Nonlinear co-integration is studied for score-driven models, using a new multivariate dynamic conditional score/generalized autoregressive score model. The model is named t-QVARMA (quasi-vector autoregressive moving average model), which is a location model for the multivariate t-distribution. In t-QVARMA, I(0) and co-integrated I(1) components of the dependent variables are included. For t-QVARMA, the conditions of the maximum likelihood estimator and impulse response functions (IRFs) are presented. A limiting special case of t-QVARMA, named Gaussian-QVARMA, is a Gaussian-VARMA specification with I(0) and I(1) components. As an empirical application, the US real gross domestic product growth, US inflation rate, and effective federal funds rate are studied for the period of 1954 Q3 to 2020 Q2. Statistical performance and predictive accuracy of t-QVARMA are superior to those of Gaussian-VAR. Estimates of the short-run IRF, long-run IRF, and total IRF impacts for the US data are reported.

This paper extracts housing boom-bust cycle signals from metropolitan statistical area (MSA)-level housing prices using a Markov-switching dynamic factor model. To mitigate the estimation bias, it utilizes high-frequency housing prices that follow the methodology of the monthly Case–Shiller house price indices. The housing bust phases specified from weekly and daily housing prices precede those based on monthly prices by approximately 2 years. MSAs with top signal-to-noise ratios offer greater marginal contributions to improvements in forecasting housing cycles than MSAs with bottom ratios for all frequencies. The results highlight the importance of indicator quality and provide evidence against “The more, the better” since incorporating more MSA-level housing prices into housing factors does not guarantee more satisfactory housing cycle forecasts.

We perform an analysis of the potential time inhomogeneity in the dependence between multiple financial time series. To this end, we use the framework of copula theory and tackle the question of whether dependencies in such a case can be assumed constant throughout time or rather have to be modeled in a time-inhomogeneous way. We focus on parametric copula models and suitable inference techniques in the context of a special copula-based multivariate time series model. A recent result due to Chan et al. (2009) is used to derive the joint limiting distribution of local maximum-likelihood estimators on overlapping samples. By restricting the overlap to be fixed, we establish the limiting law of the maximum of the estimator series. Based on the limiting distributions, we develop statistical homogeneity tests, and investigate their local power properties. A Monte Carlo simulation study demonstrates that bootstrapped variance estimates are needed in finite samples. Empirical analyses on real-world financial data finally confirm that time-varying parameters are an exception rather than the rule.

We derive asymptotic theory for the extremogram and cross-extremogram of a bivariate GARCH(1,1) process. We show that the tails of the components of a bivariate GARCH(1,1) process may exhibit power-law behavior but, depending on the choice of the parameters, the tail indices of the components may differ. We apply the theory to five-minute return data of stock prices and foreign-exchange rates. We judge the fit of a bivariate GARCH(1,1) model by considering the sample extremogram and cross-extremogram of the residuals. The results are in agreement with the independent and identically distributed hypothesis of the two-dimensional innovations sequence. The cross-extremograms at lag zero have a value significantly distinct from zero. This fact points at some strong extremal dependence of the components of the innovations.

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

In this paper we introduce an exponential continuous-time GARCH(p, q) process. It is defined in such a way that it is a continuous-time extension of the discrete-time EGARCH(p, q) process. We investigate stationarity, mixing, and moment properties of the new model. An instantaneous leverage effect can be shown for the exponential continuous-time GARCH(p, p) model.

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

A time-series consisting of white noise plus Brownian motion sampled at equal intervals of time is exactly orthogonalized by a discrete cosine transform (DCT-II). This paper explores the properties of a version of spectral analysis based on the discrete cosine transform and its use in distinguishing between a stationary time-series and an integrated (unit root) time-series.

A new representation for the characteristic function of the multivariate strictly geo-stable distribution is presented. The representation is appealing from a parametric viewpoint: its parameters have an intuitive probabilistic interpretation; and it is particularly useful for estimating the parameters of the geo-stable distribution.