Assume your research staff used regression analysis to estimate the industry dem

Question

Assume your research staff used regression analysis to estimate the industry demand curve for Product X. Qx = 10,000 - 100 Px + 0.5 Y - 1000 r (3,000) (20) (0.3) (105) Where Qx is the quantity demanded of Product X, Px is the price of X, Y is income, and r is the prime interest rate (given in decimals, e.g., 0.02 or 0.05) The standard error of each estimated coefficient is given in parentheses below it. Also, the following information is provided about the regression equation. Number of observations = 98 R2 = 0.95 F-statistic = 7.5 a. What is the number of degrees of freedom? b. What percentage of the variation in the dependent variable is explained by the equation? c. Which of the estimated coefficients are significant at the 5% level using a 2-tailed test; be sure to indicate the t-statistic for each of the coefficients. d. Perform an F test at the 5% level of the overall explanatory power of the model. e. If prices remain constant next year but income is expected to increase by 50 and interest rates rise by two percentage points (by 0.02), what is the expected change in the quantity demanded?

Explanation / Answer

a. There are 100 - 4 = 96 degrees of freedom
b. Since R2 = 0.9, 90% of the variation in Qx is explained by the equation.
c. The t-statistic for 96 degrees of freedom at the 5% level for a 2-tailed test is 1.985. The t-statistic has to be greater than 1.985 to be considered statistically significant. The t-statistic for the constant is 3.33, the t-statistic for Px is 4, the t-statistic for Y is 4.2, and the t-statistic for r is 1.11. All of the coefficients except the one for "r" are significant at the 5% level.
d. The null hypothesis is that there is no relationship be Qx and the three independent variables in the model. Using Table B.3 in your text,

F(0.05, 3, 96) = 2.68,

since F = 15 > 2.68, we can reject the null hypothesis.
e. If prices remain constant but income increases by 50 and interest rates fall by 0.02, the expected change in Qx for next year is

Qx = + 0.5 x 50 - 1000 x (-0.02) = 25 + 20 = 45

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