Testing hypotheses about interaction terms in nonlinear models
Introduction
A widely discussed contribution to econometric practice by Ai and Norton (2003) has proposed an approach to analyzing interaction effects in nonlinear single index models. The main result applies to nonlinear models such as
The authors argue that the common computation of the partial effect of the interaction term, γ12 = β12F′(.) provides no information about the interaction effect in the model Δ12 = ∂2F(.)/∂x1∂x2. They provide results for examining the magnitude and statistical significance of estimates of Δ12. This note argues that Δ12 is also difficult to interpret in terms of the relationships among the variables in the model. The difficulty is a missing element in the measure of partial effects — the ‘unit change’ in the relevant variable may itself be unreasonable. An example given below illustrates the point. We argue that graphical devices can be much more informative than the test statistics suggested by the authors.
As a corollary to this argument, we suggest that the common practice of testing hypotheses about partial effects is less informative than one might hope, and could usefully be omitted from empirical analyses. The paper proceeds to a summary of the Ai and Norton (2003) results in Section 2, some discussion of the results in Section 3 and an application in Section 4. Conclusions are drawn in Section 5.
Section snippets
Estimation and inference for interaction effects
Consider the model in Eq. (1), where F(.) is a nonlinear conditional mean function such as the normal or logistic cdf in a binary choice model, x1 and x2 are variables of interest, either or both of which may be binary or continuous, and z is a related variable or set of variables, including the constant term. For convenience, we will specialize the discussion to the a probit model,where Φ(A) is the standard normal cdf. The results will
Inference about the interaction effect
Asymptotic standard errors for the partial and interaction effects in any of these cases may be computed using the delta method, as suggested by the authors in their Eqs. (4), (5a)–(5c). They argue that inference about interaction effects should be based on Eqs. (5a), (5b), and (5c), not β12Φ′(A). In an application, they demonstrate with plots of observation specific results from Eq. (5a) and associated t ratios against the predicted probabilities for models without, then with second order
Graphical analysis of partial effects
Riphahn, Wambach and Million (RWM) (2003) constructed count data models for physician and hospital visits by individuals in the German Socioeconomic Panel (GSOEP). The data are an unbalanced panel of 7293 families, with group sizes ranging from one to seven, for a total of 27,326 family-year observations.2 We fit pooled probit models for
Conclusions
The preceding does not fault Ai and Norton's (2003) suggested calculations. Rather, we argue that the process of statistical testing about partial effects, and interaction terms in particular, produces generally uninformative and sometimes contradictory and misleading results. The mechanical reliance on statistical measures of significance obscures the economic, numerical content of the estimated model. We conclude, on the basis of the preceding, in the words of the authors, that “to improve
References (3)
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Interaction terms in logit and probit models
Economics Letters
(2003)
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