Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

Class-specific tests of spatial segregation based on nearest neighbor contingency tables

The spatial interaction between two or more classes might cause multivariate clustering patterns such as segregation or association, which can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor structure, which may result from random labeling (RL) or complete spatial randomness of points from two or more classes (which is henceforth called CSR independence ). We consider Dixon's class-specific segregation test and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-squared segregation statistic. We analyze the distributional properties and compare the empirical significant levels and power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new class-specific tests have comparable performance with the currently available tests based on NNCTs. For illustrative purposes, we use three example data sets and provide guidelines for using these tests.     

An unbiased model comparison test using cross-validation

Social scientists often consider multiple empirical models of the same process. When these models are parametric and non-nested, the null hypothesis that two models fit the data equally well is commonly tested using methods introduced by Vuong (Econometrica 57(2):307–333, 1989) and Clarke (Am J Political Sci 45(3):724–744, 2001; J Confl Resolut 47(1):72–93, 2003; Political Anal 15(3):347–363, 2007). The objective of each is to compare the Kullback–Leibler Divergence (KLD) of the two models from the true model that generated the data. Here we show that both of these tests are based upon a biased estimator of the KLD, the individual log-likelihood contributions, and that the Clarke test is not proven to be consistent for the difference in KLDs. As a solution, we derive a test based upon cross-validated log-likelihood contributions, which represent an unbiased KLD estimate. We demonstrate the CVDM test’s superior performance via simulation, then apply it to two empirical examples from political science. We find that the test’s selection can diverge from those of the Vuong and Clarke tests and that this can ultimately lead to differences in substantive conclusions.     

Test by adaptive LASSO quantile method for real-time detection of a change-point

This article proposes a test statistic based on the adaptive LASSO quantile method to detect in real-time a change in a linear model. The model can have a large number of explanatory variables and the errors don’t satisfy the classical assumptions for a statistical model. For the proposed test statistic, the asymptotic distribution under \(H_0\) is obtained and the divergence under \(H_1\) is shown. It is shown via Monte Carlo simulations, in terms of empirical sizes, of empirical powers and of stopping time detection, that the useful test statistic for applications is better than other test statistics proposed in literature. Two applications on the air pollution and in the health field data are also considered.     

Meta-analytic methods to test relative efficacy

In the context of multiple treatments for a particular problem or disorder, it is important theoretically and clinically to investigate whether any one treatment is more effective than another. Typically researchers report the results of the comparison of two treatments, and the meta-analytic problem is to synthesize the various comparisons of two treatments to test the omnibus null hypothesis that the true differences of all particular pairs of treatments are zero versus the alternative that there is at least one true nonzero difference. Two tests, one proposed by Wampold et al. (Psychol. Bull. 122:203–215, 1997) based on the homogeneity of effects, and one proposed here based on the distribution of the absolute value of the effects, were investigated. Based on a Monte Carlo simulation, both tests adequately maintained nominal error rates, and both demonstrated adequate power, although the Wampold test was slightly more powerful for non-uniform alternatives. The error rates and power were essentially unchanged in the presence of random effects. The tests were illustrated with a reanalysis of two published meta-analyses (psychotherapy and antidepressants). It is concluded that both tests are viable for testing the omnibus null hypothesis of no treatment differences.     

Nonparametric correlation integral–based tests for linear and nonlinear stochastic processes

The BDS test is the best-known correlation integral–based test, and it is now an important part of most standard econometric data analysis software packages. This test depends on the proximity ( $\varepsilon )$ and the embedding dimension ( $m)$ parameters both of which are chosen by the researcher. Although different studies (e.g., Kanzler in Very fast and correctly sized estimation of the BDS statistic. Department of Economics, Oxford University, Oxford, 1999) have been carried out to provide an adequate selection of the proximity parameter, no relevant research has yet been done on $m$ . In practice, researchers usually compute the BDS statistic for different values of $m$ , but sometimes these results are contradictory because some of them accept the null and others reject it. This paper aims to fill this gap. To that end, we propose a new simple, yet powerful, aggregate test for independence, based on BDS outputs from a given data set, that allows the consideration of all of the information contained in several embedding dimensions without the ambiguity of the well-known BDS tests.     

Mobley revisited: dynamism in the process of employee turnover

In the employee turnover literature, studies of antecedents at static points prior to the ‘stay versus leave’ decision have generally not exhibited great predictive power. Mobley (1982 Mobley, W.H. 1982. Some Unanswered Questions in Turnover and Withdrawal Research. Academy of Management Review, 7(1): 111–6. [Crossref] , [Google Scholar]) suggests that such studies require the inclusion of change, time, actions and operations. A study is accordingly reported here in which employee recall of satisfaction, commitment and withdrawal intentions at three periods prior to the stay/leave decision is examined. Measures recalled at a single time (static) are found to be unreliable or even contrary to expectation. Alternately, change in variables is found to be significantly more predictive of final turnover than static measurements of these variables, especially over a year. Different antecedents appear to dominate in the turnover process at different times. Furthermore, antecedents differ over demographic groups, suggesting that movement capital may affect turnover.     

Exogeneity in structural equation models

The practical relevance of several concepts of exogeneity of treatments for the estimation of causal parameters based on observational data are discussed. We show that the traditional concepts, such as strong ignorability and weak and super-exogeneity, are too restrictive if interest lies in average effects (i.e. not on distributional effects of the treatment). We suggest a new definition of exogeneity, KL-exogeneity. It does not rely on distributional assumptions and is not based on counterfactual random variables. As a consequence it can be empirically tested using a proposed test that is simple to implement and is distribution-free.     

How to develop a customer satisfaction scale with optimal construct validity

In this article, we investigate how to construct a customer satisfaction (CS) scale which yields optimally valid measurements of the construct of interest. For this purpose we compare three alternative methodologies for scale development and construct validation. Furthermore, we discuss a satisfaction measurement application which is consistent with Messick’s (in: Linn (ed.) Educational measurement, 1989) construct validity theory. Following the deductive design for test development and construct validation, a multi-item measure for CS with a retail bank was developed. The measure was applied in survey research ( \(N\)  = 1,689) within a Dutch retail bank. The items constituted a unidimensional scale, allowing the computation of scale scores. The tests of 11 hypotheses about scale-score characteristics demonstrated that the scale score represented the construct of CS well. Furthermore, the one-factor theory (e.g. Yi, in: Zeithaml (ed.) Review of marketing, 1990) of satisfaction/dissatisfaction was confirmed. An implication of this result is that satisfaction/dissatisfaction can be measured on one scale instead of two scales, one for satisfaction and one for dissatisfaction. The results demonstrate that the deductive design is an appropriate methodology for measure development and construct validation in applied psychological research. The article concludes that the multi-item measure is well-suited for CS measurement in retail banking and that customized satisfaction scales have advantages but also disadvantages compared to standardized scales.     

Multidimensional strong large deviation results

We establish strong large deviation results for an arbitrary sequence of random vectors under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic approximations for a multivariate tail probability of the same kind as the one obtained by Bahadur and Rao (Ann Math Stat 31:1015–1027, 1960) for the sample mean (in the one-dimensional case). The proof of our results follows the same lines as in Chaganty and Sethuraman (J Stat Plan Inference, 55:265–280, 1996). We also present three statistical applications to illustrate our results, the first one dealing with a vector of independent sample variances, the second one with a Gaussian multiple linear regression model and the third one with the multivariate Nadaraya–Watson estimator. Some numerical results are also presented for the first two applications.     

Measuring welfare effects in models with random coefficients

In economic research, it is often important to express the marginal value of a variable in monetary terms. In random coefficient models, this marginal monetary value is the ratio of two random coefficients and is thus random itself. In this paper, we study the distribution of this ratio and particularly the consequences of different distributional assumptions about the coefficients. It is shown that important characteristics of the distribution of the marginal monetary value may be sensitive to the distributional assumptions about the random coefficients. The median, however, is much less sensitive than the mean. Copyright © 2006 John Wiley & Sons, Ltd.     

Random weighting estimation of stable exponent

This paper presents a new random weighting method to estimation of the stable exponent. Assume that $X_1, X_2, \ldots ,X_n$ is a sequence of independent and identically distributed random variables with $\alpha $ -stable distribution G, where $\alpha \in (0,2]$ is the stable exponent. Denote the empirical distribution function of G by $G_n$ and the random weighting estimation of $G_n$ by $H_n$ . An empirical distribution function $\widetilde{F}_n$ with U-statistic structure is defined based on the sum-preserving property of stable random variables. By minimizing the Cramer-von-Mises distance between $H_n$ and ${\widetilde{F}}_n$ , the random weighting estimation of $\alpha $ is constructed in the sense of the minimum distance. The strong consistency and asymptotic normality of the random weighting estimation are also rigorously proved. Experimental results demonstrate that the proposed random weighting method can effectively estimate the stable exponent, resulting in higher estimation accuracy than the Zolotarev, Press, Fan and maximum likelihood methods.     

On consistency of the weighted least squares estimators in a semiparametric regression model

This paper is concerned with the semiparametric regression model \(y_i=x_i\beta +g(t_i)+\sigma _ie_i,~~i=1,2,\ldots ,n,\) where \(\sigma _i^2=f(u_i)\), \((x_i,t_i,u_i)\) are known fixed design points, \(\beta \) is an unknown parameter to be estimated, \(g(\cdot )\) and \(f(\cdot )\) are unknown functions, random errors \(e_i\) are widely orthant dependent random variables. The p-th (\(p>0\)) mean consistency and strong consistency for least squares estimators and weighted least squares estimators of \(\beta \) and g under some more mild conditions are investigated. A simulation study is also undertaken to assess the finite sample performance of the results that we established. The results obtained in the paper generalize and improve some corresponding ones of negatively associated random variables.     

Extension of some important identities in shrinkage-pretest strategies

In this paper, we establish three identities which play a crucial role in deriving the asymptotic distributional risk function and the asymptotic distributional bias of a large class of estimators of a matrix parameter. In particular, we generalize the results in Judge and Bock (The statistical implication of pre-test and Stein-rule estimators in econometrics. North Holland, Amsterdam, 1978). The established results are useful in risk analysis of a class of Stein-rule type matrix estimators.     

A test for the weights of the global minimum variance portfolio in an elliptical model

In this paper we consider the weights of the global minimum variance portfolio (GMVP). The returns are assumed to follow a matrix elliptically contoured distribution, i.e., the returns are assumed to be neither independent nor normally distributed. A test for the general linear hypothesis is given. The distribution of the test statistic is derived under the null and the alternative hypothesis. It turns out that its distribution is invariant with respect to the type of the matrix elliptical distribution, i.e., the stochastic properties of the GMVP do not depend either on the mean vector or on the distributional assumptions imposed on asset returns. In an empirical study we analyze an international diversified portfolio.     

A new approach to distribution free tests in contingency tables

We suggest an extremely wide class of asymptotically distribution free goodness of fit tests for testing independence in two-way contingency tables, or equivalently, independence of two discrete random variables. The nature of these tests is that the test statistics can be viewed as definite functions of the transformation of \(\widehat{T}_n = (\widehat{T}_{ij})=\Big (\frac{\nu _{ij}- n\hat{a}_i\hat{b}_j}{\sqrt{n\hat{a}_i\hat{b}_j}}\Big )\) where \(\nu _{ij}\) are frequencies and \(\hat{a}_i, \hat{b}_j\) are estimated marginal distributions. Our method is also applicable for testing independence of two discrete random vectors. We make some comparisons on statistical powers of the new tests with the conventional chi-square test and suggest some cases in which this class is significantly more powerful.     

Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n ^?1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ^?1 A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$ ). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.     

MULTIPLE TESTING AND HETEROGENEOUS TREATMENT EFFECTS: RE‐EVALUATING THE EFFECT OF PROGRESA ON SCHOOL ENROLLMENT

The effect of a program or treatment may vary according to observed characteristics. In such a setting, it may not only be of interest to determine whether the program or treatment has an effect on some sub‐population defined by these observed characteristics, but also to determine for which sub‐populations, if any, there is an effect. This paper treats this problem as a multiple testing problem in which each null hypothesis in the family of null hypotheses specifies whether the program has an effect on the outcome of interest for a particular sub‐population. We develop our methodology in the context of PROGRESA, a large‐scale poverty‐reduction program in Mexico. In our application, the outcome of interest is the school enrollment rate and the sub‐populations are defined by gender and highest grade completed. Under weak assumptions, the testing procedure we construct controls the familywise error rate—the probability of even one false rejection—in finite samples. Similar to earlier studies, we find that the program has a significant effect on the school enrollment rate, but only for a much smaller number of sub‐populations when compared to results that do not adjust for multiple testing. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation of the mean for critical branching process and its bootstrap approximation

This article focuses attention on the estimation of the mean of a sequence of branching process with immigration in the critical case. We get the limiting distribution of the pivot, and adopt a bootstrap procedure to bootstrap the least-square estimator with bootstrap sample size $m$ less than the size $n$ of the original sample. Under the assumptions that $m\rightarrow \infty $ and $m/n\rightarrow 0$ , the convergence in probability of the bootstrap distribution function is also established.     

Greek sovereign bond index,volatility, and structural breaks

This article investigates volatility changes in the 10-year Greek sovereign bond index returns using the multiple structural break test developed by Bai and Perron (Econometrica 66:47–78, 1998, J Appl Econ 18:1–22, 2003), which allows for endogenous identification of break dates. We find that there exists one break date in volatility, April 2010, when the European debt crisis worsened and the Greek sovereign bond was downgraded to junk status. We also obtain evidence of performance improvement in our modeling by including structural break dummies into the variance equation. We observe sharp drops in a measure of volatility persistence after incorporating the structural change. Our findings are important for not only investors who assess the volatility of sovereign bonds for portfolio risk management, but also for policy makers who wish to understand and minimize the impacts of excess volatility on the financial system in government bond markets.