Local Semi-Parametric Efficiency of the Poisson Fixed Effects Estimator Valentin Verdier∗ June 7, 2016

Abstract Hausman et al. (1984) have defined the Poisson fixed effects 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 (1999) showed that the Poisson fixed effects is actually consistent even if the distributional assumptions of the Poisson fixed effects 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 Poisson fixed effects estimator in the absence of distributional assumptions. I show that the Poisson fixed effects estimator corresponds to the optimal estimator for random coefficients models of Chamberlain (1992) 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 Poisson fixed effects 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 Poisson fixed effects estimator, should provide a useful additional justification for its use to estimate conditional mean models of panel data. Keywords: Panel Data, Non-Linear Models, Poisson Fixed Effects Estimator, Optimal Instruments JEL Classification: C01; C13; C23; C25 ∗

Department of Economics, University of North Carolina, Chapel Hill, NC 27599, United States. Tel.: +1 919966-3962. E-mail address: [email protected].

1

1

Introduction

A commonly used estimator for models of count panel data with multiplicative unobserved heterogeneity and strictly exogenous explanatory variables is the Poisson fixed effects (PFE) estimator introduced by Hausman et al. (1984). This estimator is a conditional maximum likelihood estimator which takes advantage of the assumptions of Poisson distribution and independent draws over time to derive a conditional distribution of the dependent variable that does not depend on the distribution of unobserved heterogeneity. In many applications, these distributional assumptions are likely to be violated. Wooldridge (1999) showed that the PFE estimator is consistent as long as the restriction on the conditional mean function is correctly specified, independently of whether the rest of the assumptions of the PFE model hold. The PFE estimator has become widely used in empirical work, as for instance in Rose (1990), Acemoglu and Linn (2004), Azoulay et al. (2010), or Burgess et al. (2012). Hahn (1997) showed that the PFE estimator is efficient when the distributional assumptions of the PFE model are correctly specified. Here I show that the PFE estimator achieves the semiparametric efficiency bound for estimating conditional mean models derived in Chamberlain (1992) regardless of whether the distributional assumptions of the PFE model hold or not, as long as the conditional mean of the dependent variable is equal to its conditional variance and the conditional serial correlation of the dependent variable is zero. In particular the dependent variable does not need to be a count variable for the PFE estimator to be efficient. In the next section, I present the model considered and study the asymptotically efficient estimator for this model. I then show under which conditions the PFE estimator is asymptotically efficient.

2

The Model and Estimators

As in Wooldridge (1999), I consider panel data models that specify a conditional mean function for a scalar dependent variable, yit , with strictly exogenous explanatory variables xit , multiplicative unobserved heterogeneity ci , and a column vector of unknown parameters β0 : E(yit |ci , xi ) = ci µ(xit , β0 ) ∀ i = 1, ..., n, t = 1, ..., T

2

(2.1)

where i indexes cross-sectional observations, t indexes time, µ(., .) is a known function, and xi = {xi1 , ..., xiT }. There is no restriction imposed on the relationship between the unobserved heterogeneity term ci and the explanatory variables xi , so that this model accounts for confounding unobserved factors as long as they are time constant. This model is also a special case of the random coefficients model presented in Section 4 of Chamberlain (1992). Throughout this paper I consider the case of i.i.d. cross-sectional draws and large n, fixed T asymptotics. Denote µit (β) = µ(xit , β) and µit = µit (β0 ). Wooldridge (1999) showed that the parameters in this model can be estimated from the conditional moment conditions: E(ρit (β0 )|xi ) = 0 ∀ t = 1, ..., T PT

(2.2)

yis . µis (β)

where ρit (β) = yit − µit (β) PT s=1 s=1

2.1

Asymptotically Efficient Estimation

The conditional moment conditions in (2.2) can be rewritten in the form: E(ρi (β0 )|xi ) = 0

(2.3)

0

where ρi (β) = [ρi1 (β), ..., ρiT (β)] . As in Chamberlain (1987) and Newey (2001), an optimal estimator for β0 from (2.3) can be postulated to be βˆopt from: n X

0

ˆ Di Σ+ i ρi (βopt ) = 0

(2.4)

i=1

where Di =

∂ρi E( ∂β 0 (β0 )|xi ),

1 Σi = V ar(ρi (β0 )|xi ), and Σ+ i is a symmetric generalized inverse of Σi .

Lemma 2 of Appendix A defines one particular symmetric generalized inverse of Σi , Σ− i , thereby also proving that there always exists at least one symmetric generalized inverse of Σi . Under (2.1) and standard regularity conditions, as in Newey and McFadden (1994) for instance,

1

Chamberlain (1987) considers cases where V ar(ρi (β0 )|xi ) is non-singular almost surely and Newey (2001) considers the case where V ar(ρi (β0 )|xi ) can be singular in Theorem 5.2. In our case V ar(ρi (β0 )|xi ) is shown to be singular in Lemma 1 of Appendix A.

3

βˆopt is consistent for β0 and asymptotically normal: √

d

n(βˆopt − β0 ) → N (0, Vopt ) 0

0

0

0

0

+ + + −1 −1 Vopt = E(Di Σ+ i Di ) E(Di Σi Σi Σi Di )E(Di Σi Di ) 0

−1 = E(Di Σ+ i Di )

where the last equality follows from Σ+ i being symmetric and a generalized inverse of Σi . Since (2.1) is a special case of the random coefficients model considered in Section 4 of Chamberlain (1992), the information bound for estimating β0 from (2.1) can be found in Chamberlain −1 −1 (1992), page 581, and we denote it by V0,β . The specific form of V0,β in this application is also 0 0 −1 −1 shown in the proof of Proposition 1 in Appendix A. Proposition 1 shows that Vopt = V0,β , so that 0

βˆopt is not only the optimal consistent estimator for β0 from (2.3), but also the optimal consistent estimator for β0 from (2.1). Proposition 1. For any choice of a symmetric generalized inverse of Σi , Σ+ i : −1 −1 Vopt = V0,β 0

(2.5)

so that βˆopt is asymptotically efficient for estimating β0 consistently from (2.1). Proof. Proof in Appendix A. This proposition shows that the transformation of the model from (2.1) to (2.3), which we will see is used by the PFE estimator, is not responsible for any loss of information with regard to the estimation of β0 . The next section shows under which conditions the PFE estimator coincides with βˆopt .

2.2

Conditions for Efficiency of the Poisson Fixed Effects Estimator

As shown in Wooldridge (1999), the Poisson fixed effects estimator, βˆP F E , is defined by: n X ∂pi (βˆP F E ) 0 ( ) Wi (βˆP F E )−1 ρi (βˆP F E ) = 0 ∂β 0

(2.6)

i=1

0

where pi (β) = [pi1 (β), ..., piT (β)], pit (β) =

µ (β) PT it , s=1 µis (β)

diagonal matrix with vector a for diagonal.

4

Wi (β) = diag(pi (β)), and diag(a) is the

Under (2.1) and the same regularity conditions as in Wooldridge (1999), βˆP F E is consistent for β0 and asymptotically equivalent to β˜P F E defined by: n X ∂pi (β0 ) 0 ( ) Wi (β0 )−1 ρi (β˜P F E ) = 0 ∂β 0

(2.7)

i=1

Proposition 2 shows that under the assumptions of equal conditional mean and variance and 0

0

i (β0 ) ) Wi (β0 )−1 = Di Σ− zero conditional serial correlation, ( ∂p∂β 0 i . Therefore, from Proposition 1,

when these additional assumptions on the conditional variance-covariance matrix of the dependent variable hold, the PFE estimator is asymptotically efficient in the class of estimators that are consistent for β0 under (2.1). Proposition 2. If (2.1) holds as well as: V ar(yit |ci , xi ) = ci µit Cov(yit , yit−s |ci , xi ) = 0 ∀ s = 1, ..., t

(2.8) (2.9)

then: 0

Di Σ− i =(

∂pi (β0 ) 0 ) Wi (β0 )−1 ∂β 0

(2.10)

so that βˆP F E is asymptotically efficient for estimating β0 consistently from (2.1). Proof. Proof in Appendix B.

3

Conclusion

For estimating models of panel data with multiplicative heterogeneity and strictly exogenous covariates, I showed that the Poisson fixed effects estimator is asymptotically efficient in the class of estimators consistent under restrictions on the conditional mean function of the dependent variable as long as the assumptions of equal conditional mean and variance and zero conditional serial correlation are satisfied. This result deepens our understanding of the Poisson fixed effects estimator by showing that this estimator is not only a robust conditional maximum likelihood estimator as shown in Hausman et al. (1984) and Wooldridge (1999), but is also an efficient semiparametric estimator as in Chamberlain (1992) in the special case where one believes in two restrictions on the conditional variance-covariance matrix of the dependent variable. Combined with the simplicity and robustness properties of the Poisson fixed effects estimator, and the challenge of approximating 5

optimal instruments in an unrestricted way (see e.g. Newey (1993) and subsequent papers), this new efficiency result should provide practicioners with a new justification for the use of the Poisson fixed effects estimator in empirical research. It also motivates the question of whether other parsimonious estimators exist that share the robustness properties of the Poisson fixed-effects estimator and might be efficient under more general conditions, or for a wider class of models.

References Acemoglu, D. and Linn, J. (2004). Market Size in Innovation: Theory and Evidence from the Pharmaceutical Industry. The Quarterly Journal of Economics, 119(3):1049–1090. Azoulay, P., Zivin, J. S. G., and Wang, J. (2010). Superstar Extinction. Quarterly Journal of Economics, 125(2):549–589. Burgess, R., Hansen, M., Olken, B. A., Potapov, P., and Sieber, S. (2012). The Political Economy of Deforestation in the Tropics. The Quarterly Journal of Economics, 127(4):1707–1754. Chamberlain, G. (1987). Asymptotic Efficiency in Estimation with Conditional Moment Restrictions. Journal of Econometrics, 34(3):305–334. Chamberlain, G. (1992).

Efficiency Bounds for Semiparametric Regression.

Econometrica,

60(3):567–596. Hahn, J. (1997). A Note on the Efficient Semiparametric Estimation of Some Exponential Panel Models. Econometric Theory, 13(04):583–588. Hausman, J., Hall, B. H., and Griliches, Z. (1984). Econometric Models for Count Data with an Application to the Patents-R & D Relationship. Econometrica, 52(4):909–938. Newey, W. K. (1993). 16 Efficient estimation of models with conditional moment restrictions. In Handbook of Statistics, volume Volume 11, pages 419–454. Elsevier. Newey, W. K. (2001). Conditional Moment Restrictions in Censored and Truncated Regression Models. Econometric Theory, null(05):863–888.

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Newey, W. K. and McFadden, D. (1994). Chapter 36 Large sample estimation and hypothesis testing. In Robert F. Engle and Daniel L. McFadden, editor, Handbook of Econometrics, volume Volume 4, pages 2111–2245. Elsevier. Penrose, R. (1955). A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 51(03):406–413. Rose, N. L. (1990). Profitability and Product Quality: Economic Determinants of Airline Safety Performance. Journal of Political Economy, 98(5):944–964. Wooldridge, J. M. (1999). Distribution-Free Estimation of some Nonlinear Panel Data Models. Journal of Econometrics, 90(1):77–97.

Appendix A

Efficient Estimation under Conditional Mean Restrictions PT

yis , µis (β)

Recall the notation from the body of the text so that we have ρit (β) = yit − µit (β) PT s=1 s=1

0

∂ρi ρi (β) = [ρi1 (β), ..., ρiT (β)] , Di = E( ∂β 0 (β0 )|xi ), Σi = V ar(ρi (β0 )|xi ), pit (β) =

µ (β) PT it , s=1 µis (β)

0

pi (β) =

[pi1 (β), ..., piT (β)], Wi (β) = diag(pi (β)), µit = µit (β0 ). 0

0

Introducing some new notation, let µi = [µi1 , ..., µiT ], yi = [yi1 , ..., yiT ], Σy,i = V ar(yi |xi ), hi = E(ci |xi ), ∂µit =

∂µit (β0 ), ∂β 0

0

0

0

∂µi = [(∂µi1 ) , ..., (∂µiT ) ].

We will also use M [k] to denote the k th column vector (element) of the matrix (row vector) M . Also let:

  µi1  1  Pi = PT  t=1 µit  µiT

 ... µi1    ...   ... µiT

(A.1)

so that ρi = ρi (β0 ) can be rewritten as: ρi = (I − Pi )yi Lemma 1. Σi = V ar(ρi |xi ) is singular.

7

(A.2)

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. Therefore: rank(I − Pi ) = trace(I − Pi ) = T − trace(Pi ) =T −1 since trace(Pi ) =

PT t=1 µit PT t=1 µit

= 1.

Hence I − Pi is singular. 0

0

−1 −1 −1 −1 −1 Lemma 2. Σ− i = (Σy,i − Σy,i µi (µi Σy,i µi ) µi Σy,i ) is a symmetric generalized inverse of Σi .

Proof. We can rewrite Σi as: 0

Σi = (I − Pi )Σy,i (I − Pi )

(A.3) 0

0

= Σy,i − Pi Σy,i − Σy,i Pi + Pi Σy,i Pi

(A.4)

Note that: Pi µ i = µ i

(A.5)

and: 0

0

0

0

−1 −1 −1 −1 −1 Pi Σ− i = Pi (Σy,i − Σy,i µi (µi Σy,i µi ) µi Σy,i )

=0 Therefore: 0

0

0

0

−1 −1 −1 −1 −1 Σ− i Σi = (Σy,i − Σy,i µi (µi Σy,i µi ) µi Σy,i )Σyi − 0 0

−1 −1 −1 −1 − (Σ−1 y,i − Σy,i µi (µi Σy,i µi ) µi Σy,i )Σy,i Pi + 0 0

= I − Pi

(A.6) (A.7) (A.8)

8

Therefore: 0

− − − Σ− i Σi Σi = Σi − Pi Σi

(A.9)

= Σ− i

(A.10)

Pi Pi = Pi

(A.11)

Note that:

Therefore: 0

Σi Σ− i Σi = Σi − Σi Pi

(A.12) 0

0

0

0

= Σi − Σy,i Pi + Pi Σy,i Pi + Σy,i Pi − Pi Σy,i Pi

(A.13)

= Σi

(A.14)

0

0

−1 −1 −1 −1 −1 So Σ− i = (Σy,i − Σy,i µi (µi Σy,i µi ) µi Σy,i ) is indeed a generalized inverse of Σi . It is clearly

symmetric as well. Lemma 3. The asymptotic variance of βˆopt is the same independently of which symmetric generalized inverse of Σi , Σ+ i , is used. ¯ + Proof. Let Σi and Σ+ i be two symmetric generalized inverses of Σi .

Since ρi = (I − Pi )yi , we have, for any k = 1, ..., dim(β0 ):     [k] [k] µi1 ... µi1  ∂µi1 ... ∂µi1  PT [k]      ∂µ 1 [k]  − Pt=1 it  )yi |xi )  Di = E(( PT ... ...     T 2 µ µ ) (    t=1 it  t=1 it [k] [k] ∂µiT ... ∂µiT µiT ... µiT     [k] [k] ∂µ ... ∂µ µ ... µ i1 i1  i1   i1 PT [k]      ∂µ 1 t=1 it − P   )µi = hi ( PT ... ...     T  ( t=1 µit )2   t=1 µit  [k] [k] ∂µiT ... ∂µiT µiT ... µiT PT [k] [k] t=1 ∂µit = hi (∂µi − µi PT ) t=1 µit

(A.15)

(A.16)

(A.17)

so that: PT ∂µit Di = hi (∂µi − µi Pt=1 ) T µ it t=1

(A.18)

¯ + From this expression for Di we can show that Σi Σi Di = Σi Σ+ i Di = Di by showing that, for

9

any particular choice of i, the linear system of equations in w: Σi w = Di

(A.19)

is consistent. Consistency of (A.19) follows from: PT

PT

∂µit hi (µi Pt=1 T t=1 µit

Pi Di =

− µi Pt=1 T

∂µit

t=1 µit

)

=0 and recalling Σi Σ− i = I − Pi from the previous lemma, so that: Σi Σ− i Di = (I − Pi )Di = Di − 0 = Di 0 + 0 + 0 + ¯ ¯ Therefore E(Di Σi Di ) = E(Di Σi Σi Σ+ i Di ) = E(Di Σi Di ). Hence the result of this lemma is

proved. Proof of Proposition 1. Proof. Chamberlain (1992), page 581, showed that the asymptotic information bound for estimating β0 from (2.1) is: 0

0

0

−1 −1 −1 −1 −1 V0,β = E(hi ∂µi (Σ−1 y,i − Σy,i µi (µi Σy,i µi ) µi Σy,i )∂µi hi ) 0 0

= E(hi ∂µi Σ− i ∂µi hi ) Note that: 0

0

0

0

−1 −1 −1 −1 −1 µi Σ − i = µi (Σy,i − Σy,i µi (µi Σy,i µi ) µi Σy,i )

=0 Therefore: 0

0

− Di Σ− i Di = hi ∂µi Σi ∂µi hi

(A.20)

Therefore we have: 0

−1 V0,β = E(Di Σ− i Di )

10

(A.21)

Because Σ− i is a symmetric generalized inverse of Σi as shown in Lemma 2, the estimator defined − by (2.4) with Σ+ i = Σi also has inverse asymptotic variance: 0

−1 Vopt = E(Di Σ− i Di )

(A.22)

Hence from Lemma 3, for any choice of a symmetric generalized inverse of Σi , Σ+ i , we have the result: −1 −1 V0,β = Vopt

(A.23)

ˆ Therefore, for any choice of Σ+ i , βopt is asymptotically efficient for estimating β0 consistently from (2.1). Since (2.1) implies (2.3), βˆopt is also asymptotically efficient for estimating β0 from (2.3).

B

Efficient Estimation under the Poisson Fixed Effects Assumptions

Lemma 4 provides a useful alternative characterization of Σ− i . Lemma 4. Σ− i is the unique matrix S that satisfies: SΣi S = S

(B.1)

Σi SΣi = Σi

(B.2) 0

SΣi = I − Pi

(B.3)

Σi S = I − Pi

(B.4)

− Proof. In the previous section we have shown that Σ− i satisfies (B.1), (B.2), and (B.3). Since Σi

and Σi are symmetric: Σi Σ− i = I − Pi

(B.5)

This solution is unique since for any S, S˜ satisfying these requirements2 : 0 ˜ i S˜ = S˜ S = SΣi S = S(I − Pi ) = SΣi S˜ = (I − Pi )S˜ = SΣ

2

(B.6)

This proof of uniqueness is similar to the proof of uniqueness for the Moore-Penrose pseudo inverse found in Penrose (1955).

11

Proof of Proposition 2. 0

Proof. Under (2.8) and (2.9) we have Σy,i = hi diag(µi ) + vi µi µi where vi = V ar(ci |xi ). Therefore: Σi = V ar(ρi |xi )

(B.7) 0

0

= (I − Pi )(hi diag(µi ) + vi µi µi )(I − Pi ) 0

= (I − Pi )hi diag(µi )(I − Pi ) where the last equality follows from µit µis − µis pit

PT

r=1 µir

(B.8) (B.9)

= 0.

Define: Xi = h−1 i (diag(

1 1 ) − PT J) µi t=1 µit

(B.10)

  1 ... 1    and, by an abuse of notation, ( 1 )0 = [ 1 , ..., 1 ]. We will show that where J =  ... µi µi1 µiT     1 ... 1 Σ− i = Xi . Note that: PT µit JPi = Pt=1 J T µ it t=1 =J

(B.11) (B.12)

and: diag(

1 1 J )Pi = PT µi t=1 µit

(B.13)

Therefore: 0

Σi Xi = (I − Pi )diag(µi )(I − Pi )(diag( = (I − Pi )diag(µi )(diag(

1 1 ) − PT J) µi t=1 µit

1 1 1 1 ) − PT J − PT J + PT J) µi t=1 µit t=1 µit t=1 µit

(B.14) (B.15)

= (I − Pi )(I − Pi )

(B.16)

= I − Pi

(B.17)

Since both Σi and Xi are symmetric: 0

Xi Σi = I − Pi

12

(B.18)

So in order to show that Σ− i = Xi , there only remains to show that Xi satisfies (B.1) and (B.2). For (B.1): Xi Σi Xi = Xi (I − Pi )

(B.19)

= Xi − Xi Pi

(B.20)

= Xi − h−1 i ( PT

1

t=1 µit

J − PT

1

t=1 µit

J)

= Xi

(B.21) (B.22)

For (B.2): Σi Xi Σi = (I − Pi )Σi

(B.23)

= Σi

(B.24)

where the second equality follows from the previous section of the appendix. Therefore we have shown that in this case: −1 Σ− i = hi (diag(

1 1 J) ) − PT µi t=1 µit

(B.25)

Therefore: 0

hi 0 1 1 ∂µ (diag( ) − PT J) hi i µi t=1 µit PT 0 ∂µit 0 ∂µi 0 ) − Pt=1 j =( T µi t=1 µit

Di Σ− i =

0

0

i where j = [1, ..., 1] and, by an abuse of notation, ( ∂µ µi ) = [

0

(B.26) (B.27) 0

∂µiT ∂µi1 µi1 , ..., µiT

].

Note that: PT ∂µis ∂pit ∂µit − Ps=1 µit 0 = PT T ∂β ( s=1 µis )2 t=1 µit

(B.28)

PT ∂µis ∂pit 1 ∂µit = − Ps=1 0 T µit ∂β pit s=1 µis

(B.29)

PT 0 ∂pi (β0 ) 0 ∂µi 0 −1 t=1 ∂µit 0 ) W (β ) = ( − ( ) j P i 0 T µi ∂β 0 t=1 µit

(B.30)

so that:

Therefore:

13

Hence we have shown that under (2.1), (2.8) and (2.9): 0

Di Σ− i =( Hence the asymptotic variance of

√

∂pi (β0 ) 0 ) Wi (β0 )−1 ∂β 0

(B.31)

n(βˆP F E − β0 ) is equal to Vopt and from Proposition 1 this

is equal to V0,β0 , the efficiency bound for estimating β0 consistently from (2.1).

14