In this paper the cumulated sum of squares residuals is used to construct a user-friendly misspecification test for non-linear regressions with a special emphasis on the case of non-stationary regressors. It is shown that, under the null hypothesis of correct specification, the test statistic converges to the well-known supremum of the absolute value of a Brownian Bridge. In doing so, we provide a set of (high and medium level) sufficient conditions under which the test statistic has the Brownian Bridge distribution. In this sense, we construct a general framework that allows to analyze the behavior of the cumulated sum of squares test in a wide range of models that are used in practice.

The finite sample performance of the test, in terms of size and power, is investigated through several Monte Carlo experiments. The case of non-linear models with non-stationary regressors is considered. From these simulations it is seen that the test performs quite well when compared to other existing procedures in the literature. An interesting feature of our proposal is its comparative simplicity. Joint with Bent Nielsen.