14.1 Forecasting methods

Let \(x_1, x_2, \dots , x_n\) be an observed time series. The problem is to estimate future values \(X_{n+k}\), where \(k\) is called the lead time. The forecast of \(X_{n+k}\) made at time \(n\), \(k\) steps ahead, is denoted by \(\hat x(n, k)\). Various methods of forecasting exist, and no single method is universally applicable. Every forecast is a form of extrapolation: it assumes that some characteristic of the series will continue. Forecasting methods are broadly of three kinds:

1.

Subjective: based on judgement, intuition, business knowledge or any other relevant information.

2.

Univariate: based on a model fitted only to the past observations of the series, so that \(\hat x(n, k)\) depends only on \(x_1, x_2, \dots , x_n\).

3.

Multivariate: using the past values of other series as well, for example forecasting sales from advertising, or crop yield from rainfall.

This section deals with univariate methods: exponential smoothing and its extensions, which need no formal model, and the Box–Jenkins forecasts of a fitted ARIMA model.

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