ECON 141 Midterm Exam 2 Fall 2019
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Midterm Exam 2
ECON 141
Fall 2019
Questions
Question 1 (37 points)
1.a Suppose you want to estimate the efect of government expenditure, X, on GDP, Y , where X is coded as {low, medium, high}. One expert suggests the following regression: Y = β0 + β1 D + U where
D = -1 if X = low, = 0 if X = medium and = 1 if X = high.
(i) Would this suggestion yield misleading results? Please explain.
(ii) If the answer is ”yes”, how would you do it properly?
1.b Consider the following regression model Y = β1 X + U. Will the R2 be between 0 and 1 in this case?
1.c In a ixed efect model, Yi;t = λt + ui;t where i = 1, ..., n and t = 1, ..., T with T ixed and n → ∞ . Are the ixed efects, (λt )t , consistently estimated? Please explain.
1.d Consider the ixed efect model Yi;t = αi + β1 Xi;t + ui;t where Cov(αi , Xi;t ) = 0 for all i and t.
(i) “The OLS estimator of the pooled regression will be consistent and asymptotically normal”. True or False, please explain.
(ii) “The standard errors of the OLS estimator will also be correct, provided we do not impose home- sckedasticity”. True or False, please explain.
1.e Consider a linear regression model Y = β0 + β1 X1 + β2 X2 + U. “If the t-tests for β1 = 0 and β2 = 0 do not reject each null hypothesis, then the F-test will not reject β1 = β2 = 0.” True or false, please explain.
1.f Consider the ixed efect model Yi,t = αi + β1 Xi,t + ui,t but E[ui,tjXi,1, ..., Xi,t] = 0 for each i and t ≥ 2. (notice the restriction on the residuals)
(i) Will the ixed efect estimator be consistent in this case? Please explain.
(ii) Can you construct another estimator that is consistent?
Question 2 (37 points)
Suppose you want to asses the efect of federal investment on roads on state-level GDP. Some states received the investment (Xi = 1) and some others didn’t (Xi = 0). You have two periods (before and after the policy) where Yi;t is the observed GDP of state i in period t.
Suppose the true model is given by
Yi;t = αi + 0.2 根 Xi Dt + Dt + Ui;t
where Ui;t is mean zero and independent of Xi and Dt is a dummy that takes value 1 if t = 1 and αi is a ixed-efect that measures the state time-invariant characteristics.
The results of the OLS regression are
Y(ˆ)i;1 = 1.5*** - 0.3*** 根 Xi
(the stars denote signiicance at 99 percent).
2.a Someone points out that the investment decision may not be randomly assigned. What type of corre- lation between Xi and the other variables of the model can explain the surprising result of the OLS regression?
2.b Argue that a ixed efect model can solve the potential source of the endogeneity.
Suppose the panel data estimator is given by
Yi;t = 0.2*** 根 Xi Dt + 1.01*** 根 Dt + Ui;t
2.c In view of the results, do you think the government policy is random? Please describe the potential source of endogeneity and its direction (i.e., sign of the correlation).
2.d Someone is concerned about a potential omitted variable bias because the state region (e.g., coastal, mid-west, etc) was not included. Is this a valid concern? Please explain.
2.e Someone points out that one can use this model for prediction of Yi;t. Do you agree?
Question 3 (26 points)
Consider the following panel data model
Yit = β0 + β1 Xit + αi + uit. (1)
3.a: (8 points) From equation 1 derive the following equation
Y(-)i = β0 + β1 X(-)i + αi + i. (2)
3.b: (9 points) Letβ(˜)1 be the OLS estimate of β1 in equation 2. Suppose thatX(-)i have all the same variance.
Show that
plimn!1 β(˜)1 = β1 +
with T ixed.
3.c: (8 points) Suppose further that Xit are uncorrelated across time. What does the previous equation tell you about the behavior of the asymptotic bias (i.e., ) when T increases.
2023-11-20