On goodness of fit for time series regression models

We propose a Bayesian approach to check the goodness of fit for time series regression models. The test statistics is proposed by Smith (1985) based on a sequence of random variables which are independently distributed standard normal if the model is correct. We estimate this sequence of random variables using several methods. The tests of goodness of fit are performed when either the error terms violate the Gaussian assumption, or the order is incorrect, or the model is misspecified. The methodology is illustrated using both a simulation study and three real data sets.