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From |
"Scott Merryman" <smerryman@kc.rr.com> |

To |
<statalist@hsphsun2.harvard.edu> |

Subject |
st: RE: RE: RE: interaction |

Date |
Sun, 9 Oct 2005 14:44:21 -0500 |

> -----Original Message----- > From: owner-statalist@hsphsun2.harvard.edu [mailto:owner- > statalist@hsphsun2.harvard.edu] On Behalf Of lm335@drexel.edu > Sent: Thursday, October 06, 2005 6:46 PM > To: statalist@hsphsun2.harvard.edu > 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 naïve 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/

**References**:**st: RE: RE: interaction***From:*<lm335@drexel.edu>

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