Time-Series Modeling and Decomposition

In the absence of seasonal adjustment, only the raw series is available. In such cases, it is customary to use same-month comparisons from year to year, \(z_{t}-z_{t-12}\), to assess the stage of the business cycle. The rationale is that the seasonal effect in \(z_{t}\) is approximately the same as in \(\mathscr{7} t-12\), under the assumption of slowly evolving seasonality. Same-month year ago comparisons can be expressed as the sum of the changes in the raw series between \(z_{t}\) and \(\mathscr{7} t-12\),

\(\begin{gathered}

z_{t}-z_{t-12} \equiv\left(z_{t}-z_{t-1}\right)+\left(z_{t-1}-z_{t-2}\right)+\left(z_{t-2}-z_{t-3}\right)+ \\

\ldots+\left(z_{t-11}-z_{t-12}\right) \equiv \Sigma_{j=1}^{12}\left(z_{t-j+1}-z_{t-j}\right)

\end{gathered}\)                                                      (13)

Eq. (13) shows that same-month comparison display an increase, if the increases dominate the decreases over the 13 months involved, and conversely. The timing of \(z_{t}-z_{t-12}\) is \(t-6\) , the average of \(t\) and \(t-12\) . This points out a limitation of this practise: the diagnosis provided is not timely with respect to t. Furthermore, \(z_{t}\) and \(z_{t-12}\) may contain irregular variations affecting one observation positively and the other negatively, hence conveying instability to the comparison. Moreover, for flow data the comparison is systematically distorted by trading-day variations if present.

Seasonal adjustment entails the removal of seasonality, trading-day variations and moving-holiday effects from the raw data, to produce a seasonally adjusted series, which consists of the trend-cycle and the irregular components. The irregular fluctuations in the seasonally adjusted series can be reduced by smoothing, to isolate the trend-cycle and to enable month-to-month comparisons.