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.     

The informative sample size for dynamic multiple equation systems with moving average errors

Abstract  The problem considered here, is that of finding suitable conditions for dynamic economic systems that exclude the existence of observationally equivalent structures. Here observational equivalence refers to equality of distributions or first and second moments of a small finite sample from the observable process. It is shown, that under these conditions we may act as if the lagged endogenous variables are nonrandom exogenous variables, when global identifiability is investigated.     

Semiparametric three‐step estimation methods for simultaneous equation systems

This paper proposes a new method for estimating a structural model of labour supply in which hours of work depend on (log) wages and the wage rate is considered endogenous. The main innovation with respect to other related estimation procedures is that a nonparametric additive structure in the hours of work equation is permitted. Though the focus of the paper is on this particular application, a three‐step methodology for estimating models in the presence of the above econometric problems is described. In the first step the reduced form parameters of the participation equation are estimated by a maximum likelihood procedure adapted for estimation of an additive nonparametric function. In the second step the structural parameters of the wage equation are estimated after obtaining the selection‐corrected conditional mean function. Finally, in the third step the structural parameters of the labour supply equation are estimated using local maximum likelihood estimation techniques. The paper concludes with an application to illustrate the feasibility, performance and possible gain of using this method. Copyright © 2005 John Wiley & Sons, Ltd.     

The effect of estimation method and sample size in multilevel structural equation modeling

Multilevel structural equation modeling (multilevel SEM) has become an established method to analyze multilevel multivariate data. The first useful estimation method was the pseudobalanced method. This method is approximate because it assumes that all groups have the same size, and ignores unbalance when it exists. In addition, full information maximum likelihood (ML) estimation is now available, which is often combined with robust chi‐squares and standard errors to accommodate unmodeled heterogeneity (MLR). In addition, diagonally weighted least squares (DWLS) methods have become available as estimation methods. This article compares the pseudobalanced estimation method, ML(R), and two DWLS methods by simulating a multilevel factor model with unbalanced data. The simulations included different sample sizes at the individual and group levels and different intraclass correlation (ICC). The within‐group part of the model posed no problems. In the between part of the model, the different ICC sizes had no effect. There is a clear interaction effect between number of groups and estimation method. ML reaches unbiasedness fastest, then the two DWLS methods, then MLR, and then the pseudobalanced method (which needs more than 200 groups). We conclude that both ML(R) and DWLS are genuine improvements on the pseudobalanced approximation. With small sample sizes, the robust methods are not recommended.     

Nonparametric dynamic panel data models: Kernel estimation and specification testing

Motivated by the first-differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogenous regressors. The estimators utilize the additive structure of the first-differenced model—the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the _L2$L_2$ distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well for finite samples. We apply our new method to study the relationships among economic growth, the initial economic condition and capital accumulation, and find a significant nonlinear relation between economic growth and the initial economic condition.     

Distribution-free and link-free estimation for the sample selection model

The prevalent estimation methods for the sample selection model rely heavily on parametric assumptions and are sensitive to departures from the underlying parametric assumptions [see, e.g., Goldberger (1983)]. We propose an alternative estimation method, the corrected maximum likelihood estimate, which is consistent for the slope vector in the outcome equation up to a multiplicative scalar, even through the parametric model on which the estimate is based might be misspecified. As an important corollary, it follows from our result that Olsen's (1980) corrected ordinary least squares estimate is consistent if the outcome equation is linear, without requiring Olsen's assumptions on the joint error distribution.     

Three-stage estimation procedures for the negative exponential distributions

Fixed-width confidence interval estimation problems for location parameters of negative exponential populations have been studied. Three-stage sampling procedures have been developed for both the one- and two-sample situations. Our discussions are primarily concerned with second-order expansions of various characteristics of the proposed procedures including those for the achieved coverage probability in either problem. Some simulated results are also presented to indicate the usefulness of our procedures for moderate sample sizes.     

Pooling: An experimental study of alternative testing and estimation procedures in a two-way error component model

This paper considers a two-way error component model with no lagged dependent variable and investigates the performance of various testing and estimation procedures applied to this model by means of Monte Carlo experiments. The following results were found: (1) The Chow-test performed poorly in testing the stability of cross-section regressions over time and in testing the stability of time-series regression across regions. (2) The Roy-Zellner test performed well and is recommended for testing the poolability of the data. (3) The Hausman specification test, employed to test the orthogonality assumption, gave a low frequency of Type I errors. (4) The Lagrange multiplier test, employed to test for zero variance components, did well except in cases where it was badly needed. (5) The problem of negative estimates of the variance components was found to be more serious in the two-way model than in the one-way model. However, replacing the negative variance estimates by zero did not have a serious effect on the performance of the second-round GLS estimates of the regression coefficients. (6) As in the one-way model, all the two-stage estimation methods performed reasonably well. (7) Better estimates of the variance components did not necessarily lead to better second-round GLS estimates of the regression coefficients.     

Joint estimation and testing for functional form and heteroskedasticity

This paper develops the maximum likelihood procedure to estimate the appropriate functional form in regression models with heteroskedastic errors. The analysis can then enable us to separate out the influence of non-linearity in an estimate of the transformation parameter from the influence of stabilizing the error variance. Illustrative examples have been used to show that estimation and tests for functional form and heteroskedasticity can and should be jointly considered.     

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.     

Alternative bootstrap procedures for testing cointegration in fractionally integrated processes

This paper considers alternative methods of testing cointegration in fractionally integrated processes, using the bootstrap. The investigation focuses on (a) choice of statistic, (b) use of bias correction techniques, and (c) designing the simulation of the null hypothesis. Three residual-based tests are considered, two of the null hypothesis of non-cointegration, the third of the null hypothesis that cointegration exists. The tests are compared in Monte Carlo experiments to throw light on the relative roles of issues (a)–(c) in test performance.     

Robust estimation procedures and visual display techniques for a two-way classification model

An important problem in statistics is to study the effect of one or two factors on a dependent variable. This type of problem can be formulated as a regression problem (by using dummy (0,1) variables to represent the levels of factors) and the standard least squares (LS) analysis is well-known. The least absolute value (LAV) analysis is less well known, but certainly is becoming more widely used, especially in exploratory data analysis.The purpose of this report is to present a didactic treatment of visual display methods useful in exploratory data analysis. These visual display techniques (stem- and- leaf, box- and- whisker, and two-way plots) are presented for both the least squares and the least absolute value analyses of a two-way classification model.     

Model selection test procedures in a single linear equation of a dynamic simultaneous system and their defects in small samples

A variety of asymptotically valid tests for orthogonality, serial correlation, predictive failure, and of coefficient restrictions are presented, and their rejection probabilities are assessed in linear structural models with lagged dependent and (possibly) jointly dependent variables by Monte Carlo methods. For all test procedures the small-sample distribution under the null usually deviates substantially from the asymptotic distribution; this impedes their use in a reliable model selection strategy for econometric time-series analysis. Despite the harassing dependence of type I errors on factors generally unknown to the practitioner, inconsistencies originating from specification errors or from disregarded simultaneity may be detected by particular tests in particular situations. From this study some clues emerge on how to interpret (in)significant values of the various test statistics.     

Comparisons of Bayesian estimation procedures for two-way contingency tables without interaction

Bayesian and empirical Bayesian estimation methods are reviewed and proposed for the row and column parameters in two-way Contingency tables without interaction. Rasch's multiplicative Poisson model for misreadings is discussed in an example. The case is treated where assumptions of exchangeability are reasonable a priori for the unknown parameters. Two different types of prior distributions are compared, It appears that gamma priors yield more tractable results than lognormal priors.     

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.     

Bayesian estimation and prediction based on Rayleigh sample quantiles

Ordered data arise naturally in many fields of statistical practice. Often some sample values are unknown or disregarded due to various reasons. On the basis of some sample quantiles from the Rayleigh distribution, the problems of estimating the Rayleigh parameter, hazard rate and reliability function, and predicting future observations are addressed using a Bayesian perspective. The construction of β-content and β-expectation Bayes tolerance limits is also tackled. Under squared-error loss, Bayes estimators and predictors are deduced analytically. Exact tolerance limits are derived by solving simple nonlinear equations. Highest posterior density estimators and credibility intervals, as well as Bayes estimators and predictors under linear loss, can easily be computed iteratively.     

膨胀土土体抗剪强度参数测定试样尺寸效应

文章在大量试验资料的基础上,论述测定膨胀土抗剪强度时试样尺寸对抗剪强度参数的影响,从而指出测定膨胀土抗剪强度时的合理试样直径及C、Φ值的确定方法。     

On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

Semi-nonparametric estimation and misspecification testing of diffusion models

Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [Kristensen, D., 2010. Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. Journal of Econometrics 156, 239-259] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.