*Interpreting effects in nonlinear models (based on Stolzenberg, 1979): For simplicity, disregard the error and let Y represent the systematic part of the response variable.

Suppose that Y is a function of two explanatory variables, Y ¼ f ðX1; X2Þ.

& The metric effect of X1 on Y is defined as the partial derivative ∂Y=∂X1.

& The effect of proportional change in X1 on Y is defined as X1ð∂Y=∂X1Þ.

& The instantaneous rate of return of Y with respect to X1 is ð∂Y=∂X1Þ=Y.

& The point elasticity of Y with respect to X1 is ð∂Y=∂X1ÞðX1=YÞ.

Find each of these four measures of the effect of X1 in each of the following models. Which measure yields the simplest result in each case? How can the several measures be interpreted?
How would you fit each model to data, assuming convenient forms for the errors [e.g., additive errors for models (a), (b), and (c)]?

(a) Y ¼ α þ β1X1 þ β2X2.

(b) Y ¼ α þ β1X1 þ β2X2 1 þ β3X2.

(c) Y ¼ α þ β1X1 þ β2X2 þ β3X1X2.

(d) Y ¼ expðα þ β1X1 þ β2X2).

(e) Y ¼ αXβ1 1 Xβ2 2 .