Assume that $|\phi|>1$ and {$X_k$} is the stationary solution of the non-causal AR(1) equations, $$X_k=\phi X_{k-1}+Z_k$$ where {$Z_k$} is white noise with mean $0$ and variance $\sigma^2$. Show that there is another white noise {$\hat{Z_k}$} with mean $0$ and variance $\hat{\sigma}^2$ such that {$X_k$} satisfies the causal AR(1) equations,

$$X_k=\phi^{-1} X_{k-1}+\hat{Z_k}$$.

My question is what is $\hat{\sigma}^2$?

  • $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ – gung - Reinstate Monica Mar 24 '16 at 1:58

This is Problem 3.3 from Brockwell and Davis, 2006, Time Series: Theory and Methods, 2nd ed.

It seems that Thm. 3.5.1 in the book provides an answer for the problem using theory of backshift polynomials and autocovariance generating functions. Just defining the polynomials as in the proof of Thm. 3.5.1 and calculating the generating function as in Ex. 3.5.2, the answer is found to be $$ \widehat{\sigma}^2 = \phi^{-2} \sigma^2 .$$

However, I believe that this problem and even Thm. 3.5.1 are poorly formulated. Technically, the solution to the causal AR(1) equation is not the same (I don't know what "the same" should mean here, but the solutions are not even modifications to each other) process $\{ X_t \}$ as the solution to the non-causal AR(1) equation: the time is reversed. To exhibit this, here is an alternative solution to the problem:

Rewrite the non-causal AR(1) equation $$X_t = \phi X_{t-1} + Z_t $$ as $$X_{t-1} = \phi^{-1} X_t - \phi^{-1} Z_t .$$ Now reverse the time: $$ X_t = \phi^{-1} X_{t-1} - \phi^{-1} Z_{t-1} .$$ Finally, it remains to define $\widehat{Z}_t = \phi^{-1} Z_{t-1}$. Then $\{ \widehat{Z}_t \}$ is clearly a White Noise process with $ \widehat{\sigma}^2 = \phi^{-2} \sigma^2$ and $\{ X_t \}$ satisfies the causal AR(1) equation.

However, the time has been reversed! Brockwell and Davis (2006) use this problem to conclude that "from a second-order point of view, nothing is lost by eliminating non-causal AR(1) processes from consideration". This is surely true. Whether it is OK to consider the solutions to causal and non-causal AR(1) equations as the same processes... I don't know.


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