How measurement error affects inference in linear regression
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How measurement error affects inference in linear regression Erik Meijer1
· Edward Oczkowski2 · Tom Wansbeek3
Received: 16 December 2019 / Accepted: 15 September 2020 © The Author(s) 2020
Abstract Measurement error biases OLS results. When the measurement error variance in absolute or relative (reliability) form is known, adjustment is simple. We link the (known) estimators for these cases to GMM theory and provide simple derivations of their standard errors. Our focus is on the test statistics. We show monotonic relations between the t-statistics and R 2 s of the (infeasible) estimator if there was no measurement error, the inconsistent OLS estimator, and the consistent estimator that corrects for measurement error and show the relation between the t-value and the magnitude of the assumed measurement error variance or reliability. We also discuss how standard errors can be computed when the measurement error variance or reliability is estimated, rather than known, and we indicate how the estimators generalize to the panel data context, where we have to deal with dependency among observations. By way of illustration, we estimate a hedonic wine price function for different values of the reliability of the proxy used for the wine quality variable. Keywords Measurement error · Generalized method of moments · Expert rating · Hedonic regression · Wine quality · Structural equation model JEL Classification C21 · C52 · L15 · Q11
We are grateful to Vasilis Sarafidis and two anonymous referees for their useful comments and suggestions.
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Tom Wansbeek [email protected]
1
University of Southern California, Los Angeles, USA
2
Charles Sturt University, Wagga Wagga, Australia
3
Faculty of Economics and Business, University of Groningen, Nettelbosje 2, 9747 AE Groningen, The Netherlands
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E. Meijer et al.
1 Introduction As is well known from econometric textbooks (e.g., Baltagi 2011, sec. 5.3), measurement error in one or more regressors makes OLS estimators of linear regression models inconsistent. Often, the inconsistency will cause a bias toward zero, although this does not need not be the case and the bias can be away from zero (Wansbeek and Meijer 2000, sec. 2.3). But whatever the direction of the bias, the desire “to do something” about it has spawned a huge literature since the 1930s. One strand in this literature is to limit the problem by deriving (asymptotic) bounds on the estimators, thus limiting the extent of the problem. In case the measurement error is confined to a single regressor, OLS is biased toward zero while reverse regression is biased away from zero, thus offering estimated bounds on the coefficient. This classical result (Frisch 1934) does not extend to the case of multiple mismeasured regressors, and then, outside information in the form of a bound on the measurement error covariance matrix is required to obtain estimators of bounds on the coefficients (Wansbeek and Meijer 2000, secs. 3.4 and 3.5). But, not surprisingly, the focus in the literature is on coming up with a consistent es
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