In the log-linear model (ln (y)=beta_{1}+beta_{2} x+e), the corrected predictor (hat{y}_{c}=exp left(b_{1}+b_{2} x ight) times exp left(hat{sigma}^{2}

In the log-linear model \(\ln (y)=\beta_{1}+\beta_{2} x+e\), the corrected predictor \(\hat{y}_{c}=\exp \left(b_{1}+b_{2} x\right) \times \exp \left(\hat{\sigma}^{2} / 2\right)\) is argued to have a lower mean squared error than the "normal" predictor \(\hat{y}_{n}=\exp \left(b_{1}+b_{2} x\right)\). The correction factor \(\exp \left(\hat{\sigma}^{2} / 2\right)\) depends on the regression errors having a normal distribution.

a. In exponential form, the log-linear model is \(y=\exp \left(\beta_{1}+\beta_{2} x\right) \exp (e)\). Assuming that the explanatory variable \(x\) and the random error \(e\) are statistically independent, find \(E(y)\).

b. Use the data file cps5_small for this exercise. [The data file cps 5 contains more observations and variables.] Estimate the model \(\ln (W A G E)=\beta_{1}+\beta_{2} E D U C+e\) using the first 1000 observations. Based on this regression, calculate the correction factor \(c=\exp \left(\hat{\sigma}^{2} / 2\right)\). What is this value?

c. Obtain the 1000 least squares residuals \(\hat{e}\) from the regression in (b). Calculate the correction factor \(d=\sum_{i=1}^{1000} \exp \left(\hat{e}_{i}\right) / 1000\). What is this value?

d. Using the estimates from part (b), obtain the predictions for observations 1001-1200, using \(\hat{y}_{n}=\exp \left(b_{1}+b_{2} x\right), \quad \hat{y}_{c}=c \hat{y}_{n}\), and \(\hat{y}_{d}=d \hat{y}_{n}\). Calculate the mean (average) squared forecast errors \(M S E_{n}=\sum_{i=1001}^{1200}\left(\hat{y}_{n i}-y_{i}\right)^{2} / 200, \quad M S E_{c}=\sum_{i=1001}^{1200}\left(\hat{y}_{c i}-y_{i}\right)^{2} / 200\), and \(M S E_{d}=\) \(\sum_{i=1001}^{1200}\left(\hat{y}_{d i}-y_{i}\right)^{2} / 200\). Based on this criterion, which predictor is best?