ARMA

What it does

Combines autoregressive and moving-average terms for a stationary series.

Use it when neither a pure AR nor a pure MA fits well.

Data you need

• One series
• Orders p and q

Supported data: Time series. Level: Intermediate.

Formula and method

yₜ = c + Σφᵢyₜ₋ᵢ + εₜ + Σθⱼεₜ₋ⱼ

Autoregressive and moving-average terms are estimated jointly by exact Gaussian maximum likelihood in state-space form. Stationarity and invertibility are enforced so that forecasts remain well behaved.

How to read the result

Prefer the simplest model with clean residuals.

Worked example

Illustrative run on US quarterly macro 1959-2009. These figures are the actual output of this model on that example, and they say nothing about your own data.

• AIC = 920.1512, BIC = 933.404.
• Next-period forecast: 2.2427, 95% interval -2.2318 to 6.7172.
• Last-period forecast (2011Q3): 2.794, 95% interval -2.9008 to 8.4887.
• The residuals look like white noise, which is what a well-specified model should leave behind.
• Forecast intervals widen with the horizon: uncertainty grows the further ahead you look. A forecast is an estimate, not a prediction of what will happen.

Limitations

• Results depend on the sample period

Numerical validation

Validated 16 of 16 automated checks passed, including numerical results, invalid input, missing values, small samples, constant variables, charts and the interpretation text.

Computed with python 3.13.2, numpy 2.2.5, scipy 1.14.1, pandas 2.3.3, statsmodels 0.14.4, scikit-learn 1.7.0, in your browser. A passing check confirms the listed comparison only. It does not make any result suitable for your data, and it is not a claim of equivalence with other statistical software.

Questions

Should my series be stationary?

Yes. If it is not, difference it or use ARIMA.