IV Quantile Regression for Group-level Treatments, with an Application to the Distributional Effects of Trade

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1 IV Quantile Regression for Group-level Treatments, with an Application to the Distributional Effects of Trade Denis Chetverikov Brad Larsen Christopher Palmer UCLA, Stanford and NBER, UC Berkeley September / 31

2 Motivation In many applied micro settings, researcher has data on micro-level outcomes within a group and wishes to study effect of group-level treatment Examples: Effect of law, varying at state-by-year level, on individual wages within a state-by-year cell Effect of school-level policy on student outcomes within school-level Effect of market-level regulation on outcomes of firms within market-level OLS of outcome variable on group-level treatment measures effect of treatment on average outcome in group We want effect on distribution of outcomes We develop a practical method to study effect of group-level treatment on distribution of outcomes in group 2 / 31

3 Model The response variable y ig of individual i in group g satisfies the following quantile regression model: where Q yig z ig,x g,ε g (u) = z ig γ(u) + x gβ(u) + ε g (u), u U z ig is a vector of individual-level covariates x g is a vector of group-level covariates (contains constant) ε g (u) is unobserved group-level heterogeneity γ(u) and β(u) are vectors of coefficients U is the set of quantile indices of interest. We are interested in estimating β(u) for u U. 3 / 31

4 Endogeneity and Instruments Recall the model: Q yig z ig,x g,ε g (u) = z ig γ(u) + x gβ(u) + ε g (u), u U In empirical studies, x g can be related to ε g (u) for example, the decision to introduce a new unemployment benefit in a state, x g, depends on other state-level (potentially unobservable) characteristics, ε g (u), that affect the distribution of wages, y ig. We therefore assume that there exists an instrument w g that satisfies two conditions: 1 E[w g ε g (u)] = 0 (orthogonality) 2 E[w g x g] is non-singular (relevance) 4 / 31

5 Our model is quantile extension of Hausman and Taylor (1981) Hausman and Taylor (1981) linear panel model: y ig = z ig γ + x gβ + ε g + ν ig If x g is correlated with ε g, need group-level fixed effects α g = x gβ + ε g to identify γ ( within regression ) To estimate β, can regress fixed effect estimates, α g, on x g using an instrument w g ( between regression ) Sometimes w g can be an internal instrument within variation of z ig is used to estimate γ between variation of z ig is used to instrument for x g We consider quantile analog 5 / 31

6 Comparison of our model to other quantile regression panel data models Model of Koenker (2004), Kato, Galvao, and Montes-Rojas (2012), Kato and Galvao (2011): Q yig z ig,α g (u) = z ig γ(u) + α g(u) Assuming that the number of observations per group is large, estimate γ(u) and fixed effects α g (u) jointly by running a pooled quantile regression of y ig on covariates z ig and group-level dummies We are interested in understanding dependence of α g (u) = x gβ(u) + ε g (u) on x g We will also require the number of observations per group to be large 6 / 31

7 Our setting is different from other IV quantile settings IV quantile model of Chernozhukov and Hansen (2005, 2006): i.i.d. setting, no group structure Right-hand side covariates are correlated with unobserved rank variable (random version of quantile index u) Our model: Unobserved (individual-level) rank variable is independent of covariates Unobserved group-level, separately additive, heterogeneity, ε g (u), is present Right-hand side group-level covariate of interest x g is correlated with ε g (u) 7 / 31

8 Grouped IV Quantile Estimator Recall the model: Q yig z ig,x g,ε g (u) = z ig γ(u) + x gβ(u) + ε g (u), u U }{{} α g (u) where α g (u) is the group-level fixed effect Suppose we have data on G groups and N individuals in each group, where both G and N are large Our estimator β(u) of β(u) is 1 Perform a quantile regression of y ig on z ig and a constant for each group separately to obtain estimates γ g (u) and α g (u) of γ(u) and α g (u) for each g = 1,..., G 2 Perform 2SLS estimator of α g on x g using w g as an instrument to calculate β(u) 8 / 31

9 Standard errors We show that our estimator β(u) of β(u) is asymptotically zero-mean Gaussian The standard errors for the estimator can be obtained without accounting for the first-stage estimation error (under a mild condition on the growth of the number of observations N per group see below) thus, standard errors come from the classical 2SLS estimator traditional heteroscedasticity robust and/or clustered standard erros for the 2SLS estimator can be used 9 / 31

10 Benefits of our estimator over standard quantile regression Standard quantile regression inconsistent if ε g (u) = 0, even when x g ε g (u) ε g (u) akin to left-hand side measurement error LHS measurement error biases quantile regression (Hausman 2000; Hausman, Luo, and Palmer 2014) When dimension of x g large, standard quantile regression significantly slower Over 100 times slower than ours in simulations Standard errors in quantile regression more computationally burdensome (typically require bootstrap) Ours: Traditional robust or clustered standard errors 10 / 31

11 Interaction effects of group-level treatment with micro-level covariates Interaction effect model: Q yig z ig,x g,ε g (u) = γ 0 (u) + z ig (x gβ(u) + ε g (u)) where x g, ε g correlated and z ig micro-level covariate (scalar) Estimator: 1 In each group, run quantile regression and save coefficient on z ig 2 2SLS regression of coefficients on x g, instrumenting with w g 11 / 31

12 Theoretical properties of the estimator: substantial conditions 1 Design (i) Observations are independent across groups. (ii) For all g, the pairs (z ig, y ig ) are i.i.d. across i = 1,..., N conditional on (x g, ε g ). 2 Instruments (i) E[w g ε g (u)] = 0. (ii) G 1 G g=1 E[x gw g] Q xw and G 1 G g=1 E[w gw g] Q ww. (iii) The matrices Q xw and Q ww have singular values bounded from below and from above. (iv) y ig is independent of w g conditional on (z ig, x g, α g ). (v) E[ w g 4+δ ] is finite. 3 Growth Condition G 2/3 (log N)/N / 31

13 Theoretical properties of the estimator: other regularity conditions 4 Covariates (i) Random vectors z ig and x g are bounded. (ii) All eigenvalues of E g [z 1g z 1g ] are bounded. 5 Coefficients α g (u 2 ) α g (u 1 ) C L u 2 u 1. 6 Noise (i) E[sup u U ε g (u) 4+δ ] is finite. (ii) For some (matrix-valued) function J : U U R d w d w, G 1 G g=1 E[ε g(u 1 )ε g (u 2 )w g w g] J(u 1, u 2 ) uniformly over u 1, u 2 U. (iii) ε g (u 2 ) ε g (u 1 ) C L u 2 u 1. 7 Density Some standard conditions on the density of y ig appearing in the quantile regression literature. 8 Quantile indices The set of quantile indices U is a compact set included in (0, 1). 13 / 31

14 Theoretical properties of the estimator Theorem (Main convergence result) Let Assumptions 1-8 hold. Then G( β( ) β( )) G( ), in l (U) where G( ) is a zero-mean Gaussian process with uniformly continuous sample paths and covariance function C(u 1, u 2 ) = SJ(u 1, u 2 )S where S = ( ) 1 Q xw QwwQ 1 xw Qxw Qww 1 G 1 J(u 1, u 2 ) = lim G G E[ε g (u 1 )ε g (u 2 )w g w g] g=1 G 1 Q xw = lim G G E[x g w g], g=1 G 1 Q ww = lim G G E[w g w g]. g=1 14 / 31

15 Main growth condition Theorem requires that G 2/3 (log N)/N 0 The number of observations per group is allowed to be smaller than the number of groups. This is interesting because nonlinear panel data model studies typically require at least G/N c > 0. This is achieved by employing asymptotic unbiasedness of the quantile regression estimator via the Bahadur representation: α g (u) α g (u) = 1 N G 1 G w g ( α g (u) α(u)) = 1 g=1 N G N ψ ig (u) + O P (N 3/4 ), where E[ψ ig ] = 0, and so i=1 G g=1 N i=1 which is o P (1), yielding the growth condition. ( ) G w g ψ ig (u) + O P N 3/4, 15 / 31

16 Estimation of covariance Let Ĉ(u 1, u 2 ) = ŜĴ(u 1, u 2 )Ŝ Ŝ = ( ˆQ xw ˆQ 1 ww ˆQ xw) 1 ˆQ xw ˆQ 1 ww Ĵ(u 1, u 2 ) = 1 G ˆQ xw = 1 G G g=1 ( (ˆα g (u 1 ) x g ˆβ(u 1 ))(ˆα g (u 2 ) x g ˆβ(u 2 ))w g w g G x g w g, and ˆQ ww = 1 G G w g w g. g=1 g=1 We show that Ĉ(u 1, u 2 ) is consistent for C(u 1, u 2 ) uniformly over u 1, u 2 U. Theorem (Estimating C(, )) Under the same conditions as those in Theorem 1, Ĉ(u 1, u 2 ) C(u 1, u 2 ) = o p (1) ) uniformly over u 1, u 2 U. 16 / 31

17 Simultaneous confidence bands Thus, point-wise standard errors for our estimator can be constructed using traditional heteroscedasticity robust approaches for 2SLS estimator (extension to clustered standard errors is also available) We can also construct simultaneous confidence bands covering the whole function {β j (u), u U}. Indeed, take a statistic G β j (u) β j (u) T = sup u U Ĉjj (u, u) Simultaneous confidence bands with coverage probability α are Ĉ jj (u, u) Ĉ jj (u, u) β j (u) c α, β j (u) + c α G G where c α is the (1 α)th quantile of T. 17 / 31

18 Simultaneous confidence bands: multiplier bootstrap procedure The bands above are infeasible because c α is unknown. We use the multiplier bootstrap method to estimate it: 1 Generate i.i.d. sequence of N(0, 1) random variables {e i, 1 i n} that are independent of the data 2 Define the multiplier bootstrap statistic T MB = sup u U G 1 ( e g ( α g x β(u)) ) g (Ŝw g ) j ) GĈ jj (u, u) g=1 3 Define the multiplier bootstrap estimate of c α ĉ α = (1 α) quantile of distribution of T MB given the data Using results in Chernozhukov, Chetverikov, Kato (2013, 2014a, 2014b, 2015), we can show that ĉ α is a good estimator of c α 18 / 31

19 Simultaneous confidence bands Theorem (Validity of Simultaneous Confidence Bands Based on MB Procedure) Let Assumptions 1-8 hold. In addition, suppose that all eigenvalues of J(u, u) are bounded away from zero uniformly over all u U. Then P β Ĉ j (u) ĉ jj (u,u) Ĉ 1 α G β j (u) β j (u) + ĉ jj (u,u) 1 α G 1 α. for all u U 19 / 31

20 Monte Carlo simulation Let y ig = z ig γ(u ig ) + x g β(u ig ) + ε g (u ig ) The variable x g is correlated with ε g, where x g = πw g + η g + ν g ε g (u) = uη g u/2 w g, ν g, z ig exp(0.25 N[0, 1]); u ig, η g U[0, 1] Generate data with G {25, 200}, N {25, 200} Estimate β( ) using traditional quantile regression and using grouped IV quantile regression Also examine case where x g is exogenous (η g doesn t enter first stage) and case with η g = 0 (no group-level unobservables) 20 / 31

21 Bias of grouped IV quantile estimator relative to standard quantile regression Endogenous x Exogenous x No group-level unobservables (N,G) Q reg Grouped IV Q. Reg. Q reg Grouped IV Q. Reg. Q reg Grouped IV Q. Reg. (25,25) (200,25) (25,200) (200,200) / 31

22 Several recent papers apply our estimator Example applications 1 Angrist and Lang (2004), studies Boston s Metco program, looks at impact on lower tail of student outcomes by school 2 Palmer (2012) studies effects of suburbanization at the city level on within-city distribution of outcomes 3 Larsen (2014) studies effect of occupational licensing on distribution of teacher quality 4 Backus (2015) studies question of whether competition increases productivity through weeding out less-productive firms (affecting mainly lower tail of productivity) or increasing productivity of all firms 22 / 31

23 Our application: The effect of increased import competition on the distribution of local wages Background: Wage inequality increased drastically over past 40 years Heated debates as to cause (globalization vs. skill-biased technological change vs. declining real minimum wage) Autor, Dorn, and Hanson (2013) (ADH) show local labor markets with greater emphasis on manufacturing had greater decrease in average local wage ADH instrument for Chinese import competition in US with Chinese import competition in other developed countries 23 / 31

24 Applying grouped IV quantile regression in the ADH framework A group is local labor market ( commuting zone ) ADH have micro-level data on individual wages for many workers in each group ADH compute average wage in group, regress change in group-level average wages on Chinese import competition via 2SLS Our approach: compute group-level quantiles rather than average, then follow ADH We can quantify effect of Chinese import competition on distribution of local wages 24 / 31

25 Effect of Chinese Import Competition on Conditional Wage Distribution: Full Sample Units = change in log points due to $1,000 change in Chinese imports per US worker Quantile Point Estimate ADH Estimate 95% Confidence Interval ADH 95% Confidence Interval 25 / 31

26 Effect of Chinese Import Competition on Conditional Wage Distribution: Males Only Units = change in log points due to $1,000 change in Chinese imports per US worker Quantile Point Estimate ADH Estimate 95% Confidence Interval ADH 95% Confidence Interval 26 / 31

27 Effect of Chinese Import Competition on Conditional Wage Distribution: Females Only Units = change in log points due to $1,000 change in Chinese imports per US worker Quantile Point Estimate ADH Estimate 95% Confidence Interval ADH 95% Confidence Interval 27 / 31

28 Conclusion Computationally simple estimator for effects of group-level treatment on distribution of outcomes within group When researcher has outcome data on individuals within a group, and the variable of interest varies at the group level, estimator is 1 In each group, run quantile regression and save coefficient on the constant 2 2SLS regression of coefficients on x g, instrumenting with w g If no micro-level covariates, step (1) replaced by simply computing quantile (e.g. median, 20th percentile, etc.) within group If no endogeneity, step (2) replaced by OLS Standard errors simple: standard approaches for OLS/2SLS Much faster than standard quantile regression even when both valid 28 / 31

29 Thank you 29 / 31

30 References, I 1 Angrist, J. and Lang, K. (2004). Does school integration generate peer effects? Evidence from Boston s Metco Program. American Economic Review, 94: Autor, D., Dorn, D., and Hanson, G. (2013). The China syndrome: Local labor market effects of import competition in the United States. American Economic Review, 103: Backus, M. (2014). Why is productivity correlated with competition? Working paper. 4 Chernozhukov, V., Chetverikov, D., Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. The Annals of Statistics, 41: Chernozhukov, V., Chetverikov, D., Kato, K. (2014a). Gaussian approximation of suprema of empirical processes. The Annals of Statistics, 42: Chernozhukov, V., Chetverikov, D., Kato, K. (2014b). Anti-concentration and honest, adaptive confidence bands. The Annals of Statistics, 42: Chernozhukov, V., Chetverikov, D., Kato, K. (2015). Comparison and anti-concentration bounds for maxima of gaussian random vectors. Probability Theory and Related Fields, 162: Chernozhukov, V. and Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica, 73: Chernozhukov, V. and Hansen, C. (2006). Instrumental quantile regression inference for structural and treatment effect models. Journal of Econometrics, 132: / 31

31 References, II 10 Hausman, J. (2001). Mismeasured variables in econometric analysis: Problems from the right and problems from the left. Journal of Economic Perspectives, 15: Hausman, J. and Taylor, W. (1981). Panel data and unobservable individual effects. Econometrica, 49: Hausman, J., Luo, Y., and Palmer, C. (2014). Errors in the dependent variable of quantile regression models. Working paper. 13 Kato, K. and Galvao, A. (2011). Smoothed quantile regression for panel data. Working paper. 14 Kato, K., Galvao, A., and Montes-Rojas, G. (2012). Asymptotics for panel quantile regression models with individual effects. Journal of Econometrics, 170: Koenker, R. (2004). Quantile regression for longitudinal data. Journal of Multivariate Analysis, 91: Larsen, B. (2014). Occupational licensing and quality: Distributional and heterogeneous effects in the teaching profession. Working paper. 17 Palmer, C. (2011). Suburbanization and urban decline. Working paper. 31 / 31