Use of panel data regression methods has become increasingly popular as the availability of longitudinal data sets has grown. Panel data contain repeated time series observations (T) for a large number (N) of cross sectional units (e.g., individuals, households, or firms). An important advantage of using such data is that they allow researchers to control for unobservable heterogeneity, that is, systematic differences across cross sectional units. Regressions using aggregated time series and pure cross section data are likely to be contaminated by these effects, and statistical inferences obtained by ignoring these effects could be seriously biased. When panel data are available, error components models can be used to control for these individual differences. Such a model typically assumes that the stochastic error term has two components: a time invariant individual effect which captures the unobservable individual heterogeneity and the usual random noise term. Some explanatory variables (e.g., years of schooling in the earnings equation) are likely to be correlated with the individual effects (e.g., unobservable talent or IQ). A simple treatment to this problem is the within estimator which is equivalent to least squares after transformation of the data to deviations from means.

Unfortunately, the within method has two serious defects. First, the within transformation of a model wipes out time invariant regressors as well as the individual effect, so that it is not possible to estimate the effects of time invariant regressors on the dependent variable. Second, consistency of the within estimator requires that all the regressors in a given model be strictly exogenous with respect to the random noise.

Although GMM estimators are consistent and asymptotically normally distributed under general regularity conditions, it has long been recognized that this first–order asymptotic distribution may provide a poor approximation to the finite sample distribution. In particular, GMM estimators may be badly biased, and asymptotic tests based on these estimators may have true sizes substantially different from presumed nominal sizes.

This chapter reviews these finite sample properties, from both the theoretical perspective, and from simulation evidence of Monte Carlo studies. The theoretical literature on the finite sample behavior of instrumental variables estimators and tests is seen to provide valuable insights into the finite sample behavior of GMM estimators and tests.

The chapter then considers Monte Carlo simulation evidence of the finite sample performance of GMM techniques. Such studies have often focussed on applications of GMM to estimating particular models in economics and finance, e.g., business cycle models, inventory models, asset pricing models, and stochastic volatility models. This survey reviews and summarizes the lessons from this simulation evidence.

The final section examines how this knowledge of the finite sample behavior might be used to conduct improved inference. For example, bias corrected estimators may be obtained. Also, properly implemented bootstrap techniques can deliver modified critical values or improved test statistics with rather better finite sample behavior. Alternatively, analytical techniques might be used to obtain corrected test statistics.

In recent years the GMM approach became increasingly popular for the analysis of panel data (e.g., Avery, Hansen and Hotz [1983], Arrelano and Bond [1991], Keane [1989], Lechner and Breitung [1996]). Combining popular nonlinear models used in microeconometric applications with typical panel data features like an error component structure yields complex models which are too complicated or even intractable to be estimated by maximum likelihood. In such cases the GMM approach is an attractive alternative.

A well known example is the probit model, which is one of the work horses whenever models with binary dependent variables are analyzed. Although the nonrobustness of the probit estimates to the model's tight statistical assumptions is widely acknowledged, the ease of computation of the maximum likelihood estimator (MLE)—combined with the availability of specification tests—make it an attractive choice for many empirical studies based on cross sectional data. The panel data version of the probit model allows for serial correlation of the errors in the latent equations. The problem with these types of specifications is, however, that the MLE becomes much more complicated as in the case of uncorrelated errors.

Two ways to deal with that sort of general problems have emerged in the literature. One is the simulated maximum likelihood estimation (SMLE). The idea of this technique is to find an estimator that only approximates the MLE but retains the asymptotic efficiency property of the exact MLE. SMLE uses stochastic simulation procedures to obtain approximate choice probabilities (see e.g., Börsch-Supan and Hajivassiliou [1993], or Hajivassiliou, McFadden and Ruud [1996]).

Simultaneous equations models involving limited dependent variables can have nonunique reduced forms, a problem called logical inconsistency in the econometrics literature. In response to that problem, such models can be compelled to be recursive (Maddala [1983], Amemiya [1985]) or recast in terms of the latent variables (Mallar [1977]). In labor economics and elsewhere, this approach is often contrary to structural modelling; theory involving education, childbearing, and work, for example, naturally leads to models with simultaneously related limited dependent variables. Restricting these models to be recursive is inconsistent with the theory.

It is widely believed among economists that logically inconsistent models cannot be data generating processes (see Amemiya [1974] and Maddala [1983]). However, Jovanovic [1989] showed that the structural form of a model with nonunique reduced forms can be identified. That raises the possibility that these models can produce outcomes which are random variables even if the process is logically inconsistent.

An alternative interpretation of these models is that they can generate more than one equilibrium for the endogenous variables for some values of the exogenous variables and disturbances. Viewed this way, the problem can be solved by using a selection rule (Dagsvik and Jovanovic [1991], Goldfeld and Quandt [1968]], Hamilton and Whiteman [1985]) or collapsing the possibly nonunique equilibria into one outcome for purposes of estimation (Bresnahan and Reiss [1991]).

This chapter combines the use of a selection rule to choose among alternative equilibria with the insights of Jovanovic [1989] and standard GMM estimation theory to suggest an alternative to the method of Bresnahan and Reiss [1991] to identify and estimate simultaneous equations models of limited dependent variables.

In Chapters 4 and 6 we assumed that the time horizon T, the initial values of state variables, si0, and their terminal values siT were exogenously specified. Obviously these are very restrictive assumptions. In many economic problems we want to allow some of these values to be determined endogenously (subject to constraints). For example, the optimal consumption problem (Example 6.4.1) may be modified to allow the planner to select the value of the terminal stock s(T), subject only to some constraint such that s(T) may not be less than a certain lower bound s, or even to select the economy's doomsday T after which all activities cease. Obviously, when s(T) or T is not fixed, we need additional necessary conditions to determine the new unknown (s*(T) or T*); these conditions are called the transversality conditions.

We shall look at various cases, beginning with the simplest. Section 7.8 contains a general statement synthesizing the various transversality conditions. Because there are many kinds of boundary conditions, there are also many kinds of transversality conditions. This array of special cases sometimes appears formidable to students of optimal control theory. For this reason a summary table is provided in Section 7.10. The table lists various features of control problems, and for each one gives the associated transversality condition. If a problem has several of these features, all corresponding transversality conditions apply.

Each of the following sections presents one type of problem and derives the associated transversality condition.

Problems with two-state variables

Nearly all the models hitherto encountered in this book have contained a single state variable. (Exceptions are the models of Sections 8.1, 8.3, and 9.6.) We have relied very heavily on phase diagrams in shedding light on the optimal solution. When there are two state variables, however, the (state, costate) space is four-dimensional and cannot be represented straightforwardly. It must be understood that, given the usual regularity conditions, we have in the maximum principle a set of necessary and sufficient conditions for an optimum, whatever the size of the problem, and if all functional forms and other restrictions were fully specified, we could – possibly using numerical methods – provide an explicit solution to the problem. However, since most models of interest in economic theory involve some unspecified functional forms, an explicit solution is normally unobtainable. This is why phase diagrams are such a useful device for pulling together all the pieces of information contained in the maximum principle.

Since they fail us here, we must devise other means of synthesizing the information. Unfortunately, this is often quite difficult, and in many cases a complete characterization of the solution escapes us. This is not to say that we cannot offer a partial characterization of the solution. It is the aim of this section to illustrate what can indeed be done. First note that in the models of Sections 8.1 and 8.3, the analysis was reduced to a two-dimensional phase diagram. The reader is referred to those sections.