When is a Compositional Effect not a Compositional Effect?

The question of compositional effects (that is, the effect of collective properties of a pupil body on the individual members), or Aggregated Group-Level Effects (AGLEs) as the author prefers to call them, has been the subject of considerable controversy. Some authors, e.g. Rutter et al. [Fifteen thousand hours: Secondary Schools and Their Effects on Children. London: Open Books.], Willms [Oxford Review of Education 11(1): 33–41; (1986). American Sociological Review, 51, 224–241.], Bondi [British Educational Research Journal, 17(3), 203-218.], have claimed to find such effects, while on the other hand Mortimore et al. [School Matters: the Junior Years. Wells: Open Books.] and Thomas and Mortimore [Oxford Review of Education 16(2): 137–158.] did not. Others, for example Hauser [1970], have implied that many apparent AGLEs may be spurious, while Gray et al. [Review of Research in Education, 8, 158–193.] have suggested that at least in certain circumstances such apparent effects may arise as a result of inadequate allowance for pre-existing differences. A possible statistical mechanism for this is outlined in the work of Burstein [In R. Dreeben, & J. A. Thomas (Eds.), The Analysis of Educational Productivity. Volume 1: Issues in Microanalysis, Cambridge, MASS: Ballinger, pp. 119–190] on the effect of aggregating the data when a variable is omitted from the model used. This paper suggests another way in which spurious AGLEs can arise. It shows mathematically that even if there are no omitted variables, measurement error in an explanatory variable could give rise to apparent, but spurious, AGLEs, when analysed using a multilevel modelling procedure. Using simulation methods, it investigates what the practical effects of this are likely to be, and shows that statistically significant spurious effects occur systematically under fairly standard conditions.     

Retrospective studies with survey data: Problems in social science and epidemiology research

Large-scale population surveys are a valuable source of data for epidemiologists and social scientists interested in retrospective studies of health conditions. It may not be apparent that such data sources present many problems, one of which is data reliability. It can be shown statistically that where a relation exists between two variables measured with no or minimum error, this relation vanishes if either or both variables have sizable error components. Thus such data have little utility in assessing simple relations, let alone in making causal inferences. Another problem is that unreliability or measurement error affects the sensitivity and specificity of a measuring instrument in unpredictable ways. Solutions to the problems caused by data unreliability are suggested.     

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.     

Modelling response error in school effectiveness research

Statistical modelling of school effectiveness in educational research is considered. Variance component models are generally accepted for the analysis of such studies. A shortcoming is that outcome variables are still treated as measured without an error. Unreliable variables produce biases in the estimates of the other model parameters. The variability of the relationships across schools and the effects of schools on students' outcomes differ substantially when taking the measurement error in the dependent variables of the variance component models into account. The random effects model can be extended to handle measurement error using a response model, leading to a random effects item response theory model. This extended random effects model is in particular suitable when subjects are measured repeatedly on the same outcome at several points in time.     

Measurement error in causal analysis of panel data: Attenuated versus inflated relationships

The presence of random measurement error is commonly thought to cause attenuation of statistical relationships. While this is an unquestionable truth in bivariate analysis, it cannot be generalized to the multivariate case without qualification. This paper shows that measurement error may give rise to overestimates of parameters in causal analysis whenever there is more than one independent variable and the independent variables are correlated. If the independent variables are not measured with the same amount of reliability, there may also be considerable error in estimates of the relative magnitude of their impact. Both problems are particularly serious when the amount of measurement error is large relative to some of the causal effects such as in panel analysis with lagged dependent variables.     

Efficiency of iterated WLS in the linear model with completely unknown heteroskedasticity

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is not assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way.
IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the co-variance matrix is obtained. If this estimate is minimal in some way the iteration process is stopped.
Asymptotic properties of IWLS are derived for increasing sample size n . Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. Even asymptotic efficiency with respect to WLS with the true error variances can be obtained if m is not fixed but tends to infinity with n and if the heteroskedasticity is smooth.     

Treatment effect estimation with covariate measurement error

This paper investigates the effect that covariate measurement error has on a treatment effect analysis built on an unconfoundedness restriction in which there is conditioning on error free covariates. The approach uses small parameter asymptotic methods to obtain the approximate effects of measurement error for estimators of average treatment effects. The approximations can be estimated using data on observed outcomes, the treatment indicator and error contaminated covariates without employing additional information from validation data or instrumental variables. The results can be used in a sensitivity analysis to probe the potential effects of measurement error on the evaluation of treatment effects.     

An empirical likelihood method for spatial regression

Properties of a “blockwise”empirical likelihood for spatial regression with non-stochastic regressors are investigated for spatial data on a lattice. The method enables nonparametric confidence regions for spatial trend parameters to be calibrated, even though non-random regressors introduce non-stationary forms of spatial dependence into the “blockwise” construction. Additionally, the regression results are valid in a general framework allowing for a variety of behavior in regressor variables as well as the underlying spatial error process. The same regression method also applies when the regressors are stochastic.     

Estimating panel data models in the presence of endogeneity and selection

We consider estimation of panel data models with sample selection when the equation of interest contains endogenous explanatory variables as well as unobserved heterogeneity. Assuming that appropriate instruments are available, we propose several tests for selection bias and two estimation procedures that correct for selection in the presence of endogenous regressors. The tests are based on the fixed effects two-stage least squares estimator, thereby permitting arbitrary correlation between unobserved heterogeneity and explanatory variables. The first correction procedure is parametric and is valid under the assumption that the errors in the selection equation are normally distributed. The second procedure estimates the model parameters semiparametrically using series estimators. In the proposed testing and correction procedures, the error terms may be heterogeneously distributed and serially dependent in both selection and primary equations. Because these methods allow for a rather flexible structure of the error variance and do not impose any nonstandard assumptions on the conditional distributions of explanatory variables, they provide a useful alternative to the existing approaches presented in the literature.     

A neural network demand system with heteroskedastic errors

In this paper we consider estimation of demand systems with flexible functional forms, allowing an error term with a general conditional heteroskedasticity function that depends on observed covariates, such as demographic variables. We propose a general model that can be estimated either by quasi-maximum likelihood (in the case of exogenous regressors) or generalized method of moments (GMM) if the covariates are endogenous. The specification proposed in the paper nests several demand functions in the literature and the results can be applied to the recently proposed Exact Affine Stone Index (EASI) demand system of [Lewbel, A., Pendakur, K., 2008. Tricks with Hicks: The EASI implicit Marshallian demand system for unobserved heterogeneity and flexible Engel curves. American Economic Review (in press)]. Furthermore, flexible nonlinear expenditure elasticities can be estimated.     

A class of stochastic unit-root bilinear processes: Mixing properties and unit-root test

A class of stochastic unit-root bilinear processes, allowing for GARCH-type effects with asymmetries, is studied. Necessary and sufficient conditions for the strict and second-order stationarity of the error process are given. The strictly stationary solution is shown to be strongly mixing under mild additional assumptions. It follows that, in this model, the standard (non-stochastic) unit-root tests of Phillips–Perron and Dickey–Fuller are asymptotically valid to detect the presence of a (stochastic) unit-root. The finite sample properties of these tests are studied via Monte-Carlo experiments.     

Structural changes in heterogeneous panels with endogenous regressors

This paper extends Pesaran's (Econometrica, 2006, 74, 967–1012) common correlated effects (CCE) by allowing for endogenous regressors in large heterogeneous panels with unknown common structural changes in slopes and error factor structure. Since endogenous regressors and structural breaks are often encountered in empirical studies with large panels, this extension makes Pesaran's CCE approach empirically more appealing. In addition to allowing for slope heterogeneity and cross‐sectional dependence, we find that Pesaran's CCE approach is also valid when dealing with unobservable factors in the presence of endogenous regressors and structural changes in slopes and error factor loadings. This is supported by Monte Carlo experiments.     

Testing for Heteroskedasticity and Predictive Failure in Linear Regression Models*

It is argued that, when researchers wish to carry out a Chow test of the significance of prediction errors, it is necessary to assume homoskedasticity because standard results on heteroskedasticity‐robust tests are not available. The effects of heteroskedasticity on the Chow prediction error test are examined. The implementation of tests for heteroskedasticity is discussed, with the case in which the regressors include dummy variables for prediction error tests receiving special attention. Monte Carlo results are reported.     

Consistent estimation of a regression with errors in the variables

Summary As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.     

Air pollution-mortality models: A demonstration of the effects of random measurement error

Errors of measurement have long been recognized as a chronic problem in statistical analysis. Although there is a vast statistical literature of multiple regression models estimating the air pollution-mortality relationship, this problem has been largely ignored. It is well known that pollution measures contain error, but the consequences of this error for regression estimates is not known. We use Lave and Seskin's air pollution model to demonstrate the consequences of random measurement error. We assume a range of 0% to 50% of the variance of the pollution measures is due to error. We find large differences in the estimated effects on mortality of the pollution variables as well as the other explanatory variables once this measurement error is taken into account. These results cast doubt on the usual regression estimates of the mortality effects of air pollution. More generally our results demonstrate the consequences of random measurement error in the explanatory variable of a multiple regression analysis and the misleading conclusions that may result in policy research if this error is ignored.     

The effects of sampling on measures of association between variables based on sample means

Many studies that involve people's perceptions or behaviors focus on aggregate rather than individual responses. For example, variables describing public perceptions for some set of events may be represented as mean scores for each event. Event mean scores then become the unit of analysis for each variable. The variance of these mean scores for a variable is not only a function of the variation among the events themselves, but is also due to the variation among respondents and their possible responses. This is also the case for the covariances between variables based on event mean scores. In many contexts the variance and covariance components attributable to the sampling of respondents and their responses may be large; these components can be described as measurement error. In this paper we show how to estimate variances and covariances of aggregate variables that are free of these sources of measurement error. We also present a measure of reliability for the event means and examine the effect of the number of respondents on these spurious components. To illustrate how these estimates are computed, forty-two respondents were asked to rate forty events on seven risk perception variables. Computing the variances and covariances for these variables based on event means resulted in relatively large components attributable to measurement error. A demonstration is given of how this error is removed and the resulting effect on our estimates.     

Structural measurement errors in nonseparable models

This paper considers measurement error from a new perspective. In surveys, response errors are often caused by the fact that respondents recall past events and quantities imperfectly. We explore the consequences of limited recall for the identification of marginal effects. Our identification approach is entirely nonparametric, using Matzkin-type nonseparable models that nest a large class of potential structural models. We show that measurement error due to limited recall will generally exhibit nonstandard behavior, in particular be nonclassical and differential, even for left-hand side variables in linear models. We establish that information reduction by individuals is the critical issue for the severity of recall measurement error. In order to detect information reduction, we propose a nonparametric test statistic. Finally, we propose bounds to address identification problems resulting from recall errors. We illustrate our theoretical findings using real-world data on food consumption.     

Errors of measurement and the recent acceleration in manufacturing productivity growth

Using detailed (4-digit SIC) industry data for the years 1958–1989, I examine whether the recent acceleration in manufacturing productivity can be attributed to the effects of mismeasurement of the prices of inputs and output, by testing a model linking a set of proxy variables for measurement error to a series of measures of acceleration in total factor productivity (TFP). Alternative TFP estimates are presented in order to determine if the findings are sensitive to the method of TFP calculation. The results are inconsistent with the measurement error hypothesis and invariant to the specification of the TFP equation.     

An alternative model for the analysis of variance with random effects

The standard model for the analysis of variance with random effects implies, for the case of two independent variables, that single effects must be tested not against the error, but against the interaction mean squares. This causes, in comparison with the fixed effects AV, a considerable loss of test power, particularly for the 2 × 2 table. An alternative modelling of the interaction effect is proposed which completely avoids the loss of power.     

The role of governance in mobile phones for inclusive human development in Sub-Saharan Africa

This study assesses the synergy effects of governance in mobile phone penetration for inclusive human development in Sub-Saharan Africa with data for the period 2000–2012. It employs a battery of interactive estimation techniques, namely: Fixed effects, Generalised Method of Moments and Tobit regressions. Concepts of political (voice and accountability and political stability/no violence), economic (government effectiveness and regulation quality) and institutional (corruption-control and rule of law) governance are employed. The following findings are established. The previously apparent positive correlation between mobile phones and inclusive development can be extended to a positive effect. Although political governance is overwhelmingly not significant across estimated models, the average effects from economic governance are higher relative to institutional governance. On the interactions between mobile phones and governance variables, while none are apparent in Fixed effects regressions, there are significant synergy effects in Generalised Method of Moments and Tobit estimations, notably, from: regulation quality in the former and political stability, voice and accountability and rule of law in the latter. There is consistent evidence of convergence in inclusive human development. Policy implications are discussed.