Time-Series Forecasting #1

ARIMA / SARIMA

  • Box-Jenkins
  • SARIMAX

Best for: Classical univariate forecasting Aliases: Box-Jenkins, SARIMAX

How it works

$$y_t=c+\sum_{i=1}^{p}\phi_i\,y_{t-i}+\sum_{j=1}^{q}\theta_j\,\varepsilon_{t-j}+\varepsilon_t$$

Models a (possibly differenced) series as a linear combination of its own lags and past shocks, $y_t=c+\sum_{i=1}^{p}\phi_i y_{t-i}+\sum_{j=1}^{q}\theta_j\varepsilon_{t-j}+\varepsilon_t$, where $\varepsilon_t$ is white noise. The order $(p,d,q)$ fixes the autoregressive depth, the differencing needed for stationarity, and the moving-average window; SARIMA adds seasonal $(P,D,Q)^s$ terms reflecting the same structure at lag $s$. Coefficients are fit by maximum likelihood, and orders are chosen by minimising AIC/BIC after checking stationarity and inspecting the ACF/PACF.

When to use

Univariate forecasting with strong autocorrelation/seasonality that differencing and AR/MA terms can capture.

Watch out

Stationarity is required; manual order selection is finicky; poor for multivariate or multi-seasonal data.

Common fields

Economics · demand · energy · finance