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Unformatted text preview: 12.3 Correcting for Serial Correlation w/ Strictly Exogenous Regressors The following autocorrelation correction requires all our regressors to be strictly exogenousin particular, we should have no lagged explanatory variables Assume that our error terms follow AR(1) SERIAL CORRELATION : (12.26) 1 t t t e u u + = assuming from here on in that everything is conditional on X, we can calculate variance as: (12.27) 1 ) ( 2 2  = e t u Var 12.3 Correcting for Serial Correlation w/ Strictly Exogenous Regressors If we consider a single explanatory variable, we can eliminate the correlation in the error term as follows: 2) (t ~ ) 1 ( ~ ) ( ) ( ) 1 ( 1 1 1 1 1 1 + + = + + = + + = t t t t t t t t t t t t e x y u u x x y y u x y This provides us with new error terms that are uncorrelatedNote that ytilde and xtilde are called QUASI DIFFERENCED DATA 12.3 Correcting for Serial Correlation w/ Strictly Exogenous Regressors Note that OLS is not BLUE yet as the initial y 1 is undefinedto make OLS blue and ensure the first terms errors are the same as other terms, we set 1 1 1 2 1 1 2 1 2 1 2 1 2 ~ ~ ) 1 ( ~ ) 1 ( ) 1 ( ) 1 ( ) 1 ( u x y u x y + + = + + = note that our first terms quasidifferenced data is calculated differently than all other termsnote also that this is another example of GLS estimation 12.3 Correcting for Serial Correlation w/ Strictly Exogenous Regressors Given multiple explanatory variables, we have: 1 1 11 1 2 1 1 1 1 , 1 1 , 1 1 1 1 ~ ~ ... ~ ) 1 ( ~ 2) (t ~ ... ~ ) 1 ( ~ ) ( ) ( ... ) ( ) 1 ( u x x y e x x y u u x x x x y y k k t tk k t t t t k t tk k t t t t + + + + = + + + + = + + + + = note that this GLS estimation is BLUE and will generally differ from OLSnote also that our t and F statistics are now valid and testing can be done 12.3 Correcting for Serial Correlation w/ Strictly Exogenous Regressors Unfortunately, is rarely know, but it can be estimated from the formula: t t t e u u + = 1 We then use hat to estimate: ) 1 ( ~ 2 for t ) 1 ( ~ 2) (t ~ ......
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 Spring '09
 Priemaza
 Econometrics

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