Generalized method of moments (GMM) estimation approach has a long history in the econometrics literature. Since the seminal paper of Hansen. (1982 ...
Sankhy¯ a : The Indian Journal of Statistics 2008, Volume 70-B, Part 1, pp. 34-62 c 2008, Indian Statistical Institute
Generalized Method of Moments versus Generalized Quasilikelihood Inferences in Binary Panel Data Models Brajendra C. Sutradhar Memorial University of Newfoundland, Canada
R. Prabhakar Rao and V.N. Pandit Sri Sathya Sai University, AP, India Abstract Generalized method of moments (GMM) estimation approach has a long history in the econometrics literature. Since the seminal paper of Hansen (1982, Econometrica), this GMM approach has been widely used mainly by econometricians to obtain consistent and efficient estimates for regression parameters in the class of linear dynamic dependence models. To reflect its importance on both theoretical and applied econometrics, the Journal of Business and Economic Statistics (JBES) has recently launched an interesting special issue (JBES, 2002, Vol. 20, No. 4) on this GMM estimation approach. In this paper, we propose a generalized quasilikelihood (GQL) approach, which, as compared to the GMM approach, appears to produce more efficient estimates for the regression and the dynamic dependence parameters in the non-linear regression set up. We examine this superior efficiency performance of the GQL approach in the context of binary panel data analysis, through both asymptotic and simulation studies. The GMM and GQL estimation approaches are illustrated by re-analyzing the Survey of Labour and Income Dynamics (SLID) data from Statistics Canada. AMS (2000) subject classification. Primary 62F10, secondary 62F12, 62P20. Keywords and phrases. Efficiency, estimation methods, panel data.
1
Introduction
In a binary panel data set up, repeated binary responses along with a set of multi-dimensional covariates are collected from a large number of independent individuals. When the repeated responses of an individual are also affected by a random effect, this type of panel data are usually modeled through a binary dynamic mixed model. If the repeated responses of an individual are, however, treated to be independent conditional on the random
GMM versus GQL inferences in binary panel data models 35 effect, then the data follow a binary mixed model (Breslow and Clayton (1993); Sutradhar (2004)). Let yit denote the binary response of the ith (i = 1, . . . , I) individual recorded at time t (t = 1, . . . , T ). Also, let xit be the p × 1 vector of fixed covariates corresponding to yit , and β be the p × 1 iid vector of fixed effects of xit on yit . Let γi∗ ∼N (0, σγ2 ) be the random effect of the ith individual, which along with xit influences the response yit . For γi = γi∗ /σγ , one may then write the binary mixed model as P r(yit = 1|γi ) =
exp(x′it β + σγ γi ) , 1 + exp(x′it β + σγ γi )
(1.1)
leading to the conditional independence assumption based likelihood function LI (β, σγ ) =
I Z Y i=1
∞
T Y
γi =−∞ t=1
{P r(yit = 1|γi )}yit {1 − P r(yit = 1|γi )}1−yit
×φ(γi ) dγi ,
(1.2)
where φ(·) denotes the standard normal density. Note however that the assumption that the binary responses of the ith individual, conditional on the random effect γi , are independent, does not accommodate the dynamic nature of the repeated responses. Thus, the binary logit mixed model (1.1)-(1.2) may not be appropriate to fit the binary panel data. Some econometric studies such as Heckman (1981), Amemiya (1985, p, 353), Manski (1987), and Honore and Kyriazidou (2000, p. 844) have made attempts to accommodate the dynamic nature of the repeated binary responses by using a binary dynamic logit mixed (BDLM) model given by exp(x′ β+σ γ ) 1+exp(xi1′ β+σγ γiγi ) , for i = 1, . . . , I; t = 1 i1 P r(yit = 1|γi ) = ′ exp(xit′β+θyi,t−1 +σγ γi ) , for i = 1, . . . , I; t = 2, . . . , T 1+exp(x β+θyi,t−1 +σγ γi ) it
= Fit , say.
(1.3)
which may be treated as an extension of the binary logit probability model (1.1). It is clear that obtaining the likelihood estimators of β, θ, and σγ2 in (1.3) requires the maximization of the exact likelihood function
36
Brajendra C. Sutradhar and R. Prabhakar Rao and V.N. Pandit
L(β, θ, σγ ) =
Z
∞
...
γ1 =−∞
×
"T Y t=2
Z
I Y y 1−yi1 Fi1 (x′i1 β + σγ γi i1 1 − Fi1 (x′i1 β + σγ γi )
∞
γI =−∞ i=1
y 1−yit Fit (x′it β + θyi,t−1 + σγ γi ) it 1 − Fit (x′it β + θyi,t−1 + σγ γi )
× φ(γ1 ) . . . φ(γI ) dγ1 . . . dγI ,
(1.4)
which appears to be manageable but complicated. For a distribution free random effects case, Honore and Kyriazidou (2000, p. 844) attempted to estimate β and θ, by exploiting the first differences of the responses yi1 − yi0 , yi2 − yi1 , . . ., which are approximately independent of γi . For example, for a special case with T = 4, they suggest to estimate β and θ by maximizing an approximate weighted log likelihood function ˜ = logL
I X i=1
Iδ {yi2 + yi3 = 1}Iδ {xi3 − xi4 = 0}
×ln
exp((xi2 − xi3 )β + θ(yi1 − yi4 ))yi2 1 + exp((xi2 − xi3 )β + θ(yi1 − yi4 ))
#
,
(1.5)
which seems to be very restrictive as, in longitudinal set up, it is unlikely that xi3 will be the same as xi4 to yield the indicator function value Iδ {xi3 −xi4 = 0} = 1. As a remedy to this problem due to non-stationarity, Honore and Kyriazidou (2000, eqn. 6, p. 845) further suggest to replace the indicator function Iδ {xi3 − xi4 = 0} by a kernel density function κ{(xi3 − xi4 )/bK }, where bK is the bandwidth that shrinks as K increases. This replacement however appears to be quite artificial for avoiding the technical difficulty associated with the method. In fact for larger T , the estimation problem will be much more difficult. Thus, even if one is interested for the estimation of β and θ, this semi-parametric approach of Honore and Kyriazidou (2000) appears to be impractical. Moreover, one may be interested to have an idea about the dispersion (σγ2 ) of the random effects, as this parameter affects both the mean and the variance of the binary responses. The estimation of this variance component is, however, generally difficult. In the linear dynamic mixed model set up, many econometricians such as Arellano and Bond (1991), Ahn and Schmidt (1995), Blundell and Bond
GMM versus GQL inferences in binary panel data models 37 (1998), and Imbens (2002) [see also Chamberlain (1992), Keane and Runkle (1992), and Bond, Bowsher and Windmeijer (2001)] have bypassed the estimation of the variance parameters and estimated the regression parameter β and the dynamic dependence parameter θ by using the well known generalized method of moments (GMM) due to Hansen (1982). To be specific, to bypass the variance parameters, these authors utilize the differences of the responses such as zit = yit − yi,t−1 and construct suitable moment functions for the ith individual as ψi (zi1 , . . . , ziT ; β, θ) so that E[ψi (zi1 , . . . , ziT ; β, θ)] = 0. This leads to the p + 1 dimensional moment estimating equations for β and θ given by I X ψi (zi , β, θ) = 0, (1.6) I −1 i=1
where zi ≡ [zi1 , . . . , ziT ]′ . The moment estimators obtained from (1.6) are consistent but they may be inefficient. The inefficiency of the estimators arises due to various reasons. First, the initial responses yi0 required to construct the moment equations (1.6) are not known. An arbitrary choice of these initial responses or ignoring them may cause the inefficiency of the estimates. This problem becomes serious in the non-linear panel data set up for the discrete data. Secondly, the construction of the moment equations (1.6) is not usually done by taking the variability of the functions into account. As an improvement over (1.6), some of the above mentioned authors estimate η = (β ′ , θ)′ by minimizing the quadratic form I −1
" I X i=1
#′
ψi (zi , η)
C
" I X i=1
#
ψi (zi , η) ,
(1.7)
for some positive definite m × m symmetric matrix C. The resulting estimators are referred to as the GMM estimators due to Hansen (1982). Note that even though the GMM technique is quite popular in econometric literature especially for the dynamic linear mixed model set up, there does not appear any discussion about its use in the non-linear dynamic mixed set up such as for (1.3). In this paper, we develop this GMM technique for the estimation of all parameters of the binary dynamic mixed model (1.3) including the dynamic dependence and variance component parameters. This is given in §2. In §3 we provide an alternative GQL technique for the estimation of the parameters of the binary dynamic mixed model (1.3). Note that this GQL approach has been introduced recently by Sutradhar (2004) for the estimation of the parameters of the binary or count mixed models, whereas
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Brajendra C. Sutradhar and R. Prabhakar Rao and V.N. Pandit
the present model is a generalization of the mixed model to the dynamic mixed model. An asymptotic as well as a simulation study is conducted in §4 to examine the relative performances of the GMM and GQL estimates. The GQL estimates are shown to be uniformly more efficient than the GMM estimates. The estimation approaches are also illustrated by analyzing a binary panel data collected by Statistics Canada. This is done in §5. The paper concludes in §6. 2
GMM Estimation
As pointed out in the last section, the GMM approach due to Hansen (1982) is a popular estimation approach in econometrics literature. For example, see the articles in the ‘twentieth anniversary GMM issue’ of the Journal of Business and Economic Statistics. Let α = (β ′ , θ, σγ2 ) be the (p + 2)-dimensional vector of the parameters of the dynamic mixed model (1.3). We now rewrite the quadratic function (1.7) as Qc (α) = I −1
" I X
#′
ψi (yi , α)
i=1
C
"
I X
#
ψi (yi , α) ,
i=1
(2.1)
′ , ψ , ψ ]′ is the (p + 2)-dimensional vector of moment where ψi (yi , α) = [ψ1i 2i 3i functions corresponding to β, θ, σγ2 , and C is a suitable weight matrix and must be positive definite. The GMM estimate of α will be obtained by minimizing the quadratic function (2.1). To be specific, the GMM estimating equations for β, θ and σγ2 are given by
∂ψ ′ Cψ = 0, ∂α where ψ =
(ψ1′ , ψ2 , ψ3 )′
with ψ1 = I
−1
I X
(2.2)
ψ1i , ψ2 = I
−1
i=1
i=1
I
−1
I X
ψ3i , and C is an weight matrix optimally chosen as
i=1
"
C = I −2
I X
#−1
E{ψi (yi , α)ψi′ (yi , α)}
i=1
This C matrix is constructed in §2.2.
I X
.
ψ2i , ψ3 =
GMM versus GQL inferences in binary panel data models 39 2.1. Construction of the unbiased moment functions. As far as the construction of the ψi (yi , β, θ, σγ2 ) vector is concerned, it is chosen such that E[ψi (yi , β, θ, σγ2 )] = 0. Furthermore, as the present binary dynamic mixed model contains three different types of parameters, namely, the regression parameter vector β, the scalar dynamic dependence parameter θ and the variance component of the random effects σγ2 , the ψi (yi , β, θ, σγ2 ) function will contain three components reflecting these parameters. Let ′ , ψ , ψ ]′ . Now to construct the first and the third ψi (yi , β, θ, σγ2 ) = [ψ1i 2i 3i components, one may refer to the construction of the moment functions under the binary mixed model. This is because, for the case when θ = 0, the present binary dynamic mixed model reduces to the binary mixed model which has been exploited extensively in statistics literature to analyze binary data in the generalized linear mixed model (GLMM) set up. For example, for the estimation of β and σγ2 , Jiang (1998) [see also Jiang and Zhang (2001), Sutradhar (2004)] has exploited the sufficient statistics under the conditional GLMM set up and constructed the basic distance functions as
ψ1i =
T X t=1
xit [yit − µit ], and ψ3i =
T T X X 2 [yiu yit − ρiut ], (2.3) [yit − ρitt ] + t=1
u
