ECMT6002/6702: Econometric Applications Semester 2, 2022, Final Exam

This test is worth 50 marks in total. Please answer all five (5) questions.

Note:  When performing statistical tests, always state the null and alternative hypotheses, the test statistic and its distribution under the null hypothesis, the rejection rule, level of sig- nificance and the conclusion of the test.

Statistical tables are located at the end of the test.

Question 1 [10 Marks].

(i) Under what circumstances is the quantile regression model useful? (1 mark) (ii) What is another name for the quantile regression with τ = 50%? (1 mark)

(iii) Interpret the slope coefficients in the quantile regression model when τ = 0.1 (the 10% quantile), and τ = 0.9 (the 90% quantile) [1 mark]

(iv) When are the slope coefficients in the quantile regression model likely to be the same as the

OLS estimators? [1 mark]

(v) When considering a time series regression model in levels, will the interpretation change for the slope parameters if I first difference the variables and then run the regression? Will the the interpretation change for the intercept in the models? [1 mark]

Consider the following time-series regression model:

log(yt) = β0 + β1x1,t+ β2x2,t+ γtrend + et          t = 1, · · · , T                              (1)

(vi) Write down model (1) in first difference form. [1 mark]

(vii) How do I test if yt , x1,t, and x2,t are stationary? Be clear about the test and where to get the critical statistic for the test. (1 mark)

(viii) If yt , x1,t, and x2,t  are not trend-stationary, which model is better, the model in levels or first-difference. Explain. (1 mark)

(ix) For your differenced model in (vi) add 2 lags to each explanatory variable and write down this model. How would you test the distributed lag model is appropriate? In your model with lags what is the total effect of a permanent 1 unit change in x1 on y? Is ∆y permanently affected by the permanent increase in x1 ? [2 mark]

Question 2 [10 Marks].

(i) Explain the central idea around maximum likelihood estimation (MLE)? (1 mark) (ii) What is the main problem with the Linear Probability Model (LPM)? (1 mark)

(iii) Why can MLE be used to model probabilities without the problem the LPM has? [1 mark]

(iv) Two common methods for a binary probability model are Logit and Probit. What is meant by a binary probability model? Explain the major difference between Probit and Logit. [2 marks]

(v) For a binary event, where y = 1 if event occurs and y = 0 if it doesn’t , let G(βxi) be the conditional probability of yi  = 1, given xi  . Suppose I have three random sample observa- tions: {(y1, x1 ), (y2, x2 ), (y3, x3 )}. Write down the full likelihood function and log-likelihood function for this sample of three observations? [3 marks]

(vi) Name two ways we can compare competing probit models? [1 mark]

(vii) How should you test multiple exclusions in the probit model and which distribution do we use in the test? [1 mark]

Question 3 [10 Marks].

(i) When using an IV regression due to an endogenous explanatory variable, what are the two assumptions the instrument must satisfy?  Explain if each assumption can be tested.  (2 marks)

(ii) The below expressions can be used to explain why IV bias can be much larger than OLS.

Explain why this can occur using the equations, paying special attention to the necessary IV assumptions. [2 marks]

OLS

plim1,OLS = β 1 + Corr (x, u) ×

IV

plim1,IV = β 1 +  ×

(iii) State the asymptotic variance for both the OLS and IV estimators. Explain why the asymp- totic variance is larger for the IV estimator. What effect does a weak instrument have?  [2 marks]

(iv) How do you test for exogeneity of a suspected endogenous explanatory variable? What is the logic of this test? [2 marks]

(v) If their is doubt the IV is unbiased, what are the disadvantages of using IV regression com- pared to OLS? (1 mark)

(vi) If IV and OLS are both consistent, why test for endogeneity? Explain. [1 mark]

Question 4 [10 Marks].

(i) What is a major advantage of a panel dataset over a single or pooled cross-sectional dataset. (1 mark)

You have a panel with i = 1, · · · ,n and t = 1, · · · ,T . The dependant variable is yi,t, and there are 2 explanatory variables x1,i,t and x2,i,t

(ii) Write out the fixed effects (FE) regression, where you control for unobserved time invariant variables. (2 marks)

(iii) Use the expression in (ii) to construct the "within estimator" which FEs implicitly estimates. Show all your working. (2 marks)

(iv) Is the fixed effects regression equivalent to the dummy variable regression?  Explain.  (1 mark)

(v) When is the first difference (FD) estimator and the FE estimator identical? Show that this is the case. [Hint: start with the "within form of the FE regression.] [2 marks]

(vi) In the fixed effects regression model, what variables can’t be included in the model? Is this an advantage or a disadvantage? What model overcomes this problem but what assumption is required to use it? How likely is it that this assumption holds in practice? (2 marks)

Question 5 [10 Marks]. Suppose you have a panel data set with n=1000 lawyers over T =3 years with data on annual earnings, gender, years of experience after graduation, and age. You want to estimate the gender wage gap and specify the following regression:

log(earnings)i,t = ai+ δ1 d 2 + δ2 d 3 + β 1f emalei,t+ β2 expi,t+ β3 agei,t + ui,t                       (2)

where dT is a dummy for period T, f emale is a dummy for female, exp is years of experience and age is lawyers age.

(i) Can you estimate the above model by a fixed effects regression? Why or why not? (1 mark)

(ii) What would happen if you left out ai  (implicitly setting ai  = 0 for all i) and estimated (2) using OLS with a single intercept? (1 mark)

(iii) Write down a new panel model which identifies how the gender wage gap changes over time. What effects will your expression calculate? Explain why you modelled this way. (3 marks)

We are interested in analysing the effect of the government locating an airport in Timbuktu on house prices in the suburbs on the flight path close to the runway. Rumours that a new airport would be built occurred at the start of 2004, and the airport began operating in 2005. We have data on the prices of houses sold in in 2003 (the before period) and another sample on houses that sold in 2007 (the after period). The hypothesis we wish to test is that the price of houses located on the flight path close to the airport have fallen relative to the price of other houses.

The data for each year includes the dummy variable f lightpath which is equal to one if the house is located on the direct flight path within 7kms of the airport.  House prices, for both years of data, were measured in 2003 prices. The variable rprice denotes the real house price (scaled by $100,000). The following simple regression model was estimated using only 2007 sample data:

r—price = 9.168 − 3.0902f lightpath

(0.311)   (0.4923)

n = 221       R2 = 0.169

The regression using only 2003 sample data is:

r—price = 8.2620 − 1.8725f lightpath

(0.269)   (0.3271)

n = 198       R2 = 0.098

(iv) What is the average real house price on the f lightpath in 2003 and 2007? (1 mark)

(v) What is the average real house price N OT on the f lightpath in 2003 and 2007? (1 mark)

(vi) What is the estimated effect on real house prices on the flight path from building the airport? (1 mark)

(vii) Write down a regression model that estimates the effect of the airport on house prices on the flight path. What is this model called? (2 marks)