Question

Simple Linear Regression For Questions 1 to 10, consider the following simple linear regression model: Yi = Bo + Bix; tui for i = 1,...,n. Suppose uildi~ (0,), that is, conditional on Di, ui has a distribution whose mean is 0 and variance is op. One ran this regression and got OLS estimates Bo and B1, and û; = yi-Bo-Bici. Let 33 ==2 (Yi - y)2 = 9, where y = WoW 8=(** – 7)? = 4, where i = + Use the information above to answer Questions 1 to 10. For each question, use only the information above and the information provided in the question. Do NOT use the information in other questions unless you are explicitly told to do so.

6. (4 points) Suppose n = 82. How much is σˆ 2 , the estimated variance of the error term ui?

A. 0.00625

B. 0.0125

C. 0.025

D. 0.05

7. (4 points) Suppose βˆ 1 = 0.75, SE(βˆ 1) = 0.01 and n = 52. Then the 90% confidence interval for βˆ 1 would be

A. [0.4224, 1.1994]

B. [0.6112, 0.9112]

C. [0.7336, 0.7664]

D. [0.7229, 0.7661]

8. (4 points) Suppose one tested and rejected the null hypothesis H0 : β1 = 0 against H1 : β1 > 0 at the significance level α = 0.05. Then which of the following is TRUE?

A. She/He will reject the null hypothesis H0 : β1 = 0 against H1 : β1 ̸= 0 at the significance level α = 0.05.

B. She/He will reject the null hypothesis H0 : β1 = 0 against H1 : β1 ̸= 0 at the significance level α = 0.1.

C. She/He will not reject the null hypothesis H0 : β1 = 0 against H1 : β1 ̸= 0 at the significance level α = 0.05.

D. She/He will not reject the null hypothesis H0 : β1 = 0 against H1 : β1 ̸= 0 at the significance level α = 0.1.

9. (4 points) Suppose you know that βˆ 0 = 2.5, x¯ = 2.5 and y¯ = 2. Then

A. βˆ 1 = 1.8

B. βˆ 1 = −1.8

C. βˆ 1 = 0.2

D. βˆ 1 = −0.2

10. (4 points) For the 100(1 − α)% confidence interval of βˆ 0, which of the following is TRUE?

A. The larger is α, the larger is the length of the interval.

B. The smaller is α, the larger is the length of the interval.

C. α does not affect the length of the interval.

D. The length of the interval depends on the value of βˆ 0.

Answer 1

6. The correct option would be

• C. 0.025

The residual standard error is
Στ = RSE=
, for k be the number of parameters to be estimated, ie $\sigma^2 = \frac{\sum u^2}{n-2}$ or
2 82-2
or
2 = 0.025 80
.

7. The correct option would be

• C. (0.7336, 0.7664)

For 10% significance, we have , since the df=n-k=52-2=50. The confidence interval would be $\beta_1 \pm t_{0.95,50}*SE(\beta_1)$ or $0.75 \pm 1.6759*0.01$ or $0.75 \pm 0.016759$ or , which is closest to the option C. In option C, since df>30, the t-statistic is substituted by the normal z statistic of 1.64 (which is not recommended but can be done nonetheless). In that case, the CI would be the same as in option C.

8. The correct option would be

• B. She/He will reject the null hypothesis $H_0 : \beta_0 = 0$ against $H_1 : \beta_0 \ne 0$ at the significance level $\alpha = 0.1$

The reason being that, the test for $\beta > 0$ would be one tailed, and the test would be conducted as . Now, for two tailed test at 5%, we would require . Since , the condition is not necessarily true. Hence, option A is incorrect. However, at 10%, the condition required would be , and since we already know , which implying that , the null would be rejected at 10%. Hence, option B is correct.

9. The correct option would be

• D. $\beta_1 = - 0.2$

We have $\beta_0 = \bar y - \beta_1*\bar x$ , and for the given values, we have $2.5 = 2 - \beta_1*2.5$ or $- \beta_1*2.5 = 0.5$ or $\beta_1 = - 0.2$ .

10. The correct option would be

• B. The smaller is $\alpha$ , the larger is the length of the interval.

The smaller the alpha, the smaller is the region to reject the null, and larger is the region to accept the null, and hence, larger is the CI that may include 0.