Econ 399 Chapter12a - Serial Correlation exists when errors...

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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
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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
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12.1 Serial Correlation and se Assume that our error terms follow AR(1) SERIAL  CORRELATION : (12.1) 1 t t t e u u + = - ρ -where e t  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: (12.3) x ˆ t 1 1 x t SST u + = β β
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12.1 Serial Correlation and se Computing the variance of OLS requires us to  take into account serial correlation in u t : ∑∑ ∑∑ - = - = + - = - = + + + = + = = 1 1 1 2 2 2 1 1 1 1 2 2 1 t 2 1 ) ( 2 ) ˆ ( )) ( 2 ) ( ( ) 1 ( ) ˆ ( ) x ( ) 1 ( ) ˆ ( n t t n j j t t t x x n t t n j j t t j t t t t x t x x x SST SST Var u u E x x u Var x SST Var u Var SST Var ρ σ σ β β β -Evidently this is much different than typical OLS  variance unless  ρ =0 (no serial correlation)
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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
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12.2 Testing for Serial Correlation
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