Econometrics
(NA1031)
Chap 9
Regression with Time Series Data:
Stationary Variables
1
Time series data
• Time-series data have a natural ordering according to time
• Time-series observations on a given economic unit,
observed over a number of time periods, are likely to be
correlated (called Autocorrelation or serial correlation)
• Stationary vs Nonstationary time series
• 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
2
FIGURE 9.1 The distributed lag effect
3
Dynamic relationships
• Distributed lag model (ex. Inflation and interest rate)
yt  f ( xt , xt 1 , xt 2 ,...)
• Autoregressive distributed lag (ARDL) models,
with ‘‘autoregressive’’ meaning a regression of yt on
its own lag or lags
yt  f ( yt 1 , xt , xt 1 , xt  2 )
• Model the continuing impact of change over several
periods via the error term
y  f (x )  e
t
t
t
e  g (e )
t
t 1
• In this case et is correlated with et – 1
• We say the errors are serially correlated or
autocorrelated (A shock (say an earthquake) affects
production level not only during the current period but
also in the future.)
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Finite distributed lags
yt    0 xt  1 xt 1  2 xt 2 
 q xt q  et
• Forecasting
yT 1    0 xT 1  1 xT  2 xT 1 
 q xT q 1  eT 1
• Policy analysis
• What is the effect of a change in x on y?
E ( yt ) E ( yt  s )

 s
xt  s
xt
5
Finite distributed lags
• 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
sustainedq increase after q or more periods have
elapsed  β s
s 0
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Finite distributed lags
• 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
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ASSUMPTIONS of the distributed lag model
yt    β 0 xt  β1 xt 1  β 2 xt  2 
 β q xt q  et , t  q  1,
,T
8
Serial correlation
• When cov(et, es) ≠ 0 for t ≠ s, that is
when a variable exhibits correlation over
time, we say it is autocorrelated or
serially correlated
• These terms are used interchangeably
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FIGURE 9.5 Scatter diagram for Gt and Gt-1
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Postive
Auto.
No
Auto.
et
0
et
0
et
Negative
Auto.
0
crosses line not enough
.
.
. . ..
.. . .
.
. .
...
. ..
.
..
t
crosses line randomly
. . .. . . . . .
. .
.
.
..
. .
..
.
.
.
.
.
.
crosses line too much
. .
.
. . .
.
.
.
. .
.
.
.
.
.
.
.
.t
. t
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Correlogram
• 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
• (See the text 348-9 for the formulas)
• The correlogram can also be used to check
whether the multiple regression assumption
cov(et, es) = 0 for t ≠ s is violated
12
FIGURE 9.6 Correlogram for G
13
Testing for autocorrelation
eˆ     x  eˆ  
t
1
2
t
t 1
t
• In the above regression we test whether H :   0
is true. (This is called Breusch–Godfrey test. We
could have more lags and use an F-test.)
• See the text for the Durbin-Watson test.
0
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Estimation with Serially Correlated Errors
• Three estimation procedures are
considered:
1. Least squares estimation
2. An estimation procedure that is relevant when
the errors are assumed to follow what is known
as a first-order autoregressive model
et  ρet 1  vt
3. A general estimation strategy for estimating
models with serially correlated errors (ARDL)
15
Estimation with Serially Correlated Errors
• 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 best
2. 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
• 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
16
Estimation with Serially Correlated Errors
• A model with an AR(1) error is:
yt  β1  β2 xt  et with et  ρet 1  vt
with -1 < ρ < 1
• For the vt, we have:
E  vt   0 var  vt    v2
cov  vt , vt 1   0 for t  s
• Substitution and manipulation gives (see p. 361):
yt  β1 1  ρ   β2 xt  ρyt 1  ρβ2 xt 1  vt
• This model (and hence ρ) can be estimated using
nonlinear least squares or GLS
17
A more general model
• It can be shown that:
yt  β1 1  ρ   β2 xt  ρyt 1  ρβ2 xt 1  vt
is a restricted version of:
yt  δ  θ1 yt 1  δ0 xt  δ1 xt 1  vt
This in turn is a member of a general class of
autoregressive distributed lag (ARDL) models
• Given assumptions hold one can estimate the general
ADRL model and test whether the AR(1) is a
reasonable presentation or the general model.
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Estimation with Serially Correlated Errors
• We have described three ways of
overcoming the effect of serially
correlated errors:
1. Estimate the model using least squares with
HAC standard errors
2. 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
19
Stata
• Start Stata
mkdir C:\PE
cd C:\PE
copy http://users.du.se/~rem/chap09_15.do
chap09_15.do
doedit chap09_15.do
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Assignment
• Exercise 9.15 page 388 in the textbook.
21