Specification tests for nonlinear dynamic models

We propose a new adequacy test and a graphical evaluation tool for nonlinear dynamic models. The proposed techniques can be applied in any set‐up where parametric conditional distribution of the data is specified and, in particular, to models involving conditional volatility, conditional higher moments, conditional quantiles, asymmetry, Value at Risk models, duration models, diffusion models, etc. Compared to other tests, the new test properly controls the nonlinear dynamic behaviour in conditional distribution and does not rely on smoothing techniques that require a choice of several tuning parameters. The test is based on a new kind of multivariate empirical process of contemporaneous and lagged probability integral transforms. We establish weak convergence of the process under parameter uncertainty and local alternatives. We justify a parametric bootstrap approximation that accounts for parameter estimation effects often ignored in practice. Monte Carlo experiments show that the test has good finite‐sample size and power properties. Using the new test and graphical tools, we check the adequacy of various popular heteroscedastic models for stock exchange index data.