1 ˆ ˆ for interval prediction 95 a have we ˆ ˆ

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1 0 0 ˆ ˆ for interval prediction 95% a have we ˆ ˆ given that so , ~ ˆ ˆ e se t y y y y e t e se e k n ± - = - - Usually the estimate of s 2 is much larger than the variance of the prediction, thus This prediction interval will be a lot wider than the simple confidence interval for the prediction
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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 23 Residual Analysis Information can be obtained from looking at the residuals (i.e. predicted vs. observed) Example: Regress price of cars on characteristics – big negative residuals indicate a good deal Example: Regress average earnings for students from a school on student characteristics – big positive residuals indicate greatest value-added
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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 24 Predicting y in a log model Simple exponentiation of the predicted ln( y ) will underestimate the expected value of y Instead need to scale this up by an estimate of the expected value of exp( u ) ( 29 ( 29 ( 29 ( 29 y y N u u E n ˆ l exp 2 ˆ exp ˆ follows as y predict can case In this , 0 ~ if ) 2 exp( ) exp( 2 2 σ σ σ = =
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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 25 Predicting y in a log model If u is not normal, E(exp( u )) must be estimated using an auxiliary regression Create the exponentiation of the predicted ln( y ), and regress y on it with no intercept The coefficient on this variable is the estimate of E(exp( u )) that can be used to scale up the exponentiation of the predicted ln( y ) to obtain the predicted y
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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 26 Comparing log and level models A by-product of the previous procedure is a method to compare a model in logs with one in levels. Take the fitted values from the auxiliary regression, and find the sample correlation between this and y Compare the R 2 from the levels regression with this correlation squared
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