Interaction terms in logit and probit models
Introduction
Applied economists often estimate interaction terms to infer how the effect of one independent variable on the dependent variable depends on the magnitude of another independent variable. Difference-in-difference models, which measure the difference in outcome over time for the treatment group compared to the difference in outcome over time for the control group, are examples of models with interaction terms. Although interaction terms are used widely in applied econometrics, and the correct way to interpret them is known by many econometricians and statisticians, most applied researchers misinterpret the coefficient of the interaction term in nonlinear models. A review of the 13 economics journals listed on JSTOR found 72 articles published between 1980 and 1999 that used interaction terms in nonlinear models. None of the studies interpreted the coefficient on the interaction term correctly. The recent paper by DeLeire (2000) is a welcome exception.
In linear models the interpretation of the coefficient of the interaction between two variables is straightforward. Let the continuous dependent variable y depend on two independent variables x1 and x2, their interaction, a vector of additional independent variables including the constant term independent of x1 and x2, and βs are unknown parameters. If x1 and x2 are continuous, the interaction effect of the independent variables x1 and x2 is the cross-derivative of the expected value of y
If x1 and x2 are dichotomous, then the interaction effect of a change in both x1 and x2 from zero to one is found by taking discrete differencesThe statistical significance of the interaction effect can be tested with a single t-test on the coefficient β12.
The intuition from linear models, however, does not extend to nonlinear models. To illustrate, consider a probit model similar to the previous example, except that the dependent variable y is a dummy variable. The conditional mean of the dependent variable iswhere Φ is the standard normal cumulative distribution. Suppose x1 and x2 are continuous. The interaction effect is the cross derivative of the expected value of y
However, most applied economists instead compute the marginal effect of the interaction term, which is . Perhaps this is because statistical software packages, such as Stata® 7, compute the marginal effect for any explanatory variable. However, Eq. (2) shows clearly that the interaction effect is not equal to .
There are four important implications of Eq. (2) for nonlinear models. Firstly, the interaction effect could be nonzero, even if β12=0. For the probit model with β12=0, the interaction effect isSecondly, the statistical significance of the interaction effect cannot be tested with a simple t-test on the coefficient of the interaction term β12. Thirdly, the interaction effect is conditional on the independent variables, unlike the interaction effect in linear models. (It is well known that the marginal effect of a single uninteracted variable in a nonlinear model is conditional on the independent variables.) Fourthly, the interaction effect may have different signs for different values of covariates. Therefore, the sign of β12 does not necessarily indicate the sign of the interaction effect.
In order to improve best practice by applied econometricians, we derive the formulas for the magnitude and standard errors of the estimated interaction effect in general nonlinear models. The formulas apply easily to logit, probit, and other nonlinear models. We illustrate our points with an example.
Section snippets
Estimation
We begin by introducing notation for general nonlinear models. Let y denote the raw dependent variable. Let the vector be a k×1 vector of independent variables, so . The expected value of y given x iswhere the function F is known up to β and is twice continuously differentiable. Let Δ denote either the difference or the derivative operator, depending on whether the regressors are discrete or continuous. For example, denotes the derivative if x1 is continuous and
Empirical example
To illustrate our points, we estimated a logit model to predict HMO enrolment as a function of three continuous variables—age, number of activities of daily living (a count from 0 to 6 of the number of basic physical activities a person has trouble performing), and the percent of the county population enrolled in a HMO—and their interactions (Mello et al., 2002). The data are primarily from the 1993–1996 Medicare Current Beneficiary Survey, a longitudinal survey of Medicare eligibles. There are
Conclusion
The interaction effect, which is often the variable of interest in applied econometrics, cannot be evaluated simply by looking at the sign, magnitude, or statistical significance of the coefficient on the interaction term when the model is nonlinear. Instead, the interaction effect requires computing the cross derivative or cross difference. Like the marginal effect of a single variable, the magnitude of the interaction effect depends on all the covariates in the model. In addition, it can have
References (2)
The wage and employment effects of the Americans with Disabilities Act
Journal of Human Resources
(2000)- et al.
Do Medicare HMOs still reduce health services use after controlling for selection bias?
Health Economics
(2002)
Cited by (3708)
Investment motives and performance expectations of impact investors
2024, Journal of Behavioral and Experimental FinanceTrain stations’ impact on housing prices: Direct and indirect effects
2024, Transportation Research Part A: Policy and PracticeCorporate governance reforms and voluntary disclosure: International evidence on management earnings forecasts
2024, Journal of International Accounting, Auditing and TaxationWolves at the door to the unknown: Innovation search and hedge fund activism
2024, Research PolicyMarital status, State policy environment and Foregone healthcare of same-sex families during the COVID-19 period
2024, Social Science Research