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ECMT 6002/6702: Econometric Applications 4
发布时间：2023-06-01
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ECMT 6002/6702: Econometric Applications
1 Practice problems
1. Consider the following linear regression model:
yt = β 1 + β2x2t + β3x3t + ut, t = 1, . . . , 100.
In the matrix form, we have
y = Xβ + u.
(i) Obtain the variance of the OLS estimator under the assumption that Var (u) = σ I2 . (ii) Obtain the variance of the OLS estimator when
|
\σ 2
|
0 σ 2
0 0
0 |
0 0 . . . 0 0
0 |
0 0 . . . 2σ2 0
0 |
. . . . . .
. . .
2σ2 0 |
0 0
0 0 . . . 0 |
0
0
0
2σ2) |
(1.1) |
(iii) Explain what will happen in the standard t-test to examine H0 : β2 = 0 if you ignore potential heteroskedasticity.
(Optional) In this case, can you show that Var (
j ) is bigger than that under the as- sumption Var (u) = σ 2 I?
- Hint X\ AX is nonnegative definite if A is a diagonal matrix with nonnegative entries.
(iv) Implement White’s heteroskedasticity test with 5% significance level. What is the aux- iliary regression equation? Suppose that T = 60 and TSS = 1050, ESS = 405 and RSS = 645 are obtained from the auxiliary regression. Let A be the test statistic, B be the relevant critical value and C be defined by
C =〈
−1 if H0 is rejected, (1.2)
Find the value of A + B + C .
(Note) 95% quantile of χ2 (m)
|
m=3 |
m=4 |
m=5 |
m=6 |
m=7 |
|
7.8147 |
9.4877 |
11.070 |
12.591 |
14.067 |
2. Consider the following regression model
yt = β 1 + β2 x2t + β3 x3t + ut, t = 1, . . . , 100,
wage education university
where
x3t =〈( 1 if t attended a university , (1.3)
2 Empirical application
We will consider the ECONMATH dataset again. Suppose that we have the following regression model:
yt = β 1 + β2 x2t + β3 x3t + β4 x4t + β5 x5t + ut ,
Instructions:
1. The dataset contains missing values (see Week 3 tutorial)
2. Compute the OLS estimates and report their standard errors. In R, summary(lm(y ∼ X)) can be used if X is the (T × 5) data matrix.
3. Implement White’s heteroskedasticity test and report the test result. If things are correctly done, you can detect heteroskedasticity; more specifically,
TR2 ≃ 323 (2.1)
and 95% quantile of χ2 (13) is 22.36 (why is the degrees of freedom parameter is 13?)
4. Obtain the heteroskedasticity-robust standard error (i.e., White’s standard error) of each coefficient estimate. One easy way to do this in R is using“vcovHC”function given in “sandwith”package; specifically, run vcovHC(lm(y∼X)). Then obtain the t-statistics to examle H0 : βj = 0. The results must be simiar to
1 ) = 5.16, 
2 ) = 16.30, 
3 ) = 0.478, 
4 ) = 7.82 (2.2)
5. Compare the above results with what you obtained using the usual standard errors in Week
3 tutorial.
6. Obtain the HAC robust standard error of each coefficient estimate. In R“vcovHAC”func- tion given in“sandwith”package can be used; specifically, run vcovHAC(lm(y∼X)). Then obtain the t-statistics to examle H0 : βj = 0. The results must be simiar to
1 ) = 5.16, 
2 ) = 16.89, 
3 ) = 0.495, 
4 ) = 8.15 (2.3)
7. This computing exercise is not mandatory. In this example, heteroskedasticity
