BEE1023 Introduction to Econometrics 2022
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BEE1023
Introduction to Econometrics
2022
QUESTION 1 Multiple Choice Questions [36 marks]
Provide your answers on the MCQ answer sheet. Each question carries 4 marks. There is exactly one correct answer to each question.
1. Which of the following is true of heteroskedasticity?
A. Heteroskedasticity causes bias in the OLS slope estimators only.
B. Heteroskedasticity causes bias in the OLS intercept estimator only.
C. Heteroskedasticity causes bias in all OLS estimators.
D. If MLR.1 through MLR.4 hold, heteroskedasticity causes OLS to be BLUE.
E. None of the above are correct.
2. Which of the following is true of the OLS residuals?
A. The sample mean of the residuals is always equal to zero, but their sum is not always equal to zero.
B. The sample variance of the residuals is always equal to zero, but their sample mean is not always equal to zero.
C. The sample mean of the residuals is always equal to zero, and the residuals are always uncorrelated with the regressors.
D. The residuals are also referred to as the model error term.
E. None of the above are correct.
3. Consider a regression of y on an intercept only (i.e., y = F0 + u). Which of the following statements is correct?
A. Population R-squared is equal to zero.
B. Population R-squared is equal to one.
C. Population R-squared is negative.
D. is always equal to zero.
E. None of the above are correct.
4. Consider a test of multiple linear restrictions. If the F statistic is equal to 2.05, n−k is equal to 30 and q is equal to 5, what can you say about the p-value of the test?
A. p = 0. 1
B. 0.05 ≤ p < 0. 1
C. p > 0. 1
D. We cannot make a statement about the p-value without knowing the direction of the alternative hypothesis.
E. None of the above are correct.
5. Consider the regression model
y = F0 + F1x1 + F2 x2 + F3 x1x2 + u.
where x1 is an indicator variable and x2 is a continuous variable. Which of the following is correct?
A. The model suffers from perfect collinearity; thus, MLR.3 is violated.
B. The model allows for the marginal effect of x2 on y to differ between two sub- groups of the population.
C. The model error term is heteroskedastic by construction.
D. F1 can be interpreted as the expected percentage change in y following a unit change in x2 , holding fixed x2 .
E. None of the above are correct.
6. In the regression model y = β 0 + F1 log(x) + u , 100 × F1 can be interpreted as the expected unit increase in y for a 1 percent increase in x .
A. True
B. False
7. When multicollinearity is present in linear regression, the standard errors of the estimated coefficients are inflated.
A. True
B. False
8. For a null hypothesis about a single regression coefficient, the F test and the t test will always give the same result.
A. True
B. False
9. Omitted variable bias can be eliminated by including all irrelevant variables in the regression model.
A. True
B. False
QUESTION 2 [37 marks]
A tobacco company is interested in understanding how people’s expenditure on tobacco products is related to their total expenditure and various individual characteristics.
An analyst at the company suggests the following regression model:
log(tobex) = F0 + F1 log(totalex) + F2 male + F3 asian + F4 ℎigℎed + u (A)
Where tobex is an individual’s annual expenditure (in USD) on tobacco products, totalex is an individual’s annual total expenditure (in USD), male is a dummy variable which is equal to 1 if the person is male and zero otherwise, asian is a dummy variable which is equal to 1 if the person identified themselves as Asian and zero otherwise, and ℎigℎed is a dummy variable which is equal to 1 if the person has a university degree (bachelor’s degree or higher). log(x) denotes the natural logarithm of x .
a) [4 marks] What sign would you expect F1 to have in the population? Carefully justify your answer.
Using data from the Consumer Expenditure Survey in the US, the analyst has estimated model (A). The output from the regression is provided below. Please note that some of the output has been intentionally omitted.
Source | SS df MS
------------+---------------------------------
Model | 405.575068
Residual | 895.595476
------------+---------------------------------
Total | 1301.17054
Number of obs
F( , )
Prob > F
R-squared Adj R-squared Root MSE
= 1,155
=
=
=
=
= .88248
log_tobex |
|Coefficient + | .7935756 |
Std. err. |
t |
P> |t | |
[95% conf. |
interval] |
log_totalex |
.0356541 |
22.26 |
0.000 |
.7236212 |
.8635299 |
|
male |
| -.0783078 |
.0524368 |
|
|
-.1811904 |
.0245747 |
asian |
| -.5600746 |
.1377877 |
-4.06 |
0.000 |
-.8304181 |
-.2897311 |
highed |
| -.4388331 |
.05938 |
|
|
|
|
_cons |
| -1.618049 |
.2759443 |
-5.86 |
0.000 |
-2.15946 |
-1.076639 |
b) [4 marks] Carefully interpret the estimated coefficient on log(totalex) and comment on its significance.
c) [5 marks] Carefully interpret the estimated coefficient on male and comment on its significance.
d) [4 marks] Calculate R-squared and provide an interpretation for it.
e) [5 marks] Conduct the test for significance of the regression. Follow all steps of the procedure carefully. Use the 5% level of significance.
Next, the analyst estimates the following model (using the same observations as above):
log(tobex) = F0 + F1 log(totalex) + F2 ℎigℎed + u (B)
The sum of squared residuals (SSR) in the estimation of model (B) is found to be 910.913
f) [5 marks] Use an F test to assess the null hypothesis that the coefficients on male and asian are jointly equal to zero in the population. Follow all steps of the procedure carefully. Use the 5% level of significance.
The analysist would like to test whether the effects of total expenditure and higher education on tobacco expenditure are the same for males and females.
g) [10 marks] Based on model (A), propose a new regression model that allows the researcher to test their hypothesis. Write down the appropriate null and alternative hypotheses, and explain the steps the analyst needs to take to perform this test.
QUESTION 3 [12 marks]
Assume that the true population regression function is given by
y = F0 + F1x + F2 x 2 + u (1)
where E[u|x] = 0, F0 > 0 , F1 > 0, F2 < 0 and x ≥ 0 .
A researcher mistakenly estimates the following mis-specified model:
y = F0 + F1x + u (2)
a) [6 marks] Consider the OLS estimator of F1 in model (2): = Explain why is a biased estimator of F1 . Give as much detail as possible.
b) [6 marks] If F2 < 0, what can you conclude about the direction of the bias in ? Carefully justify your answer.
QUESTION 4 [15 marks]
a) [5 marks] Explain what perfect collinearity is in multiple linear regression using an example of your choice. What (if any) problems does it pose in multiple linear regression?
b) [5 marks] Explain what multicollinearity is using another example of your choice. Carefully describe the difference between multicollinearity and perfect collinearity.
c) [5 marks] Using the following expression for the variance of in multiple linear regression, explain the impact multicollinearity has on the variance of the OLS estimator.
VaT () =
2022-08-05