I am trying to simulate time series which exhibit similar seasonal means and seasonal variances.

I believe I have some candidate simulated series of which I want to check there accuracy. My idea is to take the difference between the original series and the simulated series.

I.e. if I take an AR(2) original time series and subtract off my AR(2) simulated time series will I have white noise as a result, or would I have another AR(2) time series?

I was wondering if anyone could point me to good references on this topic. Thanks much.


Take the AR(2) process $$y_n=a_1 y_{n-1}+a_2y_{n-2}+\sigma \epsilon_n$$ where the noise terms $\epsilon_n$ are IID standard normal.

Consider a second realization of the same AR(2) process (your simulated one, assuming your model identification is perfect) : $$y'_n=a_1 y'_{n-1}+a_2y'_{n-2}+\sigma \epsilon'_n$$ with the $\epsilon'_n$ also IID standard normal and independent of the $\epsilon_n$ terms. Then if you define $w_n=y_n-y'_n$, you have: $$(y_n-y'_n)=a_1 (y_{n-1} - y'_{n-1})+a_2 (y_{n-2}-y'_{n-2})+\sigma (\epsilon_n-\epsilon'_n)$$ leading to: $$w_n=a_1 w_{n-1}+a_2 w_{n-2}+\sqrt{2}\sigma \xi_n$$ where the noise terms $\xi_n$ are IID standard normal.

So the answer is that in general the difference between AR(2) series is not white noise but rather another AR(2) series.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.