Bryan W. Brown

The Random Utility Hypothesis and Inference in Demand Systems

In this paper, the authors examine the consequences of adopting the random utility hypothesis as an approach for randomizing a system of demand equations. Random utility models are appealing since they allow the usual assumption of deterministic utility-maximizing behavior by each consumer to coexist with the apparent randomness across individuals that is exhibited by data. Their results show that the use of random utility models implies that the disturbances of the demand equations may not be homoskedastic, but must be functions of prices and/or income. Copyright 1989 by The Econometric Society.

Residual-Based Procedures for Prediction and Estimation in a Nonlinear Simultaneous System

This paper proposes the residual-based stochastic predictor as an alternative procedure for obtaining forecasts with a static nonlinear econometric model. This procedure modifies the usual Monte Carlo approach to stochastic simulations of the model in that calculated residuals over the sample period are used as proxies for disturbances instead of random draws from some assumed parametric distribution. In compar-ison with the Monte Carlo predictor, the residual-based should be less sensitive to distributional assumptions concerning disturbances in the system. It is also less demanding computationally. The large-sample asymptotic moments of the residual-based predictor are derived in this paper and compared with those of the Monte Carlo predictor. Both procedures are asymptotically unbiased. In terms of asymptotic mean squared prediction error (AMSPE), the Monte Carlo is efficient relative to the residual-based when the number of replications in the Monte Carlo simulations is large relative to sample size. This order of relative efficiency is reversed, however, when replication and sample sizes are similar. In any event, the amount by which the AMSPE of either predictor exceeds the lower bound for AMSPE is small as a percentage of the lower bound AMSPE when sample and replication sizes are at least of moderate magnitude. The paper also discusses the extension of the residual-based anld Monte Carlo procedures to the estimation of higher order moments and cumulative distribution functions of endogenous variables in the system.

The Identification Problem in Systems Nonlinear in the Variables

This paper examines the identifiability of the coefficients of a single equation in a simultaneous equation model which is nonlinear only in the variables. The concept of identifiability in this model is motivated and developed using the closely related concept of observational equivalence. This framework is then utilized to develop necessary and sufficient conditions for identifiability when the disturbances are required to be independent of the exogenous variables. The approach recommended by Fisher is shown to yield sufficient but not necessary conditions for identifiability. For several relatively common special cases the necessary and sufficient conditions are found to simplify to the familiar rank condition for identifiability in the linear model. THE SIMULTANEOUS EQUATION MODEL that is nonlinear only in the variables has enjoyed widespread application in economics. Such models are linear in the parameters and typically seem to be linear in the variables as well, when viewing a single equation. In many models the nonlinearity in the variables arises due to endogenous variables entering in different forms in different equations (logged and unlogged form, for example). In macroeconometric models nonlinearity in the variables arises when the model includes endogenous real, nominal, and price variables, which are nonlinearly related.2 Whatever the reason for the nonlinearity in the variables, it is important to determine the conditions under which the equations of such models can be identified.

Predictors in Dynamic Nonlinear Models: Large-Sample Behavior

The large-sample behavior of one-period-ahead and multiperiod-ahead predictors for a dynamic nonlinear simultaneous system is examined in this paper. Conditional on final values of the endogenous variables, the asymptotic moments of the deterministic, closed-form, Monte Carlo stochastic, and several variations of the residual-based stochastic predictor are analyzed. For one-period-ahead prediction, the results closely parallel our previous findings for static nonlinear systems. For multiperiod-ahead prediction similar results hold, except that the effective number of sample-period residuals available for use with the residual-based predictor is T/m , where T denotes sample size. In an attempt to avoid the problems associated with sample splitting, the complete enumeration predictor is proposed which is a multiperiod-ahead generalization of the one-period-ahead residual-based predictor. A bootstrap predictor is also introduced which is similar to the multiperiod-ahead Monte Carlo except disturbance proxies are drawn from the empirical distribution of the residuals. The bootstrap predictor is found to be asymptotically inefficient relative to both the complete enumeration and Monte Carlo predictors.

Stochastic specification in random production models of cost-minimizing firms

Abstract We examine the implications of additive, homoskedastic errors for models of firms cost-minimizing behavior. The premise is that some factors unobservable to the econometrician are known to the firm and must satisfy the theoretical restrictions imposed by cost minimization. We analyze additive homoskedastic errors for both the input demand model and the cost share model. We find that this simple error structure is inconsistent with rational behavior if some part of the stochastic component is known to the decision-maker. Input demand models violate nonnegativity and cannot represent homothetic technologies. For cost share systems, nonnegativity is also a potential problem but, surprisingly, concavity can be easily satisfied. An empirical application assesses the practical significance of these results.

Band Covariance Matrix Estimation Using Restricted Residuals: A Monte Carlo Analysis

Using Monte Carlo simulations, the authors examine the performance of Wald-type test statistics based on alternative versions of a heteroskedasticity consistent band covariance matrix estimator that is algorithmically constrained to be positive definite in finite samples. They find that the test statistic based on the originally proposed estimator tends to result in excessive type I errors. This problem can be alleviated to some extent by employing a quasi-maximum likelihood procedure. However, by simply using restricted, as opposed to the usual OLS residuals when constructing the band covariance matrix estimator, excessive type I errors can be substantially reduced, if not eliminated. Copyright 1995 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Stochastic Simulation, Prediction and Validation of Nonlinear Models

Many econometric models for forecasting and policy analysis consist of a statistically estimated dynamic system of nonlinear stochastic equations. The distinguishing feature of these models is the nonlinearity of the solution for the endogenous variables in terms of model disturbances. Additionally, they are dynamic — with lagged endogenous variables and/or serially correlated errors. Models of macroeconomic systems and limited dependent variables in a simultaneous setting are notable examples.

22 Stochastic simulations for inference in nonlinear errors-in-variables models

Publisher Summary This chapter focuses on the stochastic simulations for inference in nonlinear errors-in-variables models. For given values of the model parameters, the algorithm requires stochastic simulations for the numerical evaluation of the likelihood function or of first order conditions. For the limited-dependent-variable case, this entails numerical evaluation of choice probabilities, conditional on measured observations of exogenous variables. For the general case, the process revolves around evaluation of conditional expectations of endogenous variables. Most of what is available for nonlinear errors-in-variables models deals with the single-equation structural model with a linear relationship for measurement errors. The chapter considers new econometric procedures for statistical inference in nonlinear econometric models in general and limited dependent variable models in particular, where explanatory variables are measured with (possibly) nonlinear errors. In the treatment of errors in variables, stochastic simulations of the model play a central role in providing numerical approximations to conditional moments of variables appropriate to the problem at hand—estimation or prediction or specification testing. The potential uses of the stochastic simulators developed in the chapter are indicated for generalizing McFaddens (1989) method of simulated moments to nonlinear errors-in-variables models and for the numerical evaluation of maximum likelihood estimates in nonlinear errors-in-variables models.