Average and Quantile Effects in Nonseparable Panel Models

Nonseparable panel models are important in a variety of economic settings, including discrete choice. This paper gives identification and estimation results for nonseparable models under time‐homogeneity conditions that are like “time is randomly assigned” or “time is an instrument.” Partial‐identification results for average and quantile effects are given for discrete regressors, under static or dynamic conditions, in fully nonparametric and in semiparametric models, with time effects. It is shown that the usual, linear, fixed‐effects estimator is not a consistent estimator of the identified average effect, and a consistent estimator is given. A simple estimator of identified quantile treatment effects is given, providing a solution to the important problem of estimating quantile treatment effects from panel data. Bounds for overall effects in static and dynamic models are given. The dynamic bounds provide a partial‐identification solution to the important problem of estimating the effect of state dependence in the presence of unobserved heterogeneity. The impact of T, the number of time periods, is shown by deriving shrinkage rates for the identified set as T grows. We also consider semiparametric, discrete‐choice models and find that semiparametric panel bounds can be much tighter than nonparametric bounds. Computationally convenient methods for semiparametric models are presented. We propose a novel inference method that applies in panel data and other settings and show that it produces uniformly valid confidence regions in large samples. We give empirical illustrations.     

Sequential Estimation of Dynamic Discrete Games

This paper studies the estimation of dynamic discrete games of incomplete information. Two main econometric issues appear in the estimation of these models: the indeterminacy problem associated with the existence of multiple equilibria and the computational burden in the solution of the game. We propose a class of pseudo maximum likelihood (PML) estimators that deals with these problems, and we study the asymptotic and finite sample properties of several estimators in this class. We first focus on two‐step PML estimators, which, although they are attractive for their computational simplicity, have some important limitations: they are seriously biased in small samples; they require consistent nonparametric estimators of players' choice probabilities in the first step, which are not always available; and they are asymptotically inefficient. Second, we show that a recursive extension of the two‐step PML, which we call nested pseudo likelihood (NPL), addresses those drawbacks at a relatively small additional computational cost. The NPL estimator is particularly useful in applications where consistent nonparametric estimates of choice probabilities either are not available or are very imprecise, e.g., models with permanent unobserved heterogeneity. Finally, we illustrate these methods in Monte Carlo experiments and in an empirical application to a model of firm entry and exit in oligopoly markets using Chilean data from several retail industries.     

Identification of Nonseparable Models Using Instruments With Small Support

I consider nonparametric identification of nonseparable instrumental variables models with continuous endogenous variables. If both the outcome and first stage equations are strictly increasing in a scalar unobservable, then many kinds of continuous, discrete, and even binary instruments can be used to point‐identify the levels of the outcome equation. This contrasts sharply with related work by Imbens and Newey, 2009 that requires continuous instruments with large support. One implication is that assumptions about the dimension of heterogeneity can provide nonparametric point‐identification of the distribution of treatment response for a continuous treatment in a randomized controlled experiment with partial compliance.     

Nonparametric Welfare Analysis for Discrete Choice

We consider empirical measurement of equivalent variation (EV) and compensating variation (CV) resulting from price change of a discrete good using individual‐level data when there is unobserved heterogeneity in preferences. We show that for binary and unordered multinomial choice, the marginal distributions of EV and CV can be expressed as simple closed‐form functionals of conditional choice probabilities under essentially unrestricted preference distributions. These results hold even when the distribution and dimension of unobserved heterogeneity are neither known nor identified, and utilities are neither quasilinear nor parametrically specified. The welfare distributions take simple forms that are easy to compute in applications. In particular, average EV for a price rise equals the change in average Marshallian consumer surplus and is smaller than average CV for a normal good. These nonparametric point‐identification results fail for ordered choice if the unit price is identical for all alternatives, thereby providing a connection to Hausman–Newey's (2014) partial identification results for the limiting case of continuous choice.     

Risk and Rationality: Uncovering Heterogeneity in Probability Distortion

It has long been recognized that there is considerable heterogeneity in individual risk taking behavior, but little is known about the distribution of risk taking types. We present a parsimonious characterization of risk taking behavior by estimating a finite mixture model for three different experimental data sets, two Swiss and one Chinese, over a large number of real gains and losses. We find two major types of individuals: In all three data sets, the choices of roughly 80% of the subjects exhibit significant deviations from linear probability weighting of varying strength, consistent with prospect theory. Twenty percent of the subjects weight probabilities near linearly and behave essentially as expected value maximizers. Moreover, individuals are cleanly assigned to one type with probabilities close to unity. The reliability and robustness of our classification suggest using a mix of preference theories in applied economic modeling.     

Identification in Differentiated Products Markets Using Market Level Data

We present new identification results for nonparametric models of differentiated products markets, using only market level observables. We specify a nonparametric random utility discrete choice model of demand allowing rich preference heterogeneity, product/market unobservables, and endogenous prices. Our supply model posits nonparametric cost functions, allows latent cost shocks, and nests a range of standard oligopoly models. We consider identification of demand, identification of changes in aggregate consumer welfare, identification of marginal costs, identification of firms' marginal cost functions, and discrimination between alternative models of firm conduct. We explore two complementary approaches. The first demonstrates identification under the same nonparametric instrumental variables conditions required for identification of regression models. The second treats demand and supply in a system of nonparametric simultaneous equations, leading to constructive proofs exploiting exogenous variation in demand shifters and cost shifters. We also derive testable restrictions that provide the first general formalization of Bresnahan's (1982) intuition for empirically distinguishing between alternative models of oligopoly competition. From a practical perspective, our results clarify the types of instrumental variables needed with market level data, including tradeoffs between functional form and exclusion restrictions.     

Determining the Number of Factors in Approximate Factor Models

In this paper we develop some econometric theory for factor models of large dimensions. The focus is the determination of the number of factors (r), which is an unresolved issue in the rapidly growing literature on multifactor models. We first establish the convergence rate for the factor estimates that will allow for consistent estimation of r. We then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria. The theory is developed under the framework of large cross‐sections (N) and large time dimensions (T). No restriction is imposed on the relation between N and T. Simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice.     

Fixed‐Effects Dynamic Panel Models,a Factor Analytical Method

We consider the estimation of dynamic panel data models in the presence of incidental parameters in both dimensions: individual fixed‐effects and time fixed‐effects, as well as incidental parameters in the variances. We adopt the factor analytical approach by estimating the sample variance of individual effects rather than the effects themselves. In the presence of cross‐sectional heteroskedasticity, the factor method estimates the average of the cross‐sectional variances instead of the individual variances. The method thereby eliminates the incidental‐parameter problem in the means and in the variances over the cross‐sectional dimension. We further show that estimating the time effects and heteroskedasticities in the time dimension does not lead to the incidental‐parameter bias even when T and N are comparable. Moreover, efficient and robust estimation is obtained by jointly estimating heteroskedasticities.     

Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models

This paper proposes a new nested algorithm (NPL) for the estimation of a class of discrete Markov decision models and studies its statistical and computational properties. Our method is based on a representation of the solution of the dynamic programming problem in the space of conditional choice probabilities. When the NPL algorithm is initialized with consistent nonparametric estimates of conditional choice probabilities, successive iterations return a sequence of estimators of the structural parameters which we call K–stage policy iteration estimators. We show that the sequence includes as extreme cases a Hotz–Miller estimator (for K=1) and Rust's nested fixed point estimator (in the limit when K→∞). Furthermore, the asymptotic distribution of all the estimators in the sequence is the same and equal to that of the maximum likelihood estimator. We illustrate the performance of our method with several examples based on Rust's bus replacement model. Monte Carlo experiments reveal a trade–off between finite sample precision and computational cost in the sequence of policy iteration estimators.     

Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure

This paper presents a new approach to estimation and inference in panel data models with a general multifactor error structure. The unobserved factors and the individual‐specific errors are allowed to follow arbitrary stationary processes, and the number of unobserved factors need not be estimated. The basic idea is to filter the individual‐specific regressors by means of cross‐section averages such that asymptotically as the cross‐section dimension (N) tends to infinity, the differential effects of unobserved common factors are eliminated. The estimation procedure has the advantage that it can be computed by least squares applied to auxiliary regressions where the observed regressors are augmented with cross‐sectional averages of the dependent variable and the individual‐specific regressors. A number of estimators (referred to as common correlated effects (CCE) estimators) are proposed and their asymptotic distributions are derived. The small sample properties of mean group and pooled CCE estimators are investigated by Monte Carlo experiments, showing that the CCE estimators have satisfactory small sample properties even under a substantial degree of heterogeneity and dynamics, and for relatively small values of N and T.     

Instrumental Variable Estimation of Nonlinear Errors‐in‐Variables Models

This paper establishes that instruments enable the identification of nonparametric regression models in the presence of measurement error by providing a closed form solution for the regression function in terms of Fourier transforms of conditional expectations of observable variables. For parametrically specified regression functions, we propose a root n consistent and asymptotically normal estimator that takes the familiar form of a generalized method of moments estimator with a plugged‐in nonparametric kernel density estimate. Both the identification and the estimation methodologies rely on Fourier analysis and on the theory of generalized functions. The finite‐sample properties of the estimator are investigated through Monte Carlo simulations.     

Panel Data Models With Interactive Fixed Effects

This paper considers large N and large T panel data models with unobservable multiple interactive effects, which are correlated with the regressors. In earnings studies, for example, workers' motivation, persistence, and diligence combined to influence the earnings in addition to the usual argument of innate ability. In macroeconomics, interactive effects represent unobservable common shocks and their heterogeneous impacts on cross sections. We consider identification, consistency, and the limiting distribution of the interactive‐effects estimator. Under both large N and large T, the estimator is shown to be consistent, which is valid in the presence of correlations and heteroskedasticities of unknown form in both dimensions. We also derive the constrained estimator and its limiting distribution, imposing additivity coupled with interactive effects. The problem of testing additive versus interactive effects is also studied. In addition, we consider identification and estimation of models in the presence of a grand mean, time‐invariant regressors, and common regressors. Given identification, the rate of convergence and limiting results continue to hold.     

Identification and Estimation of Average Partial Effects in “Irregular” Correlated Random Coefficient Panel Data Models

In this paper we study identification and estimation of a correlated random coefficients (CRC) panel data model. The outcome of interest varies linearly with a vector of endogenous regressors. The coefficients on these regressors are heterogenous across units and may covary with them. We consider the average partial effect (APE) of a small change in the regressor vector on the outcome (cf. Chamberlain (1984), Wooldridge (2005a)). Chamberlain (1992) calculated the semiparametric efficiency bound for the APE in our model and proposed a √N‐consistent estimator. Nonsingularity of the APE's information bound, and hence the appropriateness of Chamberlain's (1992) estimator, requires (i) the time dimension of the panel (T) to strictly exceed the number of random coefficients (p) and (ii) strong conditions on the time series properties of the regressor vector. We demonstrate irregular identification of the APE when T = p and for more persistent regressor processes. Our approach exploits the different identifying content of the subpopulations of stayers—or units whose regressor values change little across periods—and movers—or units whose regressor values change substantially across periods. We propose a feasible estimator based on our identification result and characterize its large sample properties. While irregularity precludes our estimator from attaining parametric rates of convergence, its limiting distribution is normal and inference is straightforward to conduct. Standard software may be used to compute point estimates and standard errors. We use our methods to estimate the average elasticity of calorie consumption with respect to total outlay for a sample of poor Nicaraguan households.     

Inference Based on Conditional Moment Inequalities

In this paper, we propose an instrumental variable approach to constructing confidence sets (CS's) for the true parameter in models defined by conditional moment inequalities/equalities. We show that by properly choosing instrument functions, one can transform conditional moment inequalities/equalities into unconditional ones without losing identification power. Based on the unconditional moment inequalities/equalities, we construct CS's by inverting Cramér–von Mises‐type or Kolmogorov–Smirnov‐type tests. Critical values are obtained using generalized moment selection (GMS) procedures. We show that the proposed CS's have correct uniform asymptotic coverage probabilities. New methods are required to establish these results because an infinite‐dimensional nuisance parameter affects the asymptotic distributions. We show that the tests considered are consistent against all fixed alternatives and typically have power against n^−1/2‐local alternatives to some, but not all, sequences of distributions in the null hypothesis. Monte Carlo simulations for five different models show that the methods perform well in finite samples.     

Integrality in Stochastic Inventory Models

We study several finite‐horizon, discrete‐time, dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi‐period extension, and a single‐product, multi‐echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general single‐product, multi‐echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely, rolling‐horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one‐dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by establishing the integrality of both the dynamic programming and rolling‐horizon optimization models of a two‐product capacitated stochastic inventory control system.     

Time Consistency: Stationarity and Time Invariance

A sequence of experiments documents static and dynamic “preference reversals” between sooner‐smaller and later‐larger rewards, when the sooner reward could be immediate. The theoretically motivated design permits separate identification of time consistent, stationary, and time invariant choices. At least half of the subjects are time consistent, but only three‐quarters of them exhibit stationary choices. About half of subjects with time inconsistent choices have stationary preferences. These results challenge the view that present‐bias preferences are the main source of time inconsistent choices.     

Heteroskedasticity‐Robust Standard Errors for Fixed Effects Panel Data Regression

The conventional heteroskedasticity‐robust (HR) variance matrix estimator for cross‐sectional regression (with or without a degrees‐of‐freedom adjustment), applied to the fixed‐effects estimator for panel data with serially uncorrelated errors, is inconsistent if the number of time periods T is fixed (and greater than 2) as the number of entities n increases. We provide a bias‐adjusted HR estimator that is ‐consistent under any sequences (n, T) in which n and/or T increase to ∞. This estimator can be extended to handle serial correlation of fixed order.     

Instrumental Variable Estimation of Nonparametric Models

In econometrics there are many occasions where knowledge of the structural relationship among dependent variables is required to answer questions of interest. This paper gives identification and estimation results for nonparametric conditional moment restrictions. We characterize identification of structural functions as completeness of certain conditional distributions, and give sufficient identification conditions for exponential families and discrete variables. We also give a consistent, nonparametric estimator of the structural function. The estimator is nonparametric two‐stage least squares based on series approximation, which overcomes an ill‐posed inverse problem by placing bounds on integrals of higher‐order derivatives.     

Conditional Choice Probability Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity

We adapt the expectation–maximization algorithm to incorporate unobserved heterogeneity into conditional choice probability (CCP) estimators of dynamic discrete choice problems. The unobserved heterogeneity can be time‐invariant or follow a Markov chain. By developing a class of problems where the difference in future value terms depends on a few conditional choice probabilities, we extend the class of dynamic optimization problems where CCP estimators provide a computationally cheap alternative to full solution methods. Monte Carlo results confirm that our algorithms perform quite well, both in terms of computational time and in the precision of the parameter estimates.     

A Dynamic Model for Binary Panel Data With Unobserved Heterogeneity Admitting a √n‐Consistent Conditional Estimator

A model for binary panel data is introduced which allows for state dependence and unobserved heterogeneity beyond the effect of available covariates. The model is of quadratic exponential type and its structure closely resembles that of the dynamic logit model. However, it has the advantage of being easily estimable via conditional likelihood with at least two observations (further to an initial observation) and even in the presence of time dummies among the regressors.