Forecasting VARMA processes using VAR models and subspace-based state space models

VAR modelling is a frequent technique in econometrics for linear processes. VAR modelling offers some desirable features such as relatively simple procedures for model specification (order selection) and the possibility of obtaining quick non-iterative maximum likelihood estimates of the system parameters. However, if the process under study follows a finite-order VARMA structure, it cannot be equivalently represented by any finite-order VAR model. On the other hand, a finite-order state space model can represent a finite-order VARMA process exactly, and, for state-space modelling, subspace algorithms allow for quick and non-iterative estimates of the system parameters, as well as for simple specification procedures. Given the previous facts, we check in this paper whether subspace-based state space models provide better forecasts than VAR models when working with VARMA data generating processes. In a simulation study we generate samples from different VARMA data generating processes, obtain VAR-based and state-space-based models for each generating process and compare the predictive power of the obtained models. Different specification and estimation algorithms are considered; in particular, within the subspace family, the CCA (Canonical Correlation Analysis) algorithm is the selected option to obtain state-space models. Our results indicate that when the MA parameter of an ARMA process is close to 1, the CCA state space models are likely to provide better forecasts than the AR models. We also conduct a practical comparison (for two cointegrated economic time series) of the predictive power of Johansen restricted-VAR (VEC) models with the predictive power of state space models obtained by the CCA subspace algorithm, including a density forecasting analysis.

[1]  M. Aoki,et al.  State space modeling of multiple time series , 1991 .

[2]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[3]  G. Reinsel,et al.  Multivariate Reduced-Rank Regression: Theory and Applications , 1998 .

[4]  Jaime Terceiro Lomba.  Estimation of Dynamic Econometric Models with Errors in Variables , 1990 .

[5]  Lennart Ljung,et al.  System identification toolbox for use with MATLAB , 1988 .

[6]  Anthony S. Tay,et al.  Evaluating Density Forecasts with Applications to Financial Risk Management , 1998 .

[7]  Mats Viberg,et al.  Subspace-based methods for the identification of linear time-invariant systems , 1995, Autom..

[8]  Joseph E. Cavanaugh,et al.  An improved Akaike information criterion for state-space model selection , 2006, Comput. Stat. Data Anal..

[9]  Dietmar Bauer,et al.  Comparing the CCA Subspace Method to Pseudo Maximum Likelihood Methods in the case of No Exogenous Inputs , 2005 .

[10]  Kenneth F. Wallis,et al.  Density Forecasting: A Survey , 2000 .

[11]  J. Stock,et al.  Testing for Common Trends , 1988 .

[12]  David A. Heiser,et al.  On the accuracy of statistical procedures in Microsoft Excel 2007 , 1999, Comput. Stat. Data Anal..

[13]  J. Durbin,et al.  Techniques for Testing the Constancy of Regression Relationships Over Time , 1975 .

[14]  W. Larimore.  System Identification, Reduced-Order Filtering and Modeling via Canonical Variate Analysis , 1983, 1983 American Control Conference.

[15]  A Comparison of Johansen'S, Bierens' and the Subspace Algorithm Method for Cointegration Analysis , 2004 .

[16]  D. S. G. Pollock,et al.  A handbook of time-series analysis, signal processing and dynamics , 1999 .

[17]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[18]  Dietmar Bauer,et al.  Asymptotic properties of subspace estimators , 2005, Autom..

[19]  Martin Wagner.  VAR Cointegration in VARMA Models , 1999 .

[20]  Gregory C. Reinsel,et al.  Estimation and Testing for Unit Roots in a Partially Nonstationary Vector Autoregressive Moving Average Model , 1995 .

[21]  A. Stuart,et al.  Non-Parametric Statistics for the Behavioral Sciences. , 1957 .

[22]  J. Angus.  Forecasting, Structural Time Series and the Kalman Filter , 1992 .

[23]  Leo Kn usel.  On the accuracy of statistical distributions in Microsoft Excel 2003 , 1998 .

[24]  Pentti Saikkonen,et al.  Estimation and Testing of Cointegrated Systems by an Autoregressive Approximation , 1992, Econometric Theory.

[25]  D. S. G. Pollock,et al.  Recursive Estimation in Econometrics , 2002, Comput. Stat. Data Anal..

[26]  M. Wagner,et al.  The Performance of Subspace Algorithm Cointegration Analysis: A Simulation Study , 2003 .

[27]  W. Larimore.  Identification of Colinear and Cointegrated Multivariable Systems Using Canonical Variate Analysis , 2000 .

[28]  Lennart Ljung,et al.  Some facts about the choice of the weighting matrices in Larimore type of subspace algorithms , 2002, Autom..

[29]  B. D. McCullough,et al.  On the accuracy of statistical procedures in Microsoft Excel 2003 , 1999 .

[30]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[31]  Anthony S. Tay,et al.  Multivariate Density Forecast Evaluation and Calibration In Financial Risk Management: High-Frequency Returns on Foreign Exchange , 1999, Review of Economics and Statistics.

[32]  Michael P. Clements,et al.  Evaluating The Forecast of Densities of Linear and Non-Linear Models: Applications to Output Growth and Unemployment , 2000 .

[33]  E. J. Godolphin,et al.  Decomposition of time series models in state-space form , 2006, Comput. Stat. Data Anal..

[34]  M. Wagner,et al.  Estimating cointegrated systems using subspace algorithms , 2002 .

[35]  S. Johansen.  STATISTICAL ANALYSIS OF COINTEGRATION VECTORS , 1988 .

[36]  S. Siegel,et al.  Nonparametric Statistics for the Behavioral Sciences , 2022, The SAGE Encyclopedia of Research Design.

[37]  Heungsun Hwang,et al.  An Improved Method for Generalized Constrained Canonical Correlation Analysis , 2002, Comput. Stat. Data Anal..

[38]  Leo Knüsel,et al.  On the accuracy of statistical distributions in Microsoft Excel 97 , 1998 .

[39]  G. C. Tiao,et al.  An introduction to multiple time series analysis. , 1993, Medical care.

[40]  Max Donath,et al.  American Control Conference , 1993 .

[41]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .