chapter 9 regression with time series data: stationary variables
DESCRIPTION
Chapter 9 Regression with Time Series Data: Stationary Variables. Walter R. Paczkowski Rutgers University. Chapter Contents. 9.1 Introduction 9.2 Finite Distributed Lags 9 .3 Serial Correlation 9 .4 Other Tests for Serially Correlated Errors - PowerPoint PPT PresentationTRANSCRIPT
Principles of Econometrics, 4th Edition Page 1Chapter 9: Regression with Time Series Data:
Stationary Variables
Chapter 9Regression with Time Series
Data:Stationary Variables
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition Page 2Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1 Introduction9.2 Finite Distributed Lags9.3 Serial Correlation9.4 Other Tests for Serially Correlated Errors9.5 Estimation with Serially Correlated Errors9.6 Autoregressive Distributed Lag Models9.7 Forecasting9.8 Multiplier Analysis
Chapter Contents
Principles of Econometrics, 4th Edition Page 3Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1 Introduction
Principles of Econometrics, 4th Edition Page 4Chapter 9: Regression with Time Series Data:
Stationary Variables
When modeling relationships between variables, the nature of the data that have been collected has an important bearing on the appropriate choice of an econometric model– Two features of time-series data to consider:
1. Time-series observations on a given economic unit, observed over a number of time periods, are likely to be correlated
2. Time-series data have a natural ordering according to time
9.1Introduction
Principles of Econometrics, 4th Edition Page 5Chapter 9: Regression with Time Series Data:
Stationary Variables
There is also the possible existence of dynamic relationships between variables – A dynamic relationship is one in which the
change in a variable now has an impact on that same variable, or other variables, in one or more future time periods
– These effects do not occur instantaneously but are spread, or distributed, over future time periods
9.1Introduction
Principles of Econometrics, 4th Edition Page 6Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1Introduction FIGURE 9.1 The distributed lag effect
Principles of Econometrics, 4th Edition Page 7Chapter 9: Regression with Time Series Data:
Stationary Variables
Ways to model the dynamic relationship:1. Specify that a dependent variable y is a
function of current and past values of an explanatory variable x
• Because of the existence of these lagged effects, Eq. 9.1 is called a distributed lag model
9.1Introduction
9.1.1Dynamic Nature of Relationships
1 2( , , ,...)t t t ty f x x x Eq. 9.1
Principles of Econometrics, 4th Edition Page 8Chapter 9: Regression with Time Series Data:
Stationary Variables
Ways to model the dynamic relationship (Continued):2. Capturing the dynamic characteristics of time-series
by specifying a model with a lagged dependent variable as one of the explanatory variables
• Or have:
–Such models are called autoregressive distributed lag (ARDL) models, with ‘‘autoregressive’’ meaning a regression of yt on its own lag or lags
9.1Introduction
9.1.1Dynamic Nature of Relationships
Eq. 9.2 1( , )t t ty f y x
Eq. 9.3 1 1 2( , , , )t t t t ty f y x x x
Principles of Econometrics, 4th Edition Page 9Chapter 9: Regression with Time Series Data:
Stationary Variables
Ways to model the dynamic relationship (Continued):3. Model the continuing impact of change over
several periods via the error term
• In this case et is correlated with et - 1
• We say the errors are serially correlated or autocorrelated
9.1Introduction
9.1.1Dynamic Nature of Relationships
Eq. 9.4 1( ) ( )t t t t ty f x e e f e
Principles of Econometrics, 4th Edition Page 10Chapter 9: Regression with Time Series Data:
Stationary Variables
The primary assumption is Assumption MR4:
• For time series, this is written as:
– The dynamic models in Eqs. 9.2, 9.3 and 9.4 imply correlation between yt and yt - 1 or et and et
- 1 or both, so they clearly violate assumption MR4
9.1Introduction
9.1.2Least Squares Assumptions
cov , cov , 0 for i j i jy y e e i j
cov , cov , 0 for t s t sy y e e t s
Principles of Econometrics, 4th Edition Page 11Chapter 9: Regression with Time Series Data:
Stationary Variables
A stationary variable is one that is not explosive, nor trending, and nor wandering aimlessly without returning to its mean
9.1Introduction
9.1.2aStationarity
Principles of Econometrics, 4th Edition Page 12Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1Introduction
9.1.2aStationarity
FIGURE 9.2 (a) Time series of a stationary variable
Principles of Econometrics, 4th Edition Page 13Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1Introduction
9.1.2aStationarity
FIGURE 9.2 (b) time series of a nonstationary variable that is ‘‘slow-turning’’ or ‘‘wandering’’
Principles of Econometrics, 4th Edition Page 14Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1Introduction
9.1.2aStationarity
FIGURE 9.2 (c) time series of a nonstationary variable that ‘‘trends”
Principles of Econometrics, 4th Edition Page 15Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1Introduction
9.1.3Alternative
Paths Through the Chapter
FIGURE 9.3 (a) Alternative paths through the chapter starting with finite distributed lags
Principles of Econometrics, 4th Edition Page 16Chapter 9: Regression with Time Series Data:
Stationary Variables
9.1Introduction
9.1.3Alternative
Paths Through the Chapter
FIGURE 9.3 (b) Alternative paths through the chapter starting with serial correlation
Principles of Econometrics, 4th Edition Page 17Chapter 9: Regression with Time Series Data:
Stationary Variables
9.2 Finite Distributed Lags
Principles of Econometrics, 4th Edition Page 18Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider a linear model in which, after q time periods, changes in x no longer have an impact on y
– Note the notation change: βs is used to denote the coefficient of xt-s and α is introduced to denote the intercept
0 1 1 2 2t t t t q t q ty x x x x e Eq. 9.5
9.2Finite
Distributed Lags
Principles of Econometrics, 4th Edition Page 19Chapter 9: Regression with Time Series Data:
Stationary Variables
Model 9.5 has two uses:– Forecasting
– Policy analysis• What is the effect of a change in x on y?
1 0 1 1 2 1 1 1T T T T q T q Ty x x x x e Eq. 9.6
( ) ( )t t ss
t s t
E y E yx x
Eq. 9.7
9.2Finite
Distributed Lags
Principles of Econometrics, 4th Edition Page 20Chapter 9: Regression with Time Series Data:
Stationary Variables
Assume xt is increased by one unit and then maintained at its new level in subsequent periods – The immediate impact will be β0 – the total effect in period t + 1 will be β0 + β1, in
period t + 2 it will be β0 + β1 + β2, and so on • These quantities are called interim
multipliers– The total multiplier is the final effect on y of
the sustained increase after q or more periods have elapsed
0
βq
ss
9.2Finite
Distributed Lags
Principles of Econometrics, 4th Edition Page 21Chapter 9: Regression with Time Series Data:
Stationary Variables
The effect of a one-unit change in xt is distributed over the current and next q periods, from which we get the term ‘‘distributed lag model’’– It is called a finite distributed lag model of order q • It is assumed that after a finite number of
periods q, changes in x no longer have an impact on y
– The coefficient βs is called a distributed-lag weight or an s-period delay multiplier
– The coefficient β0 (s = 0) is called the impact multiplier
9.2Finite
Distributed Lags
Principles of Econometrics, 4th Edition Page 22Chapter 9: Regression with Time Series Data:
Stationary Variables
9.2Finite
Distributed Lags
9.2.1Assumptions
TSMR1.TSMR2. y and x are stationary random variables, and et is independent of current, past and future values of x.TSMR3. E(et) = 0TSMR4. var(et) = σ2
TSMR5. cov(et, es) = 0 t ≠ sTSMR6. et ~ N(0, σ2)
0 1 1 2 2β β β β , 1, ,t t t t q t q ty x x x x e t q T
ASSUMPTIONS OF THE DISTRIBUTED LAG MODEL
Principles of Econometrics, 4th Edition Page 23Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider Okun’s Law– In this model the change in the unemployment rate
from one period to the next depends on the rate of growth of output in the economy:
–We can rewrite this as:
where DU = ΔU = Ut - Ut-1, β0 = -γ, and α = γGN
9.2Finite
Distributed Lags
9.2.2An Example: Okun’s Law
1t t t NU U G G Eq. 9.8
0βt t tDU G e Eq. 9.9
Principles of Econometrics, 4th Edition Page 24Chapter 9: Regression with Time Series Data:
Stationary Variables
We can expand this to include lags:
We can calculate the growth in output, G, as:
9.2Finite
Distributed Lags
9.2.2An Example: Okun’s Law
Eq. 9.100 1 1 2 2β β β βt t t t q t q tDU G G G G e
Eq. 9.111
1
100t tt
t
GDP GDPGGDP
Principles of Econometrics, 4th Edition Page 25Chapter 9: Regression with Time Series Data:
Stationary Variables
9.2Finite
Distributed Lags
9.2.2An Example: Okun’s Law
FIGURE 9.4 (a) Time series for the change in the U.S. unemployment rate: 1985Q3 to 2009Q3
Principles of Econometrics, 4th Edition Page 26Chapter 9: Regression with Time Series Data:
Stationary Variables
9.2Finite
Distributed Lags
9.2.2An Example: Okun’s Law
FIGURE 9.4 (b) Time series for U.S. GDP growth: 1985Q2 to 2009Q3
Principles of Econometrics, 4th Edition Page 27Chapter 9: Regression with Time Series Data:
Stationary Variables
9.2Finite
Distributed Lags
9.2.2An Example: Okun’s Law
Table 9.1 Spreadsheet of Observations for Distributed Lag Model
Principles of Econometrics, 4th Edition Page 28Chapter 9: Regression with Time Series Data:
Stationary Variables
9.2Finite
Distributed Lags
9.2.2An Example: Okun’s Law
Table 9.2 Estimates for Okun’s Law Finite Distributed Lag Model
Principles of Econometrics, 4th Edition Page 29Chapter 9: Regression with Time Series Data:
Stationary Variables
9.3 Serial Correlation
Principles of Econometrics, 4th Edition Page 30Chapter 9: Regression with Time Series Data:
Stationary Variables
When is assumption TSMR5, cov(et, es) = 0 for t ≠ s likely to be violated, and how do we assess its validity? –When a variable exhibits correlation over time,
we say it is autocorrelated or serially correlated• These terms are used interchangeably
9.3Serial
Correlation
Principles of Econometrics, 4th Edition Page 31Chapter 9: Regression with Time Series Data:
Stationary Variables
9.3Serial
Correlation
9.3.1Serial
Correlation in Output Growth
FIGURE 9.5 Scatter diagram for Gt and Gt-1
Principles of Econometrics, 4th Edition Page 32Chapter 9: Regression with Time Series Data:
Stationary Variables
Recall that the population correlation between two variables x and y is given by:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
cov ,ρ
var varxy
x y
x y
Principles of Econometrics, 4th Edition Page 33Chapter 9: Regression with Time Series Data:
Stationary Variables
For the Okun’s Law problem, we have:
The notation ρ1 is used to denote the population correlation between observations that are one period apart in time– This is known also as the population autocorrelation of
order one. – The second equality in Eq. 9.12 holds because
var(Gt) = var(Gt-1) , a property of time series that are stationary
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
1 11
1
cov , cov ,ρ
varvar vart t t t
tt t
G G G GGG G
Eq. 9.12
Principles of Econometrics, 4th Edition Page 34Chapter 9: Regression with Time Series Data:
Stationary Variables
The first-order sample autocorrelation for G is obtained from Eq. 9.12 using the estimates:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
1 12
2
1
1cov ,1
1var1
T
t t t tt
T
t tt
G G G G G GT
G G GT
Principles of Econometrics, 4th Edition Page 35Chapter 9: Regression with Time Series Data:
Stationary Variables
Making the substitutions, we get:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
Eq. 9.13
12
1 2
1
T
t tt
T
tt
G G G Gr
G G
Principles of Econometrics, 4th Edition Page 36Chapter 9: Regression with Time Series Data:
Stationary Variables
More generally, the k-th order sample autocorrelation for a series y that gives the correlation between observations that are k periods apart is:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
Eq. 9.14
1
2
1
T
t t kt k
k T
tt
y y y yr
y y
Principles of Econometrics, 4th Edition Page 37Chapter 9: Regression with Time Series Data:
Stationary Variables
Because (T - k) observations are used to compute the numerator and T observations are used to compute the denominator, an alternative that leads to larger estimates in finite samples is:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
Eq. 9.15
1
2
1
1
1
T
t t kt k
k T
tt
y y y yT kr
y yT
Principles of Econometrics, 4th Edition Page 38Chapter 9: Regression with Time Series Data:
Stationary Variables
Applying this to our problem, we get for the first four autocorrelations:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
Eq. 9.16 1 2 3 40.494 0.411 0.154 0.200r r r r
Principles of Econometrics, 4th Edition Page 39Chapter 9: Regression with Time Series Data:
Stationary Variables
How do we test whether an autocorrelation is significantly different from zero?– The null hypothesis is H0: ρk = 0– A suitable test statistic is:
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
Eq. 9.17 0 0,11
kk
rZ T r NT
Principles of Econometrics, 4th Edition Page 40Chapter 9: Regression with Time Series Data:
Stationary Variables
For our problem, we have:
–We reject the hypotheses H0: ρ1 = 0 and H0: ρ2 = 0
–We have insufficient evidence to reject H0: ρ3 = 0
– ρ4 is on the borderline of being significant.–We conclude that G, the quarterly growth rate in U.S.
GDP, exhibits significant serial correlation at lags one and two
9.3Serial
Correlation
9.3.1aComputing
Autocorrelation
1 2
3 4
98 0.494 4.89, 98 0.414 4.10
98 0.154 1.52, 98 0.200 1.98
Z Z
Z Z
Principles of Econometrics, 4th Edition Page 41Chapter 9: Regression with Time Series Data:
Stationary Variables
The correlogram, also called the sample autocorrelation function, is the sequence of autocorrelations r1, r2, r3, …– It shows the correlation between observations
that are one period apart, two periods apart, three periods apart, and so on
9.3Serial
Correlation
9.3.1bThe Correlagram
Principles of Econometrics, 4th Edition Page 42Chapter 9: Regression with Time Series Data:
Stationary Variables
9.3Serial
Correlation
9.3.1bThe Correlagram
FIGURE 9.6 Correlogram for G
Principles of Econometrics, 4th Edition Page 43Chapter 9: Regression with Time Series Data:
Stationary Variables
The correlogram can also be used to check whether the multiple regression assumption cov(et, es) = 0 for t ≠ s is violated
9.3Serial
Correlation
9.3.2Serially
Correlated Errors
Principles of Econometrics, 4th Edition Page 44Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider a model for a Phillips Curve:
– If we initially assume that inflationary expectations are constant over time (β1 = INFE
t) set β2= -γ, and add an error term:
9.3Serial
Correlation
9.3.2aA Phillips Curve
1γEt t t tINF INF U U Eq. 9.18
1 2β βt t tINF DU e Eq. 9.19
Principles of Econometrics, 4th Edition Page 45Chapter 9: Regression with Time Series Data:
Stationary Variables
9.3Serial
Correlation
9.3.2aA Phillips Curve
FIGURE 9.7 (a) Time series for Australian price inflation
Principles of Econometrics, 4th Edition Page 46Chapter 9: Regression with Time Series Data:
Stationary Variables
9.3Serial
Correlation
9.3.2aA Phillips Curve
FIGURE 9.7 (b) Time series for the quarterly change in the Australian unemployment rate
Principles of Econometrics, 4th Edition Page 47Chapter 9: Regression with Time Series Data:
Stationary Variables
To determine if the errors are serially correlated, we compute the least squares residuals:
9.3Serial
Correlation
9.3.2aA Phillips Curve
Eq. 9.20 1 2t t te INF b b DU
Principles of Econometrics, 4th Edition Page 48Chapter 9: Regression with Time Series Data:
Stationary Variables
9.3Serial
Correlation
9.3.2aA Phillips Curve
FIGURE 9.8 Correlogram for residuals from least-squares estimated Phillips curve
Principles of Econometrics, 4th Edition Page 49Chapter 9: Regression with Time Series Data:
Stationary Variables
The k-th order autocorrelation for the residuals can be written as:
– The least squares equation is:
9.3Serial
Correlation
9.3.2aA Phillips Curve
1
2
1
ˆ ˆ
ˆ
T
t t kt k
k T
tt
e er
e
Eq. 9.21
0.7776 0.5279
0.0658 0.2294INF DUse
Eq. 9.22
Principles of Econometrics, 4th Edition Page 50Chapter 9: Regression with Time Series Data:
Stationary Variables
The values at the first five lags are:
9.3Serial
Correlation
9.3.2aA Phillips Curve
1 2 3 4 50.549 0.456 0.433 0.420 0.339r r r r r
Principles of Econometrics, 4th Edition Page 51Chapter 9: Regression with Time Series Data:
Stationary Variables
9.4 Other Tests for Serially Correlated
Errors
Principles of Econometrics, 4th Edition Page 52Chapter 9: Regression with Time Series Data:
Stationary Variables
An advantage of this test is that it readily generalizes to a joint test of correlations at more than one lag
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test
Principles of Econometrics, 4th Edition Page 53Chapter 9: Regression with Time Series Data:
Stationary Variables
If et and et-1 are correlated, then one way to model the relationship between them is to write:
–We can substitute this into a simple regression equation:
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test
1ρt t te e v Eq. 9.23
1 2 1β β ρt t t ty x e v Eq. 9.24
Principles of Econometrics, 4th Edition Page 54Chapter 9: Regression with Time Series Data:
Stationary Variables
We have one complication: is unknown– Two ways to handle this are:
1. Delete the first observation and use a total of T observations
2. Set and use all T observations
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test
0e
0ˆ 0e
Principles of Econometrics, 4th Edition Page 55Chapter 9: Regression with Time Series Data:
Stationary Variables
For the Phillips Curve:
– The results are almost identical– The null hypothesis H0: ρ = 0 is rejected at all
conventional significance levels–We conclude that the errors are serially
correlated
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test
i 6.219 38.67 -value 0.000
ii 6.202 38.47 -value 0.000
t F p
t F p
Principles of Econometrics, 4th Edition Page 56Chapter 9: Regression with Time Series Data:
Stationary Variables
To derive the relevant auxiliary regression for the autocorrelation LM test, we write the test equation as:
– But since we know that , we get:
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test
1 2 1ˆβ β ρt t t ty x e v Eq. 9.25
1 2 ˆt t ty b b x e
1 2 1 2 1ˆ ˆβ β ρt t t t tb b x e x e v
Principles of Econometrics, 4th Edition Page 57Chapter 9: Regression with Time Series Data:
Stationary Variables
Rearranging, we get:
– If H0: ρ = 0 is true, then LM = T x R2 has an approximate χ2
(1) distribution • T and R2 are the sample size and goodness-
of-fit statistic, respectively, from least squares estimation of Eq. 9.26
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test
1 1 2 2 1
1 2 1
ˆ ˆβ β ρˆγ γ ρ
t t t t
t t
e b b x e v
x e v
Eq. 9.26
Principles of Econometrics, 4th Edition Page 58Chapter 9: Regression with Time Series Data:
Stationary Variables
Considering the two alternative ways to handle :
– These values are much larger than 3.84, which is the 5% critical value from a χ2
(1)-distribution• We reject the null hypothesis of no
autocorrelation– Alternatively, we can reject H0 by examining
the p-value for LM = 27.61, which is 0.000
9.4Other Tests for
Serially Correlated
Errors
9.4.1A Lagrange
Multiplier Test 0e
2
2
iii 1 89 0.3102 27.61
iv 90 0.3066 27.59
LM T R
LM T R
Principles of Econometrics, 4th Edition Page 59Chapter 9: Regression with Time Series Data:
Stationary Variables
For a four-period lag, we obtain:
– Because the 5% critical value from a χ2(4)-
distribution is 9.49, these LM values lead us to conclude that the errors are serially correlated
9.4Other Tests for
Serially Correlated
Errors
9.4.1aTesting
Correlation at Longer Lags
2
2
iii 4 86 0.3882 33.4
iv 90 0.4075 36.7
LM T R
LM T R
Principles of Econometrics, 4th Edition Page 60Chapter 9: Regression with Time Series Data:
Stationary Variables
This is used less frequently today because its critical values are not available in all software packages, and one has to examine upper and lower critical bounds instead– Also, unlike the LM and correlogram tests, its
distribution no longer holds when the equation contains a lagged dependent variable
9.4Other Tests for
Serially Correlated
Errors
9.4.2The Durbin-Watson Test
Principles of Econometrics, 4th Edition Page 61Chapter 9: Regression with Time Series Data:
Stationary Variables
9.5 Estimation with Serially Correlated
Errors
Principles of Econometrics, 4th Edition Page 62Chapter 9: Regression with Time Series Data:
Stationary Variables
Three estimation procedures are considered:1. Least squares estimation2. An estimation procedure that is relevant when
the errors are assumed to follow what is known as a first-order autoregressive model
3. A general estimation strategy for estimating models with serially correlated errors
9.5Estimation with
Serially Correlated
Errors
1ρt t te e v
Principles of Econometrics, 4th Edition Page 63Chapter 9: Regression with Time Series Data:
Stationary Variables
We will encounter models with a lagged dependent variable, such as:
9.5Estimation with
Serially Correlated
Errors
1 1 0 1 1δ θ δ δt t t t ty y x x v
Principles of Econometrics, 4th Edition Page 64Chapter 9: Regression with Time Series Data:
Stationary Variables
TSMR2A In the multiple regression model Where some of the xtk may be lagged values of y, vt is uncorrelated with all xtk and their past values.
1 2 2β β βt t K K ty x x v
ASSUMPTION FOR MODELS WITH A LAGGED DEPENDENT VARIABLE9.5
Estimation with Serially
Correlated Errors
Principles of Econometrics, 4th Edition Page 65Chapter 9: Regression with Time Series Data:
Stationary Variables
Suppose we proceed with least squares estimation without recognizing the existence of serially correlated errors. What are the consequences?1. The least squares estimator is still a linear
unbiased estimator, but it is no longer best2. The formulas for the standard errors usually
computed for the least squares estimator are no longer correct• Confidence intervals and hypothesis tests
that use these standard errors may be misleading
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
Principles of Econometrics, 4th Edition Page 66Chapter 9: Regression with Time Series Data:
Stationary Variables
It is possible to compute correct standard errors for the least squares estimator: – HAC (heteroskedasticity and autocorrelation
consistent) standard errors, or Newey-West standard errors• These are analogous to the heteroskedasticity
consistent standard errors
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
Principles of Econometrics, 4th Edition Page 67Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider the model yt = β1 + β2xt + et – The variance of b2 is:
where
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
22
22
var var cov ,
cov ,var 1
var
t t t s t st t s
t s t st s
t tt t t
t
b w e w w e e
w w e ew e
w e
2t t tt
w x x x x
Eq. 9.27
Principles of Econometrics, 4th Edition Page 68Chapter 9: Regression with Time Series Data:
Stationary Variables
When the errors are not correlated, cov(et, es) = 0, and the term in square brackets is equal to one. – The resulting expression
is the one used to find heteroskedasticity-consistent (HC) standard errors
–When the errors are correlated, the term in square brackets is estimated to obtain HAC standard errors
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
22var vart tt
b w e
Principles of Econometrics, 4th Edition Page 69Chapter 9: Regression with Time Series Data:
Stationary Variables
If we call the quantity in square brackets g and its estimate , then the relationship between the two estimated variances is:
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
2 2 ˆvar varHAC HCb b g
g
Eq. 9.28
Principles of Econometrics, 4th Edition Page 70Chapter 9: Regression with Time Series Data:
Stationary Variables
Let’s reconsider the Phillips Curve model:
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
0.7776 0.5279 0.0658 0.2294 incorrect se
0.1030 0.3127 HAC se
INF DU
Eq. 9.29
Principles of Econometrics, 4th Edition Page 71Chapter 9: Regression with Time Series Data:
Stationary Variables
The t and p-values for testing H0: β2 = 0 are:
9.5Estimation with
Serially Correlated
Errors
9.5.1Least Squares
Estimation
0.5279 0.2294 2.301 0.0238 from LS standard errors
0.5279 0.3127 1.688 0.0950 from HAC standard errors
t p
t p
Principles of Econometrics, 4th Edition Page 72Chapter 9: Regression with Time Series Data:
Stationary Variables
Return to the Lagrange multiplier test for serially correlated errors where we used the equation:
– Assume the vt are uncorrelated random errors with zero mean and constant variances:
9.5Estimation with
Serially Correlated
Errors
9.5.2Estimating an
AR(1) Error Model
1ρt t te e v Eq. 9.30
20 var cov , 0 for t t v t sE v v v v t s Eq. 9.31
Principles of Econometrics, 4th Edition Page 73Chapter 9: Regression with Time Series Data:
Stationary Variables
Eq. 9.30 describes a first-order autoregressive model or a first-order autoregressive process for et
– The term AR(1) model is used as an abbreviation for first-order autoregressive model
– It is called an autoregressive model because it can be viewed as a regression model
– It is called first-order because the right-hand-side variable is et lagged one period
9.5Estimation with
Serially Correlated
Errors
9.5.2Estimating an
AR(1) Error Model
Principles of Econometrics, 4th Edition Page 74Chapter 9: Regression with Time Series Data:
Stationary Variables
We assume that:
The mean and variance of et are:
The covariance term is:
9.5Estimation with
Serially Correlated
Errors
9.5.2aProperties of an
AR(1) Error
1 ρ 1 Eq. 9.32
2
220 var
1 ρv
t t eE e e
Eq. 9.33
2
2
ρcov , , 01 ρ
kv
t t ke e k
Eq. 9.34
Principles of Econometrics, 4th Edition Page 75Chapter 9: Regression with Time Series Data:
Stationary Variables
The correlation implied by the covariance is:
9.5Estimation with
Serially Correlated
Errors
2 2
2 2
ρ corr ,
cov ,
var var
cov ,var
ρ 1 ρ
1 ρ
ρ
k t t k
t t k
t t k
t t k
t
kv
v
k
e e
e e
e e
e ee
Eq. 9.35
9.5.2aProperties of an
AR(1) Error
Principles of Econometrics, 4th Edition Page 76Chapter 9: Regression with Time Series Data:
Stationary Variables
Setting k = 1:
– ρ represents the correlation between two errors that are one period apart• It is the first-order autocorrelation for e,
sometimes simply called the autocorrelation coefficient• It is the population autocorrelation at lag one for a
time series that can be described by an AR(1) model• r1 is an estimate for ρ when we assume a series is
AR(1)
9.5Estimation with
Serially Correlated
Errors
1 1ρ corr , ρt te e Eq. 9.36
9.5.2aProperties of an
AR(1) Error
Principles of Econometrics, 4th Edition Page 77Chapter 9: Regression with Time Series Data:
Stationary Variables
Each et depends on all past values of the errors vt:
– For the Phillips Curve, we find for the first five lags:
– For an AR(1) model, we have:
9.5Estimation with
Serially Correlated
Errors
2 31 2 3ρ ρ ρt t t t te v v v v Eq. 9.37
1 2 3 4 50.549 0.456 0.433 0.420 0.339r r r r r
1 1ˆ ˆρ ρ 0.549r
9.5.2aProperties of an
AR(1) Error
Principles of Econometrics, 4th Edition Page 78Chapter 9: Regression with Time Series Data:
Stationary Variables
For longer lags, we have:
9.5Estimation with
Serially Correlated
Errors
222
333
444
555
ˆ ˆρ ρ 0.549 0.301
ˆ ˆρ ρ 0.549 0.165
ˆ ˆρ ρ 0.549 0.091
ˆ ˆρ ρ 0.549 0.050
9.5.2aProperties of an
AR(1) Error
Principles of Econometrics, 4th Edition Page 79Chapter 9: Regression with Time Series Data:
Stationary Variables
Our model with an AR(1) error is:
with -1 < ρ < 1– For the vt, we have:
9.5Estimation with
Serially Correlated
Errors
9.5.2bNonlinear Least
Squares Estimation
1 2 1β β with ρt t t t t ty x e e e v Eq. 9.38
210 var cov , 0 for t t v t tE v v v v t s Eq. 9.39
Principles of Econometrics, 4th Edition Page 80Chapter 9: Regression with Time Series Data:
Stationary Variables
With the appropriate substitutions, we get:
– For the previous period, the error is:
–Multiplying by ρ:
9.5Estimation with
Serially Correlated
Errors
9.5.2bNonlinear Least
Squares Estimation
1 2 1β β ρt t t ty x e v Eq. 9.40
1 1 1 2 1β βt t te y x Eq. 9.41
1 1 1 2 1ρ ρβ ρβt t t te e y x Eq. 9.42
Principles of Econometrics, 4th Edition Page 81Chapter 9: Regression with Time Series Data:
Stationary Variables
Substituting, we get:
9.5Estimation with
Serially Correlated
Errors
9.5.2bNonlinear Least
Squares Estimation
1 2 1 2 1β 1 ρ β ρ ρβt t t t ty x y x v Eq. 9.43
Principles of Econometrics, 4th Edition Page 82Chapter 9: Regression with Time Series Data:
Stationary Variables
The coefficient of xt-1 equals -ρβ2 – Although Eq. 9.43 is a linear function of the
variables xt , yt-1 and xt-1, it is not a linear function of the parameters (β1, β2, ρ)
– The usual linear least squares formulas cannot be obtained by using calculus to find the values of (β1, β2, ρ) that minimize Sv
• These are nonlinear least squares estimates
9.5Estimation with
Serially Correlated
Errors
9.5.2bNonlinear Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 83Chapter 9: Regression with Time Series Data:
Stationary Variables
Our Phillips Curve model assuming AR(1) errors is:
– Applying nonlinear least squares and presenting the estimates in terms of the original untransformed model, we have:
9.5Estimation with
Serially Correlated
Errors
9.5.2bNonlinear Least
Squares Estimation
1 2 1 2 1β 1 ρ β ρ ρβt t t t tINF DU INF DU v Eq. 9.44
10.7609 0.6944 0.557
0.1245 0.2479 0.090t t tINF DU e e v
se
Eq. 9.45
Principles of Econometrics, 4th Edition Page 84Chapter 9: Regression with Time Series Data:
Stationary Variables
Nonlinear least squares estimation of Eq. 9.43 is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure
9.5Estimation with
Serially Correlated
Errors
9.5.2cGeneralized
Least Squares Estimation
Principles of Econometrics, 4th Edition Page 85Chapter 9: Regression with Time Series Data:
Stationary Variables
We have the model:
– Suppose now that we consider the model:
• This new notation will be convenient when we discuss a general class of autoregressive distributed lag (ARDL) models–Eq. 9.47 is a member of this class
9.5Estimation with
Serially Correlated
Errors
9.5.3Estimating a More General
Model
1 2 1 2 1β 1 ρ β ρ ρβt t t t ty x y x v Eq. 9.46
1 1 0 1 1δ θ δ δt t t t ty y x x v Eq. 9.47
Principles of Econometrics, 4th Edition Page 86Chapter 9: Regression with Time Series Data:
Stationary Variables
Note that Eq. 9.47 is the same as Eq. 9.47 since:
– Eq. 9.46 is a restricted version of Eq. 9.47 with the restriction δ1 = -θ1δ0 imposed
9.5Estimation with
Serially Correlated
Errors
9.5.3Estimating a More General
Model
1 0 2 1 2 1δ β 1 ρ δ β δ ρβ θ ρ Eq. 9.48
Principles of Econometrics, 4th Edition Page 87Chapter 9: Regression with Time Series Data:
Stationary Variables
Applying the least squares estimator to Eq. 9.47 using the data for the Phillips curve example yields:
9.5Estimation with
Serially Correlated
Errors
9.5.3Estimating a More General
Model
1 10.3336 0.5593 0.6882 0.3200
0.0899 0.0908 0.2575 0.2499t t t tINF INF DU DU
se
Eq. 9.49
Principles of Econometrics, 4th Edition Page 88Chapter 9: Regression with Time Series Data:
Stationary Variables
The equivalent AR(1) estimates are:
– These are similar to our other estimates
9.5Estimation with
Serially Correlated
Errors
9.5.3Estimating a More General
Model
1
1
0 2
1 2
ˆ ˆ ˆδ β 1 ρ 0.7609 1 0.5574 0.3368ˆ ˆθ ρ 0.5574ˆ ˆδ β 0.6944ˆ ˆˆδ ρβ 0.5574 0.6944 0.3871
Principles of Econometrics, 4th Edition Page 89Chapter 9: Regression with Time Series Data:
Stationary Variables
The original economic model for the Phillips Curve was:
– Re-estimation of the model after omitting DUt-1 yields:
9.5Estimation with
Serially Correlated
Errors
9.5.3Estimating a More General
Model
1γEt t t tINF INF U U Eq. 9.50
10.3548 0.5282 0.4909
0.0876 0.0851 0.1921t t tINF INF DU
se
Eq. 9.51
Principles of Econometrics, 4th Edition Page 90Chapter 9: Regression with Time Series Data:
Stationary Variables
In this model inflationary expectations are given by:
– A 1% rise in the unemployment rate leads to an approximate 0.5% fall in the inflation rate
9.5Estimation with
Serially Correlated
Errors
9.5.3Estimating a More General
Model
10.3548 0.5282Et tINF INF
Principles of Econometrics, 4th Edition Page 91Chapter 9: Regression with Time Series Data:
Stationary Variables
We have described three ways of overcoming the effect of serially correlated errors:1. Estimate the model using least squares with
HAC standard errors2. Use nonlinear least squares to estimate the
model with a lagged x, a lagged y, and the restriction implied by an AR(1) error specification
3. Use least squares to estimate the model with a lagged x and a lagged y, but without the restriction implied by an AR(1) error specification
9.5Estimation with
Serially Correlated
Errors
9.5.4Summary of
Section 9.5 and Looking Ahead
Principles of Econometrics, 4th Edition Page 92Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6 Autoregressive Distributed Lag
Models
Principles of Econometrics, 4th Edition Page 93Chapter 9: Regression with Time Series Data:
Stationary Variables
An autoregressive distributed lag (ARDL) model is one that contains both lagged xt’s and lagged yt’s
– Two examples:
9.6Autoregressive Distributed Lag
Models
0 1 1 1 1t t t q t q t p t p ty x x x y y v Eq. 9.52
1 1
1
ADRL 1,1 : 0.3336 0.5593 0.6882 0.3200
ADRL 1,0 : 0.3548 0.5282 0.4909
t t t t
t t t
INF INF DU DU
INF INF DU
Principles of Econometrics, 4th Edition Page 94Chapter 9: Regression with Time Series Data:
Stationary Variables
An ARDL model can be transformed into one with only lagged x’s which go back into the infinite past:
– This model is called an infinite distributed lag model
9.6Autoregressive Distributed Lag
Models
0 1 1 2 2 3 3
0
β β β
β
t t t t t t
s t s ts
y x x x x e
x e
Eq. 9.53
Principles of Econometrics, 4th Edition Page 95Chapter 9: Regression with Time Series Data:
Stationary Variables
Four possible criteria for choosing p and q:1. Has serial correlation in the errors been
eliminated?2. Are the signs and magnitudes of the estimates
consistent with our expectations from economic theory?
3. Are the estimates significantly different from zero, particularly those at the longest lags?
4. What values for p and q minimize information criteria such as the AIC and SC?
9.6Autoregressive Distributed Lag
Models
Principles of Econometrics, 4th Edition Page 96Chapter 9: Regression with Time Series Data:
Stationary Variables
The Akaike information criterion (AIC) is:
where K = p + q + 2The Schwarz criterion (SC), also known as the Bayes information criterion (BIC), is:
– Because Kln(T)/T > 2K/T for T ≥ 8, the SC penalizes additional lags more heavily than does the AIC
9.6Autoregressive Distributed Lag
Models
2AIC ln SSE KT T
Eq. 9.54
lnSC ln
K TSSET T
Eq. 9.55
Principles of Econometrics, 4th Edition Page 97Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider the previously estimated ARDL(1,0) model:
9.6Autoregressive Distributed Lag
Models
Eq. 9.56
9.6.1The Phillips
Curve
10.3548 0.5282 0.4909 , obs 90
0.0876 0.0851 0.1921t t tINF INF DU
se
Principles of Econometrics, 4th Edition Page 98Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
Models
9.6.1The Phillips
Curve
FIGURE 9.9 Correlogram for residuals from Phillips curve ARDL(1,0) model
Principles of Econometrics, 4th Edition Page 99Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
Models
9.6.1The Phillips
Curve
Table 9.3 p-values for LM Test for Autocorrelation
Principles of Econometrics, 4th Edition Page 100Chapter 9: Regression with Time Series Data:
Stationary Variables
For an ARDL(4,0) version of the model:
9.6Autoregressive Distributed Lag
Models
Eq. 9.57
9.6.1The Phillips
Curve
1 2 3
-4
0.1001 0.2354 0.1213 0.1677
0.0983 0.1016 0.1038 0.1050
0.2819 0.7902
0.1014 0.1885
t t t t
t t
INF INF INF INF
se
INF DU
obs 87
Principles of Econometrics, 4th Edition Page 101Chapter 9: Regression with Time Series Data:
Stationary Variables
Inflationary expectations are given by:
9.6Autoregressive Distributed Lag
Models
9.6.1The Phillips
Curve
1 2 3 -40.1001 0.2354 0.1213 0.1677 0.2819Et t t t tINF INF INF INF INF
Principles of Econometrics, 4th Edition Page 102Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
Models
9.6.1The Phillips
Curve
Table 9.4 AIC and SC Values for Phillips Curve ARDL Models
Principles of Econometrics, 4th Edition Page 103Chapter 9: Regression with Time Series Data:
Stationary Variables
Recall the model for Okun’s Law:
9.6Autoregressive Distributed Lag
Models
9.6.2Okun’s Law
Eq. 9.58
1 20.5836 0.2020 0.1653 0.0700G , obs 96
0.0472 0.0324 0.0335 0.0331t t t tDU G G
se
Principles of Econometrics, 4th Edition Page 104Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
ModelsFIGURE 9.10 Correlogram for residuals from Okun’s law ARDL(0,2) model
9.6.2Okun’s Law
Principles of Econometrics, 4th Edition Page 105Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
ModelsTable 9.5 AIC and SC Values for Okun’s Law ARDL Models
9.6.2Okun’s Law
Principles of Econometrics, 4th Edition Page 106Chapter 9: Regression with Time Series Data:
Stationary Variables
Now consider this version:
9.6Autoregressive Distributed Lag
Models
9.6.2Okun’s Law
Eq. 9.59
1 10.3780 0.3501 0.1841 0.0992G , obs 96
0.0578 0.0846 0.0307 0.0368t t t tDU DU G
se
Principles of Econometrics, 4th Edition Page 107Chapter 9: Regression with Time Series Data:
Stationary Variables
An autoregressive model of order p, denoted AR(p), is given by:
9.6Autoregressive Distributed Lag
Models
9.6.3Autoregressive
Models
Eq. 9.60 1 1 2 2δ θ θ θt t t p t p ty y y y v
Principles of Econometrics, 4th Edition Page 108Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider a model for growth in real GDP:
9.6Autoregressive Distributed Lag
Models
9.6.3Autoregressive
Models
Eq. 9.61
1 20.4657 0.3770 0.2462
0.1433 0.1000 0.1029 obs = 96t t tG G G
se
Principles of Econometrics, 4th Edition Page 109Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
Models
9.6.3Autoregressive
Models
FIGURE 9.11 Correlogram for residuals from AR(2) model for GDP growth
Principles of Econometrics, 4th Edition Page 110Chapter 9: Regression with Time Series Data:
Stationary Variables
9.6Autoregressive Distributed Lag
Models
9.6.3Autoregressive
Models
Table 9.6 AIC and SC Values for AR Model of Growth in U.S. GDP
Principles of Econometrics, 4th Edition Page 111Chapter 9: Regression with Time Series Data:
Stationary Variables
9.7 Forecasting
Principles of Econometrics, 4th Edition Page 112Chapter 9: Regression with Time Series Data:
Stationary Variables
We consider forecasting using three different models:1. AR model2. ARDL model3. Exponential smoothing model
9.7Forecasting
Principles of Econometrics, 4th Edition Page 113Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider an AR(2) model for real GDP growth:
The model to forecast GT+1 is:
9.7Forecasting
9.7.1Forecasting with
an AR Model
Eq. 9.62 1 1 2 2δ θ θt t t tG G G v
1 1 2 1 1δ θ θT T T TG G G v
Principles of Econometrics, 4th Edition Page 114Chapter 9: Regression with Time Series Data:
Stationary Variables
The growth values for the two most recent quarters are:
GT = G2009Q3 = 0.8GT-1 = G2009Q2 = -0.2
The forecast for G2009Q4 is:
9.7Forecasting
9.7.1Forecasting with
an AR Model
Eq. 9.63 1 1 2 1
ˆˆ ˆ ˆδ θ θ
0.46573 0.37700 0.8 0.24624 0.2
0.7181
T T TG G G
Principles of Econometrics, 4th Edition Page 115Chapter 9: Regression with Time Series Data:
Stationary Variables
For two quarters ahead, the forecast for G2010Q1 is:
For three periods out, it is:
9.7Forecasting
9.7.1Forecasting with
an AR Model
Eq. 9.642 1 1 2
ˆˆ ˆ ˆδ θ θ0.46573 0.37700 0.71808 0.24624 0.80.9334
T T TG G G
Eq. 9.653 1 2 2 1
ˆˆ ˆ ˆδ θ θ0.46573 0.37700 0.93343 0.24624 0.718080.9945
T T TG G G
Principles of Econometrics, 4th Edition Page 116Chapter 9: Regression with Time Series Data:
Stationary Variables
Summarizing our forecasts:– Real GDP growth rates for 2009Q4, 2010Q1,
and 2010Q2 are approximately 0.72%, 0.93%, and 0.99%, respectively
9.7Forecasting
9.7.1Forecasting with
an AR Model
Principles of Econometrics, 4th Edition Page 117Chapter 9: Regression with Time Series Data:
Stationary Variables
A 95% interval forecast for j periods into the future is given by:
where is the standard error of the forecast error and df is the number of degrees of freedom in the estimation of the AR model
9.7Forecasting
9.7.1Forecasting with
an AR Model
0.975,ˆ σT j jdfG t
σ j
Principles of Econometrics, 4th Edition Page 118Chapter 9: Regression with Time Series Data:
Stationary Variables
The first forecast error, occurring at time T+1, is:
Ignoring the error from estimating the coefficients, we get:
9.7Forecasting
9.7.1Forecasting with
an AR Model
1 1 1 1 1 2 2 1 1ˆˆ ˆ ˆδ δ θ θ θ θT T T T Tu G G G G v
1 1Tu v Eq. 9.66
Principles of Econometrics, 4th Edition Page 119Chapter 9: Regression with Time Series Data:
Stationary Variables
The forecast error for two periods ahead is:
The forecast error for three periods ahead is:
9.7Forecasting
9.7.1Forecasting with
an AR Model
2 1 1 1 2 1 1 2 1 1 2ˆθ θ θT T T T T Tu G G v u v v v Eq. 9.67
23 1 2 2 1 3 1 2 1 1 2 3θ θ θ θ θT T T Tu u u v v v v Eq. 9.68
Principles of Econometrics, 4th Edition Page 120Chapter 9: Regression with Time Series Data:
Stationary Variables
Because the vt’s are uncorrelated with constant variance , we can show that:
9.7Forecasting
9.7.1Forecasting with
an AR Model
2 21 1
2 2 22 2 1
22 2 2 23 3 1 2 1
σ var σ
σ var σ 1 θ
σ var σ θ θ θ 1
v
v
v
u
u
u
2v
Principles of Econometrics, 4th Edition Page 121Chapter 9: Regression with Time Series Data:
Stationary Variables
9.7Forecasting
9.7.1Forecasting with
an AR Model
Table 9.7 Forecasts and Forecast Intervals for GDP Growth
Principles of Econometrics, 4th Edition Page 122Chapter 9: Regression with Time Series Data:
Stationary Variables
Consider forecasting future unemployment using the Okun’s Law ARDL(1,1):
The value of DU in the first post-sample quarter is:
– But we need a value for GT+1
9.7Forecasting
9.7.2Forecasting with an ARDL Model
1 1 0 1 1δ θ δ δt t t t tDU DU G G v Eq. 9.69
1 1 0 1 1 1δ θ δ δT T T T TDU DU G G v Eq. 9.70
Principles of Econometrics, 4th Edition Page 123Chapter 9: Regression with Time Series Data:
Stationary Variables
Now consider the change in unemployment– Rewrite Eq. 9.70 as:
– Rearranging:
9.7Forecasting
9.7.2Forecasting with an ARDL Model
1 1 1 0 1 1 1δ θ δ δT T T T T T TU U U U G G v
Eq. 9.71 1 1 1 1 0 1 1 1
* *1 2 1 0 1 1 1
δ θ 1 θ δ δ
δ θ θ δ δT T T T T T
T T T T T
U U U G G v
U U G G v
Principles of Econometrics, 4th Edition Page 124Chapter 9: Regression with Time Series Data:
Stationary Variables
For the purpose of computing point and interval forecasts, the ARDL(1,1) model for a change in unemployment can be written as an ARDL(2,1) model for the level of unemployment– This result holds not only for ARDL models
where a dependent variable is measured in terms of a change or difference, but also for pure AR models involving such variables
9.7Forecasting
9.7.2Forecasting with an ARDL Model
Principles of Econometrics, 4th Edition Page 125Chapter 9: Regression with Time Series Data:
Stationary Variables
Another popular model used for predicting the future value of a variable on the basis of its history is the exponential smoothing method– Like forecasting with an AR model, forecasting
using exponential smoothing does not utilize information from any other variable
9.7Forecasting
9.7.3Exponential Smoothing
Principles of Econometrics, 4th Edition Page 126Chapter 9: Regression with Time Series Data:
Stationary Variables
One possible forecasting method is to use the average of past information, such as:
– This forecasting rule is an example of a simple (equally-weighted) moving average model with k = 3
9.7Forecasting
9.7.3Exponential Smoothing
1 21ˆ
3T T T
Ty y yy
Principles of Econometrics, 4th Edition Page 127Chapter 9: Regression with Time Series Data:
Stationary Variables
Now consider a form in which the weights decline exponentially as the observations get older:
–We assume that 0 < α < 1– Also, it can be shown that:
9.7Forecasting
9.7.3Exponential Smoothing
1 21 T 1 2ˆ αy α 1 α α 1 αT T Ty y y Eq. 9.72
0α 1 α 1
s
s
Principles of Econometrics, 4th Edition Page 128Chapter 9: Regression with Time Series Data:
Stationary Variables
For forecasting, recognize that:
–We can simplify to:
9.7Forecasting
9.7.3Exponential Smoothing
2 31 2 3ˆ1 α α 1 α α 1 α α 1 αT T T Ty y y y Eq. 9.73
1ˆ ˆα 1 αT T Ty y y Eq. 9.74
Principles of Econometrics, 4th Edition Page 129Chapter 9: Regression with Time Series Data:
Stationary Variables
The value of α can reflect one’s judgment about the relative weight of current information– It can be estimated from historical information
by obtaining within-sample forecasts:
• Choosing α that minimizes the sum of squares of the one-step forecast errors:
9.7Forecasting
9.7.3Exponential Smoothing
1 1ˆ ˆα 1 α 2,3, ,t t ty y y t T Eq. 9.75
1 1ˆ ˆα + 1 αt t t t t tv y y y y y Eq. 9.76
Principles of Econometrics, 4th Edition Page 130Chapter 9: Regression with Time Series Data:
Stationary Variables
9.7Forecasting
9.7.3Exponential Smoothing
FIGURE 9.12 (a) Exponentially smoothed forecasts for GDP growth with α = 0.38
Principles of Econometrics, 4th Edition Page 131Chapter 9: Regression with Time Series Data:
Stationary Variables
9.7Forecasting
9.7.3Exponential Smoothing
FIGURE 9.12 (b) Exponentially smoothed forecasts for GDP growth with α = 0.8
Principles of Econometrics, 4th Edition Page 132Chapter 9: Regression with Time Series Data:
Stationary Variables
The forecasts for 2009Q4 from each value of α are:
9.7Forecasting
9.7.3Exponential Smoothing
1
1
ˆ ˆα 0.38 : α 1 α 0.38 0.8 1 0.38 0.403921
= 0.0536ˆ ˆα 0.8 : α 1 α 0.8 0.8 1 0.8 0.393578
= 0.5613
T T T
T T T
G G G
G G G
Principles of Econometrics, 4th Edition Page 133Chapter 9: Regression with Time Series Data:
Stationary Variables
9.8 Multiplier Analysis
Principles of Econometrics, 4th Edition Page 134Chapter 9: Regression with Time Series Data:
Stationary Variables
Multiplier analysis refers to the effect, and the timing of the effect, of a change in one variable on the outcome of another variable
9.8Multiplier Analysis
Principles of Econometrics, 4th Edition Page 135Chapter 9: Regression with Time Series Data:
Stationary Variables
Let’s find multipliers for an ARDL model of the form:
–We can transform this into an infinite distributed lag model:
9.8Multiplier Analysis
1 1 0 1 1t t p t p t t q t q ty y y x x x v Eq. 9.77
0 t 1 1 2 2 3 3α β x + β β βt t t t ty x x x e Eq. 9.78
Principles of Econometrics, 4th Edition Page 136Chapter 9: Regression with Time Series Data:
Stationary Variables
The multipliers are defined as:
9.8Multiplier Analysis
0
0
β period delay multiplier
β period interim multiplier
β total multiplier
ts
t s
s
jj
jj
ys
x
s
Principles of Econometrics, 4th Edition Page 137Chapter 9: Regression with Time Series Data:
Stationary Variables
The lag operator is defined as:
– Lagging twice, we have:
– Or:
–More generally, we have:
9.8Multiplier Analysis
1t tLy y
1 2t t tL Ly Ly y
22t tL y y
st t sL y y
Principles of Econometrics, 4th Edition Page 138Chapter 9: Regression with Time Series Data:
Stationary Variables
Now rewrite our model as:
9.8Multiplier Analysis
2 21 2 0 1 2
pt t t p t t t t
qq t t
y Ly L y L y x Lx L x
L x v
Eq. 9.79
Principles of Econometrics, 4th Edition Page 139Chapter 9: Regression with Time Series Data:
Stationary Variables
Rearranging terms:
9.8Multiplier Analysis
2 21 2 0 1 21 p q
p t q t tL L L y L L L x v Eq. 9.80
Principles of Econometrics, 4th Edition Page 140Chapter 9: Regression with Time Series Data:
Stationary Variables
Let’s apply this to our Okun’s Law model– The model:
can be rewritten as:
9.8Multiplier Analysis
Eq. 9.81 1 1 0 1 1δ θ δ δt t t t tDU DU G G v
Eq. 9.82 1 0 11 θ δ δ δt t tL DU L G v
Principles of Econometrics, 4th Edition Page 141Chapter 9: Regression with Time Series Data:
Stationary Variables
Define the inverse of (1 – θ1L) as (1 – θ1L)-1 such that:
9.8Multiplier Analysis
11 11 θ 1 θ 1L L
Principles of Econometrics, 4th Edition Page 142Chapter 9: Regression with Time Series Data:
Stationary Variables
Multiply both sides of Eq. 9.82 by (1 – θ1L)-1:
– Equating this with the infinite distributed lag representation:
9.8Multiplier Analysis
1 1 11 1 0 1 11 θ δ 1 θ δ δ 1 θt t tDU L L L G L v Eq. 9.83
Eq. 9.84 0 1 1 2 2 3 3
2 30 1 2 3
α β β β β
α β β β βt t t t t t
t t
DU G G G G e
L L L G e
Principles of Econometrics, 4th Edition Page 143Chapter 9: Regression with Time Series Data:
Stationary Variables
For Eqs. 9.83 and 9.84 to be identical, it must be true that:
9.8Multiplier Analysis
Eq. 9.85
Eq. 9.86
11
12 30 1 2 3 1 0 1
11
α= 1 θ δ
β β β β 1 θ δ δ
1 θt t
L
L L L L L
e L v
Eq. 9.87
Principles of Econometrics, 4th Edition Page 144Chapter 9: Regression with Time Series Data:
Stationary Variables
Multiply both sides of Eq. 9.85 by (1 – θ1L) to obtain (1 – θ1L)α = δ. – Note that the lag of a constant that does not
change so Lα = α– Now we have:
9.8Multiplier Analysis
11
δ1 θ α δ and α1 θ
Principles of Econometrics, 4th Edition Page 145Chapter 9: Regression with Time Series Data:
Stationary Variables
Multiply both sides of Eq. 9.86 by (1 – θ1L):
9.8Multiplier Analysis
2 30 1 1 0 1 2 3
2 30 1 2 3
2 30 1 1 1 2 1
2 30 1 0 1 2 1 1 3 2 1
δ δ 1 θ β β β β
β β β β
β θ β θ β θ
β β β θ β β θ β β θ
L L L L L
L L L
L L L
L L L
Eq. 9.88
Principles of Econometrics, 4th Edition Page 146Chapter 9: Regression with Time Series Data:
Stationary Variables
Rewrite Eq. 9.86 as:
– Equating coefficients of like powers in L yields:
and so on
9.8Multiplier Analysis
2 3 2 30 1 0 1 0 1 2 1 1 3 2 1δ δ 0 0 β β β θ β β θ β β θL L L L L L Eq. 9.89
0 0
1 1 0 1
2 1 1
3 2 1
δ = βδ β β θ0 β β θ0 β β θ
Principles of Econometrics, 4th Edition Page 147Chapter 9: Regression with Time Series Data:
Stationary Variables
We can now find the β’s using the recursive equations:
9.8Multiplier Analysis
Eq. 9.90
0 0
1 1 0 1
1 1
β = δβ δ β θβ β θ for 2j j j
Principles of Econometrics, 4th Edition Page 148Chapter 9: Regression with Time Series Data:
Stationary Variables
You can start from the equivalent of Eq. 9.88 which, in its general form, is:
– Given the values p and q for your ARDL model, you need to multiply out the above expression, and then equate coefficients of like powers in the lag operator
9.8Multiplier Analysis
Eq. 9.91
2 20 1 2 1 2
2 30 1 2 3
δ δ δ δ 1 θ θ θ
β β β β
q pq pL L L L L L
L L L
Principles of Econometrics, 4th Edition Page 149Chapter 9: Regression with Time Series Data:
Stationary Variables
For the Okun’s Law model:
– The impact and delay multipliers for the first four quarters are:
9.8Multiplier Analysis
1 10.3780 0.3501 0.1841 0.0992t t t tDU DU G G
0 0
1 1 0 1
2 1 1
3 2 1
4 3 1
ˆ ˆβ = δ 0.1841ˆ ˆ ˆ ˆβ δ β θ 0.099155 0.184084 0.350116 0.1636ˆ ˆ ˆβ β θ 0.163606 0.350166 0.0573ˆ ˆ ˆβ β θ 0.057281 0.350166 0.0201ˆ ˆ ˆβ β θ 0.020055 0.350166 0.0070
Principles of Econometrics, 4th Edition Page 150Chapter 9: Regression with Time Series Data:
Stationary Variables
9.8Multiplier Analysis
FIGURE 9.13 Delay multipliers from Okun’s law ARDL(1,1) model
Principles of Econometrics, 4th Edition Page 151Chapter 9: Regression with Time Series Data:
Stationary Variables
We can estimate the total multiplier given by:
and the normal growth rate that is needed to maintain a constant rate of unemployment:
9.8Multiplier Analysis
0
β jj
0
α βN jj
G
Principles of Econometrics, 4th Edition Page 152Chapter 9: Regression with Time Series Data:
Stationary Variables
We can show that:
– An estimate for α is given by:
– Therefore, normal growth rate is:
9.8Multiplier Analysis
1 0 10
0 1
ˆ ˆ ˆδ δ θ 0.163606ˆβ δ 0.184084 0.4358ˆ 1 0.3501161 θjj
1
δ 0.37801α 0.5817ˆ 0.6498841 θ
N0.5817G 1.3% per quarter0.4358
Principles of Econometrics, 4th Edition Page 153Chapter 9: Regression with Time Series Data:
Stationary Variables
Key Words
Principles of Econometrics, 4th Edition Page 154Chapter 9: Regression with Time Series Data:
Stationary Variables
AIC criterionAR(1) errorAR(p) modelARDL(p,q) modelautocorrelationAutoregressive distributed lagsautoregressive errorautoregressive modelBIC criterioncorrelogramdelay multiplierdistributed lag weight
Keywords
dynamic modelsexponential smoothingfinite distributed lagforecast errorforecast intervalsforecastingHAC standard errorsimpact multiplierinfinite distributed laginterim multiplierlag lengthlag operatorlagged dependent variable
LM testmultiplier analysisnonlinear least squaresout-of-sample forecastssample autocorrelationsserial correlation standard error of forecast errorSC criteriontotal multiplierT x R2 form of LM test within-sample forecasts
Principles of Econometrics, 4th Edition Page 155Chapter 9: Regression with Time Series Data:
Stationary Variables
Appendices
Principles of Econometrics, 4th Edition Page 156Chapter 9: Regression with Time Series Data:
Stationary Variables
For the Durbin-Watson test, the hypotheses are:
The test statistic is:
9AThe Durbin-Watson Test
Eq. 9A.1
0 1: 0 : 0H H
21
2
2
1
ˆ ˆ
ˆ
T
t tt
T
tt
e ed
e
Principles of Econometrics, 4th Edition Page 157Chapter 9: Regression with Time Series Data:
Stationary Variables
We can expand the test statistic as:
9AThe Durbin-Watson Test
Eq. 9A.2
2 21 1
2 2 2
2
1
2 21 1
2 2 2
2 2 2
1 1 1
1
ˆ ˆ ˆ ˆ2
ˆ
ˆ ˆ ˆ ˆ2
ˆ ˆ ˆ
1 1 2
T T T
t t t tt t t
T
tt
T T T
t t t tt t tT T T
t t tt t t
e e e ed
e
e e e e
e e e
r
Principles of Econometrics, 4th Edition Page 158Chapter 9: Regression with Time Series Data:
Stationary Variables
We can now write:
– If the estimated value of ρ is r1 = 0, then the Durbin-Watson statistic d ≈ 2• This is taken as an indication that the model
errors are not autocorrelated– If the estimate of ρ happened to be r1 = 1 then d ≈ 0• A low value for the Durbin-Watson statistic
implies that the model errors are correlated, and ρ > 0
9AThe Durbin-Watson Test
Eq. 9A.3 12 1d r
Principles of Econometrics, 4th Edition Page 159Chapter 9: Regression with Time Series Data:
Stationary Variables
9AThe Durbin-Watson Test FIGURE 9A.1 Testing for positive autocorrelation
Principles of Econometrics, 4th Edition Page 160Chapter 9: Regression with Time Series Data:
Stationary Variables
9AThe Durbin-Watson Test
9A.1The Durbin-
Watson Bounds Test
FIGURE 9A.2 Upper and lower critical value bounds for the Durbin-Watson test
Principles of Econometrics, 4th Edition Page 161Chapter 9: Regression with Time Series Data:
Stationary Variables
Decision rules, known collectively as the Durbin-Watson bounds test:– If d < dLc: reject H0: ρ = 0 and accept H1:
ρ > 0– If d > dUc do not reject H0: ρ = 0 – If dLc < d < dUc, the test is inconclusive
9AThe Durbin-Watson Test
9A.1The Durbin-
Watson Bounds Test
Principles of Econometrics, 4th Edition Page 162Chapter 9: Regression with Time Series Data:
Stationary Variables
Note that:
Further substitution shows that:
9BProperties of
the AR(1) Error
1
2 1
22 1
ρ
ρ ρ
ρ ρ
t t t
t t t
t t t
e e v
e v v
e v v
Eq. 9B.1
23 2 1
3 23 2 1
ρ ρ ρ
ρ ρ ρt t t t t
t t t t
e e v v v
e v v v
Eq. 9B.2
Principles of Econometrics, 4th Edition Page 163Chapter 9: Regression with Time Series Data:
Stationary Variables
Repeating the substitution k times and rearranging:
If we let k → ∞, then we have:
9BProperties of
the AR(1) Error
2 11 2 1ρ ρ ρ ρk k
t t k t t t t ke e v v v v Eq. 9B.3
2 31 2 3ρ ρ ρt t t t te v v v v Eq. 9B.4
Principles of Econometrics, 4th Edition Page 164Chapter 9: Regression with Time Series Data:
Stationary Variables
We can now find the properties of et:
9BProperties of
the AR(1) Error
2 31 2 3
2 3
ρE ρ ρ
0 ρ 0 ρ 0 ρ 00
t t t t tE e E v v E v E v
2 4 61 2 3
2 2 2 4 2 6 2
2 2 4 6
2
2
var var ρ var ρ var ρ var
ρ ρ ρ
1 ρ ρ ρ
1 ρ
t t t t t
v v v v
v
v
e v v v v
Principles of Econometrics, 4th Edition Page 165Chapter 9: Regression with Time Series Data:
Stationary Variables
The covariance for one period apart is:
9BProperties of
the AR(1) Error
1
2 31 2 3
2 31 2 3 4
2 3 2 5 21 2 3
2 2 4
2
2
cov ,
ρ ρ ρ
ρ ρ ρ
ρ ρ ρ
ρ 1 ρ ρ
ρ1 ρ
t t t t t
t t t t
t t t t
t t t
v
v
e e E e e
E v v v v
v v v v
E v E v E v
Principles of Econometrics, 4th Edition Page 166Chapter 9: Regression with Time Series Data:
Stationary Variables
Similarly, the covariance for k periods apart is:
9BProperties of
the AR(1) Error
2
2
ρcov , 0
1 ρ
kv
t t ke e k
Principles of Econometrics, 4th Edition Page 167Chapter 9: Regression with Time Series Data:
Stationary Variables
We are considering the simple regression model with AR(1) errors:
To specify the transformed model we begin with:
– Rearranging terms:
9CGeneralized
Least Squares Estimation
1 2 1 t t t t t ty x e e e v
1 2 1 1 2 1t t t t ty x y x v Eq. 9C.1
1 1 2 11t t t t ty y x x v Eq. 9C.2
Principles of Econometrics, 4th Edition Page 168Chapter 9: Regression with Time Series Data:
Stationary Variables
Defining the following transformed variables:
Substituting the transformed variables, we get:
9CGeneralized
Least Squares Estimation
Eq. 9C.3
1 2 1 1 1t t t t t t ty y y x x x x
1 1 2 2t t t ty x x v
Principles of Econometrics, 4th Edition Page 169Chapter 9: Regression with Time Series Data:
Stationary Variables
There are two problems:1. Because lagged values of yt and xt had to be
formed, only (T - 1) new observations were created by the transformation
2. The value of the autoregressive parameter ρ is not known
9CGeneralized
Least Squares Estimation
Principles of Econometrics, 4th Edition Page 170Chapter 9: Regression with Time Series Data:
Stationary Variables
For the second problem, we can write Eq. 9C.1 as:
For the first problem, note that:
and that
9CGeneralized
Least Squares Estimation
Eq. 9C.4 1 2 1 1 2 1( )t t t t ty x y x v
1 1 1 2 1y x e
2 2 2 21 1 1 2 11 1 1 1y x e
Principles of Econometrics, 4th Edition Page 171Chapter 9: Regression with Time Series Data:
Stationary Variables
Or:
where
9CGeneralized
Least Squares Estimation
1 11 1 12 2 1y x x e Eq. 9C.5
2 21 1 11
2 212 1 1 1
1 1
1 1
y y x
x x e e
Eq. 9C.6
Principles of Econometrics, 4th Edition Page 172Chapter 9: Regression with Time Series Data:
Stationary Variables
To confirm that the variance of e*1 is the same as
that of the errors (v2, v3,…, vT), note that:
9CGeneralized
Least Squares Estimation
22 2 2
1 1 2var( ) (1 ) var( ) (1 )1
vve e