Chapter 4 Exponential smoothing models

The basic idea with an exponential smoothing model is that your forecast of \(x\) at time \(t\) is a smoothed function of past \(x\) values.

\[\hat{x}_{t} = \alpha x_{t-1} + \alpha (1-\alpha)^2 x_{t-2} + \alpha (1-\alpha)^3 x_{t-3} + \dots\] Although this looks similar to an AR model with a constraint on the \(\beta\) terms, it is fundamentally different. There is no process model and one is not assuming that \[x_{t} = \alpha x_{t-1} + \alpha (1-\alpha)^2 x_{t-2} + \alpha (1-\alpha)^3 x_{t-3} + \dots + e_t\] The goal is to find the \(\alpha\) that minimizes \(x_t - \hat{x}_t\), i.e. the forecast error. The issues regarding stationarity do not arise because we are not fitting a stationary process model. We are not fitting a process model at all.