4.5 Measuring marginal effects in binary models
The impact of a unit change in one of the independent variables on the probability of an event is typically referred to as the marginal effect. The first step is to compute the value of the probability using equation (3) when all the values of the Xi are set at some predetermined value; ie, the marginal effect can be evaluated at different points depending on the set of Xi values chosen. Generally the mean values for the whole sample are used, but the choice is arbitrary.
The second step involves repeating the calculation of the probability with a different set of Xi values. Typically this will involve leaving all but the variable of interest at their mean values and incrementing the variable of interest by one unit.
In the above example (Case A) the probability of working with both independent variables set at their means:
In the second stage we compute the probability of working when age is increased by one year and health status remains at its mean value:
The marginal effect is then given by
From this we conclude that the marginal effect of increased age is to reduce the probability of working by one percentage point.
To this point we have been using independent variables which are continuous. What happens when there is an independent variable that itself is binary? In the current context this could be a variable such as: does the female have a partner who is working (yes or no), or is the person a migrant (yes or no); or does the person have a tertiary qualification (yes or no). The presence of such variables has two implications. To calculate the marginal effect we would have to compute the value of Z setting the binary variable to zero and then to one. We then simply need to state clearly the base for the calculations. In addition, if there are other binary variables that we wish to hold constant, then there are a number of alternatives. We could (for apparent consistency with continuous variables) use the mean which is nothing more than the raw proportion. Alternatively we could assign the variable either zero or one. There is no “correct” way, and the option chosen will depend on the particular context being analysed. In the empirical work in later sections of this report we use the mean values.
We conclude this section with a summary in Table 4-3 of the different interpretations of a logistical regression coefficient.
- Table 4-3 Interpreting logistical regression coefficients
Note:
1 There is a third case for a categorical variable Xj which is not discussed here.
