SARIMA Models

As described in ARMA Models, an ARMA(p,q) model can be expressed as

If  φ₀ = 0 (i.e. the mean of the stochastic process is zero) then this can be expressed using the lag operator as

where

Note too that an ARIMA(p, d, q) process z_i can be expressed as above but first z_i must be replaced by z_i = y_i – y_i-d. Alternatively, we can express an ARIMA(p, d, q) process y_i without constant as

We can add the constant term back in as

A seasonal ARIMA model takes the same form, but now there are additional terms that reflect the seasonality part of the model. Specifically, a SARIMA(p,d,q) × (P,D,Q)_m model without constant can be expressed as

where

Here, we have P seasonal autoregressive terms (with coefficients Φ₁, …, Φ_P), Q seasonal moving average terms (with coefficients Θ₁, …, Θ_Q) and D seasonal differencing based on m seasonal periods.

Let’s assume that the two types of differencing (corresponding to d and D) have already been done. Then a SARIMA(1,0,1) × (1,0,1)₁₂ model takes the following form:

The residuals can therefore be expressed as:

The forecast can be expressed (where the coefficients should have a hat on them):

Similarly, a SARMA(p,q) × (P,Q)_m model without constant can be expressed as

or equivalently as

And so

which is equivalent to

In the case where there is a constant term φ₀ this expression takes the form

This serves as the equation to estimate the forecast at time i (when the final ε_i is set to zero). You can also solve for ε_i to obtain an expression that can be used to estimate the residuals.

References

Greene, W. H. (2002) Econometric analysis. 5th Ed. Prentice-Hall
https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=1243286

Gujarati, D. & Porter, D. (2009) Basic econometrics. 5th Ed. McGraw Hill
http://www.uop.edu.pk/ocontents/gujarati_book.pdf

Hamilton, J. D. (1994) Time series analysis. Princeton University Press
https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis

Wooldridge, J. M. (2009) Introductory econometrics, a modern approach. 5th Ed. South-Western, Cegage Learning
https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf