We show that the order of integration of a vector autoregressive process is equal to the difference between the multiplicity of the unit root in the characteristic equation and the multiplicity of the unit root in the adjoint matrix polynomial. The equivalence with the standard I(1) and I(2) conditions (Johansen, 1996) is proved and polynomial cointegration discussed in the general setup.

This paper contains a nonlinear, nonstationary autoregressive model whose intercept changes deterministically over time. The intercept is a flexible function of time, and its construction bears some resemblance to neural network models. A modelling technique, modified from one for single hidden-layer neural network models, is developed for specification and estimation of the model. Its performa...

The structure of non-Gaussian autoregressive schemes is described. Asymptotically efficient methods for the estimation of the coefficients of the models are described under appropriate conditions, some of which relate to smoothness and positivity of the density function f of the independent random variables generating the process. The principal interest is in nonminimum phase models.

We are grateful to Bernard Hanzon for helpful comments. The research for this paper was carried out within Sonderforschungsbereich 373 at the Humboldt University Berlin and was printed using funds made available by the Deutsche Forschungsgemeinschaft.

We present an elaboration of the usual binomial AR(1) process on {0, 1, . . . , N} that allows the thinning probabilities to depend on the current state n only through the “density” n/N , a natural assumption in many real contexts. We derive some basic properties of the model and explore approaches to parameter estimation. Some special cases are considered that allow for overand underdispersion...

We consider autoregressive neural network (AR-NN) processes driven by additive noise and demonstrate that the characteristic roots of the shortcuts-the standard conditions from linear time-series analysis-determine the stochastic behavior of the overall AR-NN process. If all the characteristic roots are outside the unit circle, then the process is ergodic and stationary. If at least one charact...

Let observations y1, · · · , yn be generated from a first-order autoregressive (AR) model with positive errors. In both the stationary and unit root cases, we derive moment bounds and limiting distributions of an extreme value estimator, ρ̂n, of the AR coefficient. These results enable us to provide asymptotic expressions for the mean squared error (MSE) of ρ̂n and the mean squared prediction err...

The problem of monitoring the mean (or other aspect) of an evolving time series for deviations from some ideal state arises frequently in such fields as industrial quality control, clinical trials and air (or other) pollution monitoring. This so-called sequential detection problem has been widely studied in the case where the time series consists of independent random variables, a situation whi...

2 Moving average models Definition. The moving average model of order q, or MA(q), is defined to be Xt = t + θ1 t−1 + θ2 t−2 + · · ·+ θq t−q, where t i.i.d. ∼ N(0, σ). Remarks: 1. Without loss of generality, we assume the mean of the process to be zero. 2. Here θ1, . . . , θq (θq 6= 0) are the parameters of the model. 3. Sometimes it suffices to assume that t ∼WN(0, σ). Here we assume normality...

A binomial-type operator on a stationary Gaussian process is introduced in order to model long memory in the spatial context. Consistent estimators of model parameters are demonstrated. In particular , it is shown thatˆdN − d = OP ((Log N) 3 N), where d = (d1, d2) denotes the long memory parameter.