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.     

Person–environment fit or person plus environment: A meta-analysis of studies using polynomial regression analysis

The purpose of this study was to examine if the non-linear terms in polynomial regression analysis (PRA) explain substantial amounts of variance in assumed outcomes over and above the linear terms. This meta-analysis reviewed a total of 30 studies from 25 articles on person–environment (P-E) fit in which PRA was used. We found that none of the 30 studies did rigorous cross-validation for PRA. In addition, the overall population R² change attributable to the non-linear terms (PE, P² and E²) in PRA was only .008. However, the P-E interaction term may be more important in some situations than in others, based on the data gathered here. Specifically, the non-linear terms in PRA added to the variance explained by the linear terms (P and E) when studies examined assumed outcome variables like contextual performance or job attitudes as opposed to task performance. In summary, PRA could potentially be effective in investigating P-E fit and other human resource management-related phenomena when it is used in appropriate scenarios.     

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.     

Robustness of statistical inferences using linear models with meta-analytic correlation matrices

To examine complex relationships among variables, researchers in human resource management, industrial-organizational psychology, organizational behavior, and related fields have increasingly used meta-analytic procedures to aggregate effect sizes across primary studies to form meta-analytic correlation matrices, which are then subjected to further analyses using linear models (e.g., multiple linear regression). Because missing effect sizes (i.e., correlation coefficients) and different sample sizes across primary studies can occur when constructing meta-analytic correlation matrices, the present study examined the effects of missingness under realistic conditions and various methods for estimating sample size (e.g., minimum sample size, arithmetic mean, harmonic mean, and geometric mean) on the estimated squared multiple correlation coefficient (R²) and the power of the significance test on the overall R² in linear regression. Simulation results suggest that missing data had a more detrimental effect as the number of primary studies decreased and the number of predictor variables increased. It appears that using second-order sample sizes of at least 10 (i.e., independent effect sizes) can improve both statistical power and estimation of the overall R² considerably. Results also suggest that although the minimum sample size should not be used to estimate sample size, the other sample size estimates appear to perform similarly.     

Profiling potential green electricity tariff adopters: green consumerism as an environmental policy tool?

Green electricity tariffs are one means by which green consumers can contribute to a more sustainable future. This paper profiles potential adopters of green electricity tariffs. Potential adoption is measured in terms of respondents' willingness to pay a premium for green energy in a national survey of the UK population. Hypotheses based principally on the cognitive–behavioural literature on green consumerism and green energy markets are developed. These are tested using a broad range of variables, which are grouped into three categories (demographic, attitudinal and behavioural). Consistent with past research, the empirical analyses find that attitudinal variables best characterize potential adopters. Further, potential adopters are found to have higher income, be better informed with respect to energy matters, show concern for the environment and believe that individual actions can make a difference to environmental decay. The implications of these findings for marketing and environmental policy are explored. Copyright © 2010 John Wiley & Sons, Ltd and ERP Environment.     

Reward Contingency, Unemployment, and Functional Turnover

Based on the valence model of expectancy theory and the Cornell model of job satisfaction, this field study investigated the relationship between reward contingency, unemployment, pay satisfaction, job satisfaction, and functional turnover. The latter of which separates turnover into four categories: poor performing leavers, good performing leavers, poor performing stayers, and good performing stayers. It was conducted with a geographically dispersed sample of sales representatives (i.e., from 25 states and 66 cities), resulting in unemployment rates that ranged from 2 percent to 12 percent. The sales representatives were employed by four companies that paid different combinations of salary and commissions, ranging from mostly salary and little commission to 100 percent commission. A discriminant analysis accounted for 62 percent of the variance in functional turnover and achieved an overall classification hit rate of 67 percent across the four functional turnover groups. Follow-up univariate analyses indicated that objective reward contingency (R²=.34), state unemployment (R²=.11), state sales unemployment (R²=.08), education (R²=.09), and tenure (R²=.08) accounted for most of the variance in functional turnover. Perceived reward contingency, pay satisfaction, job satisfaction, age, and gender were not related to functional turnover.     

Efficient estimation for error component seemingly unrelated nonparametric regression models

Multivariate panel data provides a unique opportunity in studying the joint evolution of multiple response variables over time. In this paper, we propose an error component seemingly unrelated nonparametric regression model to fit the multivariate panel data, which is more flexible than the traditional error component seemingly unrelated parametric regression. By applying the undersmoothing technique and taking both of the correlations within and among responses into account, we propose an efficient two-stage local polynomial estimation for the unknown functions. It is shown that the resulting estimators are asymptotically normal, and have the same biases as the standard local polynomial estimators, which are only based on the individual response, and smaller asymptotic variances. The performance of the proposed procedure is evaluated through a simulation study and a real data set.     

Model selection in regression analysis,regression diagnostics and prior knowledge: A book review article with comments from Anthony C. Atkinson,D.R. Cox And John McDonald

Model selection of a regression model typically includes both selecting which variables to include in a model and the functional form of the relationship between the variables. In recent years regression diagnostics (a catch-all phrase that includes such topics as identifying influential observations) have been increasingly used as an aid in selecting a regression model. In evaluating the recent book Residuals and influence in regression, by Cook and Weisberg, Belsley argues that such analyses, while extremely useful, should not be used in a theoretical vacuum. Effective model building requires the analyst to make substantial use of any prior information relevant to the problem in hand. The commentators, while accepting the general thrust of Belsley's comments, are on the whole more sceptical — perhaps the prior knowledge is poor and in conflict with the data; perhaps the use of a wrong theory is more dangerous than ‘letting the data speak for themselves’.     

A Caution Regarding Rules of Thumb for Variance Inflation Factors

The Variance Inflation Factor (VIF) and tolerance are both widely used measures of the degree of multi-collinearity of the ith independent variable with the other independent variables in a regression model. Unfortunately, several rules of thumb – most commonly the rule of 10 – associated with VIF are regarded by many practitioners as a sign of severe or serious multi-collinearity (this rule appears in both scholarly articles and advanced statistical textbooks). When VIF reaches these threshold values researchers often attempt to reduce the collinearity by eliminating one or more variables from their analysis; using Ridge Regression to analyze their data; or combining two or more independent variables into a single index. These techniques for curing problems associated with multi-collinearity can create problems more serious than those they solve. Because of this, we examine these rules of thumb and find that threshold values of the VIF (and tolerance) need to be evaluated in the context of several other factors that influence the variance of regression coefficients. Values of the VIF of 10, 20, 40, or even higher do not, by themselves, discount the results of regression analyses, call for the elimination of one or more independent variables from the analysis, suggest the use of ridge regression, or require combining of independent variable into a single index.     

Developing an empirical socioeconomic typology of counties in five western states

County-level data representing a broad range of socioeconomic variables were obtained for the 190 counties in five western States-California, Colorado, Nevada, Utah and Wyoming. A factor analysis of these variables produced eight orthogonal factors: (1) Urbanism, (2) Poverty vs affluence, (3) County tax revenues and services, relative to population, (4) Proportion of aged population, (5) Social stability vs instability, (6) New housing development vs cattle ranching, (7) Crop farming, welfare and unemployment, and (8) Small business activity vs level of federal employment. Several ways of examining scores on these factors were explored and demonstrated: distributions of scores on individual factors across counties were displayed graphically; individual county profiles, comprised of scores on all eight factors, were produced and examined; and groups of counties with similar profiles (representing county “types”) were identified through cluster analysis, and their average profiles and geographic distributions displayed and discussed. Groupings of clusters which were similar in some respects and varied in others were also presented. The utility of these methods for planning activities which may involve individual counties and/or groups of counties is discussed.     

HR practices and turnover intention: the mediating roles of organizational commitment and organizational engagement in a selected region in Malaysia

This study examined the mediating effects of organizational commitment (OC) and organizational engagement (OE) on the relationship between human resource (HR) practices (career management, performance appraisal, compensation, person–job fit and job control) and turnover intention. A total of 457 employees working in various sectors in a selected region in Malaysia participated in this study. It was found that all the variables used to measure HR practices have significant effects on OC and OE. However, multiple regression analyses indicated that career management and job control did not have any significant influence on turnover intention. OC and OE were discovered to give partial mediating effects on the relationship between HR practices and turnover intention.     

Outcomes of Environmental Management Systems: the Role of Motivations and Firms’ Characteristics

This article analyzes the influence of the sources of motivation that lead companies to adopt environmental management systems (EMSs) on the outcomes of these systems. A set of hypotheses derived from an extensive review of the literature is analyzed using cluster analysis – in order to identify groups of companies – as well as correlation and regression analyses, with data obtained from a survey of 361 Spanish organizations that have environmental certification. The results reveal that, for the groups identified, companies from the holistic cluster (with high levels of both internal and external drivers) and from the internal focus cluster (with more intensive internal sources of motivation) secure greater benefits from the process of adopting an EMS. This article also sheds light on the influence on the outcomes of some variables that have been under‐researched, such as the economic resources invested in an EMS and whether or not the certified companies belong to a sector with high environmental pressure. The findings help to characterize the firms with environmental certification and may also help managers, policy makers and other stakeholders to anticipate the potential benefits of EMSs. Copyright © 2015 John Wiley & Sons, Ltd and ERP Environment     

Random group effects and the precision of regression estimates

When explanatory variable data in a regression model are drawn from a population with grouped structure, the regression errors are often correlated within groups. Error component and random coefficient regression models are considered as models of the intraclass correlation. This paper analyzes several empirical examples to investigate the applicability of random effects models and the consequences of inappropriately using ordinary least squares (OLS) estimation in the presence of random group effects. The principal findings are that the assumption of independent errors is usually incorrect and the unadjusted OLS standard errors often have a substantial downward bias, suggesting a considerable danger of spurious regression.     

The choice of functional forms for hedonic price equations: Comment

Rosen [13], Freeman [4], Halvorsen and Pollakowski [6], and others have stressed that economic theory does not suggest an appropriate functional form for hedonic price functions.¹ It consequently is reasonable to try several functional forms and utilize the multiple regression equation with the best performance. In this spirit, Halvorsen and Pollakowski [6] recommend using the Box-Cox flexible functional form for hedonic analysis and measuring best performance with a goodness of fit test. The Box-Cox methodology has also been adapted in hedonic studies by Goodman [5], Linneman [10], Blomquist and Worley [1], and Eberts and Gronberg [3].² The Box-Cox is particularly suited for testing functional forms because many familiar forms such as semilog, log linear, and translog are subsets of the flexible Box-Cox permitting nested hypothesis testing.In this note, we illustrate that the formal hypothesis testing advantage of the Box-Cox functional form is purchased at the expense of other important goals. The goal of most hedonic studies is to estimate the prices of the characteristics, to measure the response to changes in the prices, and/or to predict future expenditures. Using a best fit criterion to choose functional forms does not necessarily lead to more accurate estimates of characteristic prices. In fact, the large number of coefficients estimated with the Box-Cox functional form reduces the accuracy of any single coefficient which could lead to poorer estimates of specific prices. Second, because any negative number raised to a noninteger real power is imaginary, the traditional Box-Cox functional form is not suited to any data set containing negative numbers. Third, the Box-Cox functional form may be inappropriate for prediction. Since the mean predicted value of the untransformed dependent variable need not equal the mean of the sample upon which it is estimated, the predicted untransformed variable (housing value) will be biased. The predicted untransformed dependent variable may also be imaginary. Fourth, the nonlinear transformation results in complex estimates of slopes and elasticities which are often too cumbersome to use properly. We discuss each of these drawbacks and quantify them when possible in the remainder of this note.     

Assessing the accuracy of the aggregate law of motion in models with heterogeneous agents

This paper shows that the R² and the standard error have fatal flaws and are inadequate accuracy tests. Using data from a Krusell–Smith economy, I show that approximations for the law of motion of aggregate capital, for which the true standard deviation of aggregate capital is up to 14% (119%) higher than the implied value and which are thus clearly inaccurate, can have an R² as high as 0.9999 (0.99). Key in generating a more powerful test is that predictions of the aggregate law of motion are not updated with the aggregated simulated individual data.     

How Important are Earnings Announcements as an Information Source?

In a competitive information market, a single information source can only dominate other sources individually, not collectively. We explore whether earnings announcements constitute such a dominant source using Ball and Shivakumar's (2008) [How much new information is there in earnings?, Journal of Accounting Research, 2008, 46(5), pp. 975–1016] R ² metric: the proportion of the variation in annual returns explained by the four quarterly earnings announcement returns. We find that the earnings announcement days' R ² is 11% – higher than the corresponding R ² of days with dividend announcements, management forecasts, preannouncements, and 10-K and 10-Q filings and their amendments, and comparable to that of the four days with the largest realised absolute returns in a year. Additional analysis reveals that earnings announcements convey extreme bad news as often as management forecasts and preannouncements; for any other type of news, earnings announcements are much more frequent. We conclude that earnings announcements are an important source of new information in the equity market.     

Regression over Timescale Decompositions: A Sampling Analysis of Distributional Properties

In two previous papers, Ramsey and Lampart demonstrated that regression analyses between timescale decompositions provided important insight into the properties of economic relationships. The idea in those papers was that the relationship between any two variables, say consumption and income, was the union of the individual relationships between consumption and income at each timescale and that the regression relationship might, differ across timescales. This paper is dedicated to discovering the approximate distributional properties of the regression estimators and of the residuals in the context of such models. Sampling procedures are used to verify the distributional properties of the regression estimators at each timescale and those of the residuals. This analysis is necessary to provide the appropriate distributional information required to specify tests of hypotheses and confidence intervals.     

Bayesian clustering of distributions in stochastic frontier analysis

In stochastic frontier analysis, firm-specific efficiencies and their distribution are often main variables of interest. If firms fall into several groups, it is natural to allow each group to have its own distribution. This paper considers a method for nonparametrically modelling these distributions using Dirichlet processes. A common problem when applying nonparametric methods to grouped data is small sample sizes for some groups which can lead to poor inference. Methods that allow dependence between each group’s distribution are one set of solutions. The proposed model clusters the groups and assumes that the unknown distribution for each group in a cluster are the same. These clusters are inferred from the data. Markov chain Monte Carlo methods are necessary for model-fitting and efficient methods are described. The model is illustrated on a cost frontier application to US hospitals.     

The Relationship of the Degree of Exposure to a Technological Disaster and Emotional Response: A Structural Model Approach

The aim of this study was to analyze type and intensity of psychopathological response (depression and post-traumatic stress disorder) in a population sample from Guadalajara, Mexico, in an area affected by a series of gas explosions in 1992. The study comprised a series of interviews conducted three months after the event. Three groups were formed: subjects directly exposed to the explosions; those affected by the explosions but not directly exposed to them; and a control group. We devised a model, defining that the depressive level is affected by the socioeconomic level and the attributional style; being affected by the exposition degree to the disaster. This question was evaluated as using the structural models technique. The results indicate a close relationship between the variables defined and confirm the action of degree of exposure to the disaster. In fact, a reasonable fit for the model was only obtained in the direct exposure group; the fit was poor in the other two groups, in which results were largely similar. We conclude that the intensity of exposure to the event has a fundamental bearing on victims' psychopathological response, and on the development of posttraumatic stress disorder.