980 1985 1990 1995 2000 2005
US Unemployment Rate - Monthly
0
10
20
30
40
50
60
0.0375 0.0500 0.0625 0.0750 0.0875 0.1000
Series: UNEM
Sample 1972M01 2008M12
Observations 444
Mean 0.061547
Median 0.058000
Maximum 0.108000
Minimum 0.038000
Std. Dev. 0.014032
Skewness 0.832793
Kurtosis 3.547594
Jarque-Bera 56.86972
Probability 0.000000
FIGURE XIX - Industrial Production DIFFERENTIAL – ADF Test and KPSS Test with Constant
ANALYSIS OF THE SERIES US UNEMPLOYMENT
FIGURE XX - Graph and Descriptive Statics– US Unemployment Rate
Null Hypothesis: UNEM has a unit root
Exogenous: Constant
Lag Length: 4 (Automatic based on SIC, MAXLAG=12) t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -2.704203 0.0741
Test critical values: 1% level -3.445025
5% level -2.867904
10% level -2.570224
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(UNEM)
Method: Least Squares
Date: 12/04/11 Time: 22:28
Sample (adjusted): 1972M06 2008M12
Included observations: 439 after adjustments Coefficient Std. Error t-Statistic Prob.
UNEM(-1) -0.015537 0.005746 -2.704203 0.0071
D(UNEM(-1)) 0.017935 0.047516 0.377457 0.7060
D(UNEM(-2)) 0.229892 0.046646 4.928450 0.0000
D(UNEM(-3)) 0.208211 0.047038 4.426463 0.0000
D(UNEM(-4)) 0.145538 0.048087 3.026576 0.0026
C 0.000985 0.000362 2.717447 0.0068
R-squared 0.153888 Mean dependent var 3.64E-05
Adjusted R-squared 0.144117 S.D. dependent var 0.001804
S.E. of regression 0.001669 Akaike info criterion -9.939682
Sum squared resid 0.001206 Schwarz criterion -9.883857
Log likelihood 2187.760 Hannan-Quinn criter. -9.917657
F-statistic 15.75047 Durbin-Watson stat 2.009683
Prob(F-statistic) 0.000000
Null Hypothesis: UNEM is stationary
Exogenous: Constant
Bandwidth: 16 (Newey-West using Bartlett kernel) LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.998462
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)
Residual variance (no correction) 0.000196
HAC corrected variance (Bartlett kernel) 0.002992
KPSS Test Equation
Dependent Variable: UNEM
Method: Least Squares
Date: 12/04/11 Time: 22:29
Sample: 1972M01 2008M12
Included observations: 444 Coefficient Std. Error t-Statistic Prob.
C 0.061547 0.000666 92.42497 0.0000
R-squared 0.000000 Mean dependent var 0.061547
Adjusted R-squared 0.000000 S.D. dependent var 0.014032
S.E. of regression 0.014032 Akaike info criterion -5.692742
Sum squared resid 0.087222 Schwarz criterion -5.683517
Log likelihood 1264.789 Hannan-Quinn criter. -5.689104
Durbin-Watson stat 0.016383
FIGURE XXI – Correlogram: US unemployment Rate
FIGURE XXII – US Unemployment Rate – ADF Test and KPSS Test with Constant
FIGURE XXIII – Correlogram of the Regression of US FTSE returns on 10-years yields’ differential
TABLE 4 – Regression of US FTSE Returns on the Explanatory Variables
Dependent Variable: LN_FTSE_R
Method: Least Squares
Date: 12/03/11 Time: 10:19
Sample (adjusted): 1972M02 2008M12
Included observations: 443 after adjustments
White Heteroskedasticity-Consistent Standard Errors & Covariance Coefficient Std. Error t-Statistic Prob.
C 0.004857 0.003430 1.416232 0.1574
D_I10 -0.032386 0.006813 -4.753683 0.0000
D_LN_INDPROD 0.715542 0.929921 0.769465 0.4420
UNEM_GR -0.487517 2.065967 -0.235975 0.8136
R-squared 0.058292 Mean dependent var 0.006424
Adjusted R-squared 0.051856 S.D. dependent var 0.050559
S.E. of regression 0.049231 Akaike info criterion -3.175614
Sum squared resid 1.063983 Schwarz criterion -3.138652
Log likelihood 707.3986 Hannan-Quinn criter. -3.161037
F-statistic 9.058045 Durbin-Watson stat 1.853949
Prob(F-statistic) 0.000008
TABLE 5 – Test-Statistics for Stationarity Tests
*All regressions have been done, using White Heteroscedasticity-Consistent Standard Errors & Covariance ** Terms significance tested to 10% Significance Value.
TABLE 6 – Base-Model Regression
Model Orders Adjusted R-
Squared AIC SIC Insignificant Variables
ARMA(1;1) 0.043428 -3.164493 -3.127467 AR(1); AR(2)
ARMA(2;2) 0.066977 -3.182659 -3.127025 C; AR(1); MA(1)
ARMA(3;3) 0.084728 -3.195504 -3.121199 C
ARMA(4;4) 0.114840 -3.201451 -3.108410 C; AR(1); AR(2); AR(3);
MA(2); MA(3)
ARMA(1;2) 0.041247 -3.159976 -3.113694 AR(1);MA(1); MA(2)
ARMA(2;1) 0.054006 -3.171091 -3.124730 AR(2)
ARMA(3;2) 0.059831 -3.170899 -3.105882 C; AR(1); AR(2); AR(3);
MA(1) MA(2)
ARMA(2;3) 0.059142 -3.172062 -3.107157 C; AR(1); AR(2); MA(1)
MA(2); MA(3)
ARMA(4;3) 0.084020 -3.190214 -3.106477 C; AR(4)
ARMA(3;4) 0.071418 -3.178839 -3.095246 C; MA(4)
Dependent Variable: LN_FTSE_R
Method: Least Squares
Date: 12/03/11 Time: 12:15
Sample (adjusted): 1972M05 2008M12
Included observations: 440 after adjustments
Convergence achieved after 51 iterations
White Heteroskedasticity-Consistent Standard Errors & Covariance
MA Backcast: 1972M02 1972M04 Coefficient Std. Error t-Statistic Prob.
C 0.004780 0.006646 0.719184 0.4724
D_I10 -0.030036 0.005976 -5.026283 0.0000
AR(1) 0.819383 0.077238 10.60856 0.0000
AR(2) -0.859308 0.033825 -25.40427 0.0000
AR(3) 0.919593 0.074527 12.33906 0.0000
MA(1) -0.752423 0.101634 -7.403264 0.0000
MA(2) 0.831333 0.062602 13.27965 0.0000
MA(3) -0.816908 0.086958 -9.394252 0.0000
R-squared 0.099322 Mean dependent var 0.006491
Adjusted R-squared 0.084728 S.D. dependent var 0.050720
S.E. of regression 0.048524 Akaike info criterion -3.195504
Sum squared resid 1.017176 Schwarz criterion -3.121199
Log likelihood 711.0109 Hannan-Quinn criter. -3.166191
F-statistic 6.805529 Durbin-Watson stat 1.999248
Prob(F-statistic) 0.000000
Inverted AR Roots .94 -.06+.99i -.06-.99i
Inverted MA Roots .87 -.06+.97i -.06-.97i
Ramsey RESET Test:
F-statistic 5.490878 Prob. F(3,429) 0.0010
Log likelihood ratio 16.57872 Prob. Chi-Square(3) 0.0009
Test Equation:
Dependent Variable: LN_FTSE_R
Method: Least Squares
Date: 12/03/11 Time: 12:37
Sample: 1972M05 2008M12
Included observations: 440
Convergence achieved after 36 iterations
White Heteroskedasticity-Consistent Standard Errors & Covariance
MA Backcast: 1972M02 1972M04 Coefficient Std. Error t-Statistic Prob.
C 0.010110 0.005309 1.904307 0.0575
D_I10 -0.024872 0.009314 -2.670377 0.0079
FITTED^2 -27.45377 13.42400 -2.045125 0.0415
FITTED^3 650.9897 394.0615 1.652000 0.0993
FITTED^4 60.43975 8399.448 0.007196 0.9943
AR(1) -0.018031 0.089570 -0.201311 0.8406
AR(2) -0.049965 0.086833 -0.575417 0.5653
AR(3) 0.904243 0.085577 10.56639 0.0000
MA(1) 0.055564 0.114006 0.487382 0.6262
MA(2) 0.126752 0.109234 1.160375 0.2465
MA(3) -0.882567 0.114596 -7.701520 0.0000
R-squared 0.132627 Mean dependent var 0.006491
Adjusted R-squared 0.112409 S.D. dependent var 0.050720
S.E. of regression 0.047785 Akaike info criterion -3.219547
Sum squared resid 0.979563 Schwarz criterion -3.117377
Log likelihood 719.3002 Hannan-Quinn criter. -3.179241
F-statistic 6.559703 Durbin-Watson stat 1.982911
Prob(F-statistic) 0.000000
Inverted AR Roots .94 -.48-.85i -.48+.85i
Inverted MA Roots .90 -.48-.87i -.48+.87i
TABLE 7 –Breusch-Godfrey Serial Correlation Test to the Base Model
TABLE 8 – Ramsey RESET Test to Non-Linearity
Breusch-Godfrey Serial Correlation LM Test:
F-statistic 0.267999 Prob. F(3,429) 0.8485
Obs*R-squared 0.822584 Prob. Chi-Square(3) 0.8441
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Date: 12/03/11 Time: 12:30
Sample: 1972M05 2008M12
Included observations: 440
Presample missing value lagged residuals set to zero. Coefficient Std. Error t-Statistic Prob.
C 6.82E-05 0.005373 0.012697 0.9899
D_I10 -0.000501 0.006397 -0.078243 0.9377
AR(1) -0.014282 0.070016 -0.203983 0.8385
AR(2) -0.002178 0.031989 -0.068079 0.9458
AR(3) -0.009724 0.067217 -0.144665 0.8850
MA(1) 0.040730 0.115341 0.353127 0.7242
MA(2) 0.000334 0.053924 0.006194 0.9951
MA(3) 0.016791 0.105979 0.158440 0.8742
RESID(-1) -0.041834 0.078712 -0.531481 0.5954
RESID(-2) -0.031036 0.074404 -0.417124 0.6768
RESID(-3) 0.030455 0.069235 0.439880 0.6602
R-squared 0.001870 Mean dependent var -5.05E-05
Adjusted R-squared -0.021397 S.D. dependent var 0.048136
S.E. of regression 0.048648 Akaike info criterion -3.183740
Sum squared resid 1.015273 Schwarz criterion -3.081571
Log likelihood 711.4228 Hannan-Quinn criter. -3.143434
F-statistic 0.080352 Durbin-Watson stat 1.968838
Prob(F-statistic) 0.999935
-.4
-.2
.0
.2
.4
-.4
-.2
.0
.2
.4
1975 1980 1985 1990 1995 2000 2005
Residual Actual Fitted
FIGURE XXIV – Histogram on the Residuals of the Base Model
FIGURE XXV – Graph of the Base Model
FIGURE XXVI – Conditional Variance
TABLE 9 – ARCH Models
Model’s Orders Log Likelihood AIC Insignificant Variables
ARCH (1) 730.2019 -3.2736 AR(1); AR(2); MA(1); MA(2); MA(3)
ARCH (2) 743.9987 -3.3318 NONE
ARCH(3) 757.0368 -3.3865 AR(1); AR(3); MA(1); MA(3)
ARCH (4) 759.4547 -3.3930 AR(1); AR(2); MA(1); MA(2); MA(3)
Likelihood Test [relatively to the simplest ARCH (1)]: H0: variables added are not significant / models are indifferent in conclusions
H1: variables added are significant / distance between two models is significant
Likelihood Ratio (2) (3) (4)
Lr 730.2019 730.2019 730.2019
Lu 743.9987 757.0368 759.4547
LR 27.5936 53.6698 58.5056
m 1 2 3
Chi-sq (1%) 6.64 9.21 11.35
Chi-sq (5%) 3.84 5.99 7.82
Chi-sq (10%) 2.71 4.61 6.25
TABLE 10 – GARCH Models
Model’s Order Log Likelihood AIC SIC Insignificant Variables
GARCH (1,2) 21.869 -3.451024 -3.339567 ALL (exp d_i10)
GARCH (2,1) 13.0614 -3.431007 -3.319549 AR(1); AR(3); MA(1); MA(3)
GARCH (3,3) 30.6978 -3.457453 -3.318131 ALL (exp d_i10)
GARCH (4,3) 35.0162 -3.462722 -3.314112 AR(1); AR(2); MA(1); MA(2); MA(3)
GARCH (5,4) 34.1554 -3.451675 -3.284489 AR(1); AR(2); MA(1); MA(2); MA(3)
Likelihood Test [relatively to the most simple GARCH (1,1)]: H0: variables added are not significant / models are indifferent in conclusions
H1: variables added are significant / distance between two models is significant
Likelihood Ratio (1,2) (2,1) (3,3) (4,3) (5,4)
Lr 760.2908 760.2908 760.2908 760.2908 760.2908
Lu 771.2253 766.8215 775.6397 777.7989 777.3685
LR 21.869 13.0614 30.6978 35.0162 34.1554
m 1 1 4 5 7
Chi-sq (1%) 6.64 6.64 13.28 15.09 18.48
Chi-sq (5%) 3.84 3.84 9.49 11.07 14.07
Chi-sq (10%) 2.71 2.71 7.78 9.24 12.02
FIGURE XXVII – GARCH Best Model Output
TABLE 11 – EGARCH Models
Model’s Orders Log Likelihood AIC SIC Insignificant Variables
EGARCH (1,1) 0 -3.423291 -3.311833 NONE (exp C)
EGARCH (3,3) 31.482 -3.476659 -3.328049 NONE
EGARCH (4,3) 73.6716 -3.567999 -3.410101 NONE
EGARCH (3,4) 41.209 -3.494220 -3.336322 NONE
EGARCH (5,5) 98.7922 -3.611455 -3.425692 NONE
EGARCH (1,1) (3,3) (4,3) (3,4) (5,5)
Lr 765.124 765.124 765.124 765.124 765.124
Lu 765.124 780.865 801.9598 785.7285 814.5201
LR 0 31.482 73.6716 41.209 98.7922
m 0 4 5 5 8
Chi-sq (1%) - 13.28 15.09 15.09 20.09
Chi-sq (5%) - 9.49 11.07 11.07 15.51
Chi-sq (10%) - 7.78 9.24 9.24 13.36
FIGURE XXVIII – EGARCH Best Model Output
TABLE 12 – TGARCH Models
Model’s orders Log Likelihood AIC Insignificant Variables
TGARCH (1,1) 0 -3.457813 AR(1); AR(2); MA(1); MA(2)
TGARCH (2,2) 2.7508 -3.454974 NONE
TGARCH (3,2) 3.2274 -3.451511 AR(2); MA(2)
TGARCH (4,3) 23.4628 -3.488410 AR(1); AR(2); MA(2); MA(3)
TGARCH (4,4) 17.2008 -3.469633 NONE
TGARCH (1,1) (2,2) (3,2) (4,3) (4,4)
Lr 772.7188 772.7188 772.7188 772.7188 772.7188
Lu 772.7188 774.0942 774.3325 784.4502 781.3192
LR 0 2.7508 3.2274 23.4628 17.2008
m 0 2 3 5 6
Chi-sq (1%) - 9.21 11.34 15.09 16.81
Chi-sq (5%) - 5.99 7.82 11.07 12.59
Chi-sq (10%) - 4.6 6.25 9.24 10.64
FIGURE XXIX – Best TGARCH Model Output
TABLE 13 – EGARCH (p, q)-M with Conditional Variance
Log-Likelihood AIC Insignificant Parameters
EGARCH-M(1,1) 765,992 -3,422691 VAR; C; AR(2); MA(2)
EGARCH-M(2,2) 764,8471 -3,408396 VAR; C; AR(1); AR(2); MA(1); MA(2)
EGARCH-M(3,3) 813,7686 -3,621675 None
EGARCH-M(4,3) 816,2783 -3,628538 None
EGARCH-M(4,4) 809,5717 -3,593508 VAR; AR(2); MA(2)
TABLE 14 - EGARCH (p, q)-M with Logarithm of the Conditional Variance
Log-Likelihood AIC Insignificant Parameter
EGARCH-M(2,2) 763,7074 -3,403215
Log-VAR; C; AR(1); AR(3); MA(1); MA(3)
EGARCH-M(3,3) 783,7448 -3,485204 None
EGARCH-M(4,3) 782,5716 -3,475326
Log-VAR; C; AR(2); AR(3); MA(2); MA(3)
EGARCH-M(4,4) 803,3738 -3,565335 C; AR(2); AR(3); MA(2); MA(3)
EGARCH-M(3,4) 796,0647 -3,536658 AR(2); MA(2)
TABLE 15 – EGARCH (p, q)-M with Conditional Standard Deviation
TABLE 16 – EGARCH (p, q)-M
Log-Likelihood AIC Insignificant Parameters
EGARCH-M(2,3) 806,1241 -3,591473 AR(1); AR(2); MA(1); MA(2)
EGARCH-M(3,3) 811,8345 -3,612884 None
EGARCH-M(4,3) 798,7763 -3,548983 AR(1);AR(3); MA(1); MA(3)
EGARCH-M(4,4) 814,562 -3,616191 AR(2)
EGARCH-M(5,5) 813,6175 -3,602807 None
Log-Likelihood AIC Insignificant Parameters
EGARCH-M-VAR (4,3) 816,2783 -3,628538 None
EGARCH-M-Log-VAR(3,4)
796,0647 -3,536658 AR(2); MA(2)
EGARCH-M-STD (3,3) 811,8345 -3,612884 None
TABLE 17 – Forecasting Criteria for volatility models
MODELS ARCH(2) GARCH(4;3) EGARCH(5;5) TGARCH(4;3) EGARCH-M(4;3)
CRITERIAS
MSE 0.0189 0.0183 0.0148 0.0185 0.0184
MAE 0.0834 0.0821 0.0773 0.0825 0.0825
MAPE 139.5539 106.7275 283.1547 92.8891 105.4968