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For a ARMA process,

  • some (e.g. in Tsay's Financial Time Series) said: it is wide-sense stationary, iff all the roots of its AR characteristic polynomial are greater than 1 in magnitude. This is shown by requiring the variance of the series at any time to be nonnegative and constant.

  • some (e.g. in Brockwell's Introduction to Time Series) said: it is wide-sense stationary, iff all the roots of its AR characteristic polynomial are not equal to 1 in magnitude. It is causal, iff all the roots of its AR characteristic polynomial are greater than 1 in magnitude.

Questions

  1. According to Brockwell, it seems that a ARMA process can be wide-sense stationary while non-causal, iff all the roots of its AR characteristic polynomial are less than 1 in magnitude.

    For example, in AR(1) model $X_t = a X_{t-1} + e_t$, the root of its AR characteristic polynomial is less than 1 in magnitude, iff $|a|<1$.

    I wonder what $Var(X_t)$ in this AR(1) model is then? Is it still well defined (nonnegative)?

    How does its autocorrelation function behave as the lag increases? (not decay, but stay at 1?)

  2. I wonder how you understand the difference in the necessary and sufficient conditions fpr wide-sense stationarity given by Tsay and by Brockwell?

Thanks!

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