ECMT2150 Review Questions
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ECMT2150 Review Questions
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
Y = p0 + p1X1 + + pkXk + 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 pj , 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 
1(O)LS ? What if I also replace X1 with log(X1)?
4. Write detailed steps to testing the following hypotheses: 1. H0 : p1 = 0 vs. H1 : p1 \= 0, 2. H0 : p1 = 0,p2 = 0,p3 = 0 vs. H1 : H0 is false, and 3. H0: 2p1 = p2 + p3 vs. H1: 2p1 \= p2 + p3 (hint for 3: reparameterise the model).
5. Now focus on Y = p0 +p1X1 +U and suppose that X1 is measured with error. What are the consequences for the OLS estimator of p1 ? 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: scor ei = p0 + p1 ski ppedi + ui , i = 1, ..., n, where ski pped is the number of lec- tures missed. Do we expect the variable ski pped 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 = p0 + p1X1 + p2X2 + U, E (U |X) = 0, and Var (U |X) = a2 X1X2 . 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.
4 Instrumental Variables
1. Consider the following model Y = p0 + p1X + 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 = p0 + p1X + y1W1 + ... + ykWk + 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
pr oducti vi ty = p0 + p1D + U,
where D i = 1 if employee i participates in the training and D i = 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 i ce = p0 + p11992 +U, where 1992 = 1 if in year 1992 and 1992 = 0 in 1990. Interpret p1 . 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 = p0 + p1Xi 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) \= 0. There are two transformations that can be used to consistently estimate p1 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.
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.
2022-11-10