I am wondering if this is a valid approach:
I want to validate that certain random processes are not weakly stationary (constant mean, covariance depends on the lag, finite variance) through simulation; not necessarily with an incredibly rigorous procedure, but at keast for intuition building. I know unit root tests like the ADF are used, I am not sure if its necessarily an omnibus test for stationarity.
Statsmodel in python has functions that allow you to give parameters for an arma model and it generate time series. For each process, I want to generate 10,000 time series of length 1000, with short seasonality, such as every 5 points. (is this even a valid starting point to see properties of the random process, generate an ensemble from the generator and look at its properties?)
A few things processes I’m trying out include
- Pure Seasonal ARMA (no unit roots)
- Mixed Seasonal ARMA (no unit roots)
- Generating a deterministic sine wave and a series from ARMA, adding them together.
- Generating a deterministic sine wave and a series from ARMA, multiplying them together.
Then I can do an ensemble mean to see whether the mean is constant; if it isn’t constant, its probably not weakly stationary.
If the mean appears constant, then I would look at the autocovariance; but I am not sure how this would work. The formula for calculating autocovariance for a series seems to depend on the assumption of the series being weakly stationary, but that's what I am trying to prove.
Also getting back to the mean, I did look at a few examples:
For cases of a deterministic sine wave+ stationary AR(1), the ensemble mean tends to look like this, which seems obviously not a constant mean throughout time.
But for Pure Seasonal ARMA with no unit roots; If I have $x_{t}=0.7x_{t-5}+w_{t}, \sigma=1$ or $x_{t}=0.7x_{t-7}+w_{t}, \sigma=1$
The ensemble means look like the following graphs, where it looks like it varies, but the values are very small; is there a way to judge this?
