ECON60622 Further Econometrics Mid-term Exam 2020
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ECON60622
Further Econometrics
Mid-term Exam
March 10 2020
1. (a) Let z and u be two random variables and E[u|z] = 0. Setting z = (1,z), show that E[zu] = 0. [4 marks]
1. (b) Consider the simple linear regression model
y = β0 + β1x + u,
where E[u|z] = 0, Var[u|z] = σ 2 , z is an observable random variable. Let βˆ be the Instrumental Variables (IV) estimator of β = (β0 , β1 )/ based on the population moment condition E[zu] = 0 and a random sample {xi ,zi ,yi ; i = 1, 2, . . . ,n}. It can be shown that the formula for the IV estimator of β1 can be written as follows:
βˆ1 = β1 + 对ni=1 (zi − )(ui − )
(1)
where , and are the sample means on u, x and z respectively. Assuming E[zu] = 0 is true:
(i) Explain how the argument for the consistency of βˆ1 depends on the population correlation between x and z . [4 marks]
(ii) Are any other features of the sampling distribution of βˆ1 affected by ρx,z ? Briefly explain. [4 marks]
2. Consider the following simple linear regression model
y = β0 + xβ1 + u, (2)
Assume that equation (2) holds in the population and E[u|x] = 0.
(a) Using this model, briefly explain the difference between endogenous sample selec- tion and exogenous sample selection. [4 marks]
(b) Do endogenous sample selection and exogenous sample selection have the same consequences for the OLS estimator of the parameters of the population model in equation (2)? Justify your answer briefly but no derivations are required.
[8 marks]
3. (a) Explain the difference between repeated cross-section data and panel data. [2 marks]
3. (b) Explain what is meant by the term unobserved heterogeneity. [2 marks]
4. Kiel & McClain (1995) investigate whether the location of a new garbage incinerator had an effect on housing prices in North Andover, MA in the USA. The rumour that the incinerator would be built began spreading in 1979, and construction began in
1981. Using data on house prices for the years 1978 and 1981, the following model is estimated:
log(price) = β0 + 60 y81 + β1 ∗ nearinc + 61 ∗ y81 ∗ nearinc + error, (3)
where price is the house, y81 is a dummy variable that takes the value one if the observation is for year 1981 and is zero otherwise, and nearinc is a dummy variable that takes the value one if the house is located within three miles of the incinerator site. The output from this estimation is presented on the next page.
(a) What is the estimated impact of the construction of the incinerator on the prices of houses close to the incinerator site (that is, houses within three miles of the site)? [4 marks]
(b) Test whether the construction of the incinerator has led to a reduction in the prices of the houses located close to the incinerator site? Be sure to specify clearly your null and alternative hypotheses, and your decision rule. [8 marks]
4. The Stata output for the estimation of (3) is as follows, with some parts of the output deliberately omitted.
Source | SS df MS -------------+---------------------------------- Model | 11 .8433224 3 3 .94777412 Residual | 36 .3058203 317 . 114529402 -------------+---------------------------------- Total | 48 . 1491427 320 . 150466071 |
Number of obs F(3 , 317) Prob > F R-squared Adj R-squared Root MSE |
= = = = = = |
321 34 .47 0 .0000 0 .2460 0 .2388 .33842 |
lrprice |
| + | |
Coef . |
Std . Err . t P>|t | [95% Conf . Interval] |
y81 |
. 193094 |
.0453208 |
|
nearinc |
| |
- .3399216 |
.0545555 |
y81nrinc |
| |
- .0626498 |
.0834408 |
_cons |
| |
11 .28542 |
.0305145 |
where lrprice and y81nrinc denote respectively log(price) and y81 ∗ nearinc.
2022-08-29