Stanislav Kolenikov

The common maximum likelihood (ML) estimator for structural equation models (SEMs) has optimal asymptotic properties under ideal conditions (e.g., correct structure, no excess kurtosis, etc.) that are rarely met in practice. This paper proposes model-implied instrumental variable – generalized method of moments (MIIV-GMM) estimators for latent variable SEMs that are more robust than ML to violations of both the model structure and distributional assumptions. Under less demanding assumptions, the MIIV-GMM estimators are consistent, asymptotically unbiased, asymptotically normal, and have an asymptotic covariance matrix. They are “distribution-free,” robust to heteroscedasticity, and have overidentification goodness-of-fit J-tests with asymptotic chi-square distributions. In addition, MIIV-GMM estimators are “scalable” in that they can estimate and test the full model or any subset of equations, and hence allow better pinpointing of those parts of the model that fit and do not fit the data. An empirical example illustrates MIIV-GMM estimators. Two simulation studies explore their finite sample properties and find that they perform well across a range of sample sizes.

In this article, I discuss the need for Halton sequences and discuss the Mata implementation of scrambling of Halton sequences, providing several examples of scrambling procedures.

Heywood cases, or negative variance estimates, are a common occurrence in factor analysis and latent variable structural equation models. Though they have several potential causes, structural misspecification is among the most important. This article explains how structural misspecification can lead to a Heywood case in the population, and provides several ways to test whether a negative error variance is a symptom of structural misspecification. The authors consider z-tests based on a variety of standard errors, confidence intervals, bootstrap resampling, and likelihood-ratio-type tests. In their discussion of z-tests, the authors demonstrate which of the standard errors are consistent under different kinds of misspecification. They also introduce new tests based on the scaled chi-square difference: the test on the boundary and the signed root of the scaled chi-square difference. A simulation study assesses the performance of these tests. The authors find that signed root tests and z-tests based on the empirical sandwich and the empirical bootstrap standard errors perform best in detecting negative error variances. They outperform z-tests based on the information matrix or distribution-robust standard errors.

We deal with the problem of quantifying the degree to which parameter estimates in a structural equation model can be biased when structural relationships were not specified correctly by the researcher. We propose a framework to relate moment residuals to biases of parameter estimates and the overall noncentrality of the model. For each parameter in the model, an impact of either particular moment residual or the overall model noncentrality can be evaluated, although the latter tends to give error bounds that are rather conservative. We provide illustrative analytical and empirical examples to demonstrate the steps in application of the proposed procedures. The first example is a mildly misspecified model with causal indicators mistaken to be effect indicators. The resulting biases can be approximated very accurately by accounting for the effect of a single misfitted residual moment. The second example is a grossly misspecified model in which a mediating latent variable was erroneously omitted. In this case, the misspecification spreads to all entries of the covariance matrix, and measures based on overall noncentrality give a good indication of the magnitudes of plausible biases. The third example is an empirical study that uses Holzinger-Swineford factor analysis data that shows how the procedures can be used in practice.

In this article, I discuss the main approaches to resampling variance estimation in complex survey data: balanced repeated replication, the jackknife, and the bootstrap. Balanced repeated replication and the jackknife are implemented in the Stata svy suite. The bootstrap for complex survey data is implemented by the bsweights command. I describe this command and provide working examples.

This article describes the confa command, which fits confirmatory factor analysis models by maximum likelihood and provides diagnostics for the fitted models. Descriptions of the command and its options are given, and some illustrative examples are provided.

The last several years have seen a growth in the number of publications in economics that use principal component analysis (PCA) in the area of welfare studies. This paper explores the ways discrete data can be incorporated into PCA. The effects of discreteness of the observed variables on the PCA are reviewed. The statistical properties of the popular Filmer and Pritchett (2001) procedure are analyzed. The concepts of polychoric and polyserial correlations are introduced with appropriate references to the existing literature demonstrating their statistical properties. A large simulation study is carried out to compare various implementations of discrete data PCA. The simulation results show that the currently used method of running PCA on a set of dummy variables as proposed by Filmer and Pritchett (2001) can be improved upon by using procedures appropriate for discrete data, such as retaining the ordinal variables without breaking them into a set of dummy variables or using polychoric correlations. An empirical example using Bangladesh 2000 Demographic and Health Survey data helps in explaining the differences between procedures.