The robustness of maximum likelihood and distribution-free estimators to non-normality in confirmatory factor analysis

The present Monte Carlo compares the estimates produced by maximum likelihood (ML) and asymptotically distribution-free (ADF) methods. The study extends prior research by investigating the combined effects of sample size, magnitude of correlation among observed indicators, number of indicators, magnitude of skewness and kurtosis, and proportion of indicators with non-normal distributions. Results indicate that both ML and ADF showed little bias in estimates of factor loadings under all conditions studied. As the number of indicators in the model increased, ADF produced greater negative bias in estimates of uniquenesses than ML. In addition, the bias in standard errors for both ML and ADF estimation increased in models with more indicators, and this effect was more pronounced for ADF than ML. Increases in skewness and kurtosis resulted in greater underestimating of standard errors; ML standard errors showed greater bias than ADF under conditions of non-normality, and ML chi-square statistics were also inflated. However, when only half the indicators departed from normality, the inflation in ML chi-square decreased.     

Bias in estimating multivariate and univariate diffusions

Multivariate continuous time models are now widely used in economics and finance. Empirical applications typically rely on some process of discretization so that the system may be estimated with discrete data. This paper introduces a framework for discretizing linear multivariate continuous time systems that includes the commonly used Euler and trapezoidal approximations as special cases and leads to a general class of estimators for the mean reversion matrix. Asymptotic distributions and bias formulae are obtained for estimates of the mean reversion parameter. Explicit expressions are given for the discretization bias and its relationship to estimation bias in both multivariate and in univariate settings. In the univariate context, we compare the performance of the two approximation methods relative to exact maximum likelihood (ML) in terms of bias and variance for the Vasicek process. The bias and the variance of the Euler method are found to be smaller than the trapezoidal method, which are in turn smaller than those of exact ML. Simulations suggest that when the mean reversion is slow, the approximation methods work better than ML, the bias formulae are accurate, and for scalar models the estimates obtained from the two approximate methods have smaller bias and variance than exact ML. For the square root process, the Euler method outperforms the Nowman method in terms of both bias and variance. Simulation evidence indicates that the Euler method has smaller bias and variance than exact ML, Nowman’s method and the Milstein method.     

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.     

偏差修正的预白化HAC法在平稳过程伪回归中的应用

在传统预白化HAC法存在有限样本偏差的基础上,提出将自回归参数偏差修正法和残差调整法来减少预白化HAC法的偏差,从而降低相互独立的平稳过程之间发生伪回归的概率。通过一系列的蒙特卡罗模拟表明：第一修正的预白化HAC法确实减少了伪回归概率,且自回归参数偏差修正法减少的幅度要比残差调整法要大得多;第二相对于同方差情形而言,存在GARCH类异方差的回归中预白化HAC法具有更低的伪回归概率;第三当数据过程是AR(2)过程时,在持久性相同的情况下预白化HAC法的伪回归概率要低于相应的AR(1)数据过程。但在高于2阶的自回归数据过程的回归中,残差调整的预白化HAC的伪回归概率具有优势。在样本容量较大(T≥500)时自回归参数修正的预白化HAC法的伪回归概率很接近检验水平,但残差调整的预白化HAC法具有微弱的向下检验水平扭曲.     

Estimators and hypothesis tests for a stochastic frontier function: A Monte Carlo analysis

This paper uses Monte Carlo experimentation to investigate the finite sample properties of the maximum likelihood (ML) and corrected ordinary least squares (COLS) estimators of the half-normal stochastic frontier production function. Results indicate substantial bias in both ML and COLS when the percentage contribution of inefficiency in the composed error (denoted by *) is small, and also that ML should be used in preference to COLS because of large mean square error advantages when * is greater than 50%. The performance of a number of tests of the existence of technical inefficiency is also investigated. The Wald and likelihood ratio (LR) tests are shown to have incorrect size. A one-sided LR test and a test of the significance of the third moment of the OLS residuals are suggested as alternatives, and are shown to have correct size, with the one-sided LR test having the better power of the two.The author would like to thank Bill Griffiths, George Battese, Howard Doran, Bill Greene and two anonymous referees for valuable comments. Any errors which remain are those of the author.     

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.     

带未知异方差广义空间模型的有效估计

空间单元大小以及其它的经济特征上的差异,常常会导致空间异方差问题。本文给出了广义空间模型异方差问题的三种不同估计方法。第一种方法是将异方差形式参数化,来克服自由度的不足,使用ML估计进行实现。而针对异方差形式未知时,分别采用了基于2SLS的迭代GMM估计和更加直接的MCMC抽样方法加以解决,特别是MCMC方法表现得更加优美。蒙特卡罗模拟表明,给定异方差形式条件下, ML估计通过异方差参数化的方法依然可以获得较好的估计效果。而异方差形式未知的情况下,另外两种方法随着样本数的增大时也可以与ML的估计结果趋于一致。     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.     

Bias in dynamic panel estimation with fixed effects,incidental trends and cross section dependence

Explicit asymptotic bias formulae are given for dynamic panel regression estimators as the cross section sample size N→∞$N\to \infty$. The results extend earlier work by Nickell [1981. Biases in dynamic models with fixed effects. Econometrica 49, 1417–1426] and later authors in several directions that are relevant for practical work, including models with unit roots, deterministic trends, predetermined and exogenous regressors, and errors that may be cross sectionally dependent. The asymptotic bias is found to be so large when incidental linear trends are fitted and the time series sample size is small that it changes the sign of the autoregressive coefficient. Another finding of interest is that, when there is cross section error dependence, the probability limit of the dynamic panel regression estimator is a random variable rather than a constant, which helps to explain the substantial variability observed in dynamic panel estimates when there is cross section dependence even in situations where N is very large. Some proposals for bias correction are suggested and finite sample performance is analyzed in simulations.     

The combined inverse binomial sampling procedure

In the Combined Inverse Binomial Procedure (CIB), independent Bernoulli trials are performed one after another until for the first time either a fixed number of successes or a fixed number of failures are obtained. The sample size is the minimum of a pair of negative binomial variables. The distribution of the sample size and its expectation as well as the problems of estimation and testing of hypotheses in CIB are dealt with. It is seen that tests based on CIB procedures can lead to savings in sample size as compared to fixed sample size procedures in certain situations. It is also briefly indicated how the theory can be extended to the case of combined inverse multinomial procedures.     

Maximum likelihood methods for fitting the burr type XII distribution to multiply (progressively) censored life test data

Summary This paper develops mathematical and computational methodology for fitting, by the method of maximum likelihood (ML), the Burr Type XII distribution to multiply (or progressively) censored life test data. Mathematical expressions are given for approximating the asymptotic variances and covariances of the ML estimates (MLEs) of the parameters of the Burr Type XII distribution. A rigorous mathematical analysis is undertaken to investigate the existence and uniqueness of the MLEs for arbitrary sample data. The methodology of this paper is applied to progressively censored sample data arising in a life test experiment.     

The linkage between insurance activity and banking credit: Some evidence from dynamic analysis

This paper investigates the long-run and short-run linkages between insurance activity and banking credit for G-7 countries. To minimize the pretest bias and overcome the structural changes, we adopt the bootstrap Granger causality test applied to full sample and subsamples with a fixed window size. The Johansen cointegration test with GMM-IV estimator finds a long-run positive relation between the series. The full sample results of bootstrap Granger causality test show that there is predictive power from life insurance activity to banking credit only for France and Japan, while the short-run causal relationships between nonlife insurance activity and banking credit are country-specific. However, parameter stability test results suggest that the short-run results in full sample are unreliable. The results of rolling VAR models report that the causal linkages between them are time-varying across various subsamples. These findings offer some useful insights for achieving the co-evolution between insurance and banking credit markets.     

Jackknife estimation of stationary autoregressive models

This paper explores the properties of jackknife methods of estimation in stationary autoregressive models. Some general results concerning the correct weights for bias reduction under various sampling schemes are provided and the asymptotic properties of a jackknife estimator based on non-overlapping sub-samples are derived for the case of a stationary autoregression of order p$p$ when the number of sub-samples is either fixed or increases with the sample size at an appropriate rate. The results of a detailed investigation into the finite sample properties of various jackknife and alternative estimators are reported and it is found that the jackknife can deliver substantial reductions in bias in autoregressive models. This finding is robust to departures from normality, ARCH effects and misspecification. The median-unbiasedness and mean squared error properties are also investigated and compared with alternative methods as are the coverage rates of jackknife-based confidence intervals.     

Measurement Biases in Consumer Price Indexes

The Consumer Price Index (CPI) measures the cost of purchasing a fixed basket of goods at a fixed sample of outlets over time, and can be thought of as a practical approximation to a "true" cost-of-living index, and a measure of general inflation for the economy. In more recent times, concerns over the possibility that the U.S. CPI overstates the rate of inflation have grown. Annual changes in the CPI are used to adjust social security benefits, and wage contracts are often indexed to CPI changes. To the extent that the CPI overstates the rate of inflation individuals are being compensated for changes in the cost-of-living that have not occurred–with enormous implications for government fiscal budgets. This paper presents an up-to-date survey of the principal sources of measurement error or bias in the CPI. A number of sources of bias are examined, including the commodity substitution bias , the outlet substitution bias , and the elementary index bias . Traditional bilateral index number theory assumes that the number of goods remains constant over the pricing period and furthermore, that the goods are of unchanging quality. Changes in either of these give rise to two further biases: the new goods bias and the quality bias .     

Fixed effects estimation of structural parameters and marginal effects in panel probit models

Fixed effects estimators of nonlinear panel models can be severely biased due to the incidental parameters problem. In this paper, I characterize the leading term of a large-T expansion of the bias of the MLE and estimators of average marginal effects in parametric fixed effects panel binary choice models. For probit index coefficients, the former term is proportional to the true value of the coefficients being estimated. This result allows me to derive a lower bound for the bias of the MLE. I then show that the resulting fixed effects estimates of ratios of coefficients and average marginal effects exhibit no bias in the absence of heterogeneity and negligible bias for a wide variety of distributions of regressors and individual effects in the presence of heterogeneity. I subsequently propose new bias-corrected estimators of index coefficients and marginal effects with improved finite sample properties for linear and nonlinear models with predetermined regressors.     

The price-wage inflationary spiral in Poland

In this paper the problem of price-wage relationship modelling in the case of a mixed economy is addressed. The empirical investigation was based on Polish annual data for the period of a centrally planned system (1964–1989) and on quarterly data for the period of transition towards a market economy (1990.1–1990.3). The traditional approach proved to be inappropriate because of the variables' nonstationarity. Identification of long-run behaviour was attempted by applying the two-step Engle-Granger's, or alternatively, Johansen's maximum likelihood (ML) procedures. The ML estimator provided better estimates of cointegration vectors and, even more important, allowed as many as three to be found.The main conclusion which can be drawn from the empirical findings is that three variables: price index, average wages and labour productivity, form a multi-dimensional equilibrium space. This property of the described phenomena needs to be taken into serious account when building macroeconometric models explaining the behaviour of the Polish economy.The existence of these three cointegration vectors is troublesome because of unusual problems of interpretation. However, if it is not as a result of misspecification and/or small sample bias, it proves that much remains to be learned about the price-wage mechanisms functioning in economies having a mixed character.     

The beta-binomial convolution model for 2×2 tables with missing cell counts

This paper considers the beta-binomial convolution model for the analysis of 2×2 tables with missing cell counts. We discuss maximum-likelihood (ML) parameter estimation using the expectation–maximization algorithm and study information loss relative to complete data estimators. We also examine bias of the ML estimators of the beta-binomial convolution. The results are illustrated by two example applications.     

Poisson Regression for Clustered Data

We compare five methods for parameter estimation of a Poisson regression model for clustered data: (1) ordinary (naive) Poisson regression (OP), which ignores intracluster correlation, (2) Poisson regression with fixed cluster‐specific intercepts (FI), (3) a generalized estimating equations (GEE) approach with an equi‐correlation matrix, (4) an exact generalized estimating equations (EGEE) approach with an exact covariance matrix, and (5) maximum likelihood (ML). Special attention is given to the simplest case of the Poisson regression with a cluster‐specific intercept random when the asymptotic covariance matrix is obtained in closed form. We prove that methods 1–5, except GEE, produce the same estimates of slope coefficients for balanced data (an equal number of observations in each cluster and the same vectors of covariates). All five methods lead to consistent estimates of slopes but have different efficiency for unbalanced data design. It is shown that the FI approach can be derived as a limiting case of maximum likelihood when the cluster variance increases to infinity. Exact asymptotic covariance matrices are derived for each method. In terms of asymptotic efficiency, the methods split into two groups: OP & GEE and EGEE & FI & ML. Thus, contrary to the existing practice, there is no advantage in using GEE because it is substantially outperformed by EGEE and FI. In particular, EGEE does not require integration and is easy to compute with the asymptotic variances of the slope estimates close to those of the ML.     

Some large-concentration-parameter asymptotics for the k-class estimators

A sufficient condition is derived in this paper for the consistency and asymptotic normality of the k-class estimators (k-stochastic or nonstochastic) as the concentration parameter increases indefinitely, with the sample size either staying fixed or also increasing. It is further shown that the limited-information maximum likelihood estimator satisfies this condition. Since large sample size implies a large concentration parameter, but not vice versa, the usual conditions for consistency and asymptotic normality of the k-class estimators as the sample size increases can be inferred from the results given in this paper. But more importantly, the results in this paper shed further light on the small-sample properties of the stochastic k-class estimators and can serve as a starting point for the derivation of asymptotic approximations for these estimators as the concentration parameter goes to infinity, while the sample size either stays fixed or also goes to infinity.     

Optimum allocation in stratified sampling with replacement

In a stratified sample, when sampling is done with replacement in each stratum a better estimate of the population mean can be achieved by considering the distinct units only. An explicit expression for the variance for the mean, of a stratified sample based on the distinct units only, is obtained. Then the optimum allocation for the different stratum are obtained by minimizing this variance subject to (i) total sample size being fixed, or (ii) the expected number of distinct units being fixed. Neyman’s solutions are obtained as special cases. The solutions finally arrived at are algebraically complex, hence, numerical methods are applied. In all examples, the variance of the estimates obtained by this method are smaller than the variances obtained by Neyman’s allocation. A part of this work was supported by the Office of the Ordinance Research; U.S.A. Grant (DA-AROL(D)-31-124-G83) when the author was at University of California, Berkeley.