0

I am not an expert in statistics so sorry if it seems silly. I am trying to reproduce the results of an article which need to produce a coupled auto regressive time series with order 10:

$$ X(t+1) = \sum_{k=0}^m \alpha_k X(t-k) +\sigma \eta_X(t), \\ Y(t+1) = \sum_{k=0}^m \beta_k Y(t-k) +\sigma \eta_Y(t) + g\sum_\delta X^2(t+1-\delta). \\ $$

where $m = 10$, $\sigma$ is noise and $\delta$ is delay. The values for $\alpha_k$ and $\beta_k$ where constructed from roots of the characteristic polynomial of the the AR process, that were chosen at random on the unit circle to guarantee a stationary AR process (I don't get this part).

All I know from auto regressive process is that we always have a time series and use auto regressive to analyse the data, apply some prediction ...

How we can create some time series with given order which are coupled with some delay? I tried to replace some random number instead of $\alpha$ and $\beta$ but the time series go to huge numbers.

edit

In other words the question is how to create time stationary time series with 10 significant auto regressive term, right?

Thanks in advance for any guide.

Abolfazl
  • 101
  • What is $g$? Also yes that is why they choose on unit circle ... to prevent "explosive proces" going to huge numbers as you say. – Jesper for President Nov 02 '20 at 17:44
  • g is a coupling term and is constant. I don't know why unit circle. I tried also random numbers in [-1,1], normal random distribution with mean 0 and std =1, still the series diverge to huge numbers. – Abolfazl Nov 02 '20 at 18:54
  • 1
    Hi: If you want to generate an AR(10) that doesn't explode, you need to use AR coefficients such the roots of the associated characteristic equation are only satisfied when one is outside ( or inside depending on how one defines the characteristic equation ) the unit circle. I'm not clear on how you do that but that's how you would do it in theory. – mlofton Nov 02 '20 at 19:17
  • 1
    At https://stats.stackexchange.com/a/494445/919 I describe a general technique to create time series datasets with prescribed ACF functions. It amounts to diagonalizing the covariance matrix. Whether any term is "significant" in the usual sense (of a hypothesis test) depends on the size of the dataset as well as chance, so should we interpret "significant" as simply meaning "nonzero"? – whuber Nov 02 '20 at 19:47

0 Answers0