15.3 Forecasting seasonal models
The rules of Section 14 apply unchanged: write the model equation for time \(n + k\), replace future \(Z\)’s by \(0\), future \(X\)’s by their forecasts, and past values by the observed values and residuals. For the model of Example 15.2.3, \[ \hat x(n, 1) = x_{n-11} + \alpha (x_n - x_{n-12}) + \theta z_{n-11} . \] Two steps ahead, \(x_{n+1}\) is not yet known, so it is replaced by its forecast: \[ \hat x(n, 2) = x_{n-10} + \alpha \bigl (\hat x(n, 1) - x_{n-11}\bigr ) + \theta z_{n-10} , \] and forecasts further ahead are calculated recursively in the same way. When \(x_{n+1}\) becomes known, the one-step forecast from time \(n + 1\) is \begin {align*} \hat x(n + 1, 1) &= x_{n-10} + \alpha (x_{n+1} - x_{n-11}) + \theta z_{n-10} \\ &= \hat x(n, 2) + \alpha \bigl (x_{n+1} - \hat x(n, 1)\bigr ) = \hat x(n, 2) + \alpha z_{n+1} , \end {align*}
the updating formula of Theorem 14.4.2 with \(\psi _1 = \alpha \) (the \(\psi \) weight of this model at lag \(1\)).
Example 15.3.1. For the quarterly shop sales of Example 2.4.3, fit the SARIMA \((1, 0, 0) \times (0, 1, 0)_4\) model with a constant, \[ W_t - \mu = \alpha (W_{t-1} - \mu ) + Z_t , \qquad W_t = x_t - x_{t-4} , \] and forecast the four quarters of year 5.
Solution. The seasonal differences \(W_t = x_t - x_{t-4}\), for \(t = 5, \dots , 16\), are \[ 3, \ 4, \ 4, \ 3, \ 4, \ 4, \ 5, \ 5, \ 5, \ 6, \ 5, \ 4 , \] the year-on-year growth, with the seasonal pattern removed. Their mean is \(\hat \mu = 52/12 = 4.333\), and the deviations are \(-\frac 43\), \(-\frac 13\), \(-\frac 13\), \(-\frac 43\), \(-\frac 13\), \(-\frac 13\), \(\frac 23\), \(\frac 23\), \(\frac 23\), \(\frac 53\), \(\frac 23\), \(-\frac 13\), with sum of squares \(\frac {78}{9}\) and lag-one sum of products \(\frac {38}9\). So \(\hat \alpha = r_1 = 38/78 = 0.49\).
The last difference is \(W_{16} = 4\), so \(W_{16} - \hat \mu = -\frac 13\), and the forecasts of the differences are \(\hat W(16, k) = 4.333 + 0.49^k(-0.333)\): \(4.170\), \(4.253\), \(4.294\), \(4.314\). Adding each to the value a year earlier, \begin {align*} \hat x(16, 1) &= x_{13} + 4.170 = 31 + 4.170 = 35.17 , \\ \hat x(16, 2) &= x_{14} + 4.253 = 36 + 4.253 = 40.25 , \\ \hat x(16, 3) &= x_{15} + 4.294 = 40 + 4.294 = 44.29 , \\ \hat x(16, 4) &= x_{16} + 4.314 = 35 + 4.314 = 39.31 . \end {align*}
The forecasts keep the seasonal shape (low in Q1, high in Q3) and raise the level by about \(4.3\), the average yearly growth, with a slight dip for the first quarters because last year’s growth, \(4\), was below average.
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