estimation of ARMA(1,1) with MA parameter greater than 1

Question

I am working with the following simple ARMA(1,1) model: $$ z_{t+1} = \phi z_{t} + \theta\varepsilon_{t} + \varepsilon_{t+1} $$ In my case $\theta$ depends on some other parameters, and, therefore, I know that $\theta > 1$.

Let us generate 1000 ARMA(1,1) samples with $\phi = 0.95$, $\theta = 0.5$, $\sigma = 0.08$ and estimate the parameters.

import numpy as np
import statsmodels.api as sm
from statsmodels.tsa.arima_process import ArmaProcess

process = ArmaProcess(np.r_[1, -0.95], np.r_[1, 0.5])
y = process.generate_sample(1000, scale=0.08)
model = sm.tsa.ARMA(y, (1, 1)).fit(trend='nc', disp=0)

The result is [0.96069232 0.51912881 0.080], which is quite close to the true values.

Next, lets us try parameters $\phi = 0.95$, $\theta = 2$, $\sigma = 0.08$, i.e. change the following row in the code above:

process = ArmaProcess(np.r_[1, -0.95], np.r_[1, 2])

The result is [0.95668055 0.49700497 0.156] and there is no errors and warnings!

By default, in order to have distinguishable auto-covariance function we assume $|\theta| < 1$. Therefore, in the result the estimator is close to $\frac{1}{\theta}$. Though, the standard deviation of the variance is far from the true value.

Is there a way to estimate the case when $\theta > 1$? From what I can see, all packages use the fact that $|\theta| < 1$.

Answer 1

I suspect that the reason is that many ARMA fitting algorithms require that the ARMA system be stably invertible. In your example, the MA process is not minimum phase as the zero will be outside the unit circle. That makes the inverted system unstable (the pole will be outside the unit circle).

I don't know the details of the sm.tsa.ARMA(y, (1, 1)).fit algorithm, but I suspect that it will also be so restricted.

See, for example, this lecture by Florian Pelgrin, which contains the slide below.

In particular, note the second condition.

This means that I suspect it will be harder to find an algorithm that does the estimation for the case you're interested in.