Autoregressive exogenous, hereafter ARX, models are widely adopted in time series-related domains as they can be regarded as the combination of an autoregressive process and a predictive regression. Within a more complex structure, extant diagnostic checking methods face difficulties in remaining validity in many conditions existing in real applications, such as heteroscedasticity and error correlations exhibited between the ARX model itself and its exogenous processes. For these reasons, we propose a new serial correlation test method based on the profile empirical likelihood. Simulation results, as well as two real data examples, show that our method has a good performance in all mentioned conditions.

In this paper, we investigate the similarity property of the sample projection depth contours. It turns out that parts of the projection depth contours are of the same shape but with different sizes, behaving similarly to contours induced from the Mahalanobis depth. The major application of this investigation is that we may exploit the idea behind the similarity property to extend both the halfspace depth and zonoid depth to versions that do not vanish outside the convex hull of the data cloud, so that we can overcome the so-called ‘outside problem’ of these two depths.