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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 - PowerPoint PPT Presentation
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12 AutocorrelationSerial 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 Regressions12.1 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 seAssume that our error terms follow AR(1)
SERIAL CORRELATION :(12.1) 1 ttt euu
-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:
(12.3) x
ˆ t11
x
t
SST
u
12.1 Serial Correlation and seComputing the variance of OLS requires us to
take into account serial correlation in ut:
1
1 12
22
1
1
1 1
221
t2
1
)(2 )ˆ(
))(2)(()1
( )ˆ(
)x()1
( )ˆ(
n
t
tn
jjtt
t
xx
n
t
tn
jjttjtttt
x
tx
xxSSTSST
Var
uuExxuVarxSST
Var
uVarSST
Var
-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:
(12.11) )Var(e)|Var(e
(12.10) 0),|(2
t1t
,...21
et
ttt
u
uueE
12.2 Testing for Serial Correlation
-We adopt a null hypothesis for no serial correlation and set up an AR(1) model:
(12.13) e
(12.12) 0 :
t1
0
t-t ρuu
H
-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 TestAnother classic test for AR(1) serial correlation is the
Durbin-Watson test. The Durbin-Watston (DW) statistic is calculated from OLS residuals:
(12.15) ˆ
)ˆˆ(2
21
t
tt
u
uuDW
-It can be shown that the DW statistic is linked to the previous test for AR(1) serial correlation:
(12.16) )ˆ-2(1W D
12.2 DW TestEven 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 uhat2) 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 :
2211 tttt euuu -here we test for second order serial
correlation, or: (12.19) 0 0, : 210 HAs 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 uhat2) 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:
(12.23) )u,...,u,X|ar(u 2q-t1-ttt V
-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:
1212
44
ttt
ttt
euu
euu
-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