Lag Length Selection

An autoregressive model needs a decision on how many past periods, $p$, to include: $x_t = \phi_1 x_{t-1} + \dots + \phi_p x_{t-p} + \varepsilon_t$. Choose too few lags and genuine autocorrelation gets left in the residuals — the model is under-fit and its forecasts and standard errors are both wrong. Choose too many and the model starts fitting noise in far-back lags whose true coefficients are zero, inflating estimation error and often hurting out-of-sample forecasts even though in-sample fit looks better.

The standard fix is an information criterion — Akaike (AIC) or Bayesian (BIC) — that scores each candidate lag length by its fit penalized by the number of parameters used, then picks the $p$ that minimizes the score. BIC penalizes extra lags more heavily than AIC and so tends to select shorter, more parsimonious models, which matters because AIC is known to over-select lag length asymptotically while BIC is consistent for the true order under standard conditions.

Worked example. Fitting AR models of order 1 through 6 to a series might give AIC scores of 850, 831, 828, 829, 830, 831 — the score bottoms out at $p=3$ and creeps back up afterward, so AIC selects $p=3$: adding a fourth lag's tiny fit improvement doesn't outweigh the extra parameter's penalty.