Blocked-Error-R 2: A Conceptually Improved Definition of the Proportion of Explained Variance in Models Containing Loops or Correlated Residuals

Bentler and Raykov (2000, Journal of Applied Psychology 85: 125–131), and Jöreskog (1999a, http://www.ssicentral.com/lisrel/column3.htm, 1999b http://www.ssicentral. com/lisrel/column5.htm) proposed procedures for calculating R ² for dependent variables involved in loops or possessing correlated errors. This article demonstrates that Bentler and Raykov’s procedure can not be routinely interpreted as a “proportion” of explained variance, while Jöreskog’s reduced-form calculation is unnecessarily restrictive. The new blocked-error-R ² (beR ²) uses a minimal hypothetical causal intervention to resolve the variance-partitioning ambiguities created by loops and correlated errors. Hayduk (1996) discussed how stabilising feedback models – models capable of counteracting external perturbations – can result in an acceptable error variance which exceeds the variance of the dependent variable to which that error is attached. For variables included within loops, whether stabilising or not, beR ² provides the same value as Hayduk’s (1996) loop-adjusted-R ². For variables not involved in loops and not displaying correlated residuals, beR ² reports the same value as the traditional regression R ². Thus, beR ² provides a conceptualisation of the proportion of explained variance that spans both recursive and nonrecursive structural equation models. A procedure for calculating beR ² in any SEM program is provided.     

Does Private Money Buy Public Policy? Campaign Contributions and Regulatory Outcomes in Telecommunications

To what extent can market participants affect the outcomes of regulatory policy? In this paper, we study the effects of one potential source of influence—campaign contributions—from competing interests in the local telecommunications industry, on regulatory policy decisions of state public utility commissions. Our work is unique in that we test the effects of campaign contributions on measurable policy outcomes. This stands in stark relief against most of the existing literature, which examines potentially noisier measures of policy outcomes—such as the roll‐call votes of legislators, to examine how private money may influence public policy. By moving to more direct measures of policy effects, and using a unique new dataset, we find, in contrast to much of the literature on campaign contributions, that there is a significant effect of private money on regulatory outcomes. This result is robust to numerous alternative model specifications. We also assess the extent of omitted variable bias that would have to exist to obviate the estimated result. We find that for our result to be spurious, omitted variables would have to explain more than five times the variation in the mix of private money as is explained by the variables included in our analysis. We consider this to be very unlikely.     

The effect of latency variables on repeated measures inference applied to the measurement of risk-taking as a function of psychopathy

In numerous psychology studies, subjects are asked to perform some task a number of times, T. The effect of the choice of T on the associated inference, however, is usually not assessed. We investigate the appropriate choice of T empirically by using data collected in a study on the relationship between psychopathy and risk-taking in 90 inner city drug users enrolled in a residential treatment program. We show that, when studying this relationship, the latency variable usually discarded from the analysis behaves exponentially allowing a natural division of the study period 1, . . . , T into two distinct subperiods. These subperiods yield significantly different results—in the early period only (which we call “reactive”), subjects with high psychopathy scores exhibit lower sensitivity to reward and punishment in our risk taking experiment. The later period (which we call “stable”) shows no relationship between sensitivity to reward and punishment and psychopathic tendencies.     

A Bayesian estimator for stochastic frontier models with errors in variables

A Bayesian estimator is proposed for a stochastic frontier model with errors in variables. The model assumes a truncated-normal distribution for the inefficiency and accommodates exogenous determinants of inefficiency. An empirical example of Tobin??s Q investment model is provided, in which the Q variable is known to suffer from measurement error. Results show that correcting for measurement error in the Q variable has an important effect on the estimation results.     

Variable importance by partitioningR 2

One purpose of many regression studies is to compare the relative importance of the independent variables. Several different measures have been used to measure importance:t-values, standardized regression coefficients, elasticity, commonality analysis, increment inR ², correlation coefficients, hierarchical partitioning etc. Some of these measures have the common feature of partitioningR ² between the independent variables and assess their importance according to their contribution toR ². This paper is an attempt to clarify the advantages and disadvantages with these different methods and find out if any useful information can be gained by a partitioning ofR ².     

Cross redundancy and sensitivity in DEA models

Data envelopment analysis (DEA) measures the efficiency of each decision making unit (DMU) by maximizing the ratio of virtual output to virtual input with the constraint that the ratio does not exceed one for each DMU. In the case that one output variable has a linear dependence (conic dependence, to be precise) with the other output variables, it can be hypothesized that the addition or deletion of such an output variable would not change the efficiency estimates. This is also the case for input variables. However, in the case that a certain set of input and output variables is linearly dependent, the effect of such a dependency on DEA is not clear. In this paper, we call such a dependency a cross redundancy and examine the effect of a cross redundancy on DEA. We prove that the addition or deletion of a cross-redundant variable does not affect the efficiency estimates yielded by the CCR or BCC models. Furthermore, we present a sensitivity analysis to examine the effect of an imperfect cross redundancy on DEA by using accounting data obtained from United States exchange-listed companies.     

Local tax management in Spain: A study of the conditional efficiency of provincial tax agencies

This paper reports an efficiency analysis of local tax management by provincial tax agencies in Spain based on supramunicipal delegation. To conduct this study, we used the robust order-m conditional model that directly accounts for some socioeconomic environmental variables to estimate the efficiency scores. This is a key issue, as tax agencies do not have control over the context in which they operate, and this may have a severe impact on their performance. Our results suggest that several of the provincial contextual variables accounted for (the net property tax base, population density and inhabitants of the municipalities that have delegated management to the provincial tier of government) have a negative impact on efficiency, especially at higher variable value levels. Considering that the provincial tier of government can opt to set up specific self-governing agencies to perform these tasks, we also applied metafrontier analysis to assess their share in inefficiency. We concluded that the establishment of such self-governing agencies does not lead to higher efficiency levels.     

R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.     

The role of the elasticity of substitution in economic growth: A cross-country investigation

Although the importance of the elasticity of substitution between capital and labor (σ) has long been recognized in several branches of economics, it has not received enough attention in the growth literature. de La Grandville (1989) showed theoretically that at any stage of an economy's development, the growth rate of income per capita is increasing with σ. The higher is σ, the greater the similarity between capital and labor in the production function, and thus diminishing returns set in very slowly. To the best of our knowledge, this is the first paper that tests the hypothesis that growth rate is increasing with the value of σ at the cross-country level. We estimate σ for 90 countries from direct estimation of the normalized CES production function and then include these estimators as an explanatory variable in cross-country growth regression. We investigate the sign and significance of the coefficient of σ conditioning on country characteristics, initial conditions, and a set of policy variables. After accounting for endogeneity and the fact that σ is a “generated” regressor, we find strong support for the hypothesis. The result is robust to both Leamer's (1983) extreme value analysis and Bayesian model averaging. About a fifth to a quarter of the growth rate differential between East Asia and Sub-Saharan Africa can be explained by σ alone.     

Multivariate Prediction with Nonlinear Principal Components Analysis: Application

Gower and Blasius (Quality and Quantity, 39, 2005) proposed the notion of multivariate predictability as a measure of goodness-of-fit in data reduction techniques which is useful for visualizing and screening data. For quantitative variables this leads to the usual sums-of-squares and variance accounted for criteria. For categorical variables, and in particular for ordered categorical variables, they showed how to predict the levels of all variables associated with every point (case). The proportion of predictions which agree with the true category-levels gives the measure of fit. The ideas are very general; as an illustration they used nonlinear principal components analysis. An example of the method is described in this paper using data drawn from 23 countries participating in the International Social Survey Program (1995), paying special attention to two sets of variables concerned with Regional and National Identity. It turns out that the predictability criterion suggests that the fits are rather better than is indicated by “percentage of variance accounted for”.     

A multiple linear regression model for imprecise information

In standard regression analysis the relationship between the (response) variable and a set of (explanatory) variables is investigated. In the classical framework the response is affected by probabilistic uncertainty (randomness) and, thus, treated as a random variable. However, the data can also be subjected to other kinds of uncertainty such as imprecision. A possible way to manage all of these uncertainties is represented by the concept of fuzzy random variable (FRV). The most common class of FRVs is the LR family (LR FRV), which allows us to express every FRV in terms of three random variables, namely, the center, the left spread and the right spread. In this work, limiting our attention to the LR FRV class, we consider the linear regression problem in the presence of one or more imprecise random elements. The procedure for estimating the model parameters and the determination coefficient are discussed and the hypothesis testing problem is addressed following a bootstrap approach. Furthermore, in order to illustrate how the proposed model works in practice, the results of a real-life example are given together with a comparison with those obtained by applying classical regression analysis.     

Dose–response modelling for bivariate covariates with and without a spike at zero: theory and application to binary outcomes

In epidemiology and clinical research, there is often a proportion of unexposed individuals resulting in zero values of exposure, meaning that some individuals are not exposed and those exposed have some continuous distribution. Examples are smoking or alcohol consumption. We will call these variables with a spike at zero (SAZ). In this paper, we performed a systematic investigation on how to model covariates with a SAZ and derived theoretical odds ratio functions for selected bivariate distributions. We consider the bivariate normal and bivariate log normal distribution with a SAZ. Both confounding and effect modification can be elegantly described by formalizing the covariance matrix given the binary outcome variable Y. To model the effect of these variables, we use a procedure based on fractional polynomials first introduced by Royston and Altman (1994, Applied Statistics 43: 429–467) and modified for the SAZ situation (Royston and Sauerbrei, 2008, Multivariable model‐building: a pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables, Wiley; Becher et al., 2012, Biometrical Journal 54: 686–700). We aim to contribute to theory, practical procedures and application in epidemiology and clinical research to derive multivariable models for variables with a SAZ. As an example, we use data from a case–control study on lung cancer.     

A path-analytic investigation of the determinants of career withdrawal intentions of engineers: some HRM issues arising in a professional labour market in Singapore

Data obtained from engineers (N = 230) in Singapore were used to test a model of career withdrawal intentions. The model hypothesized personal, organizational and environmental variables as exogenous variables that affect career satisfaction and job satisfaction. These affective states in turn affect career commitment which was posited directly to affect career withdrawal intentions. The findings suggest that the model is useful in explaining career withdrawal intentions as 50 per cent (R²) of the variance was explained. As hypothesized, career commitment revealed a significant negative path to career withdrawal intentions. Some of the exogenous variables, particularly organizational variables, showed direct significant paths to career withdrawal intentions, though work–family conflict, a personal variable, approached significance. A limitation of the study, direction for future studies and implications of the findings are discussed.     

Profit, productivity and distribution: Differences across organizational forms - The case of Spanish banks

We examine variations in financial and economic performance as a function of organizational form, and, over time. The forms we consider include Spanish commercial banks, savings banks, and financial cooperatives. We decompose multilateral variation in operating profit, our measure of financial performance, into price and quantity effects. We then decompose the latter into a margin effect and productivity change. Our measure of economic performance, productivity variation, is subsequently disaggregated into technical, cost efficiency, and scale. We find that deregulation and liberalization have acted to narrow performance gaps among organizational forms; this, despite less-than compelling evidence that increased competition has contributed to this convergence. For the Spanish banking system as a whole, the margin effect appears to deliver twice as much financial benefit as do improvements in productivity. Importantly, this finding does not vary across organizational form. Such regularity has an important implication: Incentives for growth are apparently provided by a positive margin rather than by the more elusive benefits of economies of scale and improved cost efficiency.     

The coefficient of determination and simultaneous equation systems

In this paper we show that the Carter-Nagar (1977) R²'s for single structural equations and systems are in fact R² for the reduced form where the partially restricted reduced form estimation method is employed. We also show that the results of McElroy (1977) may be used to derive the Carter-Nagar system measure. If the reduced form equations are estimated by Kakwani's (1975) k-class reduced form estimator a new R² may be defined which is shown to be asymptotically equivalent to the Carter-Nagar measure.     

Effects of Using Mean Scores in Regression Models: An Example From Environmental Psychology

Multiple regression analysis with grouped data is often used as a method for exploring environmental preferences, the preferred unit for measurement in such analyses is mean scores rather than individual scores.Although this procedure allows us to reduce the potential for error in measuring different variables and, as a consequence of this, improves the reliability of the technique, it also produces some additional, undesirable effects. The latter include artificial increases in R² values which give the impression that a high degree of fit has been achieved for the regression model. Indeed, this goodness fit often appears to be better than that which could have been achieved by using individual scores. Further, given that different studies operate with differing numbers of subjects in their groups, the R² scores which result from the analyses of these groups are not directly comparable.In the following discussion, we demonstrate how any value, other than zero for correlations between variables, can be increased, at will, by simply expanding the number of subjects in each group. We present the specialised formulae used for quantifying this increase and offer a warning about the purely relative nature of any study which bases its conclusions on models of regression analysis using grouped data.     

A simple estimator for nonlinear error in variable models

We propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data, such as validation data or double measurements, are available. We assume that the marginal distributions of the measurement errors are Laplace (double exponential) with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other. Under these assumptions, we derive simple revised moment conditions in terms of the observed variables. They are used to make inference about the model parameters and the variance of the measurement error. The results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest. Our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute. Our estimation method is particularly useful in situations where no additional data are available, which is the case in many economic data sets. Simulation study demonstrates good finite sample properties of our proposed estimator. We also examine the performance of the estimator in the case where the error distribution is misspecified.     

The presence of an error term does not preclude causal inference in regression: a comment on Krause (2012)

A recent article by Krause (Qual Quant, doi:10.1007/s11135-012-9712-5, Krause (2012)) maintains that: (1) it is untenable to characterize the error term in multiple regression as simply an extraneous random influence on the outcome variable, because any amount of error implies the possibility of one or more omitted, relevant explanatory variables; and (2) the only way to guarantee the prevention of omitted variable bias and thereby justify causal interpretations of estimated coefficients is to construct fully specified models that completely eliminate the error term. The present commentary argues that such an extreme position is impractical and unnecessary, given the availability of specialized techniques for dealing with the primary statistical consequence of omitted variables, namely endogeneity, or the existence of correlations between included explanatory variables and the error term. In particular, the current article discusses the method of instrumental variable estimation, which can resolve the endogeneity problem in causal models where one or more relevant explanatory variables are excluded, thus allowing for accurate estimation of effects. An overview of recent methodological resources and software for conducting instrumental variables estimation is provided, with the aim of helping to place this crucial technique squarely in the statistical toolkit of applied researchers.     

The impact of public regulation on housing quality and on structural changes

This paper represents a theoretical investigation of profit maximizing behavior of a landlord under rent control. The situation envisioned is one that the landlord owns, free and clear, either two housing units ready to merge or one large one ready to convert into two small ones; in each case the units are already in place, such that capital costs are sunk and treated as bygones. Each unit has a technology of production of housing services with a fixed input of quantity of space (shelter) and a variable input of quality, that is affected by physical ageing (non-controllable) and by maintenance (controllable). At starting time t₀, we have a state of quality and a historical state of initial quality Q?_i, at the time the building was built, with the implicit constraint that Q_i(t) < Q?_i, for all t> t₀. The analysis addresses the general question of housing structural changes—conversion or merger—and how these changes are being accelerated under the threat of rent control.