of the estimated coeffi- cients of dummy variables in semilogarithmic regression equations. Two biased estimators have been proposed for estimating.
77
Economics Letters 10 (1982) 77-79 North-Holland Publishing Company
THE INTERPRETATION OF DUMMY VARIABLES SEMILOGARITHMIC EQUATIONS Unbiased Estimation David
E.A. GILES
Monash
University,
Received
11 March
IN
*
Clayton,
Vict. 3168, Austraha
1982
This letter considers the problem of estimating the percentage impact of a dummy variable regressor on the level of the dependent variable in a semilogarithmic regression equation. The minimum variance unbiased estimator is introduced and compared with two other previously proposed estimators.
Two recent papers by Halvorsen and Palmquist (1980) and by Kennedy (198 1) have discussed the interpretation of the estimated coefficients of dummy variables in semilogarithmic regression equations. Two biased estimators have been proposed for estimating the percentage impact of a dummy variable on the (level of) the variable being explained in the regression. Here we discuss an unbiased estimator of this impact, and show that the estimator proposed by Kennedy leads to interpretations which are negligibly different from those arising from the use of this unbiased estimator, especially in the context of the studies cited in both of the above-mentioned notes. Let the equation to be estimated be In Y=a+~b,X,+~c,D,+u, I
(1)
i
where the X, are non-stochastic * I am grateful
to Peter Hampton
0 165- 1765/82/0000-0000/$02.75
continuous
and Peter Kennedy
variables,
the D, are dummy
for their helpful
0 1982 North-Holland
comments.
78
D.E.A.
Giles / Interpretation
of dummy variables in semilogarithmrc
equations
variables, and there are v degrees of freedom. Assume that the error term in the multiplicative form of the model is log-normally distributed, so that u - N(0, ~‘1). As is noted by Halvorsen and Palmquist, the correct measure of the percentage impact of D,,, on Y is lOOg, = lOO[exp(c,) - 11. Kennedy has observed that S, = [exp( 2,) - l] is a biased estimator of g,, where E, is the least squares estimator of c,. He proposes the use of g: = exp[ C, - f p( ?,,,)I - 1, where p( t,,,) is the usual unbiased estimator of the variance of E,. The estimator gz is also biased for g,, but is less biased than is 8,. In fact, using a result from Goldberger (1968, p. 469) it may be shown easily that the minimum variance unbiased estimator of g, is
t?, =
[exp(CJl i k=lJ
(Y/2)“P(V/2)
(-+WJ)*
F(k + 421
k!
and so for sufficiently large v, gz N g,. Further, all three estimators of g, are consistent. The example [taken from the study of Hanushek and Quigley (1978)J used by Kennedy involved 375 degrees of freedom, and it may be verified easily that there is no detectable arithmetic difference between gz and 2, in this case. Large Y values are also a feature of all of the other studies mentioned by Halvorsen and Palmquist. Thus, Kennedy’s proposal carries more weight than he takes credit for - in the context in which this interpretation discussion was initiated, his estimator (gz) leads to dummy variable interpretations which differ negligibly from those that would arise if the minimum variance unbiased estimator (8,) were used. Of course, for problems involving small degrees of freedom, & still may be preferred to gz, but even in these cases the choice may be of little practical consequence. For example, from the least squares results ’ of Giles and Hampton (1978), with Y = 20, we obtain values of -21.7061, -22.0593, and -22.0594 for lOOi,, 1OOg; and lOOg, respectively as measures of the percentage impact of South Island residential location on urban out-migration in New Zealand. Clearly, from the point of view of finite sample bias, Kennedy’s estimator is to be preferred to &,, and the minimum variance unbiased
’ These results are only illustrative, and not exact, as the non-stochastic tion is not strictly satisfied in this case.
regressor
assump-
D.E.A.
Giles / Interpretation
of dummy variables in semilogmthmic
equations
19
estimator, gz, is preferable to either 2, or 2,. However, given the close numerical equivalence between g,,, and gn”, in most practical situations, computational convenience weighs heavily in favour of Kennedy’s estimator.
References Giles, D.E.A. and P. Hampton, 1978, A note on urban migration in New Zealand, Journal of Urban Economics 5, 403-408. Goldberger, A.S., 1968, The interpretation and estimation of Cobb-Douglas functions, Econometrica 35, 464-482. Halvorsen, R. and R. Palmquist, 1980, The interpretation of dummy variables in semilogarithmic equations, American Economic Review 70, 474-475. Hanushek, E.A. and J.M. Quigley, 1978, Implicit investment profiles and intertemporal adjustments of relative wages, American Economic Review 68, 67-79. Kennedy, P.E., 1981, Estimation with correctly interpreted dummy variables in semilogarithmic equations, American Economic Review 7 1, 80 1,
