Non-linear exponential smoothing and positive data
We consider the properties of nonlinear exponential smoothing state space models under various assumptions about the innovations, or error, process. Our interest is restricted to those models that are used to describe non-negative observations, because many series of practical interest are so constrained. We first demonstrate that when the innovations process is assumed to be Gaussian, the resulting prediction distribution may have an infinite variance beyond a certain forecasting horizon. Further, such processes may converge almost surely to zero; an examination of purely multiplicative models reveals the circumstances under which this condition arises. We then explore effects of using an (invalid) Gaussian distribution to describe the innovations process when the underlying distribution is lognormal. Our results suggest that this approximation causes no serious problems for parameter estimation or for forecasting one or two steps ahead. However, for longer-term forecasts the true prediction intervals become increasingly skewed, whereas those based on the Gaussian approximation may have a progressively larger negative component. In addition, the Gaussian approximation is clearly inappropriate for simulation purposes. The performance of the Gaussian approximation is compared with those of two lognormal models for short-term forecasting using data on the weekly sales of over three hundred items of costume jewelry.
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- Hyndman, R.J. & Koehler, A.B. & Ord, J.K. & Snyder, R.D., 2001. "Prediction Intervals for Exponential Smoothing State Space Models," Monash Econometrics and Business Statistics Working Papers 11/01, Monash University, Department of Econometrics and Business Statistics.
- Hyndman, R.J. & Koehler, A.B. & Snyder, R.D. & Grose, S., 2000.
"A State Space Framework for Automatic Forecasting Using Exponential Smoothing Methods,"
Monash Econometrics and Business Statistics Working Papers
9/00, Monash University, Department of Econometrics and Business Statistics.
- Hyndman, Rob J. & Koehler, Anne B. & Snyder, Ralph D. & Grose, Simone, 2002. "A state space framework for automatic forecasting using exponential smoothing methods," International Journal of Forecasting, Elsevier, vol. 18(3), pages 439-454.
- Hyndman, Rob J. & Koehler, Anne B., 2006.
"Another look at measures of forecast accuracy,"
International Journal of Forecasting,
Elsevier, vol. 22(4), pages 679-688.
- Rob J. Hyndman & Anne B. Koehler, 2005. "Another Look at Measures of Forecast Accuracy," Monash Econometrics and Business Statistics Working Papers 13/05, Monash University, Department of Econometrics and Business Statistics.
- Anne B. Koehler & Rob J. Hyndman & Ralph D. Snyder & J. Keith Ord, 2005. "Prediction intervals for exponential smoothing using two new classes of state space models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 24(1), pages 17-37.
- Ord, J.K. & Koehler, A. & Snyder, R.D., 1995. "Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models," Monash Econometrics and Business Statistics Working Papers 4/95, Monash University, Department of Econometrics and Business Statistics.
- Taylor, James W., 2003. "Exponential smoothing with a damped multiplicative trend," International Journal of Forecasting, Elsevier, vol. 19(4), pages 715-725.
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