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Modelling and forecasting interval-valued time series (ITS) have received increasing attention in statistics and econometrics. An interval-valued observation contains more information than a point-valued observation in the same time period. The previous literature has mainly considered modelling and forecasting a univariate ITS. However, few works attempt to model a vector process of ITS. In this paper, we propose an interval-valued vector autoregressive moving average (IVARMA) model to capture the cross-dependence dynamics within an ITS vector system. A minimum-distance estimation method is developed to estimate the parameters of an IVARMA model, and consistency, asymptotic normality and asymptotic efficiency of the proposed estimator are established. A two-stage minimum-distance estimator is shown to be asymptotically most efficient among the class of minimum-distance estimators. Simulation studies show that the two-stage estimator indeed outperforms other minimum-distance estimators for various data-generating processes considered.

Jeliazkov and Poirier (2008) analyze the daily incidence of violence during the Second Intifada in a statistical way using an analytical Bayesian implementation of a second-order discrete Markov process. We tackle the same data and modeling problem from our perspective as cognitive scientists. First, we propose a psychological model of violence, based on a latent psychological construct we call “build up” that controls the retaliatory and repetitive violent behavior by both sides in the conflict. Build up is based on a social memory of recent violence and generates the probability and intensity of current violence. Our psychological model is implemented as a generative probabilistic graphical model, which allows for fully Bayesian inference using computational methods. We show that our model is both descriptively adequate, based on posterior predictive checks, and has good predictive performance. We then present a series of results that show how inferences based on the model can provide insight into the nature of the conflict. These inferences consider the base rates of violence in different periods of the Second Intifada, the nature of the social memory for recent violence, and the way repetitive versus retaliatory violent behavior affects each side in the conflict. Finally, we discuss possible extensions of our model and draw conclusions about the potential theoretical and methodological advantages of treating societal conflict as a cognitive modeling problem.

The topic of volatility measurement and estimation is central to financial and more generally time-series econometrics. In this chapter, we begin by surveying models of volatility, both discrete and continuous, and then we summarize some selected empirical findings from the literature. In particular, in the first sections of this chapter, we discuss important developments in volatility models, with focus on time-varying and stochastic volatility as well as nonparametric volatility estimation. The models discussed share the common feature that volatilities are unobserved and belong to the class of missing variables. We then provide empirical evidence on “small” and “large” jumps from the perspective of their contribution to overall realized variation, using high-frequency price return data on 25 stocks in the DOW 30. Our “small” and “large” jump variations are constructed at three truncation levels, using extant methodology of Barndorff-Nielsen and Shephard (2006), Andersen, Bollerslev, and Diebold (2007), and Aït-Sahalia and Jacod (2009a, 2009b, 2009c). Evidence of jumps is found in around 22.8% of the days during the 1993–2000 period, much higher than the corresponding figure of 9.4% during the 2001–2008 period. Although the overall role of jumps is lessening, the role of large jumps has not decreased, and indeed, the relative role of large jumps, as a proportion of overall jumps, has actually increased in the 2000s.

In this study, the performance of ordinal GA-based trading strategies is evaluated under six classes of time series model, namely, the linear ARMA model, the bilinear model, the ARCH model, the GARCH model, the threshold model and the chaotic model. The performance criteria employed are the winning probability, accumulated returns, Sharpe ratio and luck coefficient. Asymptotic test statistics for these criteria are derived. The hypothesis as to the superiority of GA over a benchmark, say, buy-and-hold, can then be tested using Monte Carlo simulation. From this rigorously-established evaluation process, we find that simple genetic algorithms can work very well in linear stochastic environments, and that they also work very well in nonlinear deterministic (chaotic) environments. However, they may perform much worse in pure nonlinear stochastic cases. These results shed light on the superior performance of GA when it is applied to the two tick-by-tick time series of foreign exchange rates: EUR/USD and USD/JPY.