> -----Original Message-----
> From: [email protected] [mailto:owner-
> [email protected]] On Behalf Of [email protected]
> Sent: Thursday, October 06, 2005 6:46 PM
> To: [email protected]
> Subject: st: RE: RE: interaction
>
> Scott,
> I'm sorry if I wasn't clear about my question. Since the dependent
> variable is log transformed, the percentage increase is (exp(.29)-1)*100=
> 34 percent for group=1
Thanks. I was thinking in terms of a continuous variable, in which
ln(Y) = b1 + b2X2 + e
b2 = dln(Y)/dX2 = (dY/Y)/dx
With a dummy variable the usual interpretation is:
ln(Y) = b1 + b2D + e
Percentage change = (Y1/Y0) -1 = exp{b1 + b2*1 + e} / exp{b1 + b2*0 + e} - 1
= exp{b2} - 1
In taking a look at the issue interpreting dummy variables in loglinear
models, I came across a few papers on this topic. The following is taken
from van Garderen and Shah (2002).
Halvorsen and Palmquist (1980) pointed out that the unlike a continuous
variable, the coefficient of a dummy variable, multiplied by 100, is
not the usual percentage effect of that variable on the dependent variable
(the na�ve estimator). Instead it should be calculated as:
p = exp(b} -1
This is the case if the value of b is known. In practice, however, b is
unknown and has to be estimated. Kennedy (1981) pointed out that this
transformation results in a biased estimator for p.
If the error term is assumed to be normally distributed then the OLS
estimator of b is efficient and unbiased. Goldberger (1968) noted that the
expected value exp{b_hat} = exp{b + .5*V(b_hat)} where V(b_hat) is the
variance of b_hat.
This led Kennedy to suggest:
p = 100*(exp{b_hat - .5*V(b_hat)} - 1),
where V(b_hat) is the OLS estimate of the variance of b_hat.
Van Garderen and Shah go on to develop the Exact Minimum Variance Unbiased
Estimator of the percentage change in Y due to a change in D from 0 to 1:
p =100*{exp(b_hat)0_F_1(m; -.5*V(b_hat)) - 1},
where m = (n-k)/2, n is the sample size k is the number of regressors and
0_F_1 is the hypergeometric function.
The term 0_F_1(m; -.5*V(b_hat)) tends to exp{-.5V(b_hat)} as the sample
increases.
The authors also give the variance of Kennedy's estimator as:
V(p) = 100^2*exp{2*b_hat}*[exp{-V(b_hat)} - exp{2*V(b_hat)}
Stata does not have hypergeometric function so the exact estimator cannot be
directly calculated, but the program listed below calculates the unbiased
percentage change using Kennedy's method.
Example:
. sysuse auto, clear
(1978 Automobile Data)
. gen lnprice = ln(price)
. qui reg lnprice fore mpg gear
. disp "HP estimator =" exp(_b[fore])-1
HP estimator =.57716448
. semidum fore
Unbiased Estimated Percentage Change in Dependent Variable
Kennedy's (1981) approximation method for semilogarithmic equations
----------------------------------------------------------------------------
variable | % Change Std. Err. t P>|t| [ 95% Conf. Interval ]
---------+------------------------------------------------------------------
foreign | 56.70733 17.6987 3.20 0.002 21.40832 92.00633
----------------------------------------------------------------------------
Scott
References
Giles, D.E.A. (1982). The interpretation of dummy variables in
semilogarithmic equations. Economics Letters, 10, 77-79.
Goldberger, A.S. (1968). The interpretation and estimation of
Cobb-Douglas functions. Econometrica 36, 464-472.
Halvorsen, R. and Palmquist, R. (1980). The interpretation of dummy
variables in semilogarithmic equations. American Economic Review,
70, 474-75.
Kennedy, P. E. (1981). Estimation with correctly interpreted dummy
variables in semilogarithmic equations. American Economic Review,
71, 801.
van Garderen, K. J. (2001). Optimal prediction in loglinear models. Journal
of Econometrics, 104, 119-140.
van Garderen, K. J. and Shah, C. (2002). Exact interpretation of dummy
variables in semilogarithmic equations. The Econometrics Journal, 5,
149-159.
------------------------------------------------------------------------
*! version 1.0.0 October 9, 2005
*! Scott Merryman
program semidum
version 9.1
syntax varname, [level(integer 95)]
scalar v_hat = _se[`varlist']^2
local kennedy = 100*(exp(_b[`varlist'] -.5*v_hat)-1)
local var = 100^2*exp(2*_b[`varlist'])*[exp(-v_hat) - exp(-2*v_hat)]
local se= sqrt(`var')
local t = `kennedy'/`se'
local pvalue = 2*ttail(`=e(df_r)', abs(`t'))
local ul =`kennedy' + invttail(`=e(df_r)',(1- `level'/100)/2)*`se'
local ll =`kennedy' - invttail(`=e(df_r)',(1- `level'/100)/2)*`se'
disp ""
disp in smcl in gr "Unbiased Estimated Percentage Change in Dependent
Variable"
disp in smcl in gr "Kennedy's (1981) approximation method for
semilogarithmic equations "
disp in smcl in gr "{hline 15}{c TT}{hline 68}"
disp in smcl in gr "{ralign 14:variable}" _col(15) " {c |} " ///
_col(20) "% Change" ///
_col(30) `"Std. Err."' ///
_col(44) "t" ///
_col(52) "P>|t|" ///
_col(62) `"[ `level'% Conf. Interval ]"' ///
_n "{hline 15}{c +}{hline 68}"
di in smcl in gr "{ralign 14: `varlist' }" _col(15) " {c |} " ///
_col(18) in ye %-9.0g `kennedy' ///
_col(30) in ye %8.0g `se' ///
_col(42) in ye %5.2f `t' ///
_col(52) in ye %5.3f `pvalue' ///
_col(62) in ye %9.0g `ll' " " in ye %9.0g `ul' ///
_n in gr "{hline 15}{c BT}{hline 68}"
end
*
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
