# ECMT2150辅导、辅导Python/C++程序

ECMT2150 Review Questions
Chris Walker
1 Introduction
These are some review questions for the ECMT2150 final exam. They are not intended to be all encompassing, as
they mainly focus on important definitions and key concepts. The majority of the answers can be found in your
textbook/lecture notes.
2 Ordinary Least Squares
The first five questions relate to the following multiple linear regression model
Y =β0+β1X1+ ...+βkXk +U , E(U )= 0. (1)
The sixth and seventh questions are related to a model in Chapter 15 of Wooldridge.
1. What does it mean for an estimator to be unbiased? What is the difference between unbiasedness and consis-
tency? List the assumptions required for the OLS estimator for β j , j = 1, ...,k to be unbiased. Are these the same
as those required for consistency?
2. What is the Gauss-Markov Theorem? What must we assume about (1) for the theorem to hold true? Why is it
important?
3. Suppose I use log(Y ) rather than Y in (1). How do I interpret β?OLS1 ? What if I also replace X1 with log(X1)?
4. Write detailed steps to testing the following hypotheses: 1. H0 :β1 = 0 vs. H1 :β1 6= 0, 2. H0 :β1 = 0,β2 = 0,β3 = 0
vs. H1 : H0 is false, and 3. H0 : 2β1 =β2+β3 vs. H1 : 2β1 6=β2+β3 (hint for 3: reparameterise the model).
5. Now focus on Y =β0+β1X1+U and suppose that X1 is measured with error. What are the consequences for the
OLS estimator of β1? Remember to be specific about economic vs. statistical issues and consider a broad range
of economic issues.
6. Suppose that we take a random sample of n ECMT2150 students and estimate the following model that explains
final exam scores using OLS: scorei = β0+β1skippedi +ui , i = 1, ...,n, where skipped is the number of lec-
tures missed. Do we expect the variable skipped to be exogenous? Explain your answer (see section 15-1 for
discussion of this model).
7. Suppose that the sample is based on students who received a score of 80+ in the final exam. Do you see any
issue with this approach?
3 Heteroskedasticity
1. Explain the circumstances when the following approaches would be appropriate: 1. OLS with robust SEs, 2.
GLS, and 3. FGLS.
2. A famous test for heteroskedasticity is the Breusch-Pagan test. Explain carefully how to conduct the Breusch-
Pagan test. What are we assuming about the form of heteroskedasticity? What could we do if we are uncomfort-
able with the previous assumption?
3. Consider the following model: Y = β0 +β1X1 +β2X2 +U , E(U |X ) = 0, and Var (U |X ) = σ2X1X2. Explain why
this model suffers from heteroskedasticity? How can we transform the model so that there are homoskedastic
disturbances? Remember to show that the transformed model has homoskedastic disturbances.
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4 Instrumental Variables
1. Consider the following model Y = β0+β1X +U . We suspect that X is endogenous and propose to use Z as an
instrument for X . What assumptions must Z satisfy?
In the next questions we focus on the following structural model
Y =β0+β1X +γ1W1+ ...+γkWk +U
where we assume that X is endogenous, W1, ...,Wk are exogenous, and Z1, ...,ZR are valid instruments for X .
2. Explain why we have overidentification.
3. Explain how 2SLS uses all of the information from the instruments to estimate the parameters in the structural
model. Here you just need to explain the steps for 2SLS.
4. We know that weak instruments can lead to very poor properties for the 2SLS estimator. Explain what is meant
by the term ‘weak instrument’. How does one test for weak instruments?
The next two questions relate to the following model. Suppose a firm is interested in implementing a training
program to improve the productivity of their employees. A full-scale rollout of the program will be expensive, so
to test the efficacy of the program they randomly offer employees to participate in the program.
5. Suppose that Ti = 1 if employee i is offered to participate and Ti = 0 if not. Moreover, assume those with high
ability think the training is futile and choose not to participate in the program if offered. If we model the effect
of the program on productivity using the following model
producti vi t y =β0+β1D+U ,
where Di = 1 if employee i participates in the training and Di = 0 if not, then D is endogenous. Why? You may
assume that those who are not offered a place cannot receive the training.
6. Explain why T is a valid instrument for D . That is, why does instrumental exogeneity hold and why does instru-
mental relevance hold?
5 Panel Data and Pooled Cross-Sections
1. What is a pooled cross-section and what is a panel? How do they differ?
Here I work through example 13.3 in Wooldridge. Suppose that we have two time periods t = 1990,1992 and
I have a pooled cross-section comprised houses that includes data on house prices and whether they are within
20 miles of an incinerator that was built in 1991. We are interested in the effect of being near the incinerator on
house prices.
2. Suppose I estimate the following model using OLS restricting my sample to those near the incinerator: pr ice =
β0+β11992+U , where 1992 = 1 if in year 1992 and 1992 = 0 in 1990. Interpret β1. Does this provide a causal
effect of the incinerator on prices? Why or why not?
3. It turns out the answer to the previous question is no (I hope you can explain why!). Propose a regression model
that can be used to capture the causal effect of building the incinerator on prices. How could I obtain an esti-
mate of the causal effect without using a regression model?
Finally, we include some generic panel data questions that revolve around the following unobserved effects
model
Yi t =β0+β1Xi t + Ai +Ui t , i = 1, ...,n, t = 1, ...,T
where Ai represents time invariant unobserved heterogeneity and Ui t is the idiosyncratic error term (together
they make up a composite error term Vi t := Ai +Ui t ).
4. What is meant by the term strict exogeneity? Write down a mathematical expression and then explain what it
means intuitively (if you are unaware of the mathematical definition, then your textbook is a great resource).
5. Suppose Cov(Ai ,Xi t ) 6= 0. There are two transformations that can be used to consistently estimate β1 in the
presence of endogeneity induced by time-invariant factors. These are known as fixed effects and first-differencing.
What are these? When are they the same? Discuss how to choose between the two when they are different.
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6. If Cov(Ai ,Xi t ) = 0, then I could estimate the unobserved effects model using pooled OLS with cluster-robust
standard errors or I could use the random effects estimator. What are cluster-robust standard errors? Why
might I choose random effects over cluster robust standard errors?
7. Explain the relationship between random effects, fixed effects, and the unobserved effects model.