e, then 2ln()( ,)ywN , and wis said to have a log-normal distribution with its mean and variance specified as above. These results relate to the log-linear regression model. Let’sconsider 221212ln, and (0,),i.e.,ln( )(,)yxeeNyNx, because 12(ln)Eyx. Then 2212( )expexp22iE yx. If we want to predict E(y), we should use the corrected predictor: 222121212212ˆˆˆˆˆ( )expexpexpexp222ˆ,and are obtained from the regression output.iiiE yxbb xbb xwhere b bInstead, its natural predictor is: 12( )expinE ybb xThe natural predictor tends to systematically under-predict the value of y in a log-linear model since 2ˆ( )( ) exp(/ 2)nE yE yand 2ˆexp/ 2is always greater than zero. Example: The following income model is fitted using a survey data of 300 workers. 22ln(income)1.60940.0904ˆN=300 R0.412,0.2773EDUCFind the predicted income for a worker with 12 years of education. Solution: The natural predictor is: 12(income)expexp(1.60940.0904 12)exp(2.6943)14.7958.inEbb xThe corrected predictor is: 2ˆ2/212ˆ( )expexp/ 214.7985 16.9964iE ybb xe. We predict that the income for a worker with 12 years of education will earn £14.80 per hour if using the natural predictor, and £17.00 if we use the corrected predictor. We prefer to use the corrected predictor.

12 5. Examining When specifying a regression model, we may inadvertently choose an inadequate or incorrect functional form. This includes examining the regression results to check for homoskedasticity and serial correlation; and examining residual plots. Figure 4 Randomly scattered residuals Using plot to check for the homoskedasticity for the linear-log food expenditure model yields the following plot: Figure 5 Residual plot The well-defined quadratic pattern in the least squares residuals indicates that something is wrong with the linear model specification. The linear model has ‘‘missed’’ a curvilinear aspect of the relationshipFigure 6 Least squares residuals from a linear equation fit to quadratic data

13 Hypothesis tests and interval estimates for the coefficients rely on the assumption that the errors, and hence the dependent variable y, are normally distributed. Are they normally distributed? We can check the distribution of the residuals using a histogram. Merely checking a histogram is not a formal test, however. Many formal tests are available; a popular one is the Jarque–Bera test for normality.

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- Fall '19