A. Ronald Gallant

On the bias in flexible functional forms and an essentially unbiased form: The fourier flexible form☆

Abstract The Fourier flexible form and its derived expenditure system are introduced. Subject to smoothness conditions on the consumers true indirect utility function, the consumers true expenditure system must be of the Fourier form over the region of interest in an empirical investigation. Arbitrarily accurate finite parameter approximations of the consumers true expenditure system are obtained by dropping all high-order terms of the Fourier expenditure system past an appropriate truncation point. The resulting finite parametersystem is tractable in empirical studies. The reader who is primarily interested in applications need only read the second and fifth sections. The remainder of the article is concerned with the verification of these claims and an investigation of some aspects of the bias in Translog specifications.

Semi-nonparametric Maximum Likelihood Estimation

Often maximum likelihood is the method of choice for fitting an econometric model to data but cannot be used because the correct specific ation of (multivariate) density that defines the likelihood is unknown. In this situation, simply put the density equal to a Hermite series and apply standard finite dimensional maximum likelihood methods. Model parameters and nearly all aspects of the unknown density itself will be estimated consistently provided that the length of the series increases with sample size. The rule for increasing series length can be data dependent. The method is applied to nonlinear regression with sample selection. Copyright 1987 by The Econometric Society.

Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications

The extent to which specification error can explain rejection of the intertemporal capital asset pricing model is investigated using seminonparametric representations of the law of motion and utility. The authors find (1) consumption growth and asset returns display conditional heterogeneity, but this does not account for rejection of models assuming additively separable, constant relative risk aversion utility; (2) the model is accepted upon relaxation of the utility function in the direction of nonseparable utility; and (3) relaxation reduces overprediction of the conditional variance of consumption growth, overprediction of the conditional covariance of asset returns with consumption growth, and the equity premium. Copyright 1989 by The Econometric Society.

A unified theory of estimation and inference for nonlinear dynamic models

2. The data generation process and optimization estimators 3. Consistency of optimization estimators 4. More on near epoch dependence 5. Asymptotic mormality 6. Estimating asymptotic cavariance matrices 7. Hypothesis testing

Statistical inference for a system of simultaneous, non-linear, implicit equations in the context of instrumental variable estimation

Abstract Statistical inference for a system of simultaneous, non-linear, implicit equations is discussed. The discussion considers inference as an adjunct to two- and three-stage least squares estimation rather than in a general setting. For both of these cases the non-null asymptotic distribution of a test statistic based on the optimization criterion and a test based on the asymptotic distribution of the estimator is found; a total of four. It is argued that the tests based on the optimization criterion are to be preferred in applications. The methods are illustrated by application to hypotheses implied by the theory of demand using a translog expenditure system and data on personal consumption expenditures for durables, non-durables, and energy for the period 1947– 1971.

Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes

Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, since the transition density is generally unknown, one is forced to approximate it. The simulation-based approach suggested by Pedersen (1995) has great theoretical appeal, but previously available implementations have been computationally costly. We examine a variety of numerical techniques designed to improve the performance of this approach. Synthetic data generated by a Cox-Ingersoll-Ross model with parameters calibrated to match monthly observations of the U.S. short-term interest rate are used as a test case. Since the likelihood function of this process is known, the quality of the approximations can be easily evaluated. On datasets with 1,000 observations, we are able to approximate the maximum likelihood estimator with negligible error in well under 1 min. This represents something on the order of a 10,000-fold reduction in computational effort as compared to implementations without these enhancements. With other parameter settings designed to stress the methodology, performance remains strong. These ideas are easily generalized to multivariate settings and (with some additional work) to latent variable models. To illustrate, we estimate a simple stochastic volatility model of the U.S. short-term interest rate.

Unbiased determination of production technologies

Abstract To determine whether an industry exhibits constant returns to scale, whether the production function is homothetic, or whether inputs are separable, a common approach is to specify a cost function, estimate its parameters using data such as prices and quantities of inputs, and then test the parametric restrictions corresponding to constant returns, a homothetic technology, or separability. Statistically, such inferences are valid if the true cost function is a member of the parametric class considered, otherwise the inference is biased. That is, the true rejection probability is not necessarily adequately approximated by the nominal size of the statistical test. The use of fixed parameter flexible functional forms such as the Translog, the generalized Leontief, or the Box-Cox will not alleviate this problem. The Fourier flexible form differs fundamentally from other flexible forms in that it has a variable number of parameters and a known bound, depending on the number of parameters, on the error, as measured by the Sobolev norm, of approximation to an arbitrary cost function. Thus it is possible to construct statistical tests for constant returns, a homothetic technology, or separability which are asymptotically size α by letting the number of parameters of the Fourier flexible form depend on sample size. That is, the true rejection probability converges to the nominal size of the test as sample size tends to infinity. The rate of convergence depends on the smoothness of the true cost function; the more times is differentiable the true cost function, the faster the convergence. The method is illustrated using the data on aggregate U.S. manufacturing of Berndt and Wood (1975, 1979) and Berndt and Khaled (1979).

Original Contribution: On learning the derivatives of an unknown mapping with multilayer feedforward networks

Recently, multiple input, single output, single hidden-layer feedforward neural networks have been shown to be capable of approximating a nonlinear map and its partial derivatives. Specifically, neural nets have been shown to be dense in various Sobolev spaces. Building upon this result, we show that a net can be trained so that the map and its derivatives are learned. Specifically, we use a result of Gallants to show that least squares and similar estimates are strongly consistent in Sobolev norm provided the number of hidden units and the size of the training set increase together. We illustrate these results by an application to the inverse problem of chaotic dynamics: recovery of a nonlinear map from a time series of iterates. These results extend automatically to nets that embed the single hidden layer, feedforward network as a special case.

Seemingly unrelated nonlinear regressions

Abstract The article considers the estimation of the parameters of a set of nonlinear regression equations when the responses are contemporaneously but not serially correlated. Conditions are set forth such that the estimator obtained is strongly consistent, asymptotically normally distributed, and asymptotically more efficient than the single-equation least squares estimator. The methods presented allow estimation of the parameters subject to nonlinear restrictions across equations. The article includes a discussion of methods to perform the computations and a Monte Carlo simulation.

Imposing curvature restrictions on flexible functional forms

Abstract A general computational method for estimating the parameters of a flexible functional form subject to convexity, quasi-convexity, concavity, or quasi-concavity at a point, at several points, or over a region, is set forth and illustrated with an example.