Assume that on the basis of the criteria just listed we arrive at a model that we accept as a good model. To be concrete, let this model be
where Y = total cost of production andX = output. Equation (13.2.1) is the familiar textbook example of the cubic total cost function.
But suppose for some reason (say, laziness in plotting the scattergram) a researcher decides to use the following model:
Note that we have changed the notation to distinguish this model from the true model.
Since (13.2.1) is assumed true, adopting (13.2.2) would constitute a specification error, the error consisting in omitting a relevant variable (X3). Therefore, the error term u2i in (13.2.2) is in fact
We shall see shortly the importance of this relationship.
Now suppose that another researcher uses the following model:
Yi = ft + ft Xi + A 3 X} + ft X3 + ft X4 + U3i (13.2.4)
If (13.2.1) is the "truth," (13.2.4) also constitutes a specification error, the error here consisting in including an unnecessary or irrelevant variable in the sense that the true model assumes ft to be zero. The new error term is in fact u3i = u1i — ftX4
Now assume that yet another researcher postulates the following model:
In relation to the true model, (13.2.6) would also constitute a specification bias, the bias here being the use of the wrong functional form: In (13.2.1) Y appears linearly, whereas in (13.2.6) it appears log-linearly.
CHAPTER THIRTEEN: ECONOMETRIC MODELING 509
Finally, consider the researcher who uses the following model:
where Y* = Y* + e* and X* = X* + w*, e* and w* being the errors of measurement. What (13.2.7) states is that instead of using the true Y and X* we use their proxies, Y* and X*, which may contain errors of measurement. Therefore, in (13.2.7) we commit the errors of measurement bias. In applied work data are plagued by errors of approximations or errors of incomplete coverage or simply errors of omitting some observations. In the social sciences we often depend on secondary data and usually have no way of knowing the types of errors, if any, made by the primary data-collecting agency.
Another type of specification error relates to the way the stochastic error u* (or ut) enters the regression model. Consider for instance, the following bivariate regression model without the intercept term:
where the stochastic error term enters multiplicatively with the property that ln u* satisfies the assumptions of the CLRM, against the following model
where the error term enters additively. Although the variables are the same in the two models, we have denoted the slope coefficient in (13.2.8) by fa and the slope coefficient in (13.2.9) by a. Now if (13.2.8) is the "correct" or "true" model, would the estimated a provide an unbiased estimate of the true fa? That is, will E(a) = fa ? If that is not the case, improper stochastic specification of the error term will constitute another source of specification error.
To sum up, in developing an empirical model, one is likely to commit one or more of the following specification errors:
1. Omission of a relevant variable(s)
2. Inclusion of an unnecessary variable(s)
3. Adopting the wrong functional form
4. Errors of measurement
5. Incorrect specification of the stochastic error term
Before turning to an examination of these specification errors in some detail, it may be fruitful to distinguish between model specification errors and model mis-specification errors. The first four types of error discussed above are essentially in the nature of model specification errors in that we have in mind a "true" model but somehow we do not estimate the correct model. In model mis-specification errors, we do not know what the true model is to begin with. In this context one may recall the controversy
510 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
between the Keynesians and the monetarists. The monetarists give primacy to money in explaining changes in GDP, whereas the Keynesians emphasize the role of government expenditure to explain changes in GDP. So to speak, there are two competing models.
In what follows, we will first consider model specification errors and then examine model mis-specification errors.
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