Applied Microeconometrics. Maximilian Kasy

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Transcription:

Applied Microeconometrics Maximilian Kasy

7) Distributional Effects, quantile regression (cf. Mostly Harmless Econometrics, chapter 7)

Sir Francis Galton (Natural Inheritance, 1889): It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once.

Distributional Effects Most empirical research on treatment effects focuses on the estimation of differences in mean outcomes But methods exists for estimating the impact of a treatment on the entire distribution of outcomes: Does the intervention increase inequality? Does the intervention affect the distribution at all? Stochastic dominance? Methods for estimating distributional effects: Experiments: Compare the distributions of Y 0 and Y 1 Selection on observables: Quantile Regression Experiments with Non-compliance: Instrumental Variable Quantile Regression

Distributional Effects In an experiment with perfect compliance: Y 1, Y 0 D. To evaluate distributional effects in a randomized experiment, we can compare the distribution of the outcome for treated and untreated: F Y1 (y) = Pr(Y 1 y) = Pr(Y 1 y D = 1) = Pr(Y y D = 1) = F Y D=1 (y). Similarly, F Y0 (y) = F Y D=0 (y). We can use estimators: F Y1 (y) = 1 1{Y i y}, FY0 (y) = 1 1{Y i y}. N 1 N 0 D i =1 D i =0

Adult Women in JTPA National Study (other services) Cumulative distribution functions 0.25.5.75 1 0 20000 40000 60000 Total 30 Month Earnings cdf non assignees cdf assignees

Adult Women in JTPA National Study (other services) Cumulative distribution functions 0.25.5.75 1 Q ^ 0.50 (Y 1 ) Q ^ 0.50 (Y 0 )=10032 7152=2880 0 20000 40000 60000 Total 30 Month Earnings cdf non assignees cdf assignees

Adult Women in JTPA National Study (other services) Cumulative distribution functions 0.25.5.75 1 Q ^ 0.50 (Y 1 ) Q ^ 0.50 (Y 0 )=10032 7152=2880 Q ^ 0.75 (Y 1 ) Q ^ 0.75 (Y 0 )=21723 16484=5239 0 20000 40000 60000 Total 30 Month Earnings cdf non assignees cdf assignees

The Social Welfare Function Sometimes we want to compare the entire distributions, F Y0 and F Y1 Social Welfare Function : W (u, F ) = 0 u(y)df (y), where y is income, u(y) is utility, and F is the income distribution. We want to know if W (u, F Y1 ) W (u, F Y0 ). The problem is that u(y) is typically left unspecified However, we usually assume 1 u 0 (utility is increasing in income) 2 u 0 (utility is concave preference for redistribution)

Distributional Effects Equality (EQ) First order stochastic dominance (FSD) Second order stochastic dominance (FSD) F Y0 F Y0 F Y0 =F Y1 F Y1 F Y1 F Y1 (y) = F Y0 (y) F Y1 (y) F Y0 (y) y F Y 1 (x)dx y F Y 0 (x)dx 1 EQ implies W (u, F Y1 ) = W (u, F Y0 ) for all u 2 FSD implies W (u, F Y1 ) W (u, F Y0 ) for u 0 3 SSD implies W (u, F Y1 ) W (u, F Y0 ) for u 0 and u 0

Kolmogorov-Smirnov Test (EQ) Suppose that we have data from a randomized experiment. How can we test the null hypothesis H 0 : F Y1 = F Y0? Kolmogorov-Smirnov Statistic: ( N1 N 0 T eq = N ) 1/2 sup y F Y1 (y) F Y0 (y) If Y is continuous, then the distribution of T eq under H 0 is known If Y is not continuous (e.g., positive probability at Y = 0), we can use a bootstrap test: 1 Compute T eq in the original sample 2 Resample N 1 treated and N 0 non-treated from the pooled the samples of treated and non-treated. (In this way, we impose the null hypothesis that the distribution of Y is the same for the two groups.) Compute T eq,b for these two samples. 3 Repeat step 2 many (B) times. 4 Calculate the p-value as: p-value = 1 B B 1{T eq,b > T eq } b=1

Kolmogorov-Smirnov Test (FSD and SSD) The Kolmogorov-Smirnov bootstrap test of EQ can be easily adapted for FSD and SSD, just by changing the test statistics For first order stochastic dominance: ( ) N1 N 1/2 0 T fsd = sup ( N F Y1 (y) F Y0 (y)) y For second order stochastic dominance: ( ) N1 N 1/2 y 0 T ssd = sup ( N F Y1 (x) F Y0 (x))dx y

Conditioning on Covariates: Quantile Regression Identification Assumption 1 Assume that the θ-quantile of the distribution of Y given D and X is linear: Q θ (Y D, X ) = α θ D + X β θ. 2 D is randomized or there is selection on observables Identification Result (α θ, β θ ) = argmin (α,β) E[ρ θ (Y αd X β)] where ρ θ (λ) = (θ 1{λ < 0})λ identifies α θ, the effect of the treatment on the θ-quantile of the conditional distribution of the outcome variable: α θ = Q θ (Y D = 1, X ) Q θ (Y D = 0, X ) = Q θ (Y 1 D = 1, X ) Q θ (Y 0 D = 0, X ) = Q θ (Y 1 X ) Q θ (Y 0 X ).

Conditioning on Covariates: Quantile Regression Identification Assumption 1 Assume that the θ-quantile of the distribution of Y given D and X is linear: Q θ (Y D, X ) = α θ D + X β θ. 2 D is randomized or there is selection on observables Estimator The quantile regression estimator (Koenker and Bassett (1978)) is the sample analog: ( α θ, β θ ) = argmin (α,β) 1 N N i=1 ρ θ (Y i αd i X i β)

Conditioning on Covariates: Quantile Regression Recall: ρ 0.25 (Y α D X β) Quantile θ = 0.25 1 0.5 0 1 0 1 Y α D X β ρ 0.50 (Y α D X β) Quantile θ = 0.50 1 0.5 0 1 0 1 Y α D X β ρ 0.75 (Y α D X β) Quantile θ = 0.75 1 0.5 0 1 0 1 Y α D X β For example: ( α 0.5, β 0.5 ) = argmin (α,β) 1 N N i=1 Y i αd i X i β

Kolmogorov-Smirnov Tests with Instrumental Variables Identification Assumption 1 Independence: (Y 0, Y 1, D 0, D 1 ) Z 2 First Stage: 0 < P(Z = 1) < 1 and P(D 1 = 1) > P(D 0 = 1) 3 Monotonicity: D 1 D 0 = 1 Identification Result For any function h( ) (E h(y ) < ), E[h(Y )D Z = 1] E[h(Y )D Z = 0] E[D Z = 1] E[D Z = 0] E[h(Y )(1 D) Z = 1] E[h(Y )(1 D) Z = 0] E[(1 D) Z = 1] E[(1 D) Z = 0] = E[h(Y 1 ) D 1 > D 0 ], = E[h(Y 0 ) D 1 > D 0 ].

Kolmogorov-Smirnov Tests with Instrumental Variables Identification Result Let F Y1 D 1>D 0 (y) = E[1{Y 1 y} D 1 > D 0 ], F Y0 D 1>D 0 (y) = E[1{Y 0 y} D 1 > D 0 ]. Apply result in previous slide with h(y ) = 1{Y y} to obtain: F Y1 D 1>D 0 (y) = F Y0 D 1>D 0 (y) = E[1{Y y}d Z = 1] E[1{Y y}d Z = 0], E[D Z = 1] E[D Z = 0] E[1{Y y}(1 D) Z = 1] E[1{Y y}(1 D) Z = 0]. E[(1 D) Z = 1] E[(1 D) Z = 0] Sample counterparts can be used to estimate F Y1 D 1>D 0 (y) and F Y0 D 1>D 0 (y) Tells us how treatment affects different parts of the outcome distribution for compliers Bootstrap tests for inference

Earning for Veterans and Non-veterans 288 Journal of the American Statistical Association, March 2002 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 non veterans veterans 10,000 20,000 30,000 40,000 50,000 60,000 Annual earnings Figure 1. Empirical Distributions of Earnings for Veterans and Nonveterans. Therefore, we cannot draw causal inferences by simply com- about hypotheses (H.1) (H.3) for the subpopulation of com-

Earning for Veterans and Non-veterans (Compliers) Abadie: Bootstrap Tests for Distributional Treatment Effects 289 1 0.9 0.8 0.7 0.6 0.5 0.4 veterans 0.3 0.2 0.1 non veterans 0 10,000 20,000 30,000 40,000 50,000 60,000 Annual earnings Figure 2. Estimated Distributions of Potential Earnings for Compliers. 4. SUMMARY AND DISCUSSION OF POSSIBLE EXTENSIONS est. Estimates of cumulative distribution functions of potential outcomes for compliers show an adverse effect of military experience on the lower tail of the distribution of annual

I illustrate this method with an application to the study of Distributional the effects oftests veteran status on civilian earnings. Following Angrist (1990), use variation in veteran status induced by randomly assigned draft eligibility to identify the effects of inter- Table 1. Tests on Distributional Effects of Veteran Status on Civilian Earnings, p-values First-order Second-order Outcome Equality in stochastic stochastic variable distributions dominance dominance Annual earnings 01245 06260 07415 Weekly wages 02330 06490 07530 model estima in the (1988 promi in Pro Ano mator ble ap minim mated isoton Dykst

Quantile Regression with Instrumental Variables Identification Assumption 1 Conditional Independence of the Instrument: (Y 0, Y 1, D 0, D 1 ) Z X 2 First Stage: 0 < P(Z = 1 X ) < 1 and P(D 1 = 1 X ) > P(D 0 = 1 X ) 3 Monotonicity: P(D 1 D 0 X ) = 1 Estimate quantile regression for compliers: Estimator Using κ: Q θ (Y D, X, D 1 > D 0 ) = α θ D + X β θ ( α θ, β θ ) = argmin (α,β) N i=1 κ i ρ θ (Y i αd i X i β).

104 A. ABADIE, J. ANGRIST, AND G. IMBENS JTPA: Quantile RegressionTABLE II QUANTILE REGRESSION AND OLS ESTIMATES Dependent Variable: 30-month Earnings Quantile OLS 0.15 0.25 0.50 0.75 0.85 A. Men Training 3,754 1,187 2,510 4,420 4,678 4,806 (536) (205) (356) (651) (937) (1,055) % Impact of Training 21.2 135.6 75.2 34.5 17.2 13.4 High school or GED 4,015 339 1,280 3,665 6,045 6,224 (571) (186) (305) (618) (1,029) (1,170) Black -2,354-134 -500-2,084-3,576-3,609 (626) (194) (324) (684) (1,087) (1,331) Hispanic 251 91 278 925-877 -85 (883) (315) (512) (1,066) (1,769) (2,047) Married 6,546 587 1,964 7,113 10,073 11,062 (629) (222) (427) (839) (1,046) (1,093) Worked less than 13-6,582-1,090-3,097-7,610-9,834-9,951 weeks in past year (566) (190) (339) (665) (1,000) (1,099) Constant 9,811-216 365 6,110 14,874 21,527 (1,541) (468) (765) (1,403) (2,134) (3,896) B. Women Training 2,215 367 1,013 2,707 2,729 2,058 (334) (105) (170) (425) (578) (657) % Impact of Training 18.5 60.8 44.4 32.3 14.5 8.09 High school or GED 3,442 166 681 2,514 5,778 6,373 (341) (99) (156) (396) (606) (762) Black -544 22-60 -129-866 -1,446 (397) (115) (188) (451) (679) (869) Hispanic -1,151-31 -222-995 -1,620-1,503 (488) (130) (194) (546) (911) (992) Married -667-213 -392-758 -1,048-902 (436) (127) (209) (522) (785) (970) Worked less than 13-5,313-1,050-3,240-6,872-7,670-6,470 weeks in past year (370) (137) (289) (522) (672) (787) AFDC -3,009-398 -1,047-3,389-4,334-3,875 (378) (107) (174) (468) (737) (834) Constant 10,361 649 2,633 8,417 16,498 20,689 (815) (255) (490) (966) (1,554) (1,232)

QUANTILES OF TRAINEE EARNINGS 105 JTPA: Quantile RegressionTABLE with III IV QUANTILE TREATMENT EFFECTS AND 2SLS ESTIMATES Dependent Variable: 30-month Earnings Quantile 2SLS 0.15 0.25 0.50 0.75 0.85 A. Men Training 1,593 121 702 1,544 3,131 3,378 (895) (475) (670) (1,073) (1,376) (1,811) % Impact of Training 8.55 5.19 12.0 9.64 10.7 9.02 High school or GED 4,075 714 1,752 4,024 5,392 5,954 (573) (429) (644) (940) (1,441) (1,783) Black -2,349-171 -377-2,656-4,182-3,523 (625) (439) (626) (1,136) (1,587) (1,867) Hispanic 335 328 1,476 1,499 379 1,023 (888) (757) (1,128) (1,390) (2,294) (2,427) Married 6,647 1,564 3,190 7,683 9,509 10,185 (627) (596) (865) (1,202) (1,430) (1,525) Worked less than 13-6,575-1,932-4,195-7,009-9,289-9,078 weeks in past year (567) (442) (664) (1,040) (1,420) (1,596) Constant 10,641-134 1,049 7,689 14,901 22,412 (1,569) (1,116) (1,655) (2,361) (3,292) (7,655) B. Women Training 1,780 324 680 1,742 1,984 1,900 (532) (175) (282) (645) (945) (997) % Impact of Training 14.6 35.5 23.1 18.4 10.1 7.39 High school or GED 3,470 262 768 2,955 5,518 5,905 (342) (178) (274) (643) (930) (1026) Black -554 0-123 -401-1,423-2,119 (397) (204) (318) (724) (949) (1,196) Hispanic -1,145-73 -138-1,256-1,762-1,707 (488) (217) (315) (854) (1,188) (1,172) Married -652-233 -532-796 38-109 (437) (221) (352) (846) (1,069) (1,147) Worked less than 13-5,329-1,320-3,516-6,524-6,608-5,698 weeks in past year (370) (254) (430) (781) (931) (969) AFDC -2,997-406 -1,240-3,298-3,790-2,888 (378) (189) (301) (743) (1,014) (1,083) Constant 10,538 984 3,541 9,928 15,345 20,520 (828) (547) (837) (1,696) (2,387) (1,687)

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