We conclude this section with an example constructed to highlight the preceding points.
Table 13.2 gives hypothetical data on true consumption expenditure Y, true income X*, measured consumption Y, and measured income X. The table also explains how these variables were measured.30
Measurement Errors in the Dependent Variable Y Only
Based on the given data, the true consumption function is Y* = 25.00 + 0.6000X* (10.477) (0.0584)
whereas, if we use Y instead of Y*, we obtain
As these results show, and according to the theory, the estimated coefficients remain the same. The only effect of errors of measurement in the dependent variable is that the estimated standard errors of the coefficients tend to be larger [see (13.5.5)], which is clearly seen in (13.5.12). In passing, note that the regression coefficients in (13.5.11) and (13.5.12) are the same because the sample was generated to match the assumptions of the measurement error model.
We know that the true regression is (13.5.11). Suppose now that instead of using X* we use X. (Note: In reality X*is rarely observable.) The regression results are as follows:
These results are in accord with the theory—when there are measurement errors in the explanatory variable(s), the estimated coefficients are biased. Fortunately, in this example the bias is rather small—from (13.5.10) it is evident that the bias depends on aW/aX>, and in generating the data it was assumed that aW = 36 and aX = 3667, thus making the bias factor rather small, about 0.98 percent (= 36/3667).
We leave it to the reader to find out what happens when there are errors of measurement in both Yand X, that is, if we regress Y on X , rather than Y* on X* (see exercise 13.23).
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