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^2s 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.
Highlights
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
For the first two cases, known absolute variances and known reliabilities, we show that the t-values decrease while we move along this list. (The generality of the third case, estimated reliability or measurement error variance, defies analysis.) This greatly expands the findings in Meijer and Wansbeek (2000)
In previous papers (Meijer et al 2015, 2017), we have shown that panel data offer many additional possibilities for identification and estimation of measurement error models, compared to cross-sectional data
Summary
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. In case the measurement error is confined to a single regressor, OLS is biased toward zero while reverse regression is biased away from zero, offering estimated bounds on the coefficient. Higher moments of the variables can be used as instruments (Geary 1942; Erickson and Whited 2002) Another road to consistency lies open when the measurement error variance is known. We derive a consistent estimator of the regression coefficient and its asymptotic variance, both without and with assuming normality of the measurement error variance. 7, we investigate whether the analysis up until can be extended from a cross-sectional to a panel data context, and whether for the case of known or estimated measurement error variances or reliabilities, this makes identification and estimation easier or more difficult. Some of our special cases and extensions are new, and in particular, our main contribution to the literature is given by the results comparing the magnitudes of test statistics
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