Difference between two time series?

Question

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.

Answer 1

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.