PRACTITIONER'S CORNER*: Logarithmic Dependent Variables and Prediction Bias
| DOI | http://doi.org/10.1111/j.1468-0084.1983.mp45004006.x |
| Date | 01 November 1983 |
| Author | Peter Kennedy |
| Published date | 01 November 1983 |
PRACTITIONER'S CORNER*
Logarithmic Dependent Variables and Prediction Bias
Peter Kennedy
It is common in econometrics to find estimation undertaken via a regres-
sion in which the regressand is the logarithm of the original dependent
variable. In many such studies interest focuses exclusively on the slope
coefficients and under the usual assumptions regarding how the data
were generated, the slope coefficient estimates are best linear unbiased.
But often attention is also paid to prediction of Y, the original depend-
ent variable. In such cases the usual method of estimating Y, taking the
exponential of the estimate j of y = ln Y produces biased predictions;
although this result has been known for some time, its application to
the prediction context has not been stressed, and an examination of
recent econometric studies suggests that practitioners do not realize
that this bias can in some circumstances be sufficiently large as to
warrant an adjustment. Such circumstances are most likely to arise in
the context f simulation studies, since the variance of the predicted y
can be large when prediction is undertaken outside the bounds of the
data set.
Consider for expository purposes the estimating problem:
Y = aX13 exp(e) (1)
where X is an independent variable and a and í3 are fixed parameters.
(Generalization to the case of Y = aX1X2 .. Xf,K exp(e) is straight-
forward. It should also be clear that a similar bias exists for the semi-
logarithmic estimating form). Exp() is assumed to be a lognormally-
distributed multiplicative error term with mean one, so that the mean
of Y conditional on X is aX13, the usual interpretation attached to such
algebraic specifications. (See, e.g., Goldberger (1968a), p. 3.) If is often
the case that researchers assume that Fe O, implying that the condi-
tional expectation of Y is EY = aX13 exp (a2). However, the results to
follow do not depend on which of these two specifications is adopted.
The model (1) gives rise to the log-linear estimating form:
y = y + ßx + e
* The purpose of Practitioner's Corner is to publish brief methodological notes of interest
to applied economists. The Editors welcome submissions of this sort.
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