ECMT 6002/6702: Econometric Applications 4C/C++

Java Python ECMT 6002/6702: Econometric Applications

1 Practice problems

1. Consider the following linear regression model:

yt = β1 + 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 V ar(u) = σ2I.

(ii) Obtain the variance of the OLS estimator when

(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 V ar(β bj ) is bigger than that under the as-sumption V ar(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 T SS = 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

Find the value of A + B + C.

(Note) 95% quantile of χ 2 (m)

2. Consider the following regression model

where

We want to examine if the variance of ut gets larger along with the variable x3t

(a) Explain how to implement the Goldfeld-Quandt (GQ) test in general (ignoring the prop-erty of x3t).

(b) In order to example the hypothesis of interest using the GQ test, suppose that we split th ECMT 6002/6702: Econometric Applications 4C/C++ e samples into two subsamples of sizes T1 and T2 according to the variable x3t . Is this a reasonable approach? Why or why not?

2 Empirical application

We will consider the ECONMATH dataset again. Suppose that we have the following regression model:

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,

T R2 ≃ 60                                         (2.1)

and 95% quantile of χ 2 (13) is 22.36 (why is the degrees of freedom parameter 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

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 “sandwich” 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

7. This computing exercise is not mandatory