Obviously, the latter variance is larger than the former.27 Therefore, although the errors of measurement in the dependent variable still give unbiased estimates of the parameters and their variances, the estimated variances are now larger than in the case where there are no such errors of measurement.
27But note that this variance is still unbiased because under the stated conditions the composite error term Vi = j + ei still satisfies the assumptions underlying the method of least squares.
526 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
Errors of Measurement in the Explanatory Variable X
Now assume that instead of (13.5.1), we have the following model:
where Yi = current consumption expenditure X* = permanent income ui = disturbance term (equation error)
Suppose instead of observing Xi*, we observe
where wi represents errors of measurement in Xi*. Therefore, instead of estimating (13.5.6), we estimate
= « + fa Xi + zi where zi = ui — fawi, a compound of equation and measurement errors.
Now even if we assume that wi has zero mean, is serially independent, and is uncorrelated with ui, we can no longer assume that the composite error term zi is independent of the explanatory variable Xi because [assuming E(zi) = 0]
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