Lecture 11: TIME SERIES DATA: SERIAL CORRELATION AND HETEROSKEDASTICITY Professor Victor Aguirregabiria

1. 1

2. 2 OUTLINE Properties of OLS with Serial Correlation Testing for Serial Correlation Test for AR(1) with Strictly Exogenous Regressors Test for AR(1) with Lagged Endogenous Regressors Test for Higher Order Serial Correlation Correcting for Serial Correlation Serial Correlation-Robust Standard Errors Feasible GLS (Cochrane-Orcutt and Prais-Winsten) Heteroskedasticity Lecture 11: TIME SERIES DATA: SERIAL CORRELATION AND HETEROSKEDASTICITYProfessor Victor Aguirregabiria

3. 3 1. Properties of OLS with Serial Correlation If we have stationary data and E(ut | xt) = 0, the OLS is consistent and asymptotically normal, regardless the serial correlation of the error term. There are three main implications of serial correlation on the OLS properties. OLS is not efficient. OLS standard errors do not have the form that we have used so far. For some dynamic TS models (with regressors that depend on previous values of the error term), serial correlation of the error term implies that E(ut | xt) is not zero and OLS is inconsistent.

4. 4 Serial Correlation in Dynamic TS Models Consider the model yt = b0 + b1 xt + ut where: xt depends on lagged values of the error term. For instance, xt = yt-1 The error term is serially correlated. For instance, it is an AR(1): ut = r ut-1 + et, and et is iid. In this model, E(ut | xt) is not zero and OLS is inconsistent.

5. 5 Serial Correlation in Dynamic TS Models (cont.) However, the solution to this problem is simple. We can transform the model to obtain a dynamicallt complete model (iid error): yt = b0 + r yt-1 + b1 xt + (-rb1)xt-1 + et Now, the error term is iid and the OLS estimator is consistent.

6. 6 OLS standard errors with serial correlation Consider the model yt = b0 + b1 xt + ut where ut = r ut-1 + et, and et is iid.

7. 7 OLS standard errors with of serial correlation (cont) The first term is the variance of OLS when there is no heteroskedasticity or serial correlation. The second term is the bias in the standard error if we ignore the existence of serial correlation. If r>0 and the regressor is positively autocorrelated, the second term is positive, and we will underestimate the true s.e. if we ignore serial correlation in the error.

8. 8 2. Testing for Serial Correlation Test for AR(1) with Strictly Exogenous Regressors Suppose that our model does not contain lagged values of y. An suppose that ut = rut-1 + et, and et is iid. We want to test H0: r = 0. With strictly exogenous regressors (and stationary variables), the test is very straightforward � simply regress the residuals on lagged residuals and use a t-test.

9. 9 Testing for AR(1) Serial Correlation (continued) An alternative is the Durbin-Watson (DW) statistic, which is calculated by many packages If the DW statistic is around 2, then we can reject serial correlation, while if it is significantly < 2 we cannot reject Critical values are difficult to calculate, making the t test easier to work with

10. 10 Testing for AR(1) Serial Correlation (continued) If the regressors are not strictly exogenous, then neither the t or DW test will work Regress the residual (or y) on the lagged residual and all of the x�s The inclusion of the x�s allows each xtj to be correlated with ut-1, so don�t need assumption of strict exogeneity

11. 11 Testing for Higher Order S.C. Can test for AR(q) serial correlation in the same basic manner as AR(1) Just include q lags of the residuals in the regression and test for joint significance Can use F test or LM test, where the LM version is called a Breusch-Godfrey test and is (n-q)R2 using R2 from residual regression Can also test for seasonal forms

12. 12 3. Correcting for Serial Correlation Serial Correlation-Robust Standard Errors Though OLS is inefficient in the presence of serial correlation, we may want to use this estimator. It may be precise enough to get relevant empirical results. We do not have to make any assumption on the form of the autocorrelation. If we use OLS, we should estimate the correct standard errors robust of serial correlation and heteroskedasticity. Newey and West (1987) proposed an estimator of the OLS standard errors which is robust of serial correlation and heteroskedasticity.

13. 13 Serial Correlation-Robust Standard Errors Consider the variance of the OLS estimator of b1 Define rt as the residuals from the auxiliary regression of xt1 on xt2, � , xtk Then,

14. 14 Serial Correlation-Robust Standard Errors Newey-West estimator for this variance uses OLS residuals �t and takes into account that the stochastic process of the error term is weakly dependent and therefore the correlation is close to zero after several lags. Choose the constant g � say 1 to 3 for annual data, then

15. 15 Correcting for S.C. (continued) Consider that since yt = b0 + b1xt + ut , then yt-1 = b0 + b1xt-1 + ut-1 If you multiply the second equation by r, and subtract if from the first you get yt � r yt-1 = (1 � r)b0 + b1(xt � r xt-1) + et , since et = ut � r ut-1 This quasi-differencing results in a model without serial correlation

16. 16 Feasible GLS Estimation Problem with this method is that we don�t know r, so we need to get an estimate first Can just use the estimate obtained from regressing residuals on lagged residuals Depending on how we deal with the first observation, this is either called Cochrane-Orcutt or Prais-Winsten estimation

17. 17 Feasible GLS (continued) Often both Cochrane-Orcutt and Prais-Winsten are implemented iteratively This basic method can be extended to allow for higher order serial correlation, AR(q) Most statistical packages will automatically allow for estimation of AR models without having to do the quasi-differencing by hand