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Econ 399 Chapter12b - inparticular, (1)SERIAL CORRELATION...

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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 + = - ρ -assuming from here on in that everything is  conditional on X, we can calculate variance as: (12.27) 1 ) ( 2 2 ρ σ - = e t u Var
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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 0 1 1 1 0 1 1 0 + + - = - + - + - = - + + = - - - t t t t t t t t t t t t e x y u u x x y y u x y β β ρ ρ ρ β β ρ ρ β β This provides us with new error terms that are  uncorrelated -Note that ytilde and xtilde are called QUASI- DIFFERENCED DATA
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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 term’s  errors are the same as other terms, we set 1 1 1 0 2 1 1 2 1 2 1 0 2 1 2 ~ ~ ) 1 ( ~ ) 1 ( ) 1 ( ) 1 ( ) 1 ( u x y u x y + + - = - + - + - = - β β ρ ρ ρ β β ρ ρ -note that our first term’s quasi-differenced data is  calculated differently than all other terms -note also that this is another example of GLS  estimation
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12.3 Correcting for Serial Correlation w/  Strictly Exogenous Regressors Given multiple explanatory variables, we have: 1 1 11 1 0 2 1 1 1 0 1 , 1 1 , 1 1 1 0 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 + + + + - = + + + + - = - + - + + - + - = - - - - - β β β ρ β β β ρ ρ ρ β ρ β β ρ ρ -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
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12.3 Correcting for Serial Correlation w/  Strictly Exogenous Regressors Unfortunately,  ρ  is rarely know, but it can be  estimated from the formula: t t t e u u + = - 1 ˆ ˆ ρ We then use  ρ hat to estimate: ) ˆ 1 ( ~ 2 for t ) ˆ 1 ( ~ 2) (t ~ ... ~ ~ 2 10 0 1 1 0 0 ρ ρ β β β - = - = + + + + = x x where e x x x y t t tk k t t t Note that in this FEASIBLE GLS (FGLS), the  estimation error in  ρ hat does not affect FGLS’s  estimator’s asymptotic distribution
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Feasible GLS Estimation of the  AR(1) Model 1) Regress y on all x’s to obtain residuals uhat 2) Regress uhat t  on uhat t-1
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