Time-Series Modeling and Decomposition

A frequently applied deterministic model for trading-day variations was developed by Young,

\(y_{t}=D_{t}+u_{t}, t=1,, \ldots, n\),                                                       (22.a)

\(D_{t}=\sum_{j=1}^{7} \alpha_{j} N_{j t}\)                                                                        (22.b)

where \(u_{t} \sim W N\left(0, \sigma_{u}^{2}\right), \Sigma_{j=1}^{7} \alpha_{j}=0, \alpha_{j}, j=1, \ldots, 7\) denote the effects of the seven days of the week, Monday to Sunday, and \(N_{j t}\) is the number of times day \(j\) is present in month \(t\). Hence, the length of the month is \(N_{t}=\sum_{j=1}^{7} N_{j t}\), and the cumulative monthly effect is given by (22.b). Adding and subtracting \(\bar{\alpha}=\left(\sum_{j=1}^{7} \alpha_{j}\right) / 7\) to Eq. (22.b) yields

\(D_{t}=\bar{\alpha} N_{t}+\sum_{j=1}^{7}\left(\alpha_{j}-\bar{\alpha}\right) N_{j t}\).                                               (23)

Hence, the cumulative effect is given by the length of the month plus the net effect due to the days of the week. Since \(\Sigma_{j=1}^{7}\left(\alpha_{j}-\bar{\alpha}\right)=0\), model (23) takes into account the effect of the days present five times in the month. Model (23) can then be written as

\(D_{t}=\bar{\alpha} N_{t}+\Sigma_{j=1}^{6}\left(\alpha_{j}-\bar{\alpha}\right)\left(N_{j t}-N_{7 t}\right)\),                                 (24)

with the effect of Sunday being \(\alpha_{7}=-\sum_{j=1}^{6} \alpha_{j}\).

Deterministic models for trading-day variations assume that the daily activity coefficients are constant over the whole range of the series. Stochastic model for trading-day variations have been rarely proposed. Dagum et al. developed a model where the daily coefficients change over time according to a stochastic difference equation.