Question: "Pseudo-random number generator" for shocks to equations in ARIMAResults.simulate

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Alina S.

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Aug 24, 2022, 7:05:23 AM8/24/22
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Hi all,

I am currently writing my thesis on forecasting and using statsmodels. Right now, I am looking into the ARIMAResults.simulate function which is described here:

https://www.statsmodels.org/devel/generated/statsmodels.tsa.arima.model.ARIMAResults.simulate.html#statsmodels.tsa.arima.model.ARIMAResults.simulate

I would like to understand how this function works exactly. How does the "pseudo-random number generator" for the measurement and state shocks work? Do the shocks scale with the amplitude of my data? Could you direct me to the source? I wasn't able to find it (maybe a rookie mistake).

Thank you in advance!
Best,
Alina

Chad Fulton

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Aug 27, 2022, 12:08:15 AM8/27/22
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Hi Alina,

State space models (of which ARIMA models are a special case) have 3 fundamental sources of randomness: (1) the initial state vector, (2) the error terms in the transition equation, and (3) the error terms in the measurement equation.  All of these are assumed to be independent and multivariate normal.

So, the first step is to simulate these from the respective multivariate normal distributions using, for example, numpy.random.multivariate_normal. With these terms available, the state vector can be simulated by applying the transition equation, and the observed vector can be simulated by applying the measurement equation.

Best,
Chad

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Alina S.

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Sep 28, 2022, 5:34:04 AM9/28/22
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Hi Chad,

thank you so much for your reply. 
I have just one more question about this: 
If I have, let's say, an exponential smoothing or ARIMA model and I were to simulate a few multi-step trajectories for my time series,
can I expect the different forecasts to "fan out" for a large forecast horizon because of increasing uncertainty and error propagation?
Or should the difference between the forecasts stay more or less the same over time?
I'm still pretty new to state-space models and not quite sure I understand this.

Thanks again,
Alina
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