Hausman, Hall, and Griliches [Hausman, J., H. B. Hall, and Z. Griliches. 1984. “Econometric Models for Count Data with an Application to the Patents-R & D Relationship.” Econometrica 52 (4): 909–938.] have defined the Poisson fixed effects (PFE) estimator to estimate models of panel data with count dependent variables under distributional assumptions conditional on covariates and unobserved heterogeneity, but without any restriction on the distribution of unobserved heterogeneity conditional on covariates. Wooldridge [Wooldridge, J. M. 1999. “Distribution-Free Estimation of some Nonlinear Panel Data Models.” Journal of Econometrics 90 (1): 77–97.] showed that the PFE estimator is actually consistent even if the distributional assumptions of the PFE model are violated, as long as the restrictions imposed on the conditional mean of the dependent variable are satisfied. In this note I study the efficiency of the PFE estimator in the absence of distributional assumptions. I show that the PFE estimator corresponds to the optimal estimator for random coefficients models of Chamberlain [Chamberlain, G. 1992. “Efficiency Bounds for Semiparametric Regression.” Econometrica 60 (3): 567–596.] in the particular case where the assumptions of equal conditional mean and variance and zero conditional serial correlation are satisfied, regardless of whether the distributional assumptions of the PFE model hold. For instance the dependent variable does not need to be a count variable. This local efficiency result, combined with the simplicity and robustness of the PFE estimator, should provide a useful additional justification for its use to estimate conditional mean models of panel data.
A Efficient Estimation under Conditional Mean Restrictions
Recall the notation from the body of the text so that we have ρi(β)=[ρi1(β), …, ρiT(β)]′, Σi=Var(ρi(β0)|xi), pi(β)′=[pi1(β), …, piT(β)], Wi(β)=diag(pi(β)), μit=μit(β0).
Introducing some new notation, let Σy,i=Var(yi|xi), hi=E(ci|xi),
We will also use M[k] to denote the kth column vector (element) of the matrix (row vector) M.
so that ρi=ρi(β0) can be rewritten as:
Lemma 1: Σi=Var(ρi|xi) is singular.
Proof. Since ρi=(I–Pi)yi and Pi is a function of xi, we simply have to show that (I–Pi) is singular.
First note that Pi is idempotent, so that I–Pi is idempotent as well.
Hence I–Pi is singular.□
Lemma 2:is a symmetric generalized inverse of Σi.
Proof. We can rewrite Σi as:
So is indeed a generalized inverse of Σi. It is clearly symmetric as well.□
Lemma 3:The asymptotic variance ofis the same independently of which symmetric generalized inverse of Σi,is used.
Proof. Let and be two symmetric generalized inverses of Σi.
Since ρi=(I–Pi)yi, we have, for any k=1, …, dim(β0):
From this expression for Di we can show that by showing that, for any particular choice of i, the linear system of equations in w:
Consistency of (A.19) follows from:
and recalling from the previous lemma, so that:
Therefore Hence the result of this lemma is proved.□
Proof of Proposition 1.
Proof.Chamberlain (1992: p. 581), showed that the asymptotic information bound for estimating β0 from (1) is:
Therefore we have:
Because is a symmetric generalized inverse of Σi as shown in Lemma 2, the estimator defined by (4) with also has inverse asymptotic variance:
Hence from Lemma 3, for any choice of a symmetric generalized inverse of Σi, we have the result:
Therefore, for any choice of is asymptotically efficient for estimating β0 consistently from (1). Since (1) implies (3), is also asymptotically efficient for estimating β0 from (3).□
B Efficient Estimation under the Poisson Fixed Effects Assumptions
Lemma 4 provides a useful alternative characterization of
Lemma 4:is the unique matrix S that satisfies:
Proof. In the previous section we have shown that satisfies (B.1)–(B.3). Since and Σi are symmetric:
This solution is unique since for any S, satisfying these requirements :
Proof of Proposition 2
Proof. Under (8) and (9) we have where vi=Var(ci|xi). Therefore:
where the last equality follows from
where and, by an abuse of notation, We will show that
Since both Σi and Xi are symmetric:
So in order to show that there only remains to show that Xi satisfies (B.1) and (B.2).
where the second equality follows from the previous section of the appendix.
Therefore we have shown that in this case:
where j′=[1, …, 1] and, by an abuse of notation,
Hence we have shown that under (1), (8) and (9):
Hence the asymptotic variance of is equal to Vopt and from Proposition 1 this is equal to the efficiency bound for estimating β0 consistently from (1).□
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