All along we have assumed implicitly that the dependent variable Y and the explanatory variables, the X's, are measured without any errors. Thus, in the regression of consumption expenditure on income and wealth of households, we assume that the data on these variables are "accurate"; they are not guess estimates, extrapolated, interpolated, or rounded off in any systematic manner, such as to the nearest hundredth dollar, and so on. Unfortunately, this ideal is not met in practice for a variety of reasons, such as nonresponse errors, reporting errors, and computing errors. Whatever the reasons, error of measurement is a potentially troublesome problem, for it constitutes yet another example of specification bias with the consequences noted below.
Errors of Measurement in the Dependent Variable Y
Consider the following model:
where Y* = permanent consumption expenditure26 Xi = current income ui = stochastic disturbance term
26This phrase is due to Milton Friedman. See also exercise 13.8.
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Since Y* is not directly measurable, we may use an observable expenditure variable Yi such that
where si denote errors of measurement in Y*. Therefore, instead of estimating (13.5.1), we estimate
Y = (a + 3 Xi + Ui) + Si = a + 3 Xi + (u + Si) (13.5.3)
= a + 3 Xi + Vi where vi = ui + si is a composite error term, containing the population disturbance term (which may be called the equation error term) and the measurement error term.
For simplicity assume that E(ui) = E(si) = 0, cov (Xi, ui) = 0 (which is the assumption of the classical linear regression), and cov (Xi, si) = 0; that is, the errors of measurement in Y* are uncorrelated with Xi, and cov (ui, si) = 0; that is, the equation error and the measurement error are uncorrelated. With these assumptions, it can be seen that 3 estimated from either (13.5.1) or (13.5.3) will be an unbiased estimator of the true 3 (see exercise 13.7); that is, the errors of measurement in the dependent variable Y do not destroy the unbiasedness property of the OLS estimators. However, the variances and standard errors of 3 estimated from (13.5.1) and (13.5.3) will be different because, employing the usual formulas (see Chapter 3), we obtain
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