Checking linearity for a multivariate regression model

A linearity test for a multivariate regression model is introduced and analyzed. The proposed linearity test is based on empirical processes of the regressors marked by the residuals. Taking into account some properties of the linear combinations, it can be used a linear combination of the test statistics referred to each model or a test statistic of a model in which the response is a linear combination of the responses. Since large samples are required to obtain suitable asymptotic results a bootstrap approach is investigated. Furthermore, in order to illustrate how the proposed test works in practice, some simulation and real-case studies are given.