# ESTIMATING AVERAGE TREATMENT EFFECTS: IV AND CONTROL FUNCTIONS, II Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics

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1 ESTIMATING AVERAGE TREATMENT EFFECTS: IV AND CONTROL FUNCTIONS, II Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July Quantile Treatment Effects 2. Control Functions for Some Continuous Treatments 3. Semi- and Nonparametric CF Approaches 1

2 1. Quantile Treatment Effects Abadie, Angrist, and Imbens (2002) consider binary endogenous treatment, say D, and binary instrumental variable, say Z. The potential outcomes are Y d, d 0, 1 that is, without treatment and with treatment, respectively. The counterfactuals for treatment are D z, z 0, 1. Observed are X, Z, D 1 Z D 0 ZD 1, and Y 1 D Y 0 DY 1. AAI study treatment effects for compliers, that is, the (unobserved) subpopulation with D 1 D 0. 2

3 Assumptions: Y 1, Y 0, D 1, D 0 independent of Z conditional on X 0 P Z 1 X 1 P D 1 1 X P D 0 1 X P D 1 D 0 X 1. Under these assumptions, treatment is unconfounded for compliers: D Y 0, Y 1 D, X, D 1 D 0 D Y 0, Y 1 X, D 1 D 0 and treatment effects can be defined based on D Y X, D, D 1 D 0. 3

4 AAI focus on quantile treatment effects (Abadie looks at other distributional features): Quant Y X, D, D 1 D 0 D X. (This results in estimated differences for the quantiles of Y 1 and Y 0, not the quantile of the difference Y 1 Y 0. If the dummy variable C 1 D 1 D 0 could be observed, problem would be straightforward. Would like to use linear quantile estimation for the subpopulation C 1 because the parameters solve min, E C g Y, X, D,, where g Y, X, D,, c Y D X is the check function. 4

5 Instead, can solve min, E U g Y, X, D,,, where U Y, X, D and U P C 1 U. AAI show v U 1 D 1 v U 1 X 1 D v U X, where v U P Z 1 U, and X P Z 1 X, which can both be estimated using observed data. Two-step estimator solves min, N i 1 1 v U i 0 v U i c Y i D i X i. 5

6 Chernozhukov and Hansen (2005) show how to identify quantile functions without restricting the functional form. But monotonicity in the unosbervable (a scalar) is key. If q d, x, is the th quantile conditional on x for treatment level D d. Under the assumption that the unobservables are independent of the instruments, they show This defines moment conditions P Y q D, X, X, Z. E 1 Y q D, X, X, Z 0. Chernozhukov and Hansen (2006) consider estimation. 6

7 2. Control Function Approaches for Some Continuous Treatments Suppose y is binary but w is now a continuous treatment. Can apply MLE or a more flexibile CF approach (Rivers and Vuoung, 1988). In first stage, estimate a linear reduced form for w (on all exogenous variables) by OLS and save the residuals, v 2. In the second stage, use probit of y on any functions of x, w and add v 2. Theaverage structural function (Blundell and Powell, 2003) is consistently estimated as N ASF x, w N 1 i 1 g w, w v i2 7

8 so that v i2 gets averaged out. 8

9 In fact, can include v i2 in a very flexible way (polynomials, interactions with observables) and then average out. With ASF we can take derivatives and/or changes with respect to elements in x, w to obtain the average partial (or marginal) effects. Bootstrap standard errors. Approach is very flexible. After getting v i2 from a (flexible) linear regression, estimate favorite model of y i on functions of x i, w i, v i2 (where w i can be a vector of continuous treatments). 9

10 Example using Mroz (1987, Econometrica) data on married women s labor force participation. Treat other sources of income as endogenous, instrument with husband s education. Compare linear probability model estimated by 2SLS with Rivers-Vuong and then full MLE. 10

11 . * use mroz. * LPM first. ivreg inlf educ exper expersq age kidslt6 kidsge6 (nwifeinc huseduc) Instrumental variables (2SLS) regression Source SS df MS Number of obs F( 7, 745) 36. Model Prob F Residual R-squared Adj R-squared Total Root MSE inlf Coef. Std. Err. t P t [95% Conf. Interval nwifeinc educ exper expersq age kidslt kidsge _cons Instrumented: nwifeinc Instruments: educ exper expersq age kidslt6 kidsge6 huseduc 11

12 . * Now the CF approach; estimate reduced form for nwifeinc:. reg nwifeinc huseduc educ exper expersq age kidslt6 kidsge6 Source SS df MS Number of obs F( 7, 745) 27. Model Prob F Residual R-squared Adj R-squared Total Root MSE nwifeinc Coef. Std. Err. t P t [95% Conf. Interval huseduc educ exper expersq age kidslt kidsge _cons probit inlf nwifeinc educ exper expersq age kidslt6 kidsge6 v2hat Probit regression Number of obs 753 LR chi2(8) 229. Prob chi

13 Log likelihood Pseudo R inlf Coef. Std. Err. z P z [95% Conf. Interval nwifeinc educ exper expersq age kidslt kidsge v2hat _cons predict xbhat, xb. gen scale normalden(xbhat). sum scale Variable Obs Mean Std. Dev. Min Max scale di.3*(.037) * Very close to IV estimate,

14 . * What if we treat nwifeinc as exogenous in the probit?. probit inlf nwifeinc educ exper expersq age kidslt6 kidsge6 Probit regression Number of obs 753 LR chi2(7) 227. Prob chi Log likelihood Pseudo R inlf Coef. Std. Err. z P z [95% Conf. Interval nwifeinc educ exper expersq age kidslt kidsge _cons predict xb2, xb. gen scale2 normalden(xb2). sum scale2 Variable Obs Mean Std. Dev. Min Max 14

15 scale di.3* di.301* di.0111/ * So effect is three times as large when nwifeinc is treated as endogenous. * Joint MLE allowing nwifeinc to be endogenous:. ivprobit inlf educ exper expersq age kidslt6 kidsge6 (nwifeinc huseduc) Probit model with endogenous regressors Number of obs 753 Wald chi2(7) 200. Log likelihood Prob chi Coef. Std. Err. z P z [95% Conf. Interval inlf nwifeinc educ exper expersq

16 age kidslt kidsge _cons /lnsigma /athrho sigma rho Instrumented: nwifeinc Instruments: educ exper expersq age kidslt6 kidsge6 huseduc Wald test of exogeneity (/athrho 0): chi2(1) 2.01 Prob chi predict xbh2, xb. gen scale2 normalden(xbh2). sum scale2 Variable Obs Mean Std. Dev. Min Max scale di.297* * Very similar to CF and linear IV estimates. 16

17 This model can exploit the nature of y i (Tobit, count, ordered probit, even multinomial logit.) Then, given the estimated mean function m x, w, v 2 (including probabilities), form N ASF x, w N 1 i 1 m x, w, v i2 Some theorists find this objectionable of the model chosen for y cannot be obtained from a structural model. But this is much easier for, say, a multinomial response, y with endogenous continuous variables. And we can easily get average marginal effects. 17

18 3. Semi- and Nonparametric CF Approaches Blundell and Powell (2003, 2004) formally study semiparametric and nonparametric methods but they are restricted again to the case of w where we can write w g 2 x, z v 2 where v 2 is independent of x, z. It is this assumption that rules out discrete w. BP then study E y x, z, v 2 E y x, w, v 2 as fully nonparametric or impose a semiparatric structure, such as H x w, v 2 for unknown function H. After getting v i2, can use full nonparametric regression of y i on x i, w i, v i2 to obtain m x, w, v 2 or an index approach, such as Ichimure or Klein and Spady. 18

19 In the end, the ASF is estimated by averaging out v i2. With large dimensions, full nonparametrics can be very difficult, as can be index approaches. Instead, can use BP to justify flexible parametric approaches. If y is a fraction or binary, m x, w, v 2 f 1 x, w 1 f 2 v 2 2 f 3 x, w f 4 v 2 3 where the f j functions are specified. 19

20 This means a two-step probit QMLE can be used (after getting v i2 ) and then N ASF x, w N 1 i 1 f 1 x, w 1 f 2 v i2 2 f 3 x, w f 4 v i2 3 Opens up many possibilities that can be justified from the Blundell and Powell work. But the additivity and independence of v 2 from x, z is critical. 20

21 Chesher (2003) shows how to identify partial derivatives of structural functions without functional form restrictions, but with monotonicity restrictions. As an example, consider the system y g 1 w, x, f 1, a 2 w g 2 x, z, a 2 where g 2 is strictly increasing in a 2 and g 1 is strictly increasing in f 1. Under quantile restrictions and exclusion restrictions, the derivative of g 1 with respect to w can be identified at particular values of the observables and quantiles of the unobservables. 21

22 Allows very flexible functional forms in observables and weak assumptions about the dependence between f 1, a 2 and x, z (conditional quantile restrictions) but still only applies when w is continuous (and multivariate w harder to handle). Compared with Blundell and Powell (2003), Chesher restricts the way heterogeneity can appear in the structural equation: y 1 f 1 w xb 1 a 2 w xg 2 zb 2 2 a 2 where b 1,b 2, and g 2 are random slopes on the exogenous variables. Tradeoff is that BP require an additive error in the reduced form. 22

23 Tempting to think that when w has some discreteness that the residuals v i2 could be replaced with generalized or Pearson residuals, with coefficients that depend on x i, z i in a flexible way. Again, an astructural approach that would deliver estimates (approximately?) of average partial effects. 23

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