Econ 399 Chapter12b - 12.3 Correcting for Serial...

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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 exogenous-in 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 uncorrelated-Note 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 undefined-to 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 quasi-differenced data is calculated differently than all other terms-note 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 OLS-note 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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Econ 399 Chapter12b - 12.3 Correcting for Serial...

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