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

-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

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)

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

12.2 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

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

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

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

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

12.2 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

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

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

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

-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

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

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

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