Question

We have collected data on the annual number of cars of n - 20 different brands sold in Australia ("sales', in number of cars), as well as each brand's average retail price ("price', in dollars), their annual marketing expenditure ("mark", also in dollars), and whether or not they assemble some of their cars domestically ("domestic', dummy variable). We have estimated three models on this data set, with the following results: In(sales- A B2 ln(price) +B3 In(mark) + B4domestic + error In(sales)- A + B2 In(price) ln(sales) - RSS4.73 + error RSS= 10.69 + β3In(mark) + Adomestic + error RSS= 14.72 Test the null hypothesis that the Australian car market is completely price-driven, so that marketing and whether production is done domestically are irrelevant. Use the standard 5% significance level. F 33.79, do not reject the null hypothesis F 10.08, do not reject the null hypothesis. F 33.79, reject the null hypothesis. F 10.08, reject the null hypothesis.

Answer 1

We want to test the null hypothesis that the Australian car market is completely price-driven, so that marketing and whether production is done domestically are irrelevant. This means given that price is in the model, we want to test if the coefficients of marketing and whether production is done domestically are significant.

That is we want to test the following hypotheses

Ho : A 3 За 0 ← null hypothesis: Given that price is in the model, marketing and domestic do not explain sales Ha : at least one of β3, B4メ0 ← alternative hypothesis: At least one among marketing and domestic explain sales a 0.05 ← level of significance to test the hypotheses

We have the following Full model

in (sales) β1+ β2 In (price) β3 In (mark) + Ďadomestic + error

The Residual Sum of Square (RSS) for the full model is

RSS. = 4.73

The number of independent variables in the model is k=3

The degrees of freedom of the full model is

dff-n-k-1-20-3-1-16

We have the following reduced model (the model which has only price in it and no marketing and domestic)

In (sales)-A + β2 ln (price) + error.

The Residual Sum of Square (RSS) for the reduced model is

RSS-10.69

The number of independent variables in the model is k=1

The degrees of freedom of the reduced model is

dfr = n-k-1=20-1-1=18

The test statistics is

(RSS - RSS)/(dfr - dff) (10.69- 4.73)/(18 16) 10.08 4.73/16

The degrees of freedom for this F statistics is numerator df=18-16=2 and denominator df=16

The critical value of F fro alpha=0.05 can be obtained using F tables for alpha=0.05 and  numerator df=2 and denominator df=16. The critical value is 3.63

We will reject the null hypothesis if the test statistics is greater than the critical value.

Here, the test statistics is 10.08 and it is greater than 3.63. Hence we reject the null hypothesis

ans: F=10.08, reject the null hypothesis.

We conclude that the Australian car market is not completely price-driven, so that marketing and whether production is done domestically are relevant.