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I would like to understand how to write the equation of an ARIMA with seasonal effect

I am forecasting a financial variable using auto.arima in R. The result was an ARIMA(1 1 0)(0 1 0) 12.

So I only have 1 coefficient with value -0.4605.

enter image description here

Without the seasonal effect I know the equation would be

Yt = Yt-1  - 0.4605 * (Yt-1 - Yt-2)

So the value today is equal to the last value - beta times the lag delta.

Now, how should I include the seasonal effect?

My Data is enter image description here

Julez
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1 Answers1

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This is more of a conceptual/statistics question than a coding/programming related question. For future posts, please consider that Stack Overflow is probably not the right place for such questions. Instead you may want to consider posting on Cross Validated.

That aside, to understand the explicit algebraic form of your model, we first recognise that your ARIMA(1, 1, 0)(0, 1, 0)12 model corresponds to an AR(1) model with both a non-seasonal and seasonal difference and a seasonal period of 12 time points.

A non-seasonal AR(1) model with differencing (and zero offset) can be written as

enter image description here ,

where

enter image description here .

This is the non-seasonal model you mention in your post above.

To account for seasonality the difference is modified

enter image description here ,

where m is the seasonal period.

Expanding the first equation then gives

enter image description here .

Maurits Evers
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  • Many thanks for the suggestion and the responde Maurits! Really helpful! I have a further doubt regarding the model you shared since it does not match the result I gert from the R forecast. Estimating the value for 201901 with the equation I get 38,879,389 but the R forecast gives me 42,468,245 – Julez Jul 04 '19 at 12:22
  • @Julez The algebraic form of the model should be correct. Unfortunately there's not much I (or anyone else for that matter) can do at the moment, on account of you not providing any reproducible & minimal data (and code). – Maurits Evers Jul 04 '19 at 13:22
  • Since you're new to SO, please take a look at what you can (and should) do in order to provide a [minimal reproducible example](https://stackoverflow.com/questions/5963269/how-to-make-a-great-r-reproducible-example). – Maurits Evers Jul 04 '19 at 13:23
  • Many thanks for all the information Mautits! It was really useful, as you said that is the correct equation, the error was in the data base used that generated the error – Julez Jul 04 '19 at 15:20
  • @Julez Great I'm glad it worked out (I was trying to simulate some seasonal data to test the model). If this answers your question please consider closing the question by setting the green checkmark next to the answer. That way, you help keeping SO tidy and make it easier for other SO users to identify relevant posts. – Maurits Evers Jul 05 '19 at 00:19