# 12 Autocorrelation - PowerPoint PPT Presentation

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## 12 Autocorrelation

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### Serial Correlation exists when errors are correlated across periods-One source of serial correlation is misspecification of the model (although correctly specified ... – PowerPoint PPT presentation

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Title: 12 Autocorrelation

1
12 Autocorrelation
• Serial Correlation exists when errors are
correlated across periods
• -One source of serial correlation is
misspecification of the model (although correctly
specified models can also have autocorrelation)
• -Serial correlation does not make OLS biased or
inconsistent
• -Serial correlation does ruin OLS standard errors
and all significance tests
• -Serial correlation must therefore be corrected
for any regression to give valid information

2
12. Serial Correlation and Heteroskedasticity in
Time Series Regressions
• 12.1 Properties of OLS with Serial Correlation
• 12.2 Testing for Serial Correlation
• 12.3 Correcting for Serial Correlation with
Strictly Exogenous Regressors
• 12.5 Serial Correlation-Robust Inference after
OLS
• 12.6 Het in Time Series Regressions

3
12.1 Serial Correlation and se
• Assume that our error terms follow AR(1) SERIAL
CORRELATION

-where et are uncorrelated random variables with
mean zero and constant variance -assume that
?lt1 (stability condition) -if we assume the
average of x is zero, in the model with one
independent variable, OLS estimates
4
12.1 Serial Correlation and se
• Computing the variance of OLS requires us to take
into account serial correlation in ut

-Evidently this is much different than typical
OLS variance unless ?0 (no serial
correlation)
5
12.1 Serial Correlation Notes
• -Typically, the usual OLS formula for variance
underestimates the true variance in the presence
of serial correlation
• -this variance bias leads to invalid t and F
statistics
• -note that if the data is stationary and weakly
dependent, R2 and adjusted R2 are still valid
measures of goodness of fit
• -the argument is the same as for cross sectional
data with heteroskedasticity

6
12.2 Testing for Serial Correlation
• -We first test for serial correlation when the
regressors are strictly exogenous (ut is
uncorrelated with all regressors over time)
• -the simplest and most popular serial correlation
to test for is the AR(1) model
• -in order to the strict exogeneity assumption, we
need to assume that

7
12.2 Testing for Serial Correlation
• -We adopt a null hypothesis for no serial
correlation and set up an AR(1) model

-We could estimate (12.13) and test if ?hat is
zero, but unfortunately we dont have the true
errors -luckily, due to the strict exogeneity
assumption, the true errors can be replaced with
OLS residuals
8
Testing for AR(1) Serial Correlation with
Strictly Exogenous Regressors
• Regress y on all xs to obtain residuals uhat
• Regress uhatt on uhatt-1 and obtain OLS estimates
of ?hat
• Conduct a t-test (typically at the 5 level) for
the hypotheses
• Ho ?0 (no serial correlation)
• Ha ??0 (AR(1) serial correlation)
• Remember to report p-value

9
12.2 Testing for Serial Correlation
• -If one has a large sample size, serial
correlation could be found with a small ?hat.
• -in this case typical OLS inference will not be
far off
• -note that this test can detect ANY serial
correlation that causes adjacent error terms to
be correlated
• -correlation between ut and ut-4 would not be
picked up however
• -if the AR(1) formula suffers from HET,
Heteroskedastic-robust t statistics are used

10
12.2 Durbin-Watson Test
• Another classic test for AR(1) serial correlation
is the Durbin-Watson test. The Durbin-Watston
(DW) statistic is calculated from OLS residuals

-It can be shown that the DW statistic is linked
to the previous test for AR(1) serial correlation
11
12.2 DW Test
• Even with moderate sample sizes, (12.16) is
relatively close
• -the DW test does, however, depend on ALL CLM
assumptions
• -typically the DW test is computed for the
alternative hypothesis Ha?gt0 (since rho is
usually positive and rarely negative)
• -from (12.16) the null hypothesis is rejected if
DW is significantly less than 2
• -unfortunately the null distribution is difficult
to determine for DW

12
12.2 DW Test
• -The DW test produces two sets of critical
values, dU (for upper), and dL (for lower)
• -if DWltdL, reject H0
• -if DWgtdU, do not reject Ho
• -otherwise the tests is inconclusive
• -the DW test has an inconclusive region and
requires all CLM assumptions
• -the t test can be used asymptotically and can be
corrected for heteroskedasticity
• -Therefore t tests are generally preferred to DW
tests

13
12.2 Testing without Strictly Exogenous Regressors
• -it is often the case that explanatory variables
are NOT strictly exogenous
• -one or more xtj are correlated with ut-1
• -ie when yt-1 is an explanatory variable
• -in these cases typical t or DW tests are invalid
• -Durbins h statistic is one alternative, but
cannot always be calculated
• -the following test works for both strictly
exogenous and not strictly exogenous regressors

14
Testing for AR(1) Serial Correlation without
Strictly Exogenous Regressors
• Regress y on all xs to obtain residuals uhat
• Regress uhatt on uhatt-1 and all xt variables
obtain OLS estimates of ?hat (coefficient of
uhatt-1)
• Conduct a t-test (typically at the 5 level) for
the hypotheses
• Ho ?0 (no serial correlation)
• Ha ??0 (AR(1) serial correlation)
• Remember to report p-value

15
12.2 Testing without Strictly Exogenous Regressors
• -the different in this testing sequence is uhatt
is regressed on
• 1) uhatt-1
• 2) all independent variables
• -a heteroskedasticity-robust t statistic can also
be used if the above regression suffers from
heteroskedasticity

16
12.2 Higher Order Serial Correlation
• Assume that our error terms follow AR(2) SERIAL
CORRELATION

-here we test for second order serial
correlation, or
As before, we run a typical OLS regression for
residuals, and then regress uhatt on all
explanatory (x) variables, uhatt-1 and
uhatt-2 -an F test is then done on the joint
significance of the coefficients of uhatt-1 and
uhatt-2 -we can test for higher order serial
correlation
17
Testing for AR(q) Serial Correlation
• Regress y on all xs to obtain residuals uhat
• Regress uhatt on uhatt-1, uhatt-2,, uhatt-q and
all xt variables obtain OLS estimates of ?hat
(coefficient of uhatt-1)
• Conduct an F-test (typically at the 5 level) for
the hypotheses
• Ho ?1 ?2 ?q0 (no serial correlation)
• Ha Not H0 (AR(1) serial correlation)
• Remember to report p-values

18
12.2 Testing for Higher Order Serial Correlation
• -if xtj is strictly exogenous, it can be removed
from the second regression
• -this test requires the homoskedasticity
assumption

-but if heteroskedasticity exists in the second
equation a heteroskedastic-robust transformation
can be made as described in Chapter 8
19
12.2 Seasonal forms of Serial Correlation
• Seasonal data (ie quarterly or monthly), might
exhibit seasonal forms of serial correlation

-our test is similar to that for AR(1) serial
correlation, only the second regression includes
ut-4 or the seasonal lagged variable instead of
ut-1