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Following the incredible demonstration in Statalist by Jeff Pitblado on how to calculate - using the Delta Method - the Standard Errors for Average Marginal Effects of a Logit Model.

Q: What would be the formula to calculate the SEs for the AMEs of a multinomial logit using the Delta Method?

In particular, for the marginal effect (or better, the average partial effects) of a $k$ predictor, taking the derivative w.r.t. to $x_{ik}$ of

$$ Pr\{Y_i=j\} = P_{ij}=\frac{e^{x_i'\beta_j}}{\sum_r e^{x_i'\beta_r}} $$

will give us

$$ \frac{\partial P_{ij}}{\partial x_{ik}} = P_{ij} \left(\beta_{jk}-\sum_r P_{ir}\beta_{rk}\right) $$

which is straightforward how to compute.

If I get it right, the formula for the SEs of this AME will be

$$ SE\left(\frac{\partial P_{ij}}{\partial x_{ik}}\right) = J \hat{V} J' $$

where $\hat{V}$ is the Variance-Covariance matrix of the multinomial logit parameter estimates (and usually returned after estimation in most software like Stata or R) and $J$ is the Jacobian matrix, which is the gap in my calculations.

For the logit model, we can calculate the Jacobian in the following way in Stata:

webuse margex, clear

logit outcome i.treatment distance, nofvlabel
predict p, pr
gen dpdxb   = p*(1-p)
gen dpdx    = dpdxb*_b[distance]
gen d2pdxb2 = p*(1-p)*(1-p) - p*p*(1-p)

matrix vecaccum Jac = d2pdxb2 0b.treatment 1.treatment distance
matrix          Jac = Jac*_b[distance]/e(N)
sum dpdxb
matrix   Jac[1,3] = Jac[1,3] + r(mean)
mat list Jac 

or in `R' like this:

library(MASS)
data(birthwt)

m        <- glm(low~ smoke + age, binomial(link="logit"), data = birthwt)
p        <- m$fitted.values
dpdxb    <- p*(1-p)
dpdx     <- dpdxb*coef(m)[3]
d2pdxb2  <- p*(1-p)*(1-p) - p*p*(1-p)

Jac      <- t(d2pdxb2)%*%cbind((birthwt$smoke==0)*0, (birthwt$smoke==1)*1, birthwt$age, 1)*coef(m)[3]/length(p)
Jac[1,3] <- Jac[1,3]+mean(dpdxb)
Jac

but what would be the way to calculate the Jacobian for the multinomial logit motel?

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    $\begingroup$ Could you specify mathematically "Average Marginal Effects of a Multinomial Logit" in term of regression coefficients of logit model? $\endgroup$ – user158565 Dec 7 '18 at 4:03
  • $\begingroup$ thanks and sorry, you are right I wasn't clear. I edited my question with the formula of what I meant. In short, I want the equivalent of the SE in the margins, dydx(distance) that Jeff Pitbaldo calculates manually in his post for logit, but for multinomial logit. $\endgroup$ – Steve Dec 7 '18 at 10:30
  • $\begingroup$ Sorry I am dodging the question, but perhaps a practical solution would be to consider bootstrapping instead of delta method. $\endgroup$ – Umka Feb 4 '19 at 13:49
  • $\begingroup$ Thank you @Umka. Yes, I know that's another solution. My aim is to be didactic to myself, rather than estimate the SEs per se. I can always just use the built-in commands, but what I'm after is to understand the formulas. $\endgroup$ – Steve Feb 4 '19 at 15:52

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