計(jì)量經(jīng)濟(jì)學(xué)導(dǎo)論ch12課件

1、計(jì)量經(jīng)濟(jì)學(xué)導(dǎo)論ch12課件計(jì)量經(jīng)濟(jì)學(xué)導(dǎo)論ch12課件Testing for serial correlationTesting for AR(1) serial correlation with strictly exog. regressorsExample: Static Phillips curve (see above)Test in AR(1) model for serial correlation (with an i.i.d. series et)Replace true unobserved errors by estimated residualsReject null h

2、ypothesis of no serial correlationAnalyzing Time Series:Serial Correl. and Heterosced.Testing for serial correlationDurbin-Watson test under classical assumptionsUnder assumptions TS.1 TS.6, the Durbin-Watson test is an exact test (whereas the previous t-test is only valid asymptotically).Example: S

3、tatic Phillips curve (see above)vs.Reject if , Accept“ if Reject null hypothesis of no serial correlationUnfortunately, the Durbin-Watson test works with a lower and and an upper bound for the critical value. In the area between the bounds the test result is inconclusive.Analyzing Time Series:Serial

4、 Correl. and Heterosced.Durbin-Watson test under classTesting for AR(1) serial correlation with general regressorsThe t-test for autocorrelation can be easily generalized to allow for the possibility that the explanatory variables are not strictly exogenous:The test may be carried out in a heterosce

5、dasticity robust wayGeneral Breusch-Godfrey test for AR(q) serial correlationThe test now allows for the possibility that the strict exogeneity assumption is violated.Test forTestAnalyzing Time Series:Serial Correl. and Heterosced.Testing for AR(1) serial correCorrecting for serial correlation with

6、strictly exog. regressorsUnder the assumption of AR(1) errors, one can transform the model so that it satisfies all GM-assumptions. For this model, OLS is BLUE.Problem: The AR(1)-coefficient is not known and has to be estimatedSimple case of regression with only one explana-tory variable. The genera

7、l case works analogously. Lag and multiply byThe transformed error satis-fies the GM-assumptions.Analyzing Time Series:Serial Correl. and Heterosced.Correcting for serial correlatCorrecting for serial correlation (cont.)Replacing the unknown by leads to a FGLS-estimatorThere are two variants:Cochran

8、e-Orcutt estimation omits the first observationPrais-Winsten estimation adds a transformed first observation In smaller samples, Prais-Winsten estimation should be more efficientComparing OLS and FGLS with autocorrelationFor consistency of FGLS more than TS.3 is needed (e.g. TS.3) because the transf

9、ormed regressors include variables from different periodsIf OLS and FGLS differ dramatically this might indicate violation of TS.3Analyzing Time Series:Serial Correl. and Heterosced.Correcting for serial correlatSerial correlation-robust inference after OLSIn the presence of serial correlation, OLS

10、standard errors overstate statistical significance because there is less independent variationOne can compute serial correlation-robust std. errors after OLSThis is useful because FGLS requires strict exogeneity and assumes a very specific form of serial correlation (AR(1) or, generally, AR(q)Serial

11、 correlation-robust standard errors:Serial correlation-robust F- and t-tests are also availableThe usual OLS standard errors are normalized and then inflated“ by a correction factor. Analyzing Time Series:Serial Correl. and Heterosced.Serial correlation-robust infeCorrection factor for serial correl

12、ation (Newey-West formula)The integer g controls how much serial correlation is allowed:This term is the product of the residuals and the residuals of a regression of xtj on all other explanatory variablesg=2:g=3:The weight of higher order autocorrelations is decliningAnalyzing Time Series:Serial Co

13、rrel. and Heterosced.Correction factor for serial cDiscussion of serial correlation-robust standard errorsThe formulas are also robust to heteroscedasticity; they are therefore called heteroscedasticity and autocorrelation consistent“ (=HAC)For the integer g, values such as g=2 or g=3 are normally s

14、ufficient (there are more involved rules of thumb for how to choose g)Serial correlation-robust standard errors are only valid asymptotically; they may be severely biased if the sample size is not large enoughThe bias is the higher the more autocorrelation there is; if the series are highly correlat

15、ed, it might be a good idea to difference them first Serial correlation-robust errors should be used if there is serial corr. and strict exogeneity fails (e.g. in the presence of lagged dep. var.)Analyzing Time Series:Serial Correl. and Heterosced.Discussion of serial correlatiHeteroscedasticity in

16、time series regressionsHeteroscedasticity usually receives less attention than serial correlationHeteroscedasticity-robust standard errors also work for time seriesHeteroscedasticity is automatically corrected for if one uses the serial correlation-robust formulas for standard errors and test statis

17、ticsTesting for heteroscedasticityThe usual heteroscedasticity tests assume absence of serial correlationBefore testing for heteroscedasticity one should therefore test for serial correlation first, using a heteroscedasticity-robust test if necessaryAfter ser. corr. has been corrected for, test for

18、heteroscedasticity Analyzing Time Series:Serial Correl. and Heterosced.Heteroscedasticity in time serExample: Serial correlation and homoscedasticity in the EMHTest for serial correlation:No evidence for serial correlation Test for heteroscedasticity:Strong evidence for heteroscedasticityTest equati

19、on for the EMHNote: Volatility is higher if returns are low Analyzing Time Series:Serial Correl. and Heterosced.Example: Serial correlation anAutoregressive Conditional Heteroscedasticity (ARCH)Consequences of ARCH in static and distributed lag modelsIf there are no lagged dependent variables among

20、the regressors, i.e. in static or distributed lag models, OLS remains BLUE under TS.1-TS.5Also, OLS is consistent etc. for this case under assumptions TS.1-TS.5As explained, in this case, assumption TS.4 still holds under ARCHEven if there is no heteroscedasticity in the usual sense (the error varia

21、nce depends on the explanatory variables), there may be heteroscedasticity in the sense that the variance depends on how volatile the time series was in previous periods:ARCH(1) modelAnalyzing Time Series:Serial Correl. and Heterosced.Autoregressive Conditional HetConsequences of ARCH in dynamic mod

22、elsIn dynamic models, i.e. models including lagged dependent variables, the homoscedasticity assumption TS.4 will necessarily be violated:This means the error variance indirectly depends on explanat. variablesIn this case, heteroscedasticity-robust standard error and test statistics should be computed, or a FGLS/WLS-procedure should be appliedUsing a FGLS/WLS-procedure will also increase efficiencybecauseAnalyzing Time Series:Serial Correl. and Heterosced.Consequences of ARCH in dynamiExample: Testing for ARCH-ef