Order Selection for Time Series Models#

Stage 6 · 🏗️ Building & Forecasting Models · Lesson 14 of 18 · advanced

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Important

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The trade-off#

Every extra parameter improves the in-sample fit but risks overfitting — chasing noise that will not repeat. Order selection is the search for a model that is complex enough to fit, simple enough to generalise: the parsimony principle at the heart of Box–Jenkins.

Information criteria#

The standard tools score fit against complexity. Both the Akaike and Bayesian information criteria reward the likelihood and penalise the parameter count \(k\):

\[\mathrm{AIC} = 2k - 2\ln \hat{L}, \qquad \mathrm{BIC} = k\ln n - 2\ln \hat{L}.\]

Lower is better, and you compare candidates fitted to the same data. (The small-sample correction AICc is safer when \(n\) is not large.)

AIC versus BIC#

The two differ only in the penalty. BIC’s per-parameter cost \(\ln n\) is harsher than AIC’s \(2\) (once \(n > 7\)), so BIC favours simpler models and is preferred when parsimony matters; AIC tends to pick slightly richer models and suits predictive accuracy. When they disagree, the choice is yours to justify.

Putting it together#

In practice: pick d by differencing until stationary (ADF / KPSS), read tentative p, q off the ACF / PACF, then grid-search nearby orders and keep the lowest-criterion model that also passes diagnostics — a lower AIC means nothing if the residuals are still autocorrelated. Tools like pmdarima.auto_arima automate the search; statsmodels exposes .aic and .bic.

Hint