Consider the following general model of demand for new cars: NUMCARS_t = beta_1

Question

Consider the following general model of demand for new cars: NUMCARS_t = beta_1 + beta_2 PRICE_t + beta_3 lNCOME_t + beta_4 INTRATE, + beta_5 UNEMP_t + u_t where NUMCARS = number of new car sales per thousand population PRICE = new car price index INCOME = per-capita real disposable income (in dollars) INTRATE = interest rate UNEMP = unemployment rate Assume that the error terms are normally distributed. Using quarterly data for 10 years (40 observations), three alternative models were estimated (standard errors are in parentheses): Model A NUMCARS_t = - 7.4534 - 0.0714 PRICE_t +0.0032 INCOME_t -0.1537 INTRATE_t UNEMP_t With SSR = 23.5105, R^2 = 0.783 and R^2 = 0.758. Model B NUMCARS_t = - 10.5541 - 0.0794 PRICE_t +0.0036 INCOME_t -0.1467 INTRATE_t with SSR = 23.5502, R^2 = 0.782 and R^2 = 0.764. Model C NUMCARS_t = 15.2381 - 0.0249 PRICE_t - 0.2048 INTRATE_t, with SSR = 44.6591, R^2 == 0.587 and R^2 = 0.565. Test Model A for its overall significance at the 5% level. Formally specify the null and alternative hypotheses. Describe the null hypothesis that can be tested by comparing coefficients of determination in Models A and B. Perform the test at the 10% significance level using R^2_A and R^2_B. Can you suggest an alternative way of testing the same null hypothesis? In Model A, test the joint hypothesis beta_income = beta_unemp = 0 at 1% level using the results from Model C. What is the corresponding p-value? Which of the three models is the "best"? What criterion do you use and why? Consider the following general model of demand for new cars: NUMCARS_t = beta_1 + beta_2 PRICE_t + beta_3 lNCOME_t + beta_4 INTRATE, + beta_5 UNEMP_t + u_t where NUMCARS = number of new car sales per thousand population PRICE = new car price index INCOME = per-capita real disposable income (in dollars) INTRATE = interest rate UNEMP = unemployment rate Assume that the error terms are normally distributed. Using quarterly data for 10 years (40 observations), three alternative models were estimated (standard errors are in parentheses): Model A NUMCARS_t = - 7.4534 - 0.0714 PRICE_t +0.0032 INCOME_t -0.1537 INTRATE_t UNEMP_t With SSR = 23.5105, R^2 = 0.783 and R^2 = 0.758. Model B NUMCARS_t = - 10.5541 - 0.0794 PRICE_t +0.0036 INCOME_t -0.1467 INTRATE_t with SSR = 23.5502, R^2 = 0.782 and R^2 = 0.764. Model C NUMCARS_t = 15.2381 - 0.0249 PRICE_t - 0.2048 INTRATE_t, with SSR = 44.6591, R^2 == 0.587 and R^2 = 0.565. Test Model A for its overall significance at the 5% level. Formally specify the null and alternative hypotheses.

Explanation / Answer

(a) For overall significance test, we use F -test:

H0: regression is insignificant

Ha: regression is significant

F-stat= MSR/MSE = 32.5

&

reject H0; and conclude that regression is significant.

EXcel calculation:

Excel formula:

(b) Again we can use F-stat to compare the 2 R-sq's

H0: there is no diff

Ha: there is difference

F-stat= 1.0013

P-value= .49 >.1

Cant reject H0; no difference between R-sqs

(D)

Model B is best. Since:

1) It is significant

2) It is has the highest Adjusted R-sq which indicates best fit with least complexity.

P-value= 1.71822E-11<0.05

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