Abstract
SummaryThis paper shows how cubic smoothing splines fitted to univariate time series data can be used to obtain local linear forecasts. The approach is based on a stochastic state‐space model which allows the use of likelihoods for estimating the smoothing parameter, and which enables easy construction of prediction intervals. The paper shows that the model is a special case of an ARIMA(0, 2, 2) model; it provides a simple upper bound for the smoothing parameter to ensure an invertible model; and it demonstrates that the spline model is not a special case of Holt's local linear trend method. The paper compares the spline forecasts with Holt's forecasts and those obtained from the full ARIMA(0, 2, 2) model, showing that the restricted parameter space does not impair forecast performance. The advantage of this approach over a full ARIMA(0, 2, 2) model is that it gives a smooth trend estimate as well as a linear forecast function.
Highlights
Suppose we observe a univariate time series {yt}, t = 1, . . . , n, with non-linear trend
We present Wecker and Ansley’s state space model in the special case of cubic smoothing splines applied to spaced data
Given that the cubic spline model is a special case of an ARIMA(0,2,2) model, it is interesting to see if the restricted parameter space results in poorer forecasting performance
Summary
We are interested in forecasting the series by extrapolating the trend using a linear function estimated from the observed time series. We discuss a method for local linear extrapolation of a stochastic trend based on cubic smoothing splines. The methodology provides a smooth historical trend, a linear forecast function and prediction intervals. Forecasts are usually made using models which give most weight to recent observations, and negligible weight to the distant past This means that the smoothing parameter λ should not be too big for forecasting purposes. These relationships enable us to obtain the maximum bound for the smoothing parameter λ to ensure invertibility.
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