ACD models provide an essential step in modelling market events, which can be mod- elled most realistically as an irregularly spaced time series. The same model fitting procedure for inter-transaction durations studied in this chapter can easily be extended to the analysis of time elapsed between any other types of financial events, e.g. the time between transactions corresponding to movements in share prices above or below a certain threshold, or to the accumulation of traded volume above a certain threshold.
0 10 20 30 40 0 10 20 30 40 mixed exponential
fitted residuals (100% range)
observed residuals 0 10 20 30 40 0 10 20 30 40
mixed lognormal−gamma
fitted residuals (100% range)
0 2 4 6 8 10 0 2 4 6 8 10
fitted residuals (99% range)
observed residuals 0 2 4 6 8 10 0 2 4 6 8 10
fitted residuals (99% range)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.10
0.20
0.30
fitted residuals (50% range)
observed residuals 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.10 0.20 0.30
fitted residuals (50% range)
Figure 3.6: Parametric bootstrap version quantile-quantile plots (Darby data)
The extensive study of ACD models since its formal introduction in 1998 has resulted in many competing ACD model specifications. The necessity for model comparison between different ACD models is obvious. However, the traditional hypothesis test- ing approach can only provide limited and ad hoc solutions to the issue of evaluation of ACD models. In this chapter, we have shown how the information-theoretic ap- proach can be applied to evaluate ACD models in an objective and efficient way. Not one p-value has been calculated because of its irrelevance to an information-theoretic approach.
In this chapter, four basic ACD models and two mixed distribution ACD models are specified to fit two real stock transaction data sets, the IBM data from a mature financial market NYSE and the Darby data from an emerging Asian market KLSE. We have applied AIC and TIC to evaluate and compare these specified ACD models. The
model evaluation results using Q-statistic and bootstrap qq-plot match nicely with the AIC and TIC assessment results. A two-step model estimation strategy is proposed to facilitate the model estimation procedure and may have provided a solution for selecting an ACD model with an optimal combination of the autoregressive structure and the probabilistic error term structure.
A primary concern at the model estimation stage with ACD models is the remaining dependence structure within the estimated residuals. This is the major reason for the proposal of a L-B statistic constrained QMLE minimization objective function. In this way, the ACD model parameter estimation is still within a M-estimation framework. Hence, the information-theoretic criteria are applicable for model evaluation.
From the point of view of shedding new light on the trading process or on the market micro-structure, the proposal of a new model, mixed lognormal-gamma ACD(1,1) may not be significant. However, the result that the mixed lognormal-gamma ACD(1,1) model outperforms easily all other specified ACD models in this study, based on AIC and TIC scores, emphasizes the capability and potential usage of the information- theoretic approach in model evaluation. This is something the hypothesis testing ap- proach can never have achieved. Based on the same reason, we decided to only include basic ACD models and two mixed distribution ACD(1,1) models in this study rather than considering some more complicated existing ACD models.
The ACD model evaluation results are mainly achieved by applying the M-estimator version AIC in this study. These results support our conclusion made in Section 2.5.1 about the applicability of AIC procedure for model evaluation in general. Ideally, we would like to be able to compare the M-estimator AIC with GIC directly in model evaluation. This requires to overcome the computation difficulty in estimation of GIC’s trace term which involves twok×k matrices. The key to this problem is to be able to get the reliable numerical estimates of the first and the second derivative functions of the optimization objective function for parameter estimation and this will be a topic of future research work.
Stochastic rainfall models – a
literature review
The following five chapters present the research results of the second topic of this thesis: point process models for fine-scale rainfall time series at a single site. The study described herein further develops the original Bartlett-Lewis Pulse (BLP) model proposed by Cowpertwait et al. (2007) in terms of model structure characterization, fitting procedure, and parameterisation.
For simplicity, Cowpertwait et al. (2007) is referred to as ‘CIO2007’ (stands for the paper written by Cowpertwait, Isham, and Onof in 2007 year) for the rest of this thesis and their fitted BLP model as ‘the original BLP model’. The 60 years (01/01/1945 to 31/12/2004) of 5-minute rainfall series used in CIO2007 for fitting the original BLP model, provided by the New Zealand National Institute of Water and Atmospheric Research (NIWA), was recorded at Kelburn, a site near Wellington, New Zealand. Therefore, we refer this fine-scale rainfall data as ‘the Kelburn data’. For a comparison with the results reported in CIO2007, the Kelburn data is selected for the study of point process rainfall models in this thesis.
4.1
Introduction
Fresh water is an essential resource and rainfall is the primary source of fresh water supply. The process of precipitation is a fundamental component of the hydrological cycle and it is a complex and delicately balanced process (Shaw, 1988). However, there is a need to model the process of precipitation for various applications, e.g. short term or long term forecasting, data reduction and generation of rainfall series, scien- tific understanding, etc. The deterministic approach (primarily based on atmospheric
physics) of modelling rainfall process is generally considered as inappropriate for the following reasons: (1) the system of governing laws is not strictly deterministic or the laws are not completely understood; (2) perfect measurement of the current state of the atmosphere and its environment is impossible (Stern and Coe, 1984, discussion part). Therefore, rainfall is treated as a stochastic process in many applications.
There are a vast range of rainfall models proposed for different application pur- poses in the literature. Following Cox and Isham (1994) and Onof et al. (2000), four broad types of rainfall models may be classified, namely, models of dynamic meteorol- ogy, multi-scaling models, empirical statistical models, and point process models. The multi-scaling models will not be reviewed here for they follow a completely different av- enue in terms of the mathematical representation of rainfall. The multi-scaling models describe the spatial evolution of the rainfall process in a scale-independent fashion, e.g. multifractal cascades (Gupta and Waymire, 1993). For a review of multi-scaling models readers are referred to Foufoula-Georgiou and Krajewski (1995). The meteorological models, in which large systems of simultaneous nonlinear partial differential equations are specified to represent the physical processes, are generally used for weather fore- casting. The meteorological models thus serve a different purpose from those rainfall models of interest here in which the aim is to produce realistic simulation of the ob- served rainfall series. Therefore, this literature review is confined to the empirical statistical models and point process models, i.e. stochastic rainfall models.
Because the precipitation process is essentially a space-time process, stochastic rainfall models can be further broadly divided into two classes: temporal (single site) models and spatial-temporal models.
As early as Le Cam (1961), the space-time rainfall process has been modelled ana- lytically as a random field. In the first part of a series of three articles on the mathe- matical structure of rainfall representation, Waymire and Gupta (1981a) discussed the mathematical structure of rainfall in space and time. Whereas characterizing the highly variable temporal structure of rainfall at a single site is by no means straightforward, the challenge of modelling rainfall in both space and time is much more formidable. It is, therefore, not surprising that not as many spatial-temporal models (as temporal models) are found in the literature. Some examples of spatial-temporal models follow. Rodriguez-Iturbe and Eagleson (1987) investigated the spatial and temporal struc- ture of rainfall from storm events using point process techniques. Through detailed analysis of the space and time structure of a large number of storms observed by a 93-
station network in and around the 154-km2Walnut Gulch catchment in Arizona, USA,
they demonstrated the feasibility of modelling the spatial and temporal structure of rainstorm events using mathematical multidimensional point process techniques. Cox and Isham (1988) developed a relatively simple model, in which storm centres arrive in a homogeneous three dimensional Poisson process (two space and one time). Following each storm origin, and at the same spatial location, cell origins occur in a temporal Bartlett-Lewis process (Section 1.1.1). The aim of this model was to extend the earlier temporal model developed in Rodriguez-Iturbe et al. (1987a) to a fully spatial-temporal process in such a way that the marginal temporal process at a single spatial location of the earlier model is preserved. This model can be used to generate rainfall series over a given area for a given time scale. Partly motivated by Cox and Isham (1988)’s work, Cowpertwait (1995) proposed a generalized spatial-temporal model of rainfall based on the Neyman-Scott process in which more than one type of rain cell can be present in the same storm. By fitting this spatial-temporal model, using two cell types, to rainfall data taken from six sites in the Thames basin, Cowpertwait (1995) has shown that the model is likely to be particularly useful in hydrological studies that use hourly data. Cowpertwait’s (1995) model was implemented into a modelling package called RainSim (Burton et al., 2008). Ultimately, a full continuous space-time rainfall model enables the reproduction of most essential features of the rainfall process (Onof et al., 2000).
In the following review, we will concentrate on temporal models of rainfall modelling at short time scales up to and including daily series, since these relate to the models in this research. These models may be useful in typical hydrological applications that include the design of stormwater sewerage systems for which sub-hourly data are likely of particular importance, and the estimation of the design flood hydrograph, typically based on hourly to daily rainfall (Onof et al., 2000). We focus on the evolution of the Poisson-based point process models: from Poisson white noise model to the recent developments of temporal Poisson cluster process models. The Poisson white noise model is the simplest Poisson-based point process rainfall model, and thus forms a starting point for the review of the Poisson-based rainfall models. One objective of this research is to improve the model performance of the Bartlett-Lewis pulse (BLP) model. This literature review provides the background information about why and how the BLP model was developed and indicates a direction in which the model may be improved.