Econometrics-I-10

Econometrics-I-10 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 10. Prediction in the Classical Regression Model
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Forecasting Objective:  Forecast Distinction:  Ex post vs. Ex ante forecasting Ex post: RHS data are observed Ex ante: RHS data must be forecasted Prediction vs. model validation.   Within sample prediction “Hold out sample”
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Prediction Intervals Given x 0  predict y 0 . Two cases:         Estimate E[y| x 0 ] =  β′ x 0       Predict y 0  =  β′ x 0  +  ε 0    Obvious predictor,  b’x 0 + estimate of  ε 0 .   Forecast  ε 0  as 0, but allow for  variance.  Alternative:  When we predict y 0  with  b x 0 , what is the 'forecast error?'        Est.y 0   -  y 0  =  b x 0   -   β′ x 0   -   ε 0 , so the variance of the forecast error is               x 0 Var[ b -  β ] x 0   +  σ 2 How do we estimate this?  Form a confidence interval.  Two cases:       If  x 0  is a vector of constants, the variance is just  x 0  Var[ b x 0 .  Form  confidence interval as usual.       If  x 0  had to be estimated, then we use a random variable.  What is the  variance of the product?  (Ouch!)  One possibility:  Use bootstrapping. 
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Forecast Variance       Variance of the forecast error is              σ 2  +  x 0 ’ Var[ b ] x 0   =  σ 2  +  σ 2 [x 0 ’ ( X’X ) -1 x 0 ]       If the model contains a constant term, this is
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Econometrics-I-10 - Applied Econometrics William Greene...

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