Assume we have the following ARMA(1, 1) model:

$$ z_{t+1} = \phi z_{t} + \theta \varepsilon_{t} + \varepsilon_{t+1}, $$ where $\varepsilon_{t}$ are i.i.d. with $var(\varepsilon_{t}) = \sigma^2$.

A standard identifiability condition asks the coefficient $\theta$ to be less than 1. Nevertheless, if in my model $\theta > 1$. For example, in the problem considered here:

Superposition of random walk and autoregressive process

Therefore, $\theta$ is parametrised in a way that it must be greater than 1.

How can I estimate the parameters $\phi$, $\theta$ and $\sigma^2$ with the restriction $\phi < 1$ and $\theta > 1$?

I am thinking about some transformation of $z_{t}$ in order to be able to use standard algorithms.

  • 1
    $\begingroup$ There is no true restriction on the MA coefficient in an MA(1). There's an invertibility condition that warrants it but that's only if you want to write it as $AR(\infty)$. Invertibility condition is not as important as stationarity condition. The $\phi < 1$ restriction is correct ( stationarity condition ) which means you need an algorithm that checks if you're over the boundary and, if so, brings you back. It's obviously easiest to use already written software that's been tested. $\endgroup$
    – mlofton
    Nov 5, 2019 at 15:06
  • $\begingroup$ dear @mlofton, about stationarity I understand. Though, all standard python algorithms use estimate with restriction $|\theta| < 1$. How to estimate with my constraint? $\endgroup$
    – ABK
    Nov 5, 2019 at 15:25
  • $\begingroup$ maybe do a grid search on $\theta$ from -1 to 1.0 by say .001 and maximize the likelhood ? Not sure why python does that except to meet the invertibility constraint ? Like I said I don't think searching in that range is the right thing to do but maybe someone else who's written ARIMA estimation algorithms can comment or tell me it is the right thing to do ? I'm opened to being corrected and learning. $\endgroup$
    – mlofton
    Nov 5, 2019 at 16:58

1 Answer 1


I think I have got the solution. The acf for ARMA(1,1) is $$ \rho(1) = \frac{(\phi + \theta)(1 + \phi\theta)}{1 + 2\phi\rho + \theta^2} $$ and for $\tau >1$ $$ \rho(\tau) = \phi^{\tau-1}\rho(1). $$ Therefore, the process $z_{t}$ has the same acf for both $\theta := \theta$ and $\theta := \frac{1}{\theta}$, i.e.

$$ z_{t+1} = \phi z_{t} + \frac{1}{\theta}\varepsilon_{t} + \varepsilon_{t+1} $$


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