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ECMT 6002/6702: Econometric Applications
发布时间:2023-06-02
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ECMT 6002/6702: Econometric Applications
1 Practice problems
1. Consider the following linear regression model:
yt = α + βxt + ut, t = 1, . . . , 100.
Consider the following hypotheses:
H0 : β = 0 vs H1 = β 0
The Wald statistic for the above hypothesis is defined as
s2 / 对12(xt - )2 ,
where s2 = 对 t(2) .
(i) What is the limiting distribution of the above statistic if H0 is true? (Optional) We know under standard assumptions,
|
→d N (0, 1) |
Can you obtain the limiting distribution of the Wald statistic based on the above result?
(ii) Show that RRSS and URSS are given as follows:
T
RRSS = 工(yt - y¯t )2
t=1
T T
URSS = 工(yt - y¯ + - xt)2 = 工 t(2)
(Restricted) : y = α + ut .
(Unrestricted) : y = α + βxt + ut .
(iii) Show that T (RRSS − URSS)/URSS is equivalent to (1.2).
(iv) Suppose that RRSS = 1105, URSS = 900, T = 100. We want to implement the Wald test with 95% significance level. Let A be the Wald statistic, B be the critical value and C be defined by
C =〈
(
Find the value of A + B + C .
(Note) 95% quantile of χ2 (m)
1 if H0 is rejected,
−1 if H0 is not rejected.
m=1 |
m=2 |
m=3 |
m=4 |
m=5 |
3.84 |
5.99 |
7.81 |
9.48 |
11.07 |
(v) Suppose that RRSS = 1200, URSS = 900, T = 100. We want to implement the LM test with 95% significance level. Let A be the LM statistic, B be the relevant critical value and C be defined by
C =〈
(
1 if H0 is rejected,
−1 if H0 is not rejected.
Find the value of A + B + C .
(vi) Write down the auxiliary regression for the White’s heteroskedasticity test. Assuming that R2 from the auxiliary regression is given by 0.3, compute the test statistic (A) and find relevant critical value (B ), and then find the value of A + B . (Note : White’s test is
already based on the asymptotic properties of the OLS estimators. So you can do this as in the Week 3 lecture note. I would recommend you to check how many variables will
be included in the auxiliarly regression for the White’s heteroskedasticity test in a more general case where k regressors.)
2 Empirical application
We will consider the housing pricing example given in Wooldridge’s textbook. Suppose that we have the following regression model:
yt = β 1 + β2 x2t + β3 x3t + β4 x4t + ut ,
Instructions:
1. Compute the OLS estimates.
2. Examine H0 : β2 = 0.1, β3 = 0.01 using the Wald, LR, LM tests (with 95% significance level).
- Construct the restricted/unrestricted models and compute URSS and RRSS, and the compute the statistics. They must be close to
Wald = 159.87, LR = 91.13, LM = 56.75.
- To compute the critical value, you can use qchisq(0.95,2) in R.
3. Examine H0 : β2 + β3 = 0.1 using the Wald, LR, LM tests (with 95% significance level).
- Note that the restricted model can be written as
yt − 0.1x3t = β 1 + β2 (x2t − x3t) + β4x4t + ut .
- The results must be close to
Wald = 3.7299, LR = 3.6530, LM = 3.5782.
- To compute the critical value, you can use qchisq(0.95,1) in R.
4. This computing exercise is not mandatory.