John C. Chao Department of Economics University of Maryland Spring 2017 Due: 2/28/2017 Economics 423 Econometrics II Problem Set 1 Question 1 Suppose that Yi = 0 + 1 Xi + ui , where (Xi , ui ) are i.i.d., and Xi is a Bernoulli random variable with Pr (Xi = 1) = 0.20. Moreover, when Xi = 1, ui N (0, 4) and when Xi = 0, ui N (0, 1). (a) Show that the following assumptions are satised for this model (i) E [ui |Xi ] = 0 with probability one. (ii) Large outliers are unlikely; more precisely, E Xi4 < and E u4i < . (b) Does the error term of this model exhibit conditional heteroskedasticity? Does the error term of this model exhibit unconditional heteroskedasticity? Justify your answers with the needed calculations. (c) Derive an explicit expression for the (approximate) large sample variance of the OLS estimator 1 in this case. (Hint: Evaluate the terms of the more general formula 2 1 1 V ar [(Xi X ) ui ] , X = E [Xi ] , n [V ar (Xi )]2 for the specic model considered here.) 1 Question 2 Consider the regression model with a single regressor: Yi = 0 + 1 Xi + ui . Suppose that the following assumptions are satised with respect to this model. 1. E [ui |Xi ] = 0 with probability one. 2. (Xi , ui ), i = 1, ..., n, are independent and identically distributed (i.i.d.) draws from their joint distribution, with V ar (Xi ) > 0 and V ar (ui ) > 0. 3. Large outliers are unlikely, i..e., E Xi4 < and E u4i < (a) Show that in this case Xi is a valid instrument. That is, show that the conditions Cov (Zi , Xi ) = 0 (instrument relevance), Cov (Zi , ui ) = 0 (instrument exogeneity) are satised with Zi = Xi . (b) Show that the IV (or TSLS) estimator constructed using Zi = Xi in this case is identical to the OLS estimator. 2 Question 3 Consider the instrumental variable regression model. Yi = 0 + 1 Xi + 2 Wi + ui , where Xi is correlated with ui and Zi is a possible instrument. Suppose that the following conditions hold. 4. E [ui |Wi ] = 0 with probability one. 5. (Xi , Wi , Zi , ui ), i = 1, ..., n, are independent and identically distributed (i.i.d.) draws from their joint distribution. 6. Large outliers are unlikely; more precisely, E Xi4 < , E Wi4 < , E Zi4 < , and E u4i < . Which IV assumption is not satistied when (a) Zi is independent of (Yi , Xi , Wi )? (b) Zi = Wi ? (c) Wi = 1 for all i? (d) Zi = Xi ? 3 Question 4 During the 1880s, a cartel known as the Joint Executives Committee (JEC) controlled the rail transport of grain from the Midwest to eastern cities in the United States. The cartel preceded the Sherman Antitrust Act of 1890, and it legally operated to increase the price of grain above what would have been the competitive price. From time to time, cheating by members of the cartel brought about a temporary collapse of the collusive price-setting agreement. In this exercise, you will use variations in supply associated with the cartel's collapses to estimate the elasticity of demand for rail transport of grain. The data set needed to do this problem can be downloaded from the Econ 423 course space on ELMS (URL:elms.umd.edu) as JEC.dta (Stata format) or JEC.xls (Excel format). A detailed decription of the data is contained in the le JEC Description.pdf which can also be downloaded from the Econ 423 course space on ELMS. Suppose that the demand curve for rail transport of grain is specied as 12 ln (Qi ) = 0 + 1 ln (Pi ) + 2 Icei + 2+j Seasj,i + ui , j=1 where Qi is the total tonnage of grain shipped in week i, Pi is the price of shipping a ton of grain by rail, Icei is a binary variable that is equal to 1 if the Great Lakes are not navigable because of ice, and Seasj,i is a binary variable that captures seasonal variation in demand. Icei is included because grain could also be transported by ship when the Great Lakes were navigable. (a) Estimate the demand equation by OLS. What is the estimated value of the demand elasticity and its standard error? (b) Explain why the interaction of supply and demand could make the OLS estimator of the elasticity biased? (c) Consider using the variable cartel as instrumental variable for ln (P ). Use economic reasoning to argue whether cartel plausibly satises the two conditions for a valid instrument. (d) Estimate the rst-stage regression. Is cartel a weak instrument? (e) Estimate the demand equation by instrumental variable regression. What is the estimated demand elasticity and its standard error? (f) Does the evidence suggest that the cartel was charging the prot-maximizing monopoly price? Explain. (Hint: What should a monopolist do if the price elasticity is less than 1?) 4 Question 5 How does fertility aect labor supply? That is, how much does a woman's labor supply fall when she has an additional child? In this exercise you will estimate this eect using data for married women from the 1980 U.S. Census. The data set can be downloaded from the Econ 423 course space on ELMS as fertility.dta (Stata format) or fertility.xlsx (Excel format). The data set contains information on married women aged 21-35 with two or more children. More information about the data set is given in the le Fertility Description.pdf which can also be downloaded from the Econ 423 course space on ELMS. (a) Regress weekworked on the indicator variable morekids using OLS. On average, do women with more than two children work less than women with two children? How much less? (b) Explain why the OLS regression estimated in (a) is inappropriate for estimating the causal eect of fertility (morekids) on labor supply (weeksworked). (c) The data set contains the variable samesex, which is equal to 1 if the rst two children are of the same sex (boy boy or girl girl) and equal 0 otherwise. Are couples whose rst two children are of the same sex more likely to have a third child? Is the eect large? Is it statistically signicant? (d) Explain why samesex is a valid instrument for the instrumental variable regressin of weeksworked on morekids. (e) Is samesex a weak instrument? (f) Estimate the regression of weeksworked on morekids using samesex as an instrument. How large is the fertility eect on labor supply? (g) Do the results change when you include the variables agem1, black, hispan, and othrace in the labor supply regression (treating these variable as exogenous)? Explain why or why not. 5