I've come across a question where I have an MA(1) process like so:

$X_t = b_t - 0.4 b_{t-1}$ (where $b_t$ is a white noise process and $t$ is the time index)

The question asks me to find $\phi_{11}$ (Phi) and $\phi_{22}$

From what my notes say:

  • $\phi$ is from the AR(1) process.
  • There exists a technique to convert an MA(1) process into an infinite AR process

However, I cannot find any information or techniques for converting an MA(1) process into an AR(1) process or even for using the $\theta$ (Theta) value as a means for converting into the $\phi$ value.

Is there something I am missing in this time series problem?

For some extra info:

  • I have calculated the mean of the MA(1) process
  • I also have the variance
  • I also have autocorrelations for $k=0$, $k=1$

Any help will be much appreciated.


1 Answer 1


Ok, it seems I have come across something called the PACF of the MA(1) process.

I think I may have confused the Phi in the AR(1) process with this other Phi. I'm not even sure they are related.

The formulae are:

Phi11 = Rho1 = -Theta1 / 1 + Theta1^2

Phi22 = - (Rho1^2 / 1 - Rho1^2)

  • $\begingroup$ $\phi_i$'s and $\phi_{ii}$'s are related. $\endgroup$
    – Glen_b
    Commented Mar 7, 2014 at 0:52

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