## Semilogarithmic Regression with Dummy Regressor

In Section 9.10 we noted that in models of the type ln Yi = 01 + 02 Di (1)

the relative change in Y (i.e., semielasticity), with respect to the dummy regressor taking values of 1 or 0, can be obtained as (antilog of estimated 02) - 1 times 100, that is, as

The proof is as follows: Since ln and exp (= e) are inverse functions, we can write (1) as:

Now when D = 0, e02Di = 1 and when D = 1, e02Di = e02. Therefore, in going from state 0 to state 1, ln Yi changes by (e02 - 1). But a change in the log of a variable is a relative change, which after multiplication by 100 becomes a percentage change. Hence the percentage change is (e02 - 1) x 100, as claimed. (Note: lne e = 1, that is, the log of e to base e is 1, just as the log of 10 to base 10 is 1. Recall that log to base e is called the natural log and that log to base 10 is called the common log.)

This example is adapted from Peter Kennedy, A Guide to Econometrics, 4th ed., MIT Press, Cambridge, Mass., 1998, p. 347. 