In this exercise we explore the relationship between total household expenditures and expenditures on telephone services. Use the data file malawi_small (malawi has more observations).

a. Using observations for which PTELEPHONE \(>0\), create the variable \(\ln (\) TELEPHONE \()=\) \(\ln (\) PTELEPHONE \(\times\) TOTEXP \()\). Plot \(\ln (\) TELEPHONE \()\) versus \(\ln (\) TOTEXP \()\) and include the least squares fitted line.

b. Based on the OLS regression of \(\ln (\) TELEPHONE) on \(\ln (T O T E X P)\) what is the estimated elasticity of telephone expenditures with respect to total expenditure. Compute a 95\% interval estimate for the elasticity. Based on the estimates, would you classify telephone services as a necessity or a luxury?

c. Test for the presence of heteroskedasticity in the regression in part (b). What do you conclude?

d. Estimate the model PTELEPHONE \({ }_{i}=\beta_{1}+\beta_{2} \ln \left(\right.\) TOTEXP \(\left._{i}\right)+e_{i}\) by OLS. Test the null hypothesis that \(\beta_{2} \leq 0\) against \(\beta_{2}>0\) using the \(5 \%\) level of significance.

e. Calculate the elasticity of telephone expenditures with respect to total expenditure at the sample median of total expenditures. The expression for an elasticity in such a model was derived in Exercise 4.12. Use your software to compute a 95\% interval estimate for the elasticity. Compare the estimated elasticity to that in (b).

f. Test for the presence of heteroskedasticity in the regression in part (d). What do you conclude?

g. Estimate the model in (d) using FGLS with \(\ln \left(\right.\) TOTEXP \(\left._{i}\right)\) being the variable that may be associated with the heteroskedasticity. Using the conventional FGLS standard errors, test the null hypothesis that \(\beta_{2} \leq 0\) against \(\beta_{2}>0\) using the \(5 \%\) level of significance.

h. Repeat part (g) but using FGLS with robust standard errors.

i. Summarize your findings about the elasticity of telephone services expenditure with respect to total expenditure.