Covariances for Bivariate Selection Model Second-Step Estimates

Transcription

1 Eon 240B, Fall 2001 Daniel MFadden Covarianes for Bivariate Seletion Model Seond-Step Estimates Consider the bivariate latent variable model with normal disturbanes, (1) y * = + C, w * = zα + σν, where x and z are vetors of exogenous variables, not neessarily all distint, α and β are parameter vetors, again not neessarily all distint, and σ is a positive parameter. The interpretation of y * is latent desired hours of work, and of w * is latent log potential wage. The disturbanes C and ν have a standard bivariate normal distribution C 0 (2) ~, ν 0, 1 ρ ρ 1 with zero means, unit varianes, and orrelation ρ. There is a nonlinear observation rule determined by the appliation that maps the latent variables into observations. A typial rule might be Observe y = 1 and w = w * if y * > 0; observe y = -1 and nominally w = 0 when y * 0. This ould orrespond, for example, to an appliation where the event of working (y = 1) or not working (y = 0) is observed, but the wage is observed only if the individual works (y * > 0). The event of working is given by a probit model, P(yx) = Φ(y). The onditional log likelihood of w given partiipation is ρ wzα (3) l 2 wzα σ (α,β,σ,ρ) = -log σ + log φ( ) + log Φ - log Φ(). σ 1ρ 2 See Chapter 2 for the derivation of this likelihood. Writing (4) w = zα + σe(ν x,z,y=1) + ξ for the observations with partiipation defines a disturbane ξ that by onstrution has onditional mean zero, given x,z, in the seleted sample. Using the property that the onditional expetation of ν given y = 1 equals the onditional expetation of ν given C, integrated over the onditional density of C given y = 1, plus the property of the normal that dφ(c)/dc = -Cφ(C), one has 1

2 (5) E{wz,y=1} = zα + σe{νy=1} = zα + σ E{νC}φ(C)dC/Φ() = zα + σρ Cφ(C)dC/Φ() = zα + σρφ()/φ() zα + λm(), where λ = σρ and M() = φ()/φ() is alled the inverse Mill's ratio. Further, using the relationship (6) E(ν 2 C) = Var(νC) + {E(νC)} 2 = 1 - ρ 2 + ρ 2 C 2, and the integration-by-parts formula (7) C 2 φ(c)dc = - Cφ(C)dC = -φ() + φ(c)dc = -φ() + Φ(), one obtains (8) E{(w-zα) 2 z,y=1} = σ 2 E{ν 2 y=1} = σ 2 E{ν 2 C}φ(C)dC/Φ() Then, = σ 2 {1 - ρ 2 + ρ 2 C 2 }φ(c)dc/φ() = σ 2 {1 - ρ 2 + ρ 2 - ρ 2 φ()/φ()} = σ 2 {1 - ρ 2 φ()/φ()} = σ 2 {1 - ρ 2 M()}. (9) E [w zα E{wzαz,y1}] 2 z,y1 = E{(w-zα) 2 z,y=1} - [E{w-zαz,y=1}] 2 = σ 2 {1 - ρ 2 φ()/φ() - ρ 2 φ() 2 /Φ() 2 } = σ 2 {1 - ρ 2 M()[ + M()}. Then (4) is a regression equation that with β known so that M() is a known transformation of observations has E(ξ x,z,y=1) = 0 for the partiipant sub-population, but E(ξ 2 x,z,y=1) given by (9) and heterosedasti.. The Hekman two-step proedure estimates (4) by OLS, with a onsistent estimator for β obtained from the probit seletion model plugged into the inverse Mills ratio formula. This regression estimates only the produt λ σρ, but onsistent estimates of σ and ρ ould be obtained in a further step: The OLS regression does not orret for heterosedastiity or the presene of earlier stage estimates. It is nevertheless onsistent, but printed out standard errors are not onsistent. The general theory of two-step GMM estimation an be used to get onsistent estimates of the ovariane matrix for the estimates in (4). Suppose a random sample of observations (x,y,z,w), with x and z interpreted as explanatory variables, y as the dependent variable determining seletion, and w as the dependent variable determining the ontinuous outome, given seletion. Suppose in the first stage one estimates the parameter vetor β by b satisfying (10) 0 = E h(b ;x,y), 2

3 where E denotes empirial expetation (or sample average) and h(β;x,y) = β logφ(y) is the sore of the marginal likelihood of y. Suppose in the seond stage one estimates a parameter vetor θ = (α,λ) with λ = σρ by oeffiients t = (a, ) obtained by applying OLS to the model w = zα + λm(xb ) + ζ, where M(xb ) = φ(xb )/Φ(xb ) is an inverse Mills ratio evaluated at xb. This regression orresponds to the moments z (11) 0 = E g(t,b ;x,y,z,w) E M(xb ) (w-za - M(xb )). ote that the g moments do not depend on y; this will simplify formulas later. Make a Taylor's expansion of both the first-stage and the seond-stage moment onditions around the true β, α, and λ. Suppress the x,y,z,w arguments to simplify notation: 0 E h(β) A 0 (12) = - (b - β) - (t - θ) + o p, 0 E g(θ,β) B C where A = -plim E β h(β), B = -plim E β g(θ,β), and C = -plim E θ g(θ,β). The term E h(β) E g(θ,β) is asymptotially normal by a entral limit theorem, with a ovariane matrix Ω hh Ω hg Ω gh Ω», with Ω hh = plim E h(β)h(β), Ω hg = plim E h(β)g(θ,β), and Ω gg = plim E g(θ,β)g(θ,β). Solve the first blok of equations and substitute the solution (b - β) = A -1 E h(β) + o p into the seond blok to obtain (13) 0 = {E g(θ,β) - BA -1 E h(β)} - C (t - θ) + o p. The term in braes on the right-hand-side of this expression has an asymptoti ovariane matrix (14) Ω gg - BA -1 Ω hg - Ω gh A -1 B + BA -1 Ω hh A -1 B. 3

4 Then, (t - θ) = C -1 {E g(θ,β) - BA -1 E h(β)} + o p has asymptoti ovariane matrix (15) aov(t ) = C -1 {Ω gg - BA -1 Ω hg - Ω gh A -1 B + BA -1 Ω hh A -1 B}C -1 This is the general formula for ovarianes of seond-step estimators, and there will be some simplifiation for the appliation. In general, if B = 0, there is no orretion; this is the blok diagonality ase where θ an be estimated onsistently even if the estimator of β is not onsistent. In the bivariate seletion problem, letting m() = dm()/d(), z (16) B = -E β (w-zα-λm()) M() zxm() z (17) = -λe - E x,z E w x,z (w-zα-λm()) = -λe M()xm() m()x zxm() M()xm(). Then B = 0 if λ = σρ = 0; i.e., if ρ = 0 so that seletion is at random and is independent of the determination of the latent w. A useful impliation of this result is that for testing the hypothesis that ρ = 0, one is in the blok diagonal ase under the null, so that the ovariane orretion for first-stage estimation does not enter the determination of standard errors. Further, the w regression is homosedasti when ρ = 0. Then, a onventional T-test for λ = 0 an be arried out using the standard OLS statistis, without any orretions for the estimation of β in the inverse Mills ratio or for heterosedastiity. It will be useful to work out the form of the C matrix, z zz zm() (18) C = -E θ (w-zα-λm()) =. M() M() M() 2 ote that C is just the XX matrix for the seond-stage regression. If the regression were homosedasti, then C would equal Ω gg. However, it is not, and Ω gg is the array appearing in the enter of White's robust ovariane matrix estimator for regressions with heterosedastiity of unknown form. If g does not depend on y, then Ω gh = E x,z E wx,z {ge yx h} = 0. This is true for the bivariate seletion appliation. Then, one has the simplifiation (19) aov(t ) = C -1 {Ω gg + BA -1 Ω hh A -1 B}C -1 The fat that the first stage of estimation in the bivariate seletion problem was maximum marginal likelihood gives a further simplifiation. One has h(β;x,y) = β logφ(y), and hene A = - E ββ logφ(y). The information equality for maximum likelihood then implies A = E( β logφ(y))( β logφ(y)) = Ω hh, and the ovariane matrix for t simplifies further to 4

5 (20) aov(t ) = C -1 {Ω gg + B(Ω hh ) -1 B}C -1. All the terms of this ovariane matrix ould be estimated from sample analogs, omputed at the onsistent estimates. The following table summarizes onsistent estimators for the various ovariane terms; reall that E denotes empirial expetation (sample average): Matrix Estimator C -E θ g(t,b ) B -E β g(t,b ) A -E β h(b ) Ω hh E h(b )h(b ) Ω gh E g(t,b )h(b ) E g(t,b )g(t,b ) Ω gg 5