First-differencing in panel data models with incidental functions
Summary
This note discusses a class of models for panel data that accommodate between-group heterogeneity that is allowed to exhibit positive within-group variance. Such a set-up generalizes the traditional fixed-effect paradigm in which between-group heterogeneity is limited to univariate factors that act like constants within groups. Notable members of the class of models considered are non-linear regression models with additive heterogeneity and multiplicative-error models suitable for non-negative limited dependent variables. The heterogeneity is modelled as a non-parametric nuisance function of covariates whose functional form is fixed within groups but is allowed to vary freely across groups. A simple approach to perform inference in such situations is based on local first-differencing of observations within a given group. This leads to moment conditions that, asymptotically, are free of nuisance functions. Conventional generalized method of moments procedures can then be readily applied. In particular, under suitable regularity conditions, such estimators are consistent and asymptotically normal, and asymptotically valid inference can be performed using a plug-in estimator of the asymptotic variance.
1. INTRODUCTION
represents log-input factors such as capital and labour, α0 is the corresponding vector of elasticities, and 
is total factor productivity, which will typically be correlated with the inputs, rendering the ordinary least-squares estimator of α0 inconsistent. To estimate the elasticities from within-group variation, total factor productivity is decomposed as 
, where 
is assumed to be orthogonal to the production inputs but 
can be correlated with the 
. In this case, a within-group transformation will sweep out 
, after which least-squares can be applied to estimate α0. The inclusion of fixed effects in this manner has become standard practice in applied work.However, there are good reasons to believe that unobserved heterogeneity goes beyond what can be captured by such location parameters. In the production-function example, it seems natural that managerial ability depends on such things as experience, education and sector-specific characteristics. As such, ability itself is the outcome of a production process, and it can be difficult to justify that it remains constant over the sampling period. A more appropriate way to control for managerial ability then could have 
for some latent function 
that maps drivers 
, such as experience and schooling, into ability. Similarly, in matching models, 
could represent the match-efficiency parameter. In the context of the labour market, Sedláček (2014) finds empirical evidence that matching efficiency is procyclical and is, at least partially, driven by the hiring standards of firms. Moreover, the matching literature has argued that the efficiency parameter should be endogenous to the agents' optimization behaviour, rather than exogenously determined.
This note suggests a simple way to conduct inference on common parameters in panel-data models with non-parametric incidental functions. Besides the linear set-up just described, the approach can equally be used for models with multiplicative errors, such as models for count data, and for logit models, for example. In either case, staying true to the fixed-effect paradigm, the aim is to estimate a finite-dimensional parameter while controlling for between-group heterogeneity in a non-parametric manner. The difference with the traditional fixed-effect view, however, is that the heterogeneity is allowed to vary both within and between groups. This view on unobserved heterogeneity is different from the one taken in recent work on the linear random-coefficient model (Arellano and Bonhomme, 2012, and Graham and Powell, 2012) and, as such, can serve as a useful complement.
2. LOCAL FIRST-DIFFERENCING
2.1. Incidental Functions
Consider a panel dataset consisting of two observations on n units. Restricting attention to two observations is without loss of generality. We let 
denote the outcome variables for unit i, and let 
and 
denote observable covariates. The distinction between the variables 
and 
will become clear below.
(2.1)
and vector of slope coefficients α0. Applications of this model are widespread. When 
, an ordinary least-squares regression of 
on 
is known to yield a consistent point estimator of α0 as 
. Indeed,
has full rank. When the covariates are not strictly exogenous, the above moment condition can be replaced by 
for a vector of instrumental variables 
. A leading case would be a dynamic model where 
and 
contains further lags of the outcome variable; see, e.g., Arellano and Bond (1991).
, as additional regressors. This delivers a specification of the form
(2.2)
can be flexible polynomials specifications or other non-linear transformations of the controls and, of course, can include interactions with 
. The choice of functional form is up to the researcher, and linearity is popular because of the resulting ease of computation via multiple regression. An approach that would prevent functional-form misspecification in the effect of the control variables would be to work with the partially linear model
(2.3)
to be non-parametric, it is restricted to be identical across i. Recent empirical work has stressed the presence of excess heterogeneity across agents in microeconometric models. Guvenen (2009), Browning et al. (2010) and Browning and Carro (2010, 2014), for example, provide extensive discussions and empirical evidence on this. An alternative extension of the Robinson framework that stays true to the fixed-effect tradition would be
(2.4)
are unit-specific non-parametric functions, and the usual location parameter 
has been absorbed into it. A special case of 2.4 that has received some attention recently is the standard linear random-coefficient model (Swamy, 1970, Chamberlain, 1992b, and Arellano and Bonhomme, 2012). Another is the varying-coefficient model (Hastie and Tibshirani, 1993). None the less, the motivation for allowing for excess heterogeneity is clearly different in these cases.
and 
can be consistently estimated. Clearly, this is not possible under asymptotics where the number of observations per unit is held fixed. Suppose that 
for some function 
for which the expectation 
exists for all v in a neighbourhood of zero. Then, provided that 
is a constant,
has full rank. Indeed,
is fairly weak. Suppose that 
is continuously differentiable. Then, its derivative, say 
, is locally bounded. Hence, 
, with v restricted to the neighbourhood 
, 
, satisfies the required Lipschitz-type smoothness condition. When the support of 
is discrete, we require that 
. An estimator of α0 would be
. This estimator is 
-consistent and asymptotically normal under standard moment assumptions. When 
is continuous, the event 
has probability zero and 
-consistent estimation will not be possible. However, under suitable regularity conditions, we can still perform asymptotically valid inference on α0 via 
on redefining 
as
.1,2 Here, the convergence rate of 
will be reduced to 
. We provide regularity conditions and more detailed asymptotic theory below. In either case, the approach consists of simply constructing 
for each i and then performing a weighted least-squares regression of 
on 
with weight 
. This estimator is similar in spirit to the one considered for sample-selection models by Kyriazidou (1997, 2001).The 
can be seen as incidental functions, as opposed to the incidental parameters 
in the conventional set-up in 2.1. Furthermore, the 
can be seen as draws from a distribution that depends on 
but which is left unspecified. The approach just described does not estimate these functions but, rather, differences them out by focusing on the population of “stayers” (Chamberlain, 1984), that is, on units for which 
lies in a shrinking neighbourhood of zero. As such, this approach could be called local first-differencing. Of course, a prerequisite to identification is that the support of 
and the support of 
are not disjoint. The leading example where this requirement would be violated is when the 
include time dummies or time trends. Such aggregate time effects are commonly used in applied work. Of course, they can easily be included in the traditional way, that is, by including them in a linear fashion and assigning them homogeneous coefficients.
to be predetermined as opposed to strictly exogenous, and can accommodate discrete components in both 
and 
. None the less, as in Hoderlein and White (2012) and Arellano and Bonhomme (2012), allowing for feedback toward the 
is complicated, because the distribution of the transitory shocks, 
, can change after conditioning on the event 
.2.2. Non-linear Specifications
The applicability of local first-differencing is not limited to the linear model. Indeed, any fixed-effect model where heterogeneous intercepts can be accommodated can be extended to allow for incidental functions. The literature on panel data models is large, and we do not attempt to give a complete overview here. A survey is provided by Arellano and Honoré (2001).
and 
but to maintain additivity of the incidental function, as in
. Here,
independent of 
. An application of the conditional-likelihood argument shows that
. The optimal unconditional moment condition in the sense of Chamberlain (1987) equals
In each of the examples just mentioned, it is easy to construct a generalized method of moments (GMM) estimator in which the usual moment condition is complemented with the kernel weight 
as described above. We provide asymptotic theory in the next section.
There are several other models that could be extended to allow for incidental functions. Some interesting examples are truncated and censored regression models (Honoré, 1992), as well as general transformation models and generalized regression models (Abrevaya, 1999, 2000). The resulting estimators would have similar asymptotic properties. However, they are M-estimators rather than GMM estimators, and the associated criterion functions are characterized by a certain degree of non-smoothness. As such, they will not fit exactly the generic set-up entertained below.
3. ASYMPTOTIC THEORY
is identified through the moment condition
is a non-negative bandwidth sequence that is o(1) and k is a kernel function. Regularity conditions on 
and k are collected in Assumption 3.3. A GMM estimator of α0 based on 
is then given by
denotes a given positive-definite weight matrix. This section provides distribution theory for 
in the form of a consistency result and an asymptotic-normality result. The proofs are given in the Appendix.Some elementary regularity conditions are collected in Assumption 3.1.
Assumption 3.1.
is a compact set and α0 is interior to it. m is twice continuously differentiable in α with derivatives 
and 
. The distribution of 
is absolutely continuous and the associated density function is strictly positive in a neighbourhood of zero.
denote the Euclidean and Frobenius norms. To state sufficient conditions for consistency, let
.
Assumption 3.2.For all 
, 
and 
are finite, 
is bounded in v, and 
is continuous in v in a neighbourhood of zero.
Assumption 3.3.
is a bounded and symmetric sth-order kernel function.
The conditions in Assumptions 3.2 and 3.3 are conventional. We refer to Li and Racine (2007) for a definition, examples and a discussion of kernel functions that satisfy Assumption 3.3.
The consistency result is stated in Theorem 3.1.
Theorem 3.1.Let Assumptions 3.1–3.3 hold. Suppose that 
for W0 non-stochastic and positive definite. Then 
.
, we need an additional set of conditions. We let
Assumption 3.4.For all 
, 
is finite, 
is bounded in v, and 
is continuous in v in a neighbourhood of zero. 
is bounded. 
is continuous in v in a neighbourhood of zero and 
is bounded. 
is s-times continuously differentiable with bounded derivatives.
Let 
and 
. Theorem 3.2 gives the asymptotic distribution of 
.
Theorem 3.2.Let Assumptions 3.1–3.4 hold. Suppose that Σ has maximal column rank, that Δ is positive definite, and that 
for W0 non-stochastic and positive definite. Then
and 
.
ACKNOWLEDGMENTS
I am grateful to Jaap Abbring, Manuel Arellano, Stefan Hoderlein and three referees.
APPENDIX A
PROOFS OF THEOREMS
. Given identification, the regularity conditions in Assumption 3.1, and the fact that 
, we only need to verify
is finite, and k is bounded, Lemma 2.9 in Newey and McFadden (1994) further states that it suffices to prove that 
for all 
. Fix 
. By the triangle inequality,
by the law of large numbers. Dominated convergence implies that
. Thus, 
. This holds for any 
, and so consistency has been shown. 
and (b) 
. The asymptotic distribution of the estimator then follows from the linearization
, 
and the bias term is asymptotically negligible. The leading term satisfies the conditions of Lyapunov's central limit theorem. To see this, write
and
. This establishes (a). To verify (b), we can proceed as in the proof of Theorem 3.1. In particular, Lemma 2.9 of Newey and McFadden (1994) can again be applied. By the moment conditions in Assumption 3.4, we have that 
An application of the bounded convergence theorem similarly shows that 
. Uniform convergence of the Jacobian matrix follows and the proof is complete. 
REFERENCES
. An automated approach to selecting the bandwidth is to estimate it jointly with α0, as in Härdle et al. (1993). See the Online Appendix for details and simulation experiments.
and indicator functions for the discrete elements.
