Using Subspace Methods for Estimating ARMA Models for Multivariate Time Series with Conditionally Heteroskedastic Innovations

This paper deals with the estimation of linear dynamic models of the ARMA type for the conditional mean for time series with conditionally heteroskedastic innovation process widely used in modelling financial time series. Estimation is performed using subspace methods which are known to have computational advantages as compared to prediction error methods based on criterion minimization. These advantages are especially strong for high dimensional time series. The subspace methods are shown to provide consistent estimators. Moreover asymptotic equivalence to prediction error estimators in terms of the asymptotic variance is proved. Also order estimation techniques are proposed and analyzed. The estimators are not efficient as they do not model the conditional variance. Nevertheless, they can be used to obtain consistent estimators of the innovations. In a second step these estimated residuals can be used in order to levitate the problem of specifying the variance model in particular in the multi-output case. This is demonstrated in an ARCH setting, where it is proved that the estimated innovations can be used in place of the true innovations for testing in a linear least squares context in order to specify the structure of the ARCH model without changing the asymptotic distribution.Multivariate models, conditional heteroskedasticity, ARMA systems, subspace methods

A Canonical Form for Unit Root Processes in the State Space Framework

In this paper we develop a canonical state space representation for rational stochastic processes containing unit roots with integer integration orders at arbitrary points on the unit circle. It is shown that the state space framework, which is -- in a certain sense made precise in the paper -- equivalent to the ARMA framework, is very suitable for the analysis of unit roots and cointegration issues. The advantages become especially prominent for systems with higher integration orders at the various roots on the unit circle. A unique state space representation is constructed that clearly reveals the integration and cointegration properties. The canonical form given in the paper can be used to construct a parameterization of the class of all rational processes with a given state space unit root structure, which is defined in the papercanonical form; state space representation; unit roots; cointegration

On Polynomial Cointegration in the State Space Framework

This paper deals with polynomial cointegration, i.e. with the phenomenon that linear combinations of a vector valued rational unit root process and lags of the process are of lower integration order than the process itself (for definitions see Section 2). The analysis is performed in the state space representation of rational unit root processes derived in Bauer and Wagner (2003). The state space framework is an equivalent alternative to the ARMA framework. Unit roots are allowed to occur at any point on the unit circle with arbitrary integer integration order. In the paper simple criteria for the existence of non-trivial polynomial cointegrating relationships are given. Trivial cointegrating relationships lead to the reduction of the integration order simply by appropriate differencing. The set of all polynomial cointegrating relationships is determined from simple orthogonality conditions derived directly from the state space representation. These results are important for analyzing the structure of unit root processes and their polynomial cointegrating relationships and also for the parameterization for system sets with given cointegration properties.Unit roots; polynomial cointegration; state space representation

The Performance of Subspace Algorithm Cointegration Analysis: A Simulation Study

This paper presents a simulation study that assesses the finite sample performance of the subspace algorithm cointegration analysis developed in Bauer und Wagner (2002b). The method is formulated in the state space framework, which is equivalent to the VARMA framework, in a sense made precise in the paper. This implies applicability to VARMA processes. The paper proposes and compares six different tests for the cointegrating rank. The simulations investigate four issues: the order estimation, the size performance of the proposed tests, the accuracy of the estimation of the cointegrating space and the forecasting performance. of the state space models estimated by the proposed method. The simulations are performed on a set of trivariate processes with cointegrating ranks ranging from zero to three as well as on processes of output dimension four and cointegrating rank two. We analyze the influence of the sample size on the results as well as the sensitivity of the results with respect to stable poles approaching the unit circle. All results are compared to benchmark results obtained by applying the Johansen procedure on VAR models fitted to the data. The simulations show advantages of subspace algorithm cointegration analysis for the small sample performance of the tests for the cointegrating rank in many cases. However, we find that the accuracy of the subspace algorithm based estimation of the cointegrating space is unsatisfactory for the four-dimensional simulated systems. The forecasting performance is grosso modo comparable to the results obtained by applying the Johansen methodology on VAR approximations, although for very small sample sizes the forecasts based on VAR approximations outperform the subspace forecasts. The appendix provides critical values for the test statisticsState space representation; cointegration; subspace algorithms; simulation study

Asymptotic Properties of Pseudo Maximum Likelihood Estimates for Multiple Frequency I(1) Processes

In this paper we derive (weak) consistency and the asymptotic distribution of pseudo maximum likelihood estimates for multiple frequency I(1) processes. By multiple frequency I(1) processes we denote processes with unit roots at arbitrary points on the unit circle with the integration orders corresponding to these unit roots all equal to 1. The parameters corresponding to the cointegrating spaces at the different unit roots are estimated super-consistently and have a mixture of Brownian motions limiting distribution. All other parameters are asymptotically normally distributed and are estimated at the standard square root of T rate. The problem is formulated in the state space framework, using the canonical form and parameterization introduced by Bauer and Wagner (2002b). Therefore the analysis covers vector ARMA processes and is not restricted to autoregressive processes.state space representation; unit roots; cointegration; pseudo maximum likelihood estimation

Using Subspace Methods for Estimating ARMA Models for Multivariate Time Series with Conditionally Heteroskedastic Innovations

This paper deals with the estimation of linear dynamic models of the ARMA type for the conditional mean for time series with conditionally heteroskedastic innovation process widely used in modelling financial time series. Estimation is performed using subspace methods which are known to have computational advantages as compared to prediction error methods based on criterion minimization. These advantages are especially strong for high dimensional time series. The subspace methods are shown to provide consistent estimators. Moreover asymptotic equivalence to prediction error estimators in terms of the asymptotic variance is proved. Also order estimation techniques are proposed and analyzed. The estimators are not efficient as they do not model the conditional variance. Nevertheless, they can be used to obtain consistent estimators of the innovations. In a second step these estimated residuals can be used in order to levitate the problem of specifying the variance model in particular in the multi-output case. This is demonstrated in an ARCH setting, where it is proved that the estimated innovations can be used in place of the true innovations for testing in a linear least squares context in order to specify the structure of the ARCH model without changing the asymptotic distribution