Statistical inference for autoregressive conditional duration models

The class of nonlinear time series models known as autoregressive conditional duration [ACD] models plays a central role for modeling durations, also known as waiting-times. These waiting-times/durations are positive random variables which are defined in relation to an irregularly observed time series of events occurring at random points in time. For example, the waiting time for the change in the share-price of an asset to exceed a certain threshold, or the duration between consecutive financial transactions such as share trading. This thesis focusses on the ACD family of models for such durations or waiting times. The ACD model describes a duration process {Zi } by Zi = Ψiεi , where Ψi is the expected duration conditional on the past information, and εi is an independent and identically distributed [iid] positive error term with E(εi ) = 1. An ACD model is characterized by two specifications: (i) the specification of the conditional mean functionΨi , and (ii) the specification of the distribution of the error term εi . A number of parametric models have been proposed for (i) and (ii). A large proportion of these models have complicated probabilistic structures. As a consequence, assessing the goodness-of-fit of such models is a non-trivial task. However, the methodology for inference in ACD models is still in the early stages of development, and hence the literature on testing for the goodness or lack of fit of such parametric models is relatively scant. Thus, there is a need for developing new methodologies for testing parametric specifications for the ACD class of models. This thesis contributes to this broad area of statistical inference. Our results and methodology are presented in detail in Chapters 3, 4 and 5 of this thesis. The main objectives and contributions of this thesis can be summarized as follows. (a) Martingale transformation techniques have been successfully adapted in the literature to develop asymptotically distribution free specification tests for regression and autoregressive models. We extend this transformation technique to multiplicative models in Chapter 3, in order to develop an asymptotically distribution free test for the specification of the conditional mean of an ACD model. (b) In Chapter 4 we introduce a new parametric form which nests all linear specifications of the conditional mean function of an ACD model, and develop a class of tests for checking its goodness-of-fit. In the process of deriving the asymptotic distributions of the proposed test statistics, we show that a quasi maximum likelihood estimator [QMLE] is root-n consistent and asymptotically normal for the proposed family of models. Further, a residual-based bootstrap procedure is proposed for computing the critical values of the test statistics, and its asymptotic validity is established. (c) Finally, in Chapter 5 a new class of tests is developed for fitting a parametric form for the error distribution of an ACD model. The critical values of the test statistics are obtained via a parametric bootstrap algorithm and its asymptotic validity is established. The tests proposed in this thesis were evaluated via simulation studies, comparing them with some of the best available tests from the existing literature. In these simulations, the tests proposed in this thesis exhibited a good overall performance. In each chapter, the proposed testing procedures are illustrated using empirical examples.