Patterns › Time series

ARIMA fit (manual or auto)

An ARIMA(p, d, q) model — or seasonal SARIMA(P, D, Q)[s] — fits the Box-Jenkins family to a time series, with manual or automatic order selection.

What is ARIMA fit (manual or auto)?

ARIMA decomposes a series into autoregressive (AR), integrated (I = differencing), and moving-average (MA) components. The (p, d, q) order specifies how many of each. Seasonal SARIMA adds (P, D, Q) on the seasonal lag for cyclical patterns.

Two workflows: manual Box-Jenkins (look at ACF/PACF, pick (p, q), check residuals, iterate) or automatic order selection (searches over candidate orders by AIC). Auto is faster and usually correct; manual is more interpretable when you understand the series.

When should I use ARIMA fit (manual or auto)?

  • Univariate time-series forecasting.
  • Modelling stationary series with autocorrelation.
  • Difference until stationary (d > 0 picks the differencing order).

What data does it need?

Numeric series + (p, d, q) order + optional seasonal (P, D, Q, period).

What does it report?

Coefficient table, σ², logLik, AIC, BIC, Ljung-Box test on residuals as a white-noise check.

What does it assume?

  • Stationarity after differencing.
  • Residuals are white noise (check via residual ACF + Ljung-Box).
  • Constant variance — log-transform if not.

How do I interpret the result?

Significant Ljung-Box on residuals ⇒ the model didn't capture all the structure; revisit (p, d, q).

Compare candidate models by AIC — differences < 2 are negligible.

See also

References

  • Box, Jenkins, Reinsel & Ljung (2015). Time Series Analysis: Forecasting and Control, 5th ed.