12 Autocorrelation Serial Correlation exists when errors are

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 |ρ|<1 (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 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, R 2 and adjusted R 2 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 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 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: 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, d. U (for upper), and d. L (for lower) -if DW<d. L, reject H 0 -if DW>d. U, 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 : -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 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 H 0 (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