Nonparametric Test for Logistic Regression with Application to Italian Enterprises’ Propensity for Innovation

In this work, a nonparametric method is proposed to jointly test the significance of the regression coefficient estimates in a logistic regression model and identify which explanatory variables are effective in predicting the binary response. The motivating example is related to the factors affecting the propensity of Italian Small Medium Enterprises (SMEs) to innovate. The explanatory variables of the model represent firms’ characteristics, such as size and age, and the possible effect of the sector of economic activity is taken into account by including a set of binary variables as control factors. The dependent variable indicates whether a company, in the period under study, introduced at least one product or process innovation. Therefore, it is also dichotomous, and the logistic regression model is appropriate for representing the relationship between explanatory variables and dependent variable. Specifically, the logit transformation of the firm’s propensity to innovate, i.e., the probability that a company randomly chosen from the population of Italian SMEs has introduced an innovation or, equivalently, the proportion of innovative companies among the Italian SMEs, is expressed as a linear function of the predictors (explanatory and control variables). The proposed test is based on the permutation approach and satisfies important statistical properties, proved in a simulation study. The test is more flexible and robust than the classic parametric approach, and is preferable to typical stepwise regression procedures for the selection of a parsimonious and effective model.