Structural equation modeling with small sample sizes using two-stage ridge least-squares estimation

Strong consistency and weak distributional convergence to highly non-Gaussian limits are established for closed-form, two stage least squares (TSLS) estimators for a class of ARCH(p) models. Conditions for these results include (relatively) mild moment existence criteria that are supported empirically by many (high frequency) financial returns. These conditions are not shared by competing closed-form estimators like OLS. Identification of these TSLS estimators depends on asymmetry, either in the model's rescaled errors or in the conditional variance function. Monte Carlo studies reveal TSLS estimation to sizably outperform quasi maximum likelihood estimation in (relatively) small samples. This outperformance is most pronounced when returns are heavily skewed.

The asymptotically distribution free (ADF) method is often used to estimate parameters or test models without a normal distribution assumption on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance versus correlation structure analysis. The behaviors of 3 test statistics frequently used to evaluate structural equation models with nonnormally distributed variables, χ2 test TAGLS and its small-sample variants TYB and TF(AGLS) were compared. Results showed that the ADF method in correlation structure analysis with test statistic TAGLS performs much better at small sample sizes than the corresponding test for covariance structures. In contrast, test statistics TYB and TF(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. It is proposed that excessively large and variable condition numbers of weight matrices are a cause of poor behavior of ADF test statistics in small samples, and results showed that these condition numbers are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.

Strong consistency and (weak) distributional convergence to highly non-Gaussian limits are established for closed-form, two stage least squares (TSLS) estimators of linear and threshold ARCH (p) models, with special attention paid to the ARCH (1) and threshold ARCH (1) cases. Conditions supporting these results include (relatively) mild moment existence criteria that enjoy empirical support. These conditions are not shared by competing estimators like OLS. Identification of the TSLS estimators depends on asymmetry, either in the model’s rescaled errors or in the conditional variance function. Monte Carlo studies reveal TSLS estimation can sizably outperform quasi maximum likelihood (QML) in small samples and even best recently proposed two-step estimators specifically designed to enhance the efficiency of QML.

This article develops and examines the small sample properties of alternative estimators when the data is suspected to contain measurement error. Two types of estimators are considered: (1) a linear combination of the OLS and instrumental variable (IV) estimators and (2) a method of choosing between the OLS and IV estimators on the basis of sample information. Large sample approximations are used to derive optimal procedures that are then evaluated by Monte Carlo experiments to obtain the mean square error for small and moderate size samples.

In this paper, we consider two‐stage least squares (2SLS) and simple instrumental variable (IV) type estimation of dynamic panel data models with both individual‐specific effects and heterogeneous time trend when both N and T tend to infinity. We consider the forward orthogonal deviations (FOD) proposed by (Hayakawa, et al. Econometric Reviews, 2019. Vol. 38, pp. 1055–1088) and the double first difference (2FD) to remove both the individual‐specific effects and heterogeneous trend. As the main theoretical contribution, we establish the asymptotic properties of the 2SLS estimation of the lag coefficient and find that the 2SLS estimation using FOD and optimal 2SLS estimation using 2FD are asymptotically biased of order , while the 2SLS based on 2FD using non‐optimal weighting matrix is asymptotically biased of order . We also establish the asymptotic unbiasedness of the simple IV estimation using first differenced lagged dependent variable as instrument, and establish the invalidity of using level lagged dependent variable as instrument for the simple IV estimation. Monte Carlo simulations confirm our findings in this paper.

In the last several years, sociologists have expanded their interest in simultaneous equation models beyond recursive systems. (See, for example, Duncan and others, 1971; Gurr and Duvall, 1973; Hawkes, 1975; Hibbs, 1973; Kohn and Schooler, 1973; Waite and Stolzenberg, 1976.) The purpose of this chapter is to present some results of potential use in sociology for nonrecursive models estimated by two-stage least squares (2SLS). Two-stage least-squares estimation is widely employed in the

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

Correct detection of measurement bias could help researchers revise models or refine psychological scales. Measurement bias detection can be viewed as a variable-selection problem, in which biased items are optimally selected from a set of items. This study investigated a number of regularization methods: ridge, lasso, elastic net (enet) and adaptive lasso (alasso), in comparison with maximum likelihood estimation (MLE) for detecting various forms of measurement bias in regard to a continuous violator using restricted factor analysis. Particularly, complex structural equation models with relatively small sample sizes were the study focus. Through a simulation study and an empirical example, results indicated that the enet outperformed other methods in small samples for identifying biased items. The alasso yielded low false positive rates for non-biased items outside of a high number of biased items. MLE performed well for the overall estimation of biased items.

In the general structural equation model only the direction of the vector of coefficients of the endogenous variables is determined.The traditional normalization rule defines the coefficients that are of interest but should not be embodied in the estimation procedure: we show that the properties of the traditionally defined ordinary least squares and two stage least squares estimators are distorted by their dependence on the normalization rule.Symmetrically normalized analogues of these estimators are defined and are shown to have essentially similar properties to those of the limited information maximum likelihood estimator.

In this chapter, we bring together the notions of measurement error discussed in Chaps. 3 and 4 with the structural modeling of simultaneous relationships presented in Chap. 6. We demonstrate that a bias is introduced when estimating the relationship between two variables measured with error if that measurement error is ignored. We then present a methodology for estimating the parameters of structural relationships between variables that are not observed directly: analysis of covariance structures. We focus on the role of the measurement model as discussed in Chap. 4 with the confirmatory factor analytic model.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

Node self-localization is a key research topic for wireless sensor networks (WSNs). There are two main algorithms, the triangulation method and the maximum likelihood (ML) estimator, for angle of arrival (AOA) based self-localization. The ML estimator requires a good initialization close to the true location to avoid divergence, while the triangulation method cannot obtain the closed-form solution with high efficiency. Here, we develop a set of efficient closed-form AOA based self-localization algorithms using auxiliary variables based methods. First, we formulate the self-localization problem as a linear least squares problem using auxiliary variables. Based on its closed-form solution, a new auxiliary variables based pseudo-linear estimator (AVPLE) is developed. By analyzing its estimation error, we present a bias compensated AVPLE (BCAVPLE) to reduce the estimation error. Then we develop a novel BCAVPLE based weighted instrumental variable (BCAVPLE-WIV) estimator to achieve asymptotically unbiased estimation of locations and orientations of unknown nodes based on prior knowledge of the AOA noise variance. In the case that the AOA noise variance is unknown, a new AVPLE based WIV (AVPLE-WIV) estimator is developed to localize the unknown nodes. Also, we develop an autonomous coordinate rotation method to overcome the tangent instability of the proposed algorithms when the orientation of the unknown node is near <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\pi/2$</tex> </formula> . We also derive the Cramér-Rao lower bound of the ML estimator. Extensive simulations demonstrate the new algorithms achieve much higher localization accuracy than the triangulation method and avoid local minima and divergence in iterative ML estimators.

Purpose– The purpose of this paper is to expand understanding of the determinants of adoption innovation in SME context by empirically examining the effect of corporate governance structure on manager's innovative behavior. This was done through exploring whether ownership structure affects managers’ innovative behavior and if so, whether the effect is mediated by board composition.Design/methodology/approach– Using a sample of 197 managers within Tunisian SMEs, hypotheses were tested through structural equation modeling and especially using covariance structure analysis (or LISREL method) with Analysis of Moment Structures (AMOS) 18.0 software and maximum likelihood estimation method.Findings– The paper found that ownership structure is significantly associated with manager's innovative behavior. Further analysis arising from introducing outsiders’ representation on the board as a mediating variable reveals that the relationship is fully mediated by this variable.Practical implications– This study gives insights to policy makers who are interested in improving board efficacy in emerging economies such as Tunisia. Indeed, the study results should encourage nominating committees and seniors to reflect warily on an effective structuring of board composition by ensuring a certain priority for innovation activities in the firm.Originality/value– This paper extends the understanding of how ownership structure shapes strategic decisions such as innovation in emerging markets. In addition, it fills the literature void by introducing the board composition as a mediating concept between ownership structure and innovative behavior, which was neglected by previous researchers.

In this paper I examine graphical comparisons of one-dimensional (or marginal) distribution functions of alternative estimators. It is shown that areas under the c.d.f. (cumulative distribution function) curve can be given a decision-theoretic interpretation as risk under a bounded absolute-error loss function. I also show that by a simple rescaling of the graph's axes, graphical areas are created which can be interpreted as risk under bounded squared-error loss. The bounded loss functions are applied to compare graphically and numerically the risk of exact distributions of the limited-information maximum likelihood and two-stage least-squares estimators in a simultaneous equations model.

Statistical theories of goodness-of-fit tests in structural equation modeling are based on asymptotic distributions of test statistics. When the model includes a large number of variables or the population is not from a multivariate normal distribution, the asymptotic distributions do not approximate the distribution of the test statistics very well at small sample sizes. A variety of methods have been developed to improve the accuracy of hypothesis testing at small sample sizes. However, all these methods have their limitations, specially for nonnormal distributed data. We propose a Monte Carlo test that is able to control Type I error with more accuracy compared to existing approaches in both normal and nonnormally distributed data at small sample sizes. Extensive simulation studies show that the suggested Monte Carlo test has a more accurate observed significance level as compared to other tests with a reasonable power to reject misspecified models.