Probit Marginal Effects and Changes in Predictors
Summary
The document asks how to attribute a change in predicted success probability to changes in individual predictors in a Probit model. It gives the derivative of the probability with respect to one predictor: the predictor’s coefficient multiplied by the standard normal density evaluated at the full linear index. Because that density depends on all predictors, comparing marginal effects across two dates can produce a nonzero difference even when the predictor of interest did not change.
The answer clarifies that a marginal effect for one predictor must hold the other regressors constant. A comparison that lets other inputs vary does not isolate that predictor’s contribution. The discussion offers this conceptual correction but does not provide a complete decomposition of the total probability change across several jointly changing predictors. Such an attribution would require specifying a method for varying the inputs and handling their interactions through the nonlinear link.
Key ideas
- A Probit model maps its linear predictor through the standard normal cumulative distribution function.
- The marginal effect of a predictor depends on the full set of regressor values.
- To isolate one predictor’s marginal effect, hold the other regressors fixed.
- A difference in marginal effects across dates does not by itself attribute the probability change to one input.
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# Interpretation of coefficients of a Probit model
# Interpretation of coefficients of a Probit model
The exact problem I am trying to solve is as follows. I have a Probit specification:
$$ P_t = \Phi(\beta^T x_t) $$
where $\Phi$ is a standard normal CDF and $x$ is a matrix of independent variables measured at time $t$. $P$ is the probability of success.
I am trying to calculate the marginal effect of change in each $x$ over the probability. Formally:
I have two estimates estimated at different time points: $$ P_t = \Phi(\beta^T x_t) \quad ... (1)$$ and $$ P_{t+1} = \Phi(\beta^T x_{t+1}) \quad ... (2)$$
I want to get the weighted impact of each change in $x$ that is $\Delta x = x_{t+1}-x_t$ over the difference in the probability of success $\Delta P = P_{t+1} - P_t$. In simple linear regression, this is easy as the marginal impact was the coefficient multiplied by the absolute change in the variable.
I have already tried to calculate the marginal probability effect:
$$\frac {\partial P_t} {\partial x_{it}} = \beta_i \phi(\beta^T x_t)$$
and then take a simple difference: $$\frac {\partial P_{t+1}} {\partial x_{i \ t+1}} x_{t+1}- \frac {\partial P_t} {\partial x_{it}} x_t$$ where $\phi$ is the standard normal PDF. However, the problem with this approach is that the marginal effect is a function of every independent variable. Suppose, $x_{it} = x_{i \ t+1}$ then the marginal impact would be non-zero if any other $x_j$ changed. This should intuitively be zero as the driver did not change at all.
Any insights or sources would be much appreciated. Thanks in anticipation.
## Answer by Richard Hardy (score 0)
https://quant.stackexchange.com/a/74742
You have done a good analysis that tells you all you need to know. Just note that the marginal effect should keep all of the other regressors constant (ensure that ceteris is actually paribus). Otherwise, what you get from $$ \frac {\partial P_{t+1}} {\partial x_{i \ t+1}} x_{t+1}- \frac {\partial P_t} {\partial x_{it}} x_t $$ is not a marginal effect of $x_{i}$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.