marginal-effectstata
I ran a regression where the dependent variable is winning (1=win)
Given that my regression is probit I want to understand the coefficient.
I've done margins, dydx() for my independent variable (average marginal effects). This yielded a result of -.41.
What does this mean? Does it mean that the probability of winning goes down by .41 percentage points? and if so, when does it go down by that much?
I just want a lay person's way to explain this .41 value.
This has been answered before but I will try to include a very simple explanation which can hopefully get you on the right track.
A logit regression model, linking the probability of a dependent variable $y$ to some vector of independent variables $X$ is written as follows
$$Pr(y=1) = \Lambda(X\beta)$$ where $\Lambda()$ represents a logistic c.d.f.
The marginal effect can be though of as the impact a change in some variable $x_j$ has on the response probability $Pr(y=1)$ and can be written as.
$$\frac{\partial Pr(y=1)}{\partial x_j} = \beta_j \lambda(X\beta) $$ where $\lambda$ is the p.d.f of a logistic function (the first derivative of $\Lambda$ w.r.t its argument)
Notice that for different values of X, you get a different values of $\lambda(XB)$, giving you different marginal effects.
To calculate the average marginal effect, you take the average of the logistic p.d.f for all the values of X in your sample and multiply it by your coefficient $\beta_j$.
$$\frac{\partial Pr(y=1)}{\partial x_j} = \beta_j E[\lambda(X\beta)] $$
Aside Note: This is different than the marginal effect at the average.
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 $x$:
$$AME =\frac{\sum_i^N\exp\{\alpha +\beta \cdot x_{it} + \gamma_i\} \cdot \beta}{N}$$
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 $\gamma_i=0$.
Best Answer
The average marginal effect gives you an effect on the probability, i.e. a number between 0 and 1. It is the average change in probability when x increases by one unit. Since a probit is a non-linear model, that effect will differ from individual to individual. What the average marginal effect does is compute it for each individual and than compute the average. To get the effect on the percentage you need to multiply by a 100, so the chance of winning decreases by 41 percentage points.