On Diagnostic Checking Autoregressive Conditional Duration Models with Wavelet-Based Spectral Density Estimators

Abstract

There has been an increasing interest recently in the analysis of financial data that arrives at irregular intervals. An important class of models is the autoregressive Conditional Duration (ACD) model introduced by Engle and Russell (Econometrica 66:1127–1162, 1998, [22]) and its various generalizations. These models have been used to describe duration clustering for financial data such as the arrival times of trades and price changes. However, relatively few evaluation procedures for the adequacy of ACD models are currently available in the literature. Given its simplicity, a commonly used diagnostic test is the Box-Pierce/Ljung-Box statistic adapted to the estimated standardized residuals of ACD models, but its asymptotic distribution is not the standard one due to parameter estimation uncertainty. In this paper we propose a test for duration clustering and a test for the adequacy of ACD models using wavelet methods. The first test exploits the one-sided nature of duration clustering. An ACD process is positively autocorrelated at all lags, resulting in a spectral mode at frequency zero. In particular, it has a spectral peak at zero when duration clustering is persistent or when duration clustering is small at each individual lag but carries over a long distributional lag. As a joint time-frequency decomposition method, wavelets can effectively capture spectral peaks and thus are expected to be powerful. Our second test checks the adequacy of an ACD model by using a wavelet-based spectral density of the estimated standardized residuals over the whole frequency. Unlike the Box-Pierce/Ljung-Box tests, the proposed diagnostic test has a convenient asymptotic “nuisance parameter-free” property—parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistic. Moreover, it can check a wide range of alternatives and is powerful when the spectrum of the standardized duration residuals is nonsmooth, as can arise from neglected persistent duration clustering, seasonality, calender effects and business cycles. For each of the two new tests, we propose and justify a suitable data-driven method to choose the finest scale—the smoothing parameter in wavelet estimation. This makes the methods fully operational in practice. We present a simulation study, illustrating the merits of the wavelet-based procedures. An application with tick-by-tick trading data of Alcoa stock is presented.