Recursive estimation of simultaneous equation models

A recursive instrumental variable estimator is derived. For simultaneous equation estimation, the choice of the instruments is discussed. A computationally simple and asymptotically efficient recursive estimator is proposed in this context.     

Coefficients of correlation for simultaneous equation systems

This paper presents measures of correlation for use with either a single equation within a simultaneous system or for the whole system, which specifically account for the identifying restrictions. The measures lie between zero and one and have the same interpretation as the familiar R² used with classical least squares. When used in conjunction with several common coefficient estimators they have straightforward limiting distributions so that asymptotic tests of significance are possible.     

Semiparametric estimation of consumer demand systems in real expenditure

Microdata concerning consumer demand typically show considerable variation in real expenditures, but very little variation in prices. We propose a semiparametric strategy for the consumer demand problem in which expenditure share equations are estimated nonparametrically in the real expenditure direction and estimated parametrically (with fixed or varying coefficients) in price directions. In our model, Engel curves are unrestricted: demands may have any rank. Because the demand model is derived from a cost function, it may be restricted to satisfy integrability and used for consumer surplus calculations. Since real expenditure is unobserved, but rather estimated under the model, we face a semiparametric model with a nonparametrically generated regressor. We show efficient convergence rates for parametric and nonparametric components. We illustrate the feasibility of our proposed strategy using Canadian expenditure and price data: Engel curves display curvature which cannot be encompassed by standard parametric models. We also find that the rationality restriction of Slutsky symmetry is rejected in the fixed‐coefficients model, but not in the varying‐coefficients model. Copyright © 2009 John Wiley & Sons, Ltd.     

Method-of-moment view of linear simultaneous equation systems

In this paper, we review the modern method-of-moment-based approaches to identification and estimation of linear simultaneous equation systems. First, we present the rank condition for the structural form (SF) parameter identification. The rank condition comes naturally and is much easier to understand than that in the conventional reduced-form-based indirect approach. Then, we show how to estimate all SF parameters jointly (in a single step) with method-of-moment estimators. As it turns out, using only unconditional moments, but not any conditional moments, greatly simplifies the identification and estimation issues, and makes light work of conveying the essential ideas involved.     

The coefficient of determination and simultaneous equation systems

In this paper we show that the Carter-Nagar (1977) R²'s for single structural equations and systems are in fact R² for the reduced form where the partially restricted reduced form estimation method is employed. We also show that the results of McElroy (1977) may be used to derive the Carter-Nagar system measure. If the reduced form equations are estimated by Kakwani's (1975) k-class reduced form estimator a new R² may be defined which is shown to be asymptotically equivalent to the Carter-Nagar measure.     

Evaluating estimation methods for ordinal data in structural equation modeling

This study examined the performance of two alternative estimation approaches in structural equation modeling for ordinal data under different levels of model misspecification, score skewness, sample size, and model size. Both approaches involve analyzing a polychoric correlation matrix as well as adjusting standard error estimates and model chi-squared, but one estimates model parameters with maximum likelihood and the other with robust weighted least-squared. Relative bias in parameter estimates and standard error estimates, Type I error rate, and empirical power of the model test, where appropriate, were evaluated through Monte Carlo simulations. These alternative approaches generally provided unbiased parameter estimates when the model was correctly specified. They also provided unbiased standard error estimates and adequate Type I error control in general unless sample size was small and the measured variables were moderately skewed. Differences between the methods in convergence problems and the evaluation criteria, especially under small sample and skewed variable conditions, were discussed.     

Large sample estimation and testing procedures for dynamic equation systems

Joint two-step estimation procedures which have the same asymptotic properties as the maximum likelihood (ML) estimator are developed for the final equation, transfer function and structural form of a multivariate dynamic model with normally distributed vector-moving average errors. The ML estimator under fixed and known initial values is obtained by iterating the procedure until convergence. The asymptotic distribution of the two-step estimators is used to construct large sample testing procedures for the different forms of the model.     

Large sample estimation and testing procedures for dynamic equation systems

In 1980, Palm and Zellner presented a number of joint or system estimation and testing procedures for the final equations, transfer functions and structural equations of a multivariate dynamic model with normally distributed vector moving average errors. In this paper, we show that there is a flaw in their argument which renders most of the procedures invalid. Possible alternative procedures are described.     

Nonparametric methods and local‐time‐based estimation for dynamic power law distributions

This paper introduces nonparametric econometric methods that characterize general power law distributions under basic stability conditions. These methods extend the literature on power laws in the social sciences in several directions. First, we show that any stationary distribution in a random growth setting is shaped entirely by two factors: the idiosyncratic volatilities and reversion rates (a measure of cross‐sectional mean reversion) for different ranks in the distribution. This result is valid regardless of how growth rates and volatilities vary across different economic agents, and hence applies to Gibrat's law and its extensions. Second, we present techniques to estimate these two factors using panel data. Third, we describe how our results imply predictability as higher‐ranked processes must on average grow more slowly than lower‐ranked processes. We employ our empirical methods using data on commodity prices and show that our techniques accurately describe the empirical distribution of relative commodity prices. We also show that rank‐based out‐of‐sample forecasts of future commodity prices outperform random‐walk forecasts at a 1‐month horizon.     

On univariate time series methods and simultaneous equation econometric models

Systematic testing of the implications of the structural assumptions for the properties of the final equations and transfer functions associated with a dynamic econometric model, as proposed by Zellner and Palm (1974–1975), proved to be useful in model building. This paper contains several remarks on the use of univariate time series methods to empirically check out the implications of a linear dynamic simultaneous equation model.     

Application of one‐step method to parameter estimation in ODE models

In this paper, we study application of Le Cam's one‐step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non‐linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one‐step method starts from a preliminary ‐consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n →∞ ) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one‐step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary ‐consistent estimator that we use depends on non‐parametric smoothing, and we provide a data‐driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one‐step method for practical use is pointed out.     

A ridge-like method for simultaneous estimation of simultaneous equations

In the context of full information estimation in a linear simultaneous equations model, this paper considers a ridge-like modification of the 3SLS estimator. The proposed method is particularly desirable where the square matrix of the 3SLS normal equationsis singular (or near-singular) leading to non-existence (or poor performance) of the estimator. Furthermore, the type of solution suggested here does seem to result in the existence of the finite sample moments of the estimator even when the degrees of over identification are as low as zero (just identified models). This paper considers only a simple scalar form of the ‘ridge-matrix” with a relatively simple choice of the modifying scalar that preserves the asymptotic properties of the 3SLS estimator. A value of this scalar is derived which minimizes an appropriatequadratic risk criterion. The approximate quadratic risk function is based upon the asymptotic approximation of the relevant moments in the manner of Nagar (1959). A range of risk reducing values of the ‘ridge-scalar” is also given.     

A note on simultaneous equation estimation techniques applications with a regional econometric model

A large-scale regional econometric model is estimated using six estimation techniques, including Iterated Instrumental Variables and Iterated Two-Stage Least-Squares. Following estimation the model is simulated and a seventh technique called PANGLOSS is derived. The seven techniques are then compared in ex post and ex ante tests.     

Feasible estimation of firm‐specific allocative inefficiency through Bayesian numerical methods

Both the theoretical and empirical literature on the estimation of allocative and technical inefficiency has grown enormously. To minimize aggregation bias, ideally one should estimate firm and input‐specific parameters describing allocative inefficiency. However, identifying these parameters has often proven difficult. For a panel of Chilean hydroelectric power plants, we obtain a full set of such parameters using Gibbs sampling, which draws sequentially from conditional generalized method of moments (GMM) estimates obtained via instrumental variables estimation. We find an economically significant range of firm‐specific efficiency estimates with differing degrees of precision. The standard GMM approach estimates virtually no allocative inefficiency for industry‐wide parameters. Copyright © 2009 John Wiley & Sons, Ltd.     

Semiparametric Estimator for Binary‐outcome Sample Selection: Prejudice Matters in Election

a semiparametric estimator for binary‐outcome sample‐selection models is proposed that imposes only single index assumptions on the selection and outcome equations without specifying the error term distribution. I adopt the idea in Lewbel (2000) using a ‘special regressor’ to transform the binary response Y so that the transformed Y becomes linear in the latent index, which then makes it possible to remove the selection correction term by differencing the transformed Y equation. There are various versions of the estimator, which perform differently trading off bias and variance. A simulation study is conducted, and then I apply the estimators to US presidential election data in 2008 and 2012 to assess the impact of racial prejudice on the elections, as a black candidate was involved for the first time ever in the US history.     

A number‐of‐modes reference rule for density estimation under multimodality

We consider kernel density estimation for univariate distributions. The question of interest is as follows: given that the data analyst has some background knowledge on the modality of the data (for instance, ‘data of this type are usually bimodal’), what is the adequate bandwidth to choose? We answer this question by extending Silverman's idea of ‘normal‐reference’ to that of ‘reference to a Gaussian mixture’. The concept is illustrated in the light of real data examples.     

Analysis of Internet topology with a three‐level components model

The vertical structure of the Internet is considered as having three‐level components: backhyphen‐bone‐level interconnection, mid‐level transit, and local‐level access. This paper considers single and cross mergers between an integrated provider and an entrant in the different area. As a result of these mergers, cross entry, in which both integrated providers merge with the retail entrants in the other areas, is more socially desirable than single entry, in which only one firm merges, which is, in turn, preferred to no entry. Copyright © 2005 John Wiley & Sons, Ltd.     

Quantile regression methods for recursive structural equation models

Two classes of quantile regression estimation methods for the recursive structural equation models of Chesher [2003. Identification in nonseparable models. Econometrica 71, 1405–1441.] are investigated. A class of weighted average derivative estimators based directly on the identification strategy of Chesher is contrasted with a new control variate estimation method. The latter imposes stronger restrictions achieving an asymptotic efficiency bound with respect to the former class. An application of the methods to the study of the effect of class size on the performance of Dutch primary school students shows that (i) reductions in class size are beneficial for good students in language and for weaker students in mathematics, (ii) larger classes appear beneficial for weaker language students, and (iii) the impact of class size on both mean and median performance is negligible.     

Testing for serial correlation in simultaneous equation models: Some further results

In this paper approximate counterparts of the exact tests earlier proposed by the authors are examined. Type 1 error probabilities and test powers are estimated and compared using Monte Carlo experiments. The effect on the Type 1 error probabilities of the misspecification which results when serial correlation is present elsewhere in the system is also investigated.