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The objective of this paper is to explore the issue of whether the numbers of regimes and variables in aggregate time series are similar to those in individual time series. Equal and value weighted methods of aggregation are considered. A Monte Carlo simulation is carried out with different settings to investigate possible sources of changes those could affect the characteristic of aggregate time series. The results show that the numbers of regimes and variables in aggregate time series is a function of those of individual time series, regardless of the aggregation method. When combining two individual time series (e.g., one series has two regimes and one lag, while another time series has three regimes and one lag), for instance, the numbers of regimes and variables in aggregate time series would be two and one, respectively.

We propose a convolution-based approach to the estimation of nonlinear autoregressive processes. The model allows for state-dependent autocorrelation, that is different persistence of the shocks in different phases of the market and dependent innovations, that is drawn from different distributions in different phases of the market.

In this paper, we apply the maximum entropy test developed for a goodness of fit in iid samples by [11] to autoregressive time series models including non-stationary unstable models. Its asymptotic distribution is derived under the null hypothesis. A bootstrap version of the test is also discussed and its performance is evaluated through Monte Carlo simulations. A real data analysis is conducted for illustration.

Recently, various variants of the self-organsing map SOM have been proposed for modelling and predicting time series. However, most of them are based on lattice structure. In this paper, a hybrid neural model combining neural gas NG and mixture autoregressive models is developed for forecasting foreign exchange FX rates. It takes advantage of some NG features i.e. neighbourhood rankings and incorporates mixture autoregressive models, for effectively modelling and forecasting non-stationary and nonlinear time series. Experiments on FX rates are presented and the results show that the proposed model performs significantly better than other methods, in terms of normalised root-mean-squared-error and correct trend prediction percentage.

The objective of this paper is to extensively discuss the theory behind the multivariate autoregressive prediction technique used elsewhere for forecasting Universal Time (UT1-UTC) and to characterise its performance depending on input geodetic and geophysical data. This method uses the bivariate time series comprising length-of-day and the axial component of atmospheric angular momentum data and needs to be combined with a least-squares extrapolation of a polynomial-harmonic model. Two daily length-of-day time series, i.e. EOPC04 and EOPC04_05 spanning the time interval from 04.01.1962 to 02.05.2007, are utilised. These time series are corrected for tidal effects following the IERS Conventions model. The data on the axial component of atmospheric angular momentum are processed to gain the 1-day sampling interval and cover the time span listed above. The superior performance of the multivariate autoregressive prediction in comparison to autoregressive forecasting is noticed, in particular during El Nino and La Nina events. However, the accuracy of the multivariate predictions depends on a particular solution of input length-of-day time series. Indeed, for EOPC04-based analysis the multivariate autoregressive predictions are more accurate than for EOPC04_05-based one. This finding can be interpreted as the meaningful influence of smoothing on forecasting performance.

In this chapter, we discuss dynamic models for discrete-valued data and quote processes. As illustrated in Chap. 4, the time series of the number of events in a given time interval yields a counting process and provides an alternative way to characterize the underlying point process. Section 13.1 presents a class of univariate autoregressive models for count data based on dynamic parameterizations of the conditional mean function in a Poisson distribution. Moreover, we discuss extensions thereof, such as the Negative Binomial distribution and Double Poisson distribution.

Agent's behaviour can be described as time series of agent's parameters and actions.The comparative analysis shows AR and ARMA methods are preferable in low volatility of the environment while GARCH methods have better predictive characteristics in case of heteroskedasticity.

An autoregressive model is defined for fuzzy random variables under the concept of Frechet variance and covariance as well as Gaussian fuzzy random variable. In some special case, by using the Hukuhara difference between fuzzy sets, the conditions for stationary solution of a p-order autoregressive process ( AR(p)) are extended to the case of fuzzy data in the manner of conventional stochastic setting.

The self-organizing mixture autoregressive (SOMAR) model regards a time series as a mixture of regressive processes. A self-organizing algorithm is used with the LMS algorithm to learn the parameters of these regressive models. The self-organizing map is used to simplify the mixture as a winner-take-all selection of local models, combined with an autocorrelation coefficient based measure as the similarity measure for identifying correct local models. The SOMAR has been shown previously being able to uncover underlying autoregressive processes from a mixture. This paper proposes a generalized SOMAR that fully considers the mixing mechanism and individual model variances that make modeling and prediction more accurate for non-stationary time series. Experiments on both benchmark and financial time series are presented. The results demonstrate the superiority of the proposed method over other time-series modeling techniques on a range of performance measures.

The mixture autoregressive (MAR) model regards a time series as a mixture of linear regressive processes. A self-organizing algorithm has been used together with the LMS algorithm for learning the parameters of the MAR model. The self-organizing map has been used to simplify the mixture as a winner-takes-all selection of local models, combined with an autocorrelation coefficient based measure as the similarity measure for identifying correct local models and has been shown previously being able to uncover underlying autoregressive processes from a mixture. In this paper the self-organizing network is further generalized so that it fully considers the mixing mechanism and individual model variances in modeling and prediction of time series. Experiments on both benchmark time series and several financial time series are presented. The results demonstrate the superiority of the proposed method over other time-series modeling techniques on a range of performance measures including mean-square-error, prediction rate and accumulated profits.

We derive a vector autoregressive (VAR) representation from the dynamic dividend discount model to predict stock returns. This valuation approach with time-varying expected returns is augmented with macroeconomic variables that should explain time variation in expected returns and cash flows. The VAR is estimated by a Bayesian approach to reduce some of the statistical problems of earlier studies. This model is applied to forecasting the returns of a portfolio of large German firms. While the absolute forecasting performance of the Bayesian vector-autoregressive model (BVAR) is not significantly different from a naive no-change forecast, the predictions of the BVAR are better than alternative time-series models. When including past stock returns instead of macroeconomic variables, the forecasting performance becomes superior relative to the naive no-change forecast especially over longer horizons.

In the past few years, various variants of the self-organising map (SOM) have been proposed to extend its ability for modelling time-series or temporal sequence. Most of them, however, have little connection to, or are over-simplified, autoregressive (AR) models. In this paper, a new extension termed, self-organising mixture autoregressive (SOMAR) network is proposed to topologically cluster time-series segments into underlying generating AR models. It uses autocorrelation values as the similarity measure between the model and the time-series segments. Such networks can be used for modelling nonstationary time-series. Experiments on predicting artificial time-series (Mackey-Glass) and real-world data (foreign exchange rates) are presented and results show that the proposed SOMAR network is a viable and superior to other SOM-based approaches.

The previous chapter presented a statistical approach to analyse the relations between time series: starting with univariate models, we asked for relations that might exist between two time series. Subsequently, the approach was extended to situations with more than two time series. Such a procedure where models are developed bottom up to describe relations is hardly compatible with the economic approach of theorising where – at least in principle – all relevant variables of a system are treated jointly. For example, starting out from the general equilibrium theory as the core of economic theory, all quantities and prices in a market are simultaneously determined. This implies that, apart from the starting conditions, everything depends on everything, i.e. there are only endogenous variables. For example, if we consider a single market, supply and demand functions simultaneously determine the equilibrium quantity and price.

All models discussed so far use the conditional expectation to describe the mean development of one or more time series. The optimal forecast, in the sense that the variance of the forecast errors will be minimised, is given by the conditional mean of the underlying model. Here, it is assumed that the residuals are not only uncorrelated but also homoscedastic, i.e. that the unexplained fluctuations have no dependencies in the second moments. However, BENOIT MANDELBROT (1963) already showed that financial market data have more outliers than would be compatible with the (usually assumed) normal distribution and that there are ‘volatility clusters’: small (large) shocks are again followed by small (large) shocks. This may lead to ‘leptokurtic distributions‘, which – as compared to a normal distribution – exhibit more mass at the centre and at the tails of the distribution. This results in ‘excess kurtosis’, i.e. the values of the kurtosis are above three.

By reference to a preliminary analysis of the Australian blowfly data, the paper addresses the questions of ‘when’, ‘how’ and ‘what’ in non-linear time series modelling. Intercourse between population dynamics and statistical analysis is emphasised throughout.

This chapter deals with dynamic models for financial duration processes. As already discussed in Chapter 2, duration approaches are the most common way to model point processes since they are easy to estimate and allow for straightforward computations of forecasts. In Section 5.1, we discuss autoregressive models for log durations as a natural starting point. In Section 5.2, we present the basic form of the autoregressive conditional duration (ACD) model proposed by Engle and Russell (1997, 1998). Because it is the most common type of autoregressive duration model and is extensively considered in recent econometrics literature, we discuss the theoretical properties and estimation issues in more detail. Section 5.3 deals with extensions of the basic ACD model. In this section, we discuss generalizations with respect to the functional form of the basic specification. Section 5.4 is devoted to specification tests for the ACD model. Here, we focus on (integrated) conditional moment tests as a valuable framework to test the conditional mean restriction implied by the ACD model. Finally, in Section 5.5 we illustrate several applications of the ACD model. The first application deals with the evaluation of different ACD specifications based on trade and price durations by using the testing framework considered in Section 5.4. A further objective is to empirically test the market microstructure hypotheses derived in Chapter 3. Moreover, we apply the ACD model to quantify illiquidity risks on the basis of excess volume durations.

The linear models presented so far in cross-section data and in time series assume a constant variance and covariance function, that is to say, the homoscedasticity is assumed.

This first chapter is devoted to a general introduction into the Markov-switching vector autoregressive (MS-VAR) time series model. In Section 1.2 we present the fundamental assumptions constituting this class of models. The discussion of the two components of MS-VAR processes will clarify their on time invariant vector auto-regressive and Markov-chain models. Some basic stochastic properties of MS-VAR processes are presented in Section 1.3. Finally, MS-VAR models are compared to alternative non-normal and non-linear time series models proposed in the literature. As most non-linear models have been developed for univariate time series, this discussion is restricted to this case. However, generalizations to the vector case are also considered.