Jeffrey M. Wooldridge Michigan State University

Transcription

1 Fractional Response Models with Endogenous Explanatory Variables and Heterogeneity Jeffrey M. Wooldridge Michigan State University 1. Introduction 2. Fractional Probit with Heteroskedasticity 3. Fractional Probit with an Endogenous Explanatory Variable 4. Linear Unobserved Effects Models with Unbalanced Panels 5. Nonlinear UE Models with Unbalanced Panels 1

2 1. Introduction A fractional response y satisfies 0 y 1, possibly with P y 0 1orP y 1 0 (or both). Assume y is the variable we would like to explain in terms of covariates, x x 1,...,x K. (No data censoring, but y may be a corner solution.) Focus here is on mean response. If x is exogenous, goal is to estimate E y x. 2

3 Can always use a linear model for E y x, but it is at best an approximation. Papke and Wooldridge (1996, Journal of Applied Econometrics): Model E y x using models of the form G x for 0 G nonindex forms). 1(or 3

4 So-called fractional response models (fractional probit, fractional logit) easily estimated using glm, and robust inference is trivial (and very important: MLE standard errors are too large). For panel data, can use xtgee. Papke and Wooldridge (2008, Journal of Econometrics) show how to use correlated random effects approaches to estimate fractional response models for panel data. But for balanced panels. Wooldridge (2005, Rothenberg Festschrift; 2010, MIT Press) considers models with continuous endogenous explanatory variables (EEVs). Proposes two-step control function approach. 4

5 Papke and Wooldridge (2008): heterogeneity and continuous EEV. Combination of CRE and control function methods for fractional probit. But balanced panel, and only two-step estimators. What if we want a one-step quasi-mle (which simplifies inference and may have better finite-sample properties)? So y 1 is a fractional response and y 2 a continuous EEV. Wooldridge (2011, unpublished) shows that the ivprobit log-likelihood identifies the (scaled) parameters under correct specification of E y 1 y 2, z 1, a 1 where a 1 is the omitted variable. (and y 2 follows classical linear model). 5

6 What if y 1 is a fractional response and y 2 abinaryeev?two-step forbidden regression is not valid. Wooldridge (2011) shows the biprobit log likelihood identifies the (scaled) parameters if E y 1 y 2, z 1, a 1 is correctly specified (and y 2 follows a probit). Neither ivprobit nor biprobit allow y 1 to be a fractional response. Neither does cmp (Roodman, 2009). Bottom line: Many existing Stata commands could be used to estimate flexible fractional response models allowing for endogeneity and unbalanced panel by removing the data checks on the response variable. 6

7 2. Fractional Probit with Heteroskedasticity Let x x 1, x 2,...,x K. Fractional probit model is E y x 0 x 0 1 x 1... K x K Might want more flexibility. If P y 0 0, could use a two-part model. But can directly make model for E y x more flexible, for example, E y x 0 x exp z /2 where z 1 M is a function of x 1, x 2,...,x K that does not include a constant. 7

8 The j and h are consistently estimated using the Bernoulli quasi-mle if E y x is correctly specified. As usual, need to use robust inference because y is not binary. (The conditional mean may be misspecified, anyway.) Ideally, just type hetprobit y x1... x2, het(z1 z2... zm), robust But y is turned into a binary response. Can easily test H 0 : 0 with robust Wald statistic. 8

9 clear capture program drop frac_het program frac_het version 11 args llf xb zg quietly replace llf $ML_y1*log(normal( xb *exp(- zg ))) /// (1 - $ML_y1)*log(1 - normal( xb *exp(- zg ))) end ml model lf frac_het (prate mrate ltotemp age sole) /// (mrate ltotemp age sole, nocons), vce(robust) ml max 9

10 Number of obs 4075 Wald chi2(4) 152. Log pseudolikelihood Prob chi Robust prate Coef. Std. Err. z P z [95% Conf. Interval eq1 mrate ltotemp age sole _cons eq2 mrate ltotemp age sole

11 . test [eq2] ( 1) [eq2]mrate 0 ( 2) [eq2]ltotemp 0 ( 3) [eq2]age 0 ( 4) [eq2]sole 0 chi2( 4) Prob chi * Usual fractional probit (could use glm): capture program drop frac_probit program frac_probit version 11 args llf xb quietly replace llf $ML_y1*log(normal( xb )) /// (1 - $ML_y1)*log(1 - normal( xb )) end ml model lf frac_probit (prate mrate ltotemp age sole), vce(robust) ml max 11

12 Number of obs 4075 Wald chi2(4) 695. Log pseudolikelihood Prob chi Robust prate Coef. Std. Err. z P z [95% Conf. Interval mrate ltotemp age sole _cons

13 Should do a comparison of average partial effects between ordinary fractional probit and heteroskedastic fractional probit. The hetprobit quasi-mle is needed for nonlinear CRE panel models with unbalanced panels. 13

14 3. Fractional Probit with an Endogenous Explanatory Variable Adapted from Wooldridge (2011, unpublished). Set up endogeneity as an omitted variable problem, and start by assuming y 2 is continuous: E y 1 z, y 2, a 1 x 1 1 a 1. y 2 z 2 v 2, where x 1 is a general nonlinear function of z 1, y 2, a 1 is an omitted factor thought to be correlated with y 2 but independent of the exogenous variables z. 14

15 The average partial effects in this model are obtained from the average structural function (ASF): ASF x 1 E a1 x 1 1 a 1 x 1 a1 where a1 1 / 1 2 a1 1/2. Happily, these are precisely the parameters that are identified. If a 1, v 2 is jointly normal, a two-step control function method is valid (Wooldridge, 2005). Note that the distribution of y 1 is not further restricted. 15

16 (i) Regress y i2 on z i and obtain the residuals, v i2. (ii) Use probit of y i1 on x i1, v i2 to estimate parameters with different scales, say e1 and e1. (Can implement as a generalized linear model. ) The average structural function (ASF) is consistently estimated as N ASF y 2, z 1 N 1 i 1 x 1 e1 e1 v i2, and this can be used to obtain APEs with respect to y 2 or z 1 (Wooldridge, 2005). 16

17 What about a quasi-liml approach? Can show that E y 1 y 2, z x 1 r1 1 / 2 y 2 z /2 and so we can plug this mean function into the Bernoulli quasi-log likelihood. This gives q 1 y 1, y 2, z, 1, 2. Identify 2 and 2 using the Gaussian QLL, which gives q 2 y 2, z, 2. The same objective function we get for MLE with y 1 binary can be used when y 1 is fractional continuous or otherwise. In other words, ivprobit could be easily modified and use robust inference. 17

18 A similar argument holds when y 2 is binary and follows a probit model: y 2 1 z 2 v 2 0 v 2 z ~ Normal 0, 1 Can show that E y 1 y 2, z has the same form as the response probability in the so-called bivariate probit model. 18

19 For example, E y 1 y 2 1, z z 2 x 1 r1 1 v /2 dv 2 So for q 2 y 2, z, 2 we use the usual probit log-likelihood. For q 1 y 1, y 2, z, 1, 2 we use the Bernoulli QLL associated with bivariate probit. So if y 1 were allowed to be fractional, biprobit with a robust option could be used. 19

20 4. Linear Unobserved Effects Models with Unbalanced Panels Model for a random draw i has T potential time periods: E u it x i1,...,x it, c i 0. y it x it c i u it, t 1,...,T Given access to a balanced random sample, the zero conditional mean assumption is sufficient for FE to be consistent (as N, T fixed) and N -asymptotically normal, provided all elements of x it have some time variation. 20

21 Let s it : t 1,...,T be a sequence of selection indicators : s it 1 if and only if observation i, t is used. These are generally outcomes of random variables. T The number of time periods available for unit i is T i r 1 s ir ; this is properly viewed as random. 21

22 Fixed Effects on the Unbalanced Panel The time-demeaned data uses a different number of time periods for different i. Let T 1 ÿ it y it T i r 1 T 1 ẍ it x it T i r 1 s ir y ir s ir x ir 22

23 The FE estimator is then FE N N 1 i 1 T t 1 1 s it ẍ it ẍ it N N 1 i 1 T t 1 s it ẍ it ÿ it, A sufficient condition for consistency of FE on the unbalanced panel is an extension of the usual strict exogeneity assumption: E u it x i, s i, c i 0, t 1,...,T s i s i1,...,s it 23

24 Both the covariates and selection are strictly exogenous conditional on c i. Rules out selection in any time period depending on the shocks in any time period. That is, the condition is generally violated if Cov s ir, u it 0 for any r, t pair. Importantly, it allows s it to depend on c i in an unrestricted way. xtreg allows unbalanced panels and properly computes standard errors and test statistics. 24

25 Random Effects on the Unbalanced Panel The quasi-time-demeaning value for unit i is i T i c2 / u2 1/2. Now define y it y it iȳ i 1 T where ȳ i T i r 1 s ir y ir, and similarly for x it. Then, RE is POLS of y it on x it using the s it 1 data points. 25

26 Useful equivalence result (Wooldridge, 2010, unpublished). Define T 1 x i T i r 1 s ir x ir and consider either POLS or RE estimation of the following equation on the unbalanced panel: y it x it x i v it Then POLS RE FE. Generally, POLS RE 26

27 Must be careful in constructing x i; only use periods where all variables are observed (s it 1). Must now include the time averages of year dummies because these are no longer constants in an unbalanced panel. Same result holds when add any other time-constant covariates. Implies that the CRE is robust even with unbalanced panels. Basis for robust Hausman test. H 0 : 0. Use RE with all time-constant controls included. 27

28 Heterogeneous Slopes Suppose the population model is E y it x i, a i, b i a i x it b i, so, in the population, x it : t 1,...,T is strictly exogenous conditional on a i, b i. Define a i c i, b i d i and write y it x it c i x it d i u it where E u it x i, a i, b i E u it x i, c i, d i 0 for all t. 28

29 Assume that selection may be related to x i, a i, b i but not the idiosyncratic shocks: E u it x i, a i, b i, s i 0, t 1,...,T. Multiply population equation by the selection indicator: s i y it s it s it x it s it c i s it x it d i s it u it Find an estimating equation by conditioning on s it, s it x it : t 1,...,T. 29

30 Let h i h it : t 1,...,T s it, s it x it : t 1,...,T and consider E s i y it h i s it s it x it s it E c i h i s it x it E d i h i and then make assumptions concerning E c i h i and E d i h i. We might choose w i T i, x i as the exchangeable functions satisfying E c i h i E c i w i, E d i h i E d i w i. 30

31 A flexible specification with g ir 1 T i r : T E c i T i, x i r 1 T r g ir r r 1 T T E d i T i, x i g ir r r r 1 r 1 g ir x i r r g ir x i r I K r, where the r are the expected values of x i given r time periods observed and r is the fraction of observations with r time periods: r E x i T i r, r E 1 T i r 31

32 This formulation is identical to running separate regressions for each T i : y it on 1, x it, x i, x i r x it,fors it 1 where r N 1 r N i 1 1 T i r x i and N r is the number of observations with T i r. The coefficient on x it, r, is the APE given T i r. Average these across r to obtain the overall APE. Cannot identify the APE for T i 1. 32

33 A simple test of the null that the r do not change. Augmented equation is y it x it 1 T i 2 x it T i T 1 x it T 1 c i u it where the base group is T i T. Use FE on the unbalanced panel and obtain a fully robust test of H 0 : 2 0,..., T 1 0 This is like a Chow test where the slopes are allowed to differ by the number of available time periods for each unit. 33

34 . use meap94_98. xtset schid year. egen tobs sum(1), by(schid). tab tobs tobs Freq. Percent Cum , , , Total 7, gen tobs4 tobs 4. gen tobs3 tobs 3. gen tobs3_lavgrexp tobs3*lavgrexp. gen tobs4_lavgrexp tobs4*lavgrexp 34

35 . xtreg math4 lavgrexp lunch lenrol y95 y96 y97 y98, fe cluster(distid)... (Std. Err. adjusted for 467 clusters in distid Robust math4 Coef. Std. Err. t P t [95% Conf. Interval lavgrexp lunch lenrol y y y y _cons sigma_u sigma_e rho (fraction of variance due to u_i)

36 . xtreg math4 lavgrexp tobs3_lavgrexp tobs4_lavgrexp lunch lenrol y95 y96 y97 y98, fe cluster(distid)... (Std. Err. adjusted for 467 clusters in distid Robust math4 Coef. Std. Err. t P t [95% Conf. Interval lavgrexp tobs3_lavg~p tobs4_lavg~p lunch lenrol y y y y _cons sigma_u sigma_e rho (fraction of variance due to u_i)

37 . test tobs3_lavgrexp tobs4_lavgrexp ( 1) tobs3_lavgrexp 0 ( 2) tobs4_lavgrexp 0 F( 2, 466) 2.37 Prob F * Might get away with using the pooled equations. 37

38 5. Nonlinear UE Models with Unbalanced Panels Adapted from Wooldridge (2010, unpublished). Interested in E y it x it, c i, where 0 y it 1 and c i is unobserved heterogeneity. (Binary response as special case.) Again, unbalanced panel. Assume strictly exogenous covariates conditional on c i and ignorable selection: E y it x i, c i, s i E y it x it, c i, t 1,...,T. 38

39 Do not model serial correlation. Make inference robust. Specify models for D c i s it, s it x it : t 1,...,T. Let w i be a vector of known functions of s it, s it x it : t 1,...,T that act as sufficient statistics, so that D c i s it, s it x it : t 1,...,T D c i w i 39

40 For simplicity, take E y it x i, c i E y it x it, c i x it c i, t 1,...,T where x it can include time dummies or other aggregate time variables. Assume that selection is conditionally ignorable for all t, that is, E y it x i, c i, s i E y it x i, c i. 40

41 All that is left is to specify a model for D c i w i for suitably chosen functions w i of s it, s it x it : t 1,...,T. Simplest is the time average on the selected periods, x i, and the number of time periods, T i. A specification linear in x i but with intercept and slopes different for each T i is T E c i w i r 1 T r 1 T i r r 1 1 T i r x i r 41

42 At a minimum, should let the variance of c i change with T i : Var c i w i exp T 1 r 1 1 T i r r If we also maintain that D c i w i is normal, then we obtain the following: E y it x it, w i, s it 1 x T it r 1 T r g ir r 1 g ir x i r 1/2 g ir r T exp r 2 where g ir 1 T i r. No difficulty in adding g ir x i for r 1,...,T to the variance function. 42

43 Can use heteroskedastic probit software provided the response variable can be fractional. The explanatory variables at time t are 1, x it, g i1,...,g it, g i1 x i,...,g it x i and the explanatory variables in the variance are simply the dummy variables g i2,...,g it, or also add g i1 x i,...,g it x i. Might want to impose restrictions, such as constant slopes on x i. 43

44 The average partial effects are easy to obtain from the estimated average structural function : N ASF x t N 1 i 1 x T t r 1 T rg ir r 1 T exp r 2 g ir r g ir x i r 1/2, where the coefficients with ^ are from the pooled heteroskedastic fractional probit estimation. The functions of T i, x i are averaged out, leaving the result a function of x t. Take derivatives or changes with respect to x tj. 44

45 . use meap94_98 xtset schid year panel variable: time variable: delta: schid (unbalanced) year, 1994 to 1998, but with gaps 1 unit. tab tobs number of time periods Freq. Percent Cum , , , Total 7, gen tobs3 tobs 3. gen tobs4 tobs 4. replace math4 math4/100 (7150 real changes made) 45

46 . capture program drop frac_het.. program frac_het 1. version args llf xb zg 3. quietly replace llf $ML_y1*log(normal( xb *exp(- zg ))) /// (1 - $ML_y1)*log(1 - normal( xb *exp(- zg ))) 4. end.. end of do-file. ml model lf frac_het (math4 lavgrexp lunch lenrol y95 y96 y97 y98 lavgrexpb lunchb lenrolb y95b y96b y97b y98b tobs3 tobs4) (tobs3 tobs4, nocons), vce(cluster schid). ml max 46

47 Log pseudolikelihood Prob chi (Std. Err. adjusted for 1683 clusters in schid Robust math4 Coef. Std. Err. z P z [95% Conf. Interval eq1 lavgrexp lunch lenrol y y y y lavgrexpb lunchb lenrolb y95b y96b y97b y98b tobs tobs _cons eq2 tobs tobs

48 . ml model lf frac_probit (math4 lavgrexp lunch lenrol y95 y96 y97 y98 lavgrexpb lunchb lenrolb y95b y96b y97b y98b tobs3 tobs4), vce(cluster schid. ml max Log pseudolikelihood Prob chi (Std. Err. adjusted for 1683 clusters in schid Robust math4 Coef. Std. Err. z P z [95% Conf. Interval lavgrexp lunch lenrol y y y y lavgrexpb lunchb lenrolb y95b y96b y97b y98b tobs tobs _cons