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
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
12.1 Serial Correlation and se
Assume that our error terms follow AR(1) SERIAL
CORRELATION :
ut  ut 1  et
(12.1)
-where et are uncorrelated random variables with
mean zero and constant variance
-assume that |ρ|<1 (stability condition)
-if we assume the average of x is zero, in the
model with one independent variable, OLS
estimates:
ˆ1  1
xu


t t
SSTx
(12.3)
12.1 Serial Correlation and se
Computing the variance of OLS requires us to
take into account serial correlation in ut:
1 2
ˆ
Var ( 1 )  (
) Var ( x t ut )
SSTx
n 1 n t
1
Var ( ˆ1 )  (
) 2 ( xt2Var (ut )  2 xt xt  j E (ut ut  j ))
SSTx
t 1 j 1
Var ( ˆ1 ) 

2
SSTx
 2(

n 1 n t
2
SSTx
)  xt xt  j
t
2
t 1 j 1
-Evidently this is much different than typical OLS
variance unless ρ=0 (no serial correlation)
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
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:
E (et | ut 1 , ut 2,... )  0
(12.10)
Var(e t | ut 1 )  Var(e t )  
2
e
(12.11)
12.2 Testing for Serial Correlation
-We adopt a null hypothesis for no serial
correlation and set up an AR(1) model:
H0 :   0
ut  ρut-1  e t
(12.12)
(12.13)
-We could estimate (12.13) and test if ρhat is
zero, but unfortunately we don’t have the true
errors
-luckily, due to the strict exogeneity assumption,
the true errors can be replaced with OLS
residuals
Testing for AR(1) Serial Correlation
with Strictly Exogenous Regressors:
1) Regress y on all x’s to obtain residuals uhat
2) Regress uhatt on uhatt-1 and obtain OLS
estimates of ρhat
3) Conduct a t-test (typically at the 5% level) for
the hypotheses:
Ho: ρ=0 (no serial correlation)
Ha: ρ≠0 (AR(1) serial correlation)
Remember to report p-value
12.2 Testing for Serial Correlation
-If one has a large sample size, serial correlation
could be found with a small ρhat.
-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
12.2 Durbin-Watson Test
Another classic test for AR(1) serial correlation is
the Durbin-Watson test. The DurbinWatston (DW) statistic is calculated from
OLS residuals:
(uˆ  uˆ

DW 
 uˆ
t 1
2
t
)
2
(12.15)
t
-It can be shown that the DW statistic is linked to
the previous test for AR(1) serial correlation:
DW  2(1 - ˆ ) (12.16)
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:ρ>0 (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
12.2 DW Test
-The DW test produces two sets of critical values,
dU (for upper), and dL (for lower)
-if DW<dL, reject H0
-if DW>dU, 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
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
-Durbin’s h statistic is one alternative, but cannot
always be calculated
-the following test works for both strictly
exogenous and not strictly exogenous
regressors
Testing for AR(1) Serial Correlation
without Strictly Exogenous Regressors:
1) Regress y on all x’s to obtain residuals uhat
2) Regress uhatt on uhatt-1 and all xt variables
obtain OLS estimates of ρhat (coefficient of
uhatt-1)
3) Conduct a t-test (typically at the 5% level) for
the hypotheses:
Ho: ρ=0 (no serial correlation)
Ha: ρ≠0 (AR(1) serial correlation)
Remember to report p-value
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
12.2 Higher Order Serial Correlation
Assume that our error terms follow AR(2) SERIAL
CORRELATION :
ut  1ut 1   2ut 2  et
-here we test for second order serial correlation,
or:
H 0 : 1  0,  2  0 (12.19)
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:
Testing for AR(q) Serial Correlation
1) Regress y on all x’s to obtain residuals uhat
2) Regress uhatt on uhatt-1, uhatt-2,…, uhatt-q and
all xt variables obtain OLS estimates of ρhat
(coefficient of uhatt-1)
3) Conduct an F-test (typically at the 5% level)
for the hypotheses:
Ho: ρ1= ρ2=…= ρq=0 (no serial correlation)
Ha: Not H0 (AR(1) serial correlation)
Remember to report p-values
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:
Var(u t | X t , u t -1 ,..., u t -q )  
2
(12.23)
-but if heteroskedasticity exists in the second
equation a heteroskedastic-robust
transformation can be made as described in
Chapter 8
12.2 Seasonal forms of
Serial Correlation
Seasonal data (ie: quarterly or monthly), might
exhibit seasonal forms of serial correlation:
ut   4ut 4  et
ut  12ut 12  et
-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