Evaluating financial time series models for irregularly spaced data: A spectral density approach

Engle and Russellʹs autoregressive conditional duration (ACD) models have been proven successful in modelling financial data that arrive at irregular intervals. In practice, evaluating these models represents a crucial step. The spectral density is widely used in engineering and applied mathematics. Here, we advocate its use when testing for the so-called ACD effects, and for evaluating the adequacy of ACD models. Two classes of test statistics for duration clustering and one class of test statistics for the adequacy of ACD models are proposed. We adapt Hongʹs [Consistent testing for serial correlation of unknown form. Econometrica 1996;64:837–64; One-sided testing for conditional heteroskedasticity in time series models. Journal of Time Series Analysis 1997;18:253–77] approach in the context of evaluating ACD models. In particular, we justify rigorously the asymptotic distributions, which are all standard normal, of the test statistics in the ACD framework. When testing for ACD effects, the second class of test statistics exploits the one-sided nature of the alternative hypothesis and we discuss in which circumstances these test statistics should be more powerful. Using a particular kernel function, the classes based on integrated measures provide generalized versions of the classical Box–Pierce/Ljung–Box test statistics, which are popular procedures among practitioners. However, we obtain more powerful test procedures in many situations, using nonuniform kernels. Important aspects of the paper are a simulation study illustrating the merits of the proposed procedures in the ACD context, and applications with financial data.