Economics 508 Roger Koenker Lecture 2 Transformations and the Specification of Econometric Models
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چکیده
What relationship does (â, b̂) bear to (α, β) in the original model and can we hope to say anything reasonable having made this initial specification error? In Figure 1, we can examine a specific version of this situation in which (α, β) = (1, .5) and the variance of ui is quite small. Clearly we don’t do a very good job of estimating the curve represented by the observed points by the line indicating the least squares fit, but it is useful to look at this more carefully.∗ On a more optimistic note it might appear that the slope of the linear fit might provide a decent approximation to the tangent of the curve at a point roughly corresponding to x̄. Figure 2 illustrates this phenomenon on the elasticity scale. Were we to estimate the log-linear model we would have an easily interpreted constant elasticity estimate. However, since we have estimated the model in the linear form, the implied elasticity of y with respect to x varies as we move along the fitted line. More explicitly, the elasticity is defined as
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