Solved – fixed effect and marginal effect in negative binomial

In the panel NB model, the expected number of events is

$$E[y vert x]=exp{alpha +beta x + gamma_i}.$$

By default, margins, dydx(.) gives you the effect on the index function (the part inside $exp{.}$), so you just get back the reported coefficients since the index function is linear.

By altering the predict() option, you can get average marginal effects (AMEs) on the number of events, which is the average of the derivatives of the expression above with respect to $x$:

$$AME =frac{sum_i^Nexp{alpha +beta cdot x_{it} + gamma_i} cdot beta}{N}$$

However, Stata will evaluate each derivative as if the fixed effect $gamma_i$ was zero for all $i$. This happens because the xtnbreg, fe model is estimated by a method known as conditional maximum likelihood. Here the FE for each individual are conditioned out of the likelihood function. This means that they are not available to be plugged in since these nuisance parameters are eliminated to make estimation easier/feasible. However, I believe they sum up to zero, so the AME can be interpreted as the AME as if everyone had the typical FE and their own covariates. I cannot find this explicitly stated in the documentation, so take that with a grain of salt.

Here's some Stata code showing this:

. webuse airacc, clear  . xtset airline        panel variable:  airline (balanced)  . /* Index function coefficients*/ . xtnbreg i_cnt inprog, exposure(pmiles) fe nolog  Conditional FE negative binomial regression     Number of obs     =         80 Group variable: airline                         Number of groups  =         20                                                  Obs per group:                                                               min =          4                                                               avg =        4.0                                                               max =          4                                                  Wald chi2(1)      =       2.11 Log likelihood  = -174.25143                    Prob > chi2       =     0.1463  ------------------------------------------------------------------------------        i_cnt |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------       inprog |  -.0984214   .0677414    -1.45   0.146    -.2311921    .0343492        _cons |  -3.414208   1.006801    -3.39   0.001    -5.387501   -1.440915   ln(pmiles) |          1  (exposure) ------------------------------------------------------------------------------  . margins, dydx(inprog)  Average marginal effects                        Number of obs     =         80 Model VCE    : OIM  Expression   : Linear prediction, predict() dy/dx w.r.t. : inprog  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------       inprog |  -.0984214   .0677414    -1.45   0.146    -.2311921    .0343492 ------------------------------------------------------------------------------  . margins, dydx(inprog) predict(xb) // same as above  Average marginal effects                        Number of obs     =         80 Model VCE    : OIM  Expression   : Linear prediction, predict(xb) dy/dx w.r.t. : inprog  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------       inprog |  -.0984214   .0677414    -1.45   0.146    -.2311921    .0343492 ------------------------------------------------------------------------------  . /* AME assuming fixed effects is zero */ . margins, dydx(inprog) predict(nu0)  Average marginal effects                        Number of obs     =         80 Model VCE    : OIM  Expression   : Predicted number of events (assuming u_i=0), predict(nu0) dy/dx w.r.t. : inprog  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------       inprog |  -9.695908   11.62238    -0.83   0.404    -32.47535    13.08354 ------------------------------------------------------------------------------  . gen double ME = exp(_b[_cons] + _b[inprog]*inprog + 1*ln(pmiles)) * _b[inprog]  . sum ME      Variable |        Obs        Mean    Std. Dev.       Min        Max -------------+---------------------------------------------------------           ME |         80   -9.695907    1.776251  -12.94475  -6.500259 

I am using a logarithmic offset for miles flown to make events proportional to miles flown, and treating inprog as if it was continuous rather than binary (since that is more similar to your situation). The AME is 9.7 fewer injury incidents from participating in an experimental safety training program across all airlines given their miles flown history.

You can also still interpret the index function coefficient as a kind of semi-elasticity. Here a one unit change in inprog is associated with a $100 cdot beta=-9.84 %$ drop in events. This does not require you to fix $gamma_i=0$.