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From Tsay's Financial Time Series, there is a SARIMA model: enter image description here

He said that:

enter image description here

I can't understand

  • why deterministic seasonality is special case of the multiplicative seasonal model, and

  • what the deterministic component in the SARIMA model (2.41) is when $\Theta=1$.

I really appreciate your help. Thanks!

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One way to see it is the following. Suppose that the true data generating process is an ARIMA(0,1,1) with deterministic seasonality:

$$ (1 - B)x_t = (1 - \theta B) a_t + \sum_{i=1}^s\gamma_i SD_{t,i} \,, \quad a_t \sim \hbox{NID} (0, \sigma^2) \,, $$

where $SD$ are seasonal dummies where each column vector takes on the value 1 for a given season and 0 otherwise.

Taking seasonal differences in both sides of the equation the model becomes:

$$ (1 - B^s)(1 - B)x_t = (1 - B^s)(1 - \theta B) a_t \,. $$

The seasonal dummies, $SD$, are gone because they turn zero after taking seasonal differences on them. In the case with $\theta=1$ we get the airline model. In practice, having two unit roots shared by the AR and MA parts can be troublesome numerically. That's why it is generally said that deterministic seasonality can be approximated by the airline model as the seasonal MA coefficient approaches one, for example for $\theta=0.9$.

A similar reasoning can be followed to see that the airline model with a MA coefficient $\theta$ close to unity can accommodate a deterministic linear trend.

Observe that an estimate of $\theta$ close to unity will not necessarily imply the presence of a deterministic seasonal pattern. The reason is that the argument given above still holds when $\gamma_1=\gamma_2=\dots=\gamma_s$, that is, when $\sum_{i=1}^s\gamma_i SD_{t,i}$ collapses to a constant intercept $\alpha$. Thus, the airline model also encompasses models without seasonality.

It's not straightforward to extract the seasonal pattern from the airline model. It requires decomposing the ARIMA model and obtain the corresponding filter that returns the seasonal component.

If seasonality is deterministic, using seasonal dummies is a more natural and convenient approach than fitting the airline model. However, the point made in the excerpt shown in the question is that the airline model is a relatively simple yet flexible model that can be helpful as a general tentative model. With macroeconomic data, this model often performs fairly well. In the large-scale study by Fischer and Planas [1], the airline model was chosen in 60% of the cases.

[1] Fischer, B. and Planas, C. (2000) Journal of Official Statistics, Vol.16, No.2, 2000. pp. 173–184 Large Scale Fitting of Regression Models with ARIMA Errors. URL: http://www.jos.nu/Articles/abstract.asp?article=162173.

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