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 ofX tobe0. (5pt)
(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,wecanapplyttestandtestH0 :β1 =0andH0 :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 β1 denotes the estimator of β1
from the simple regression, and β1 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 β1 and β1 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 β1 and β1 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(β1) and se(β1) 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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