Econometrics Problem Set WISE, Xiamen University Spring 207 Conceptual Questions. (SW 2.) This question refers to the panel data regressions summarized in the following table: Dependent variable: ln(q cigarettes i,995 ) ln(q cigarettes i,985 ) Regressor () (2) (3) ln(p cigarettes i,995 ) ln(p cigarettes i,985 ) 0.94.34.20 (0.2) (0.23) (0.20) ln(inc cigarettes i,995 ) ln(inc cigarettes i,985 ) 0.53 0.43 0.46 (0.34) (0.30) (0.3) Intercept 0.2 0.02 0.05 (0.07) (0.07) (0.06) Instrumental variable(s) Sales tax Cigarette-specific tax Both sales tax and cigarette-specific tax First stage F -statistic 33.70 07.20 88.60 Overidentifying restrictions 4.93 J -test and p-value (0.026) (a) Suppose that the federal government is considering a new tax on cigarettes that is estimated to increase the retail price by $0.0 per pack. If the current price per pack is $2.00, use the regression in column () to predict the change in demand. Construct a 95% confidence interval for the change in demand. Solution: The change in the regressor, ln(p cigarettes i,995 ) ln(p cigarettes i,985 )), from a $0.0 per pack increase in the retail price is ln 2.0 ln 2.00 0.0488. The expected percentage change in cigarette demand is 0.94 0.0488 00% 4.5872%. The 95% confidence interval is ( 0.94±.96 0.2) 0.0488 00% [ 6.60%, 2.58%]. (b) Suppose that the United States enters a recession and income falls by 2%. Use the regression in column () to predict the change in demand. Solution: The change in demand is 0.53-2%-.06%. If income falls by 2%, demand will decline by.06%. (c) Recessions typically last less than one year. Do you think that the regression in column () will provide a reliable answer to the question in (b)? Why or why not?
Solution: No, the time interval used in this regression covers 0 years, the effect of the recession will disappear during this long period. So it is not reliable. (d) Suppose that the F -statistic in column () was 3.7 instead of 33.7. Would the regression provide a reliable answer to the question posed in (a)? Why or why not? Solution: No, the two stage least square regression results suffer from weak instruments since the first stage regression F -statistic is below 0. So it is not reliable. 2. (SW 2.4) Consider TSLS estimation with a single included endogenous variable and a single T SLS instrument. Show that ˆ s ZY s ZX. Solution: As we know T SLS ˆ s ˆXY, and ˆX s 2ˆX i ˆπ 0 + ˆπ Z i. Then: s ˆXY ˆπ s ZY ( ˆX i ˆX)(Yi Ȳ ) (ˆπ 0 + ˆπ Z i ˆπ 0 ˆπ Z)(Yi Ȳ ) ˆπ (Z i Z)(Y i Ȳ ) and, s 2ˆX ˆπ 2 s 2 Z. ( ˆX i X) 2 2 (ˆπ 0 + ˆπ Z i ˆπ 0 ˆπ Z) ˆπ (Z 2 i Z) 2 Thus, we have since ˆπ s ZX. s 2 Z ˆβ T SLS ˆπ s ZY ˆπ 2 s 2 Z s ZY ˆπ s 2 Z s ZY s ZX, 3. (SW 2.5) Consider the instrumental variable regression model Y i β 0 + X i + β 2 W i + u i Page 2
where X i is correlated with u i and Z i is an instrument. Suppose that E(u i W i ) 0; (X i, W i, Z i, Y i ) are i.i.d. draws from their joint distribution; and the random variables X, W, Z, and Y have nonzero finite fourth moments. Which IV assumption is not satisfied (if any) when: (a) Z i is independent of (Y i, X i, W i )? Solution: Recall that instrument validity requires ( ˆX i,..., ˆX ki, W i,..., W ri, ) are not perfectly multi-collinear (where ˆX ji is the predicted value of X ji from the population regression of X ji on the instruments (Z s) and the included exogenous regressors (W s)). corr(z i, u i )... corr(z mi, u i ) 0. If Z i is independent of (Y i, X i, W i ) then Z i has no relation with X i, violating instrument relevance. (b) Z i W i? Solution: If Z i W i then Z is not a valid instrument. ˆX will be perfectly collinear with W, violating instrument relevance. (Alternatively, the first stage regression suffers from perfect multicollinearity.) (c) W i for all i? Solution: If W i for all i then W is perfectly co-linear with the constant term, violating instrument relevance. Alternatively, you could say that W i has a zero fourth moment. (d) Z i X i? Solution: If Z i X i, cov(z i, u i ) 0 by assumption, violating instrument exogeneity. 4. (SW 2.7) In an instrumental variable regression model with one regressor, X i, and two instruments, Z i and Z 2i, the value of the J-statistic is J 8.2. (a) Does this suggest that E(u i Z i, Z 2i ) 0? Explain. Solution: Under the null hypothesis of instrument exogeneity, the J-statistic is distributed as a χ 2 random variable, with a % critical value of 6.63. Thus the statistic is significant, and instrument exogeneity E(u i Z i, Z 2i ) 0 is rejected. Page 3
(b) Does this suggest that E(u i Z i ) 0? Explain. Solution: The J-test suggests that E(u i Z i, Z 2i ) 0, but doesn t provide evidence about whether the problem is with Z or Z 2 or both. (c) Does this suggest that E(X i Z i, Z 2i ) E(X i )? Explain. Solution: No, the J-statistic is a test of exogeneity. It says nothing about relevance. (d) Does this suggest that E(X i Z i ) E(X i )? Explain. Solution: No, the J-statistic is a test of exogeneity. It says nothing about relevance. 5. (SW 2.8) Consider a product market with a supply function Q s i β 0 + P i + u s i, a demand function Q d i γ 0 + u d i and a market equilibrium condition Q s i Q d i, where u s i and u d i are mutually independent i.i.d. random variables, both with a mean of zero. (a) Show that P i and u s i are correlated. Solution: Since in equilibrium Q s i Q d i Q i, β 0 + P i + u s i γ 0 + u d i. Thus, Thus, P i γ 0 β 0 + ud i u s i. cov(p i, u s u) γ 0 β 0 + ud i u s i cov( γ 0 β 0 + ud i u s i, u s i ) σ2 u s (b) Show that the OLS estimator of is inconsistent. Solution: Recall that p ˆ + σ P u s σ 2 P Page 4
Since σp 2 σ2 + σ 2 u d u d, β 2 p ˆ σu 2 s σu 2 +. s σ2 u d (c) How would you estimate β 0, and γ 0? Solution: Note that in equilibrium Q i γ 0 + u d i. Thus γ 0 can be estimated as the sample average of Q. Furthermore, cov(q i, u s i ) cov(γ 0 + u d i, u s i ) 0. We can rewrite the demand equations as P i β 0 / + Q i / u s i /. Since, cov(q i, u s i ) 0, regressing P on Q will give an unbiased estimate of /. This, and the estimated intercept can be used to estimate β 0. This is a form of indirect least squares. In a sense, we can use Q as an instrument to estimate the demand equation as Q is correlated with P but uncorrelated with the u s i (demand is completely inelastic). 6. Consider the regression model Y i β 0 + X i +u i where X i π 0 +π Z i +v i, cov(x i, u i ) 0, cov(z i, X i ) 0 and cov(z i, u i ) 0. T SLS (a) Is ˆ an unbiased estimator of if E(v i Z i ) 0? Solution: Yes, the proof of unbiasedness shown in class makes no assumption regarding the E(v i Z i ). This can be seen by the following: cov(y i, Z i ) cov(β 0 + X i + u i, Z i ) cov(x i, Z i ). (b) Now consider the regression model, Y i β 0 + X i + β 2 W i + u i where X i π 0 + π Z i + π 2 W i + v i, cov(x i, u i ) 0, cov(z i, X i ) 0 and cov(z i, u i ) cov(w i, u i ) 0. T SLS i. Assume that ˆ is calculated by omitting W i in both the first and second stage of the regression. Under what conditions is the TSLS estimator a consistent estimator of? Solution: Instrument relevance is assured by the assumption that cov(z i, X i ) 0. Instrument exogeneity requires that cov(w i, Z i ) 0. This can be seen by the following: cov(y i, Z i ) cov(β 0 + X i + β 2 W i + u i, Z i ) cov(x i, Z i ) + β 2 cov(w i, Z i ). Since sample covariance is a consistent estimator of population covariance, then if cov(w i, Z i ) 0 s Y Z s XZ p cov(y, Z) cov(x, Z) cov(x, Z) + β 2 cov(w, Z) cov(x, Z). Page 5
ii. What are the advantages of including exogenous regressors (W s) in a TSLS regression? Solution: Including W increases the R 2 in the first stage of the regression, allowing to be more precisely estimated. Including W can make an otherwise invalid instrument valid by ensuring its exogeneity. Empirical Questions For these empirical exercises, the required datasets and a detailed description of them can be found at www.wise.xmu.edu.cn/course/gecon/written.html. 7. (SW E2.) During the 880s, a cartel known as the Joint Executive 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 890, 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 pricesetting agreement. In this exercise, you will use variations in supply associated with the cartel s collapse to estimate the elasticity of demand for rail transport of grain. Suppose the demand curve for rail transport of grain is specified as 2 ln(q i ) β 0 + ln(p i ) + β 2 Ice i + β 2+j Seas j,i + u i where Q i is the total tonnage of grain shipped in week i, P i is the price of shipping a ton of grain by rail, Ice i is a binary variable that is equal to if the Great Lakes are not navigable because of ice, and Seas i is a binary variable that captures seasonal variation in demand. Ice is included because grain could also be transported by ship when the Great Lakes were navigable. Use the dataset JEC to carry out the following exercises. j Solution: The below code initialises the dataset. # load AER package library(aer) # load CartelStability dataset data(cartelstability) # Create log variables CartelStability$lq<-log(CartelStability$quantity) CartelStability$lp<-log(CartelStability$price) ## Page 6
The following table summarizes the regressions used to answer the questions. Regressor OLS 2SLS ln(p rice) 0.639 0.867 (0.073) (0.34) Ice 0.448 0.423 (0.35) (0.35) Seas and Intercept Not Shown Not Shown First Stage F -statistic 83.0 Robust standard errors in parentheses; 328 observations. significant at p <.0; p <.00 (a) Estimate the demand equation by OLS. What is the estimated value of the demand elasticity and its standard error? Solution: # OLS regression and results ma<-lm(lq~lp+ice+factor(season),datacartelstability) coeftest(ma, vcovvcovhc(ma, type"hc")) ## From the OLS column of the above table, the estimated demand elasticity is 0.639479, and its heteroskedasticity consistent standard error is 0.073606. (b) Explain why the interaction of supply and demand could make the OLS estimator of the elasticity biased. Solution: The OLS estimator of the elasticity suffers from simultaneous causality bias, because the regression model suffers from endogeneity (ln(p i )) and the regression error term are correlated, which is caused by the fact that price and quantity are determined by the interaction of demand and supply. (c) Consider using the variable cartel as an instrumental variable for ln(p ). Use economic reasoning to argue whether cartel plausibly satisfies the two conditions for a valid instrument. Solution: When considering using cartel as an instrumental variable for ln(p ), you need to consider () whether cartel is correlated with ln(p ) and (2) whether it is uncorrelated with the regression error terms. We find that cartel is a plausible IV variable, because on one hand cartel determines the selling price (condition is satisfied), and on the other hand cartel does not affect consumer demand (condition 2 is satisfied). (d) Estimate the first-stage regression. Is cartel a weak instrument? Page 7
Solution: # First stage F statistic md<-lm(lp ~ cartel+ice+factor(season),datacartelstability) linear.hypothesis(md, "cartelyes0", vcovvcovhc(md, type "HC")) ## In the first-stage, we check the relevance of cartel and ln(p ). Because the F - statistics 82.66 > 0, we decide that cartel is not a weak instrument. (e) Estimate the demand equation by instrument variable regression. What is the estimated demand elasticity and its standard error? Solution: # IV regression me<-ivreg(lq~lp+ice+factor(season) cartel+ice+factor(season), datacartelstability) coeftest(me, vcov vcovhc(me, type "HC")) ## From the instrumental variable regression shown in the 2SLS column of the above table, the estimated demand elasticity is -0.86, and the heteroskedasticity consistent standard error is 0.34. (f) Does the evidence suggest that the cartel was charging the profit-maximizing monopoly price? Explain. (Hint What should a monopolist do if the price elasticity is less than?) Solution: If demand price elasticity is less than, then a % increase in price will cause demand to decrease by less than %, hence total revenue will increase, total cost will fall and profit would increase. This suggests that the firms are not acting collusively as a single monopolist since such a monopoly would charge a lower price. Page 8