Time Series Forecasting: The Case for the Single Source of Error State Space Approach

The state space approach to modelling univariate time series is now widely used both in theory and in applications. However, the very richness of the framework means that quite different model formulations are possible, even when they purport to describe the same phenomena. In this paper, we examine the single source of error [SSOE] scheme, which has perfectly correlated error components. We then proceed to compare SSOE to the more common version of the state space models, for which all the error terms are independent; we refer to this as the multiple source of error [MSOE] scheme. As expected, there are many similarities between the MSOE and SSOE schemes, but also some important differences. Both have ARIMA models as their reduced forms, although the mapping is more transparent for SSOE. Further, SSOE does not require a canonical form to complete its specification. An appealing feature of SSOE is that the estimates of the state variables converge in probability to their true values, thereby leading to a formal inferential structure for the ad-hoc exponential smoothing methods for forecasting. The parameter space for SSOE models may be specified to match that of the corresponding ARIMA scheme, or it may be restricted to meaningful sub-spaces, as for MSOE but with somewhat different outcomes. The SSOE formulation enables straightforward extensions to certain classes of non-linear models, including a linear trend with multiplicative seasonals version that underlies the Holt-Winters forecasting method. Conditionally heteroscedastic models may be developed in a similar manner. Finally we note that smoothing and decomposition, two crucial practical issues, may be performed within the SSOE framework.

[1]  R. Snyder.  Exponential Smoothing: A Prediction Error Decomposition Principle , 2004 .

[2]  P. D. Jong.  Smoothing and Interpolation with the State-Space Model , 1989 .

[3]  R. Snyder.  A Pedant's Approach to Exponential Smoothing , 2005 .

[4]  D. Mayne,et al.  On the discrete time matrix Riccati equation of optimal control , 1970 .

[5]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[6]  M. Nerlove,et al.  On the Optimality of Adaptive Forecasting , 1964 .

[7]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[8]  David Q. Mayne,et al.  “On the discrete time matrix Riccati equation of optimal control-a correction” , 1971 .

[9]  Arnold L. Sweet,et al.  Computing the variance of the forecast error for the holt‐winters seasonal models , 1985 .

[10]  A. Koehler,et al.  Exponential Smoothing Model Selection for Forecasting , 2006 .

[11]  S. Zionts,et al.  Programming with linear fractional functionals , 1968 .

[12]  Catherine S. Forbes,et al.  Bayesian Exponential Smoothing , 2000 .

[13]  Steven C. Hillmer,et al.  An ARIMA-Model-Based Approach to Seasonal Adjustment , 1982 .

[14]  H. Theil,et al.  Some Observations on Adaptive Forecasting , 1964 .

[15]  Richard A. Davis,et al.  Introduction to time series and forecasting , 1998 .

[16]  J. Muth.  Optimal Properties of Exponentially Weighted Forecasts , 1960 .

[17]  R. Brown.  Statistical forecasting for inventory control , 1960 .

[18]  Irène Gijbels,et al.  Understanding exponential smoothing via kernel regression , 1999 .

[19]  Antonio García-Ferrer,et al.  On trend extraction models: Interpretation, empirical evidence and forecasting performance , 1992 .

[20]  A. Harvey,et al.  Unobserved component time series models with Arch disturbances , 1992 .

[21]  S. Koopman,et al.  Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models , 1997 .

[22]  Rob J Hyndman,et al.  Prediction Intervals for Exponential Smoothing State Space Models , 2001 .

[23]  J. Ord,et al.  A New Look at Models For Exponential Smoothing , 2001 .

[24]  Anne B. Koehler,et al.  Forecasting models and prediction intervals for the multiplicative Holt-Winters method , 2001 .

[25]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[26]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[27]  Tommaso Proietti,et al.  A Beveridge-Nelson smoother , 2000 .

[28]  H. Akaike.  Markovian Representation of Stochastic Processes and Its Application to the Analysis of Autoregressive Moving Average Processes , 1974 .

[29]  J. Ord,et al.  Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models , 1997 .

[30]  R. Lawton.  How should additive Holt–Winters estimates be corrected? , 1998 .

[31]  M. Bartlett.  A comment on D. V. Lindley's statistical paradox , 1957 .

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

[33]  Siem Jan Koopman,et al.  Signal extraction and the formulation of unobserved components models , 2000 .

[34]  W. Bell,et al.  Signal Extraction for Nonstationary Time Series , 1984 .

[35]  Melvin J. Hinich,et al.  Time Series Analysis by State Space Methods , 2001 .

[36]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

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

[38]  Kenneth F. Wallis,et al.  Prediction theory for autoregressivemoving average processes , 1998 .

[39]  Anne B. Koehler,et al.  Prediction Intervals for ARIMA Models , 2001 .

[40]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[41]  C. Nelson,et al.  A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’☆ , 1981 .

[42]  Peter R. Winters,et al.  Forecasting Sales by Exponentially Weighted Moving Averages , 1960 .

[43]  Eric Zivot,et al.  Why Are the Beveridge-Nelson and Unobserved-Components Decompositions of GDP So Different? , 2003, Review of Economics and Statistics.