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360 Chapter 8 Nonlinear Regression Functions

In a regression model, Y depends both on X and on the error term u. Because u is
random, it is conventional to evaluate the elasticity as the percentage change not
of Y but of the predicted component of Y—that is, the percentage change in
E(Y ͉ X). Accordingly, the elasticity of E(Y ͉ X) with respect to X is

dE(Y ͉ X) * X = d ln E(Y ͉ X).
dX E(Y͉ X) d ln X

The elasticities for the linear model and for the three logarithmic models summarized in
Key Concept 8.2 are given in the table below.

Case Population Regression Elasticity of E(Y|X ) with
Model Respect to X
linear b1X
Y = b0 + b1X + u b0 + b1X
linear-log
log-linear Y = b0 + b1ln(X ) + u b1
log-log ln(Y) = b0 + b1X + u b0 + b1ln(X)
ln(Y) = b0 + b1ln(X) + u
b1X

b1

The log-log specification has a constant elasticity, but in the other three specifications,

the elasticity depends on X.

We now derive the expressions for the linear-log and log-linear models. For the linear-

log model, E(Y ͉ X ) = b0 + b1 ln(X). Because dln(X)>dX = 1>X, applying the chain rule
yields dE(Y ͉ X)>dX = b1>X . Thus the elasticity is dE(Y ͉ X)>dX * X>E(Y ͉ X) =
(b1>X) * X>[b0 + b1ln(X)] = b1>[b0 + b1ln(X)], as is given in the table. For the log-linear
model, it is conventional to make the additional assumption that u and X are independently

distributed, so the expression for E(Y ͉ X) given following Equation (8.25) becomes
E(Y ͉ X) = ceb0+b1X, where c = E(eu) is a constant that does not depend on X because of
the additional assumption that u and X are independent. Thus dE(Y ͉ X)>dX = ceb0+b1Xb1
and the elasticity is dE(Y ͉ X)>dX * X>E(Y ͉ X) = ceb0+b1Xb1 * X>(ceb0+b1X) = b1X. The
derivations for the linear and log-log models are left as Exercise 8.11.