Consider a model for household expenditure as a function of household income using the 2013 data from the Consumer Expenditure Survey, cex5_small. The data file cex5 contains more observations. Our attention is restricted to three-person households, consisting of a husband, a wife, plus one other. In this exercise, we examine expenditures on a staple item, food. In this extended example, you are asked to compare the linear, log-log, and linear-log specifications.

a. Calculate summary statistics for the variables: FOOD and INCOME. Report for each the sample mean, median, minimum, maximum, and standard deviation. Construct histograms for both variables. Locate the variable mean and median on each histogram. Are the histograms symmetrical and "bell-shaped" curves? Is the sample mean larger than the median, or vice versa? Carry out the Jarque-Bera test for the normality of each variable.

b. Estimate the linear relationship FOOD \(=\beta_{1}+\beta_{2} I N C O M E+e\). Create a scatter plot FOOD versus INCOME and include the fitted least squares line. Construct a \(95 \%\) interval estimate for \(\beta_{2}\). Have we estimated the effect of changing income on average FOOD relatively precisely, or not?

c. Obtain the least squares residuals from the regression in (b) and plot them against INCOME. Do you observe any patterns? Construct a residual histogram and carry out the Jarque-Bera test for normality. Is it more important for the variables FOOD and INCOME to be normally distributed, or that the random error \(e\) be normally distributed? Explain your reasoning.

d. Calculate both a point estimate and a \(95 \%\) interval estimate of the elasticity of food expenditure with respect to income at \(I N C O M E=19,65\), and 160 , and the corresponding points on the fitted line, which you may treat as not random. Are the estimated elasticities similar or dissimilar? Do the interval estimates overlap or not? As INCOME increases should the income elasticity for food increase or decrease, based on Economics principles?

e. For expenditures on food, estimate the \(\log\)-log relationship \(\ln (\) FOOD \()=\gamma_{1}+\gamma_{2} \ln (\) INCOME \()+e\). Create a scatter plot for \(\ln (F O O D)\) versus \(\ln (I N C O M E)\) and include the fitted least squares line. Compare this to the plot in (b). Is the relationship more or less well-defined for the log-log model relative to the linear specification? Calculate the generalized \(R^{2}\) for the log-log model and compare it to the \(R^{2}\) from the linear model. Which of the models seems to fit the data better?

f. Construct a point and \(95 \%\) interval estimate of the elasticity for the log-log model. Is the elasticity of food expenditure from the log-log model similar to that in part (d), or dissimilar? Provide statistical evidence for your claim.

g. Obtain the least squares residuals from the log-log model and plot them against \(\ln (I N C O M E)\). Do you observe any patterns? Construct a residual histogram and carry out the Jarque-Bera test for normality. What do you conclude about the normality of the regression errors in this model?
h. For expenditures on food, estimate the linear-log relationship FOOD \(=\alpha_{1}+\alpha_{2} \ln (\) INCOME \()+e\). Create a scatter plot for FOOD versus \(\ln\) (INCOME) and include the fitted least squares line. Compare this to the plots in (b) and (e). Is this relationship more well-defined compared to the others? Compare the \(R^{2}\) values. Which of the models seems to fit the data better?

i. Construct a point and \(95 \%\) interval estimate of the elasticity for the linear-log model at INCOME = 19,65 , and 160 , and the corresponding points on the fitted line, which you may treat as not random. Is the elasticity of food expenditure similar to those from the other models, or dissimilar? Provide statistical evidence for your claim.

j. Obtain the least squares residuals from the linear-log model and plot them against \(\ln (\) INCOME). Do you observe any patterns? Construct a residual histogram and carry out the Jarque-Bera test for normality. What do you conclude about the normality of the regression errors in this model?

k. Based on this exercise, do you prefer the linear relationship model, or the log-log model or the linear-log model? Explain your reasoning.