Stastic exam (may be required using R)

GBA6230, Midterm Exam. Deadline: 3/21 8:00 PM. Please submit

your answers in a single pdf file along with your R code file via Blackboard.

If you are using other software than R-studio, please also submit your code

for your software.

1. True or False. Explain your answer in detail. Your score will be based on your

explanation.

(a) E(u|X1,X2) = 0 implies E(u) = 0. It also implies that (1) u is uncorre- lated with X1 and X2; and (2) X1 and X2 is uncorrelated. (5 pt)

(b) In order for our regression estimators to be unbiased, we need the variance

of X to be as small as possible. In the best scenario, we want the variance

of X to be 0. (5 pt)

(c) R2 measures how much of the variation in data can be explained by linear

regression model, and it never increases when we try to control more X in

the model. (5 pt)

(d) E(u|X1,X2) = 0 implies E(u) = 0. It also implies that u is uncorrelated with X1 and X2 and X1 and X2 is uncorrelated. (5 pt)

(e) Suppose we are interested in testing the null hypothesis: H0 : β1 =

0 and β2 = 0, we can apply t test and test H0 : β1 = 0 and H0 : β2 = 0

separately. (5 pt)

(f) When there are 3 groups in the sample, we should define 3 dummy vari-

ables and use all of them in the regression model to control all the group

differences. (5 pt)

2. Consider the following two models relates education to wage:

log(wage) = β0 + β1educ + u

log(wage) = β0 + β1educ + β2sibs + e

where wage denotes monthly wage; educ is the education level measured by

year; and sibs is the number of siblings. Let β̃1 denotes the estimator of β1

from the simple regression, and β̂1 denotes the estimator from the multiple

regression.

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(a) Suppose educ and sibs are positively correlated in the sample, and sibs

has negative effects on log(wage), would you expect β̃1 and β̂1 to be very

different? If yes, which one will be larger? Explain your answer in detail.

(5 pt)

(b) Suppose educ and sibs are positively correlated in the sample, and sibs has

no effects on log(wage), would you expect β̃1 and β̂1 to be very different?

If yes, which one will be larger? Explain your answer in detail. (5 pt)

(c) In the same circumstance in part (b), would you expect se(β̃1) and se(β̂1)

to be very different? If yes, which one will be larger? Explain your answer

in detail. (5 pt)

3. Use wage1 data for this question. Consider the following model,

log(wage) = β0 + β1educ + β2exper + β3tenure + β4educ∗ tenure + u

(a) Holding other factors fixed, what is the marginal effect of educ to log(wage)

based on the estimation result? (5 pt)

(b) State the null hypothesis that the educ has no effect on log(wage) against

the alternative hypothesis that it has effect. (5 pt)

(c) Test the hypothesis in part (b). Explain your answer in detail. (5 pt)

4. Use ceosal1 data for this question. Consider the following model that links

CEO’s salary to the type of industry, company’s sales and roe,

log(salary) = β0 + β1finance + β2consprod + β3utility + β4sales + β5roe + u

where we have 4 types of industry in the data: industrial, financial, consumer

products, and utilities industries. finance, consprod, and utility are binary

variables indicating the financial, consumer products, and utilities industries.

(a) Which industry is the base group? (5 pt)

(b) Compute the approximate percentage difference in estimated salary be-

tween the industrial and utilities industries, holding sales and roe fixed. Is

the difference statistically significant at the 1% level? (5 pt)

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(c) Compute the approximate percentage difference in estimated salary be-

tween the utilities and finance industries, holding sales and roe fixed. (5

pt)

(d) Test whether the difference in part (c) is significant at 5% level. Explain

your answer in detail. (5 pt)

5. Use hprice1 data for this question. Consider the following model that links

house price to its square feet, lot size, and number of bedrooms,

price = β0 + β1sqrft + β2lotsize + β3bdrms + u

(a) Test whether sqrft has the same effect as lotsize on price at 5% level.

Report the results under traditional standard error and robust standard

error. Do you find different conclusions? Explain your answer in detail. (5

pt)

(b) We have two types of house in the data, colonial style and non-colonial

style. Define colonial as a dummy variable for colonial style house. Con-

sider the following model,

price = β0 + δ0colonial + β1sqrft + δ1sqrft∗ colonial + β2lotsize

+δ2lotsize∗ colonial + β3bdrms + δ3bdrms∗ colonial + u

Explain what does the null hypothesis, H0 : δ0 = δ1 = δ2 = δ3 = 0 imply?

(5 pt)

(c) Test the null hypothesis in (b). Report the results from traditional F-test

and the robust F-test. Do you find different conclusions? Explain your

answer in detail. (5 pt)

(d) Perform Breusch-Pagan and White tests on the model in part (b). What

are your conclusions based on these two tests? (5 pt)

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