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.     

Some robust exact results on sample autocorrelations and tests of randomness

Several exact results on the second moments of sample autocorrelations, for both Gaussian and non-Gaussian series, are presented. General formulae for the means, variances and covariances of sample autocorrelations are given for the case where the variables in a sequence are exchangeable. Bounds for the variances and covariances of sample autocorrelations from an arbitrary random sequence are derived. Exact and explicit formulae for the variances and covariances of sample autocorrelations from a Gaussian white noise are given. It is observed that the latter results hold for all spherically symmetric distributions. A simulation experiment, with Gaussian series, indicates that normalizing each sample autocorrelation with its exact mean and variance, instead of the usual approximate moments, can improve considerably the accuracy of the asymptotic N(0,1) distribution to obtain critical values for tests of randomness. The exact second moments of rank autocorrelations are also studied.     

Estimating time variation in measurement error from data revisions: an application to backcasting and forecasting in dynamic models

Over time, economic statistics are refined. This implies that data measuring recent economic events are typically less reliable than older data. Such time variation in measurement error affects optimal forecasts. Measurement error, and its time variation, are of course unobserved. Our contribution is to show how estimates of these can be recovered from the variance of revisions to data using a behavioural model of the statistics agency. We illustrate the gains in forecasting performance from exploiting these estimates using a real‐time dataset on UK aggregate expenditure data. Copyright © 2009 John Wiley & Sons, Ltd.     

Area Disparities in Britain: Understanding the Contribution of People vs. Place Through Variance Decompositions

This article considers methods for decomposing wage variation into individual and group specific components. We discuss the merits of these methods, which are applicable to variance decomposition problems generally. The relative magnitudes of the measures depend on the underlying variances and covariances, and we discuss how to interpret them, and how they might relate to structural parameters of interest. We show that a clear‐cut division of variation into area and individual components is impossible. An empirical application to the British labour market demonstrates that labour market area effects contribute very little to the overall variation of wages in Britain.     

Variable selection for additive partially linear models with measurement error

Variable selection for additive partially linear models with measurement error is considered. By the backfitting technique, we first propose a variable selection procedure for the parametric components based on the smoothly clipped absolute deviation (SCAD) penalization, and one-step spare estimates for parametric components are also presented. The resulting estimates perform asymptotic normality as well as an oracle property. Then, two-stage backfitting estimators are also presented for the nonparametric components by using the local linear method, and the structures of asymptotic biases and covariances of the proposed estimators are the same as those in partially linear model with measurement error. The finite sample performance of the proposed procedures is illustrated by simulation studies.     

Forecasting realized variance measures using time-varying coefficient models

This paper considers the problem of forecasting realized variance measures. These measures are highly persistent estimates of the underlying integrated variance, but are also noisy. Bollerslev, Patton and Quaedvlieg (2016), Journal of Econometrics 192(1), 1–18 exploited this so as to extend the commonly used heterogeneous autoregressive (HAR) by letting the model parameters vary over time depending on the estimated measurement error variances. We propose an alternative specification that allows the autoregressive parameters of HAR models to be driven by a latent Gaussian autoregressive process that may also depend on the estimated measurement error variance. The model parameters are estimated by maximum likelihood using the Kalman filter. Our empirical analysis considers the realized variances of 40 stocks from the S&P 500. Our model based on log variances shows the best overall performance and generates superior forecasts both in terms of a range of different loss functions and for various subsamples of the forecasting period.     

On the optimum number of levels in the one-way model with random effects

The optimal number of levels is studied for the one-way random model with normally distributed effects. The optimum criteria used are based on the variances of the traditional analysis of variance estimators of the variance components. Exact solutions are compared to earlier results based on lower bounds of the sampling variances. Comparisons are also made to the large-sample variances of the estimates based on restricted maximum likelihood. Received February 2002     

Tests of international CAPM with time-varying covariances

We perform maximum-likelihood estimation of a model of international asset pricing based on CAPM. We test the restrictions imposed by CAPM against a more general asset pricing model. The ‘betas’ in our CAPM vary over time as the supplies of assets change and as the conditional covariances or returns on those assets change. We let the covariances change over time as a function of macroeconomic data, and an alternative model allows the covariances to follow a multivariate ARCH process. We also can identify a modified CAPM model with measurement error. We find that the estimated CAPM performs much better when variances are not constant over time. Nonetheless, CAPM is rejected in favour of the lessrestricted model of asset pricing.     

Comparisons of aggregate variable forecasts using aggregate and disaggregate models

When forecasting aggregate variables, a choice must often be made to either add up individual forecasts made at a disaggregate level or to simply forecast at the aggregate level. The presence of heterogeneity introduces aggregation bias and makes the disaggregates approach more preferable, while the presence of data and specification errors introduces relatively large variances in the disaggregate forecasts, making the aggregate approach more preferable. It is suggested that the mean square error is useful in evaluating the combined effects of heterogeneity and specification and data errors, and in facilitating comparisons between aggregate and disaggregate approaches to aggregate variable forecasting.     

Bayesian estimation of a random coefficient model

This paper carries out a Bayesian analysis of the Hildreth-Houck (1968) random coefficient model and applies it to some cross-section production function data. Posterior distributions for mean coefficients, actual coefficients, variances and variance ratios are derived. The variance ratio posteriors are largely uninformative but they do lead to relatively informative densities on the variances, and the problem of negative variance estimates, obtained with previous techniques, is overcome. Posterior densities for the mean coefficients are not extremely sensitive to the variance ratios.     

Gains in efficiency from joint estimation of systems of autoregressive-moving average processes

The paper examines gains in efficiency from joint estimation of systems of ARMA processes where cross-correlation is due to contemporaneous correlation among disturbances. The asymptotic variance of joint estimates is derived and it involves only variances and covariances among purely AR processes corresponding to the AR and MA parts of the constituent processes. Small sample gains are evaluated by Monte Carlo methods. Application of joint estimation to two short-term interest rates is shown to result in more accurate post-sample predictions relative to both univariate models and the FMP econometric model.     

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.     

A note on the covariance matrix of the maximum likelihood estimator in constrained multivariate linear regression

A useful result concerning variances and covariances of a linear function of a random matrix is applied to find the variance–covariance matrix of the maximum likelihood estimator in multivariate linear regression subject to zero constraints.     

Restricted maximum likelihood estimation under Eisenhart model Ill

For a balanced two-way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non-negativity requirements of the variance components by L ee and K apadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non-negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.     

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.     

An assessment of equivalence between Internet and paper-based surveys: evidence from collectivistic cultures

Little research exists that addresses the equivalence in collectivistic cultures of paper- versus Internet-based surveys. This study addressed this gap and examined the measurement equivalence of individual innovativeness scales between Internet surveys and paper-based surveys within a collectivistic culture (with China serving as our example). The study analyzed and compared survey data from both paper and web-based surveys using confirmatory factor analysis. The assessment of invariance included the levels of configural, metric, scalar, and covariance invariance. The means and variance of latent variables were also compared. The results show that measurements are invariant at the two levels (configural and metric), and the covariances between latent variables are also equivalent, but the mean and variance differences of latent variables are apparent. The results indicate that when conducting research in collectivistic cultures and collecting data from distinct survey modes, researchers should concern themselves with the potential of extreme response patterns and the inclination of social desirability responding, as well as considering the measurement invariance across survey modes.     

Adjusting for measurement error in the value added model: evidence from Portugal

The multilevel value added approach to measuring school effectiveness is now widely used. We propose a method to adjust for measurement error to investigate the extent to which this changes school effect estimates. It is applied to longitudinal data collected in the region of Cova da Beira (NUT III) for 1st, 3rd, 5th, 7th and 8th grades. Three different variance component models are considered, depending on the predictor variables included. Assuming measurement error occurs in explanatory and/or response variables, corrections are made for different values of the coefficient of reliability. Moreover, models are fitted under the assumption of either independent or correlated measurement errors.     

A general analysis of bias in the estimated standard errors of least squares coefficients

It is well known that when errors in the usual regression model are not independently distributed with equal variances, the application of ordinary least squares leads to calculated variances of the coefficient estimates which are biased and inconsistent. The nature of this bias has been investigated extensively, but the existing literature is limited in two significant ways. First, derivations of exact expressions for the bias have been restricted to special cases and, except for the simplest of these, the expressions derived are almost unmanageably complex. Second, for general error specifications, attention has been focused exclusively on deriving bounds for the bias, which are usually wide and do not allow even the probable direction of any bias to be determined. This paper derives an asymptotic expression for the bias which allows both its sign and approximate magnitude to be described easily in most regression problems. This expression is then used to investigate the bias in the cases of serial correlation of an arbitrary degree, variance components models and approximation of a non-linear relationship with a linear specification.     

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.