Consider the following expenditure share equation where WFOOD is the proportion of household total expenditure allocated to food, TOTEXP is total weekly household expenditure in British pounds ( \(£)\), and \(N K\) is the number of children in the household. Conditions MR1-MR5 are assumed to hold. We will be using data from the file london5.

a. For a household with the median total expenditure of \(£ 90\), show that the change in \(E(\) WFOOD \(\mid\) TOTEXP,\(N K)\) from adding an extra child is \(\beta_{3}+\beta_{4} \ln (90)\).

b. For a household with two children, show that the change in \(E(W F O O D \mid T O T E X P, N K)\) from an increase in total expenditure from \(£ 80\) /week to \(£ 120\) /week is \(\beta_{2} \ln (1.5)+2 \beta_{4} \ln (1.5)\).

c. For a household with two children and total expenditure of \(£ 90 /\) week, show that

d. Consider the following three statements:

A. \(\beta_{3}+\beta_{4} \ln (90)=0.025\)

B. \(\beta_{2} \ln (1.5)+2 \beta_{4} \ln (1.5)=-0.04\)

C. \(\beta_{1}+\beta_{2} \ln (90)+2 \beta_{3}+2 \beta_{4} \ln (90)=0.37\) 

We will be concerned with using \(F\) and \(\chi^{2}\) tests to test the following three null hypotheses: \(H_{0}^{(1)}: \mathrm{A}\) is true; \(H_{0}^{(2)}: \mathrm{A}\) and \(\mathrm{B}\) are true; \(H_{0}^{(3)}: \mathrm{A}\) and \(\mathrm{B}\) and \(\mathrm{C}\) are true. The alternative hypothesis in each case is that \(H_{0}^{(i)}\) is not true.
What are the relationships between the \(F\) and \(\chi^{2}\) tests for each of the three hypotheses? For \(H_{0}^{(1)}\), what is the relationship between the \(t\) and \(F\) tests?

e. Find the \(p\)-values for the \(F\) and \(\chi^{2}\) tests for \(H_{0}^{(1)}, H_{0}^{(2)}\), and \(H_{0}^{(3)}\), first using the first 100 observations in london5, then using the first 400 observations, and then using all 850 observations.

f. Comment on how changing the sample size, and adding more hypotheses, affects the results of the tests. Are there any dramatic differences between the \(F\)-test outcomes and the \(\chi^{2}\)-test outcomes?

Transcribed Image Text:

WFOOD = B + Bln(TOTEXP) + 3NK + B [NK In(TOTEXP)] +e