7.4.2 Deterministic Time Trends

7.4.2 Deterministic Time Trends

Sometimes a deterministic trend component is used to predict the level of a variable. For example, linear, quadratic, and exponential trends take the forms

$$\begin{array}{rl}& {\mu }_t={\mu }_0+{\beta }_{0}t,\\ & \\ & {\mu }_t={\mu }_0+{\beta }_{0}t+{\beta }_1{t}^2,\\ & \\ & {\mu }_t=\exp ({\mu }_0+{\beta }_{0}t).\end{array}\text{\hspace{1em}(7.39)}$$

These are global trends and so are unlikely to provide accurate descriptions of the future value of most time series at long forecast horizons. Even if a variable’s past can be well approximated by one of these trend models, it is common to find that the dynamics changes as the variable reaches a saturation point or other dynamics start to dominate at future points in time.

7.4.3 Holt–Winters Procedure

The Holt–Winters procedure is commonly used to forecast time series that could contain both a deterministic trend and a seasonal component. The additive version of this approach models the outcome variable as the sum of a local mean, a seasonal factor, and an error term. The multiplicative version factors the outcome into the product of a local mean, and a seasonal component and adds an error term. In both cases the local mean at time t is updated from the the local mean at time t − 1 plus a local trend.

Following Chatfield (1978), let $m_t$ be an estimate of the deseasonalized local mean at time t, while seast is an estimate of the seasonal component at time t, and ${\mathrm{trend}}_t$ is the estimated trend term for period t (the amount by which the deseasonalized mean is expected to change from t − 1 to t). Here S is again the seasonal frequency, i.e., 4 for quarterly data, 12 for monthly data, etc. Given a set of estimates for period $t-1$ $m_{t-1},{\mathtt{seas}}_{t-1},{\mathtt{trend}}_{t-1}$ , the updating equations for the multiplicative Holt–Winters procedure take the form

$$\begin{array}{r}\begin{array}{rl}& m_t=\alpha \frac{y_t}{s{\mathrm{eas}}_{t-S}}+(1-\alpha )(m_{t-1}+{\mathrm{trend}}_{t-1}),\\ & \\ & {\mathrm{seas}}_t=\beta \frac{y_t}{m_t}+(1-\beta ){\mathrm{seas}}_{t-S},\\ & \\ & {\mathrm{trend}}_t=\gamma (m_t-m_{t-1})+(1-\gamma ){\mathrm{trend}}_{t-1}.\end{array}\end{array}\text{\hspace{1em}(7.40)}$$

Here $\alpha ,\beta ,\gamma$ are a set of smoothing coefficients that can either be estimated or set in advance. Multistep forecasts from this model can now be generated from (7.40) as (for $h=1,2,\dots )$

$$f_{t+h\mid t}=(m_t+h\times {\mathrm{trend}}_t){\mathrm{seas}}_{t+h-S}.\text{\hspace{1em}(7.41)}$$

The additive Holt–Winters model leaves the trend equation unchanged, but alters the first two equations to

$$\begin{array}{r}\begin{array}{rl}& m_t=\alpha (y_t-{\mathsf{seas}}_{t-S})+(1-\alpha )(m_{t-1}+{\mathrm{trend}}_{t-1}),\\ & \\ & {\mathsf{seas}}_t=\beta (y_t-m_t)+(1-\beta ){\mathsf{seas}}_{t-S},\end{array}\end{array}\text{\hspace{1em}(7.42)}$$

with forecasts generated as

$$f_{t+h\mid t}=m_t+h\times {\mathrm{trend}}_t+{\mathrm{seas}}_{t+h-S}.\text{\hspace{1em}(7.43)}$$

How well this method works will of course depend on the extent to which the assumed form of the trend and seasonality is aligned with the data and also on how well the smoothing parameters $\alpha ,\beta ,\gamma$ can be estimated. See Chatfield (1978) for further discussion.

7.5 EXPONENTIAL SMOOTHING AND UNOBSERVED COMPONENTS

Exponential smoothing methods provide an alternative approach to ARMA models for predicting a variable using only its own past values. These methods are essentially automated rules for updating forecasts based on the arrival of a new observation and a forecast error each period. The simplest and most common example of such an updating rule is

$$f_{t+1|t}=f_{t|t-1}+(1-\delta )(y_t-f_{t|t-1}),\text{\hspace{1em}(7.44)}$$

where $0<\delta <1$ is a constant and the process is initialized at $f_0$ . Here the previous forecast, $f_{t\mid t-1}$ , is updated with some portion (1 − δ) of the previous period’s forecast error, $e_{t|t-1}=y_t-f_{t|t-1}$ . For a book length examination of forecasting with exponential smoothing, see Hyndman et al. (2008).