Q 1 Food expenditure and its determinants have been extensively researched in social science. We intend to estimate the link between food spending and some of its factors in this exercise. The data file food.xlsx contains 200 observations for the following variables from a cross-section of households. Foodexp: Weekly expenditure on food (excluding restaurants) in dollars. Income: Weekly household income in dollars. Children: Number of dependent children living in the household. Retired: A binary (0/1) indicator of whether head of the household is retired {ret.-1). (a) Estimate the following two models and present your summary report for both models. What do you conclude about the fit of the two models? [4 marks] (1) Foodexp = a + a₂ Income + e; Foodexp = Y₁+ y₂log (Income) + u₂ (b) Now estimate the following regression model and answer all the following questions. Foodexp: = Bo + B₁ log(Income;) + B₂ Children; + Retired; + Gi Estimate the model using Grefl and provide the summary results (Gretl: Model →Ordinary Least Squares and then select the "Foodexp" as the dependent variable and "log(Income", "Children" and "Retired" as Regressors →OK.) (A summary results should include fitted equation with coefficients, standard error, t-statistic, p-value, sample size, F-statistic and R- squared). [4 marks] (c) Does the sign of the slope coefficients agree with your expectations? Comment. [4 marks] (d) Comment on the statistical significance of the estimates of the variables, Income, Children and Retired at 5% significance level. (No need to carry out hypothesis tests) [4 marks] (e) Test the overall validity of the regression model at the 5% significance level. State the hypotheses, restricted and unrestricted model, test statistics and its distribution when null hypothesis is true, critical value and your conclusion. [4 marks] (f) Construct 95% confidence interval for B₁, the slope of the log(Income) variable and interpret your results. [4 marks] (g) Based on your answer in part (f), without performing a hypothesis test, would you reject the hypothesis Ho: B₁ = 90, H₁: ₁90. Clearly states your conclusion? [4 marks] (h) Graph the residuals of least squares against log(Income) and describe the pattern. Do you find any evidence against the violation of any multiple regression assumptions? Explain. [4 marks] (i) Test for the existence of heteroscedasticity at the 5% significance level. Use the White's test (Squares only) and attach your Grefl results. Clearly states all steps in your test; null and alternative hypotheses, the auxiliary regression and the test statistic, critical value, your decision and the conclusion. [4 marks] (i) Based on your findings in part (i), is the model in part (b), valid? How would you rectify the problem? Attach your Gretl output. Compare your results with the output in part (b). Comment. [2 marks] (k) Now run the following regression model: Foodexp = a₁ + a₂ Income + a₂ Children, + a₂ Retired; + e¡ Compare your model that with part (a). Which model would And Why? [8 marks] you choose?