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Predicting a non-causal stationary autoregressive process

1

1

$begingroup$

Consider the unique stationary solution to the following relation
$$X_t = theta X_t-1 + Z_t$$
where $(Z_t)_t in mathbbZ$ is a white noise series and $lvert theta rvert > 1$.
I am trying to find out the projection of $X_t$ onto the closure (in $L^2$) of the linear span of $X_t-1, X_t-2,ldots$, which I denote by $Pi_t-1 X_t$.
Clearly,

$$Pi_t-1 X_t = Pi_t-1 left(theta X_t-1 + Z_tright) = theta X_t-1 + Pi_t-1 Z_t$$
Since $lvert theta rvert > 1$, the stationary solution is purely non-causal. So $Pi_t-1 Z_t neq 0$.

I don't think starting with the projection equations is possible as I don't really know what is in the closure, i.e. I cannot say
$$Pi_t-1 X_t = sum_igeq 1 a_i X_t-i$$
since I would be missing the limit points. I feel like I am forgetting a crucial fact that is applicable here. Any help appreciated.

stochastic-processes time-series projection

share|cite|improve this question

asked Apr 2 at 9:43

CalculonCalculon

3,1501824

$endgroup$

add a comment | 

1

1

$begingroup$

Consider the unique stationary solution to the following relation
$$X_t = theta X_t-1 + Z_t$$
where $(Z_t)_t in mathbbZ$ is a white noise series and $lvert theta rvert > 1$.
I am trying to find out the projection of $X_t$ onto the closure (in $L^2$) of the linear span of $X_t-1, X_t-2,ldots$, which I denote by $Pi_t-1 X_t$.
Clearly,

$$Pi_t-1 X_t = Pi_t-1 left(theta X_t-1 + Z_tright) = theta X_t-1 + Pi_t-1 Z_t$$
Since $lvert theta rvert > 1$, the stationary solution is purely non-causal. So $Pi_t-1 Z_t neq 0$.

I don't think starting with the projection equations is possible as I don't really know what is in the closure, i.e. I cannot say
$$Pi_t-1 X_t = sum_igeq 1 a_i X_t-i$$
since I would be missing the limit points. I feel like I am forgetting a crucial fact that is applicable here. Any help appreciated.

stochastic-processes time-series projection

share|cite|improve this question

asked Apr 2 at 9:43

CalculonCalculon

3,1501824

$endgroup$

add a comment | 

$begingroup$

Consider the unique stationary solution to the following relation
$$X_t = theta X_t-1 + Z_t$$
where $(Z_t)_t in mathbbZ$ is a white noise series and $lvert theta rvert > 1$.
I am trying to find out the projection of $X_t$ onto the closure (in $L^2$) of the linear span of $X_t-1, X_t-2,ldots$, which I denote by $Pi_t-1 X_t$.
Clearly,

$$Pi_t-1 X_t = Pi_t-1 left(theta X_t-1 + Z_tright) = theta X_t-1 + Pi_t-1 Z_t$$
Since $lvert theta rvert > 1$, the stationary solution is purely non-causal. So $Pi_t-1 Z_t neq 0$.

I don't think starting with the projection equations is possible as I don't really know what is in the closure, i.e. I cannot say
$$Pi_t-1 X_t = sum_igeq 1 a_i X_t-i$$
since I would be missing the limit points. I feel like I am forgetting a crucial fact that is applicable here. Any help appreciated.

stochastic-processes time-series projection

share|cite|improve this question

asked Apr 2 at 9:43

CalculonCalculon

3,1501824

$endgroup$

share|cite|improve this question

asked Apr 2 at 9:43

CalculonCalculon

3,1501824

share|cite|improve this question

asked Apr 2 at 9:43

CalculonCalculon

3,1501824

asked Apr 2 at 9:43

CalculonCalculon

3,1501824

asked Apr 2 at 9:43

CalculonCalculon

3,1501824