Time series data y,=Bo+B Brit ◆2. Further issues Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Time Series Data yt = b0 + b1 xt1 + . . .+ bk xtk + ut 2. Further Issues
Testing for Ar(1) Serial Correlation Want to be able to test for whether the errors are serially correlated or not e Want to test the null that p=0 in u,=pu, I + t=2., n, where u, is the model error term and e is iid o With strictly exogenous regressors, the test Is very straightforward - - simply regress the residuals on lagged residuals and use a t-test Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Testing for AR(1) Serial Correlation Want to be able to test for whether the errors are serially correlated or not Want to test the null that r = 0 in ut = rut-1 + et , t =2,…, n, where ut is the model error term and et is iid With strictly exogenous regressors, the test is very straightforward – simply regress the residuals on lagged residuals and use a t-test
Testing for Ar(1) Serial Correlation(continued) o 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 Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 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
Testing for Ar(1) Serial Correlation(continued) e If the regressors are not strictly exogenous, then neither the t or dw test will work e Regress the residual (or y)on the lagged residual and all of the x's The inclusion of the x' s allows each x to be correlated with u,), so dont need assumption of strict exogeneity Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 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
Testing for Higher Order SC Q Can test for AR(q serial correlation in the same basic manner as AR(1) Just include g 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-qR2 using R2 from residual regression Can also test for seasonal forms Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 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