ARIMA

What it does

Models a differenced series with AR and MA terms, with diagnostics, forecasts, intervals, AIC and BIC.

Use it for non-seasonal series that need differencing.

Data you need

• One series
• Orders p, d and q

Supported data: Time series. Level: Intermediate.

Formula and method

φ(L)(1−L)ᵈyₜ = c + θ(L)εₜ

The series is differenced d times, and an ARMA(p, q) is estimated by state-space maximum likelihood. Forecasts of the differenced series are integrated back to the original scale, and their intervals widen with the horizon.

How to read the result

Compare AIC and BIC across orders and check residual diagnostics.

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 = 2,191.81, BIC = 2,205.04.
• Next-period forecast: 13,031.53, 95% interval 12,926.02 to 13,137.04.
• Last-period forecast (2011Q3): 13,367.95, 95% interval 12,852.23 to 13,883.68.
• 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 17 of 17 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

How many differences should I use?

Usually one is enough. Use unit-root tests, and avoid over-differencing.