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From |
May Boggess <mboggess@stata.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Question on mfx and regress |

Date |
06 Apr 2004 12:16:37 -0500 |

On Tuesday, Joao asked: > Could any one help me understand why the coefficients of the natural log of > a dependent variable are close, but not equivalent to the marginal effect of > the same dependent variable estimated with the eydx option (dlogy/dx). The short answer is because we are differentiating two different functions. Let's start with the semilog model. The linear predictor is the expected value of the response E[ln(Y)]. Since this is a linear function of the independent variables, the derivatives are just the coefficients of the model. So, for example, the coefficient of x_i can be written d(E[ln(Y)])/dx_i. --------------(1) Note that this is a constant function of the independent variables, meaning, we obtain the same answer regardless of where we evaluate the function. Now let's consider the partial elasticities of the linear model. This time the linear predictor of the model is the expected value E[Y] and to get the partial elasticity we take the log of this and then differentiate: d(ln(E[Y]))/dx_i. --------------(2) So the difference between (1) and (2) is that E[ln(Y)] is not the same function as ln(E[Y]). We can calculate the partial elasticity (2) by hand easily. Writing f for E[Y] and using the chain rule we obtain: d(ln(f))/dx = d(ln(f))/df * df/dx, and since f is the linear predictor, d(ln(f))/dx = 1/f * b = b/f. where b is the coefficient of x. Note that this is not a constant function of the independent variables, because the linear predictor f varies according to where I evaluate it. Let's check all of this with an example: clear sysuse auto replace wei=wei/1000 replace pri=pri/10000 replace len=len/100 regress price wei len sum wei scalar meanw=r(mean) sum len scalar meanl=r(mean) scalar xb=meanw*_b[weight] + meanl*_b[length]+_b[_cons] di "weight: eydx = " _b[weight]/xb di "length: eydx = " _b[length]/xb mfx, predict(xb) nose eydx gen lnprice=ln(price) regress lnprice wei len We see that -mfx- agrees with our hand calculation of eydx. We could verify that the partial elasticities are not a constant function of the independent variables by using the option -at- on -mfx-. We also see that the partial elasticities are not the same as the coefficients of the semilog model, which is because taking logs, estimating the model then differentiating, is not the same as estimating the model, taking logs, then differentiating. For those interested, there is an FAQ on the -mfx- option -eyex-: http://www.stata.com/support/faqs/stat/mfx_eyex.html --May mmb@stata.com * * 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: Question on mfx and regress***From:*"Joao Pedro W. de Azevedo" <jazevedo@provide.com.br>

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