Question

3. (Based on Stock & Watson “Introduction to Econometrics” 6th ed., Exercise 5.7.) Suppose that (Yi , Xi) satisfy the assumptions SLR.1 - SLR.4. A random sample of size 2 n = 250 is drawn and yields Yˆ = 5.4+3.2X, n = 250, R 2 = 0.26. (3.1) (1.5)

(a) Test H0 : β1 = 0 against H1 : β1 6= 0 at the significance level 5%.

(b) Construct a 95% confidence interval for β1.

(c) Suppose you learned that Yi and Xi were independent. Would you be surprised? Explain.

(d) Suppose that Yi and Xi are independent and many samples of size n = 250 are drawn, regression estimated, and (a) and (b) answered. In what fraction of the samples would H0 from (a) be rejected? In what fraction of samples would the value β1 = 0 be included in the confidence interval from (b)?

Answer 1

(a) The hypothesis being tested is:

H0: β1 = 0

H1: β1 ≠ 0

t = 3.2/1.5 = 2.13

The p-value is 0.0339.

Since the p-value (0.0339) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that β1 ≠ 0.

(b) The 95% confidence interval for β1 is:

= β1 t*se

= 3.2 1.97*1.5

= 0.25, 6.15

The 95% confidence interval for β1 is between 0.25 and 6.15.

(c) No, because the assumptions of the test are met.

(d) The fraction of the samples where H0 from (a) be rejected = 0.95*250 = 237 = 237/250

The fraction of samples would the value β1 = 0 be included in the confidence interval from (b) = 0.05*250 = 13 = 13/250

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