Question

12.3 Suppose you fit the multiple regression model y = Bo + B1x1 + Bzxz + Bzxz + e to n = 30 data points and obtain the following result: ŷ = 3.4 - 4.6x + 2.7x2 + .93xz The estimated standard errors of B, and Bs are 1.86 and .29, respectively. a. Test the null hypothesis Ho: B2 = 0 against the alternative hypothesis He: B2 + 0. Use a = .05. b. Test the null hypothesis Ho: Bz = 0 against the alterna- tive hypothesis H : B3 + 0. Use a = .05. c. The null hypothesis Ho: B2 = 0 is not rejected. In con- trast, the null hypothesis Ho: B3 = 0 is rejected. Explain how this can happen even though B2 > Bz.

Answer 1

Solution:

a. The null and alternative hypotheses are:

0 = id: H

H: B270

Under the null hypothesis, the test statistic is:

se 82

p-value = 2 x P(t > 1.452) = 0.1585

Since the p-value is greater than the significance level, we, therefore, fail to reject the null hypothesis.

b. The null and alternative hypotheses are:

H : B3 = 0

На : Bg #О

Under the null hypothesis, the test statistic is:

+= $3 = 0.93 = 3.207 t=se, 0.29 0.93 = 3.207 se as

p-value = 2 x P(t > 3.207) = 0.0035

Since the p-value is less than the significance level, we, therefore, reject the null hypothesis.

c. The reason
H : B3 = 0
is rejected because the standard error of $\small \hat{\beta_{3}}$ is around three times the $\small \hat{\beta_{3}}$ , while the standard error of $\small \hat{\beta_{2}}$ is around 1.5 times the $\small \hat{\beta_{2}}$