How to Test Residuals for White Noise and Choose ARMA Models with AIC/BIC

White‑Noise Test for Residuals

ARMA model identification and estimation assume that the random disturbance term is white noise; therefore, if the estimated model is correct, its residuals should form a white‑noise series. When sample residuals are not white noise, the model identification or estimation is flawed and must be redone. In practice, the main check is whether the residual series exhibits autocorrelation.

The Q‑test statistic can be used for this purpose: for a given significance level, compute Q‑statistics at various lags and compare them with critical values from the appropriate distribution table. If a Q‑value exceeds its critical value, the hypothesis that the residuals are white noise is rejected, indicating that the estimated model should be reconsidered.

AIC and BIC Model Selection Criteria

When identifying ARMA models, multiple parameter sets may pass the identification tests. Increasing the orders of the AR and MA components improves goodness‑of‑fit but reduces degrees of freedom, creating a trade‑off between model simplicity and fit.

The common model‑selection criteria are Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (BIC, also called SBC). Both are computed as:

$$\text{AIC} = 2k - 2\ln(L)$$

$$\text{BIC} = k\ln(n) - 2\ln(L)$$

where k is the number of estimated parameters (including any constant term), n is the number of usable observations, and RSS is the residual sum of squares. Smaller AIC or BIC values indicate a better model. Adding lag terms that do not improve explanatory power will increase RSS only slightly while raising k, causing AIC/BIC to increase. Comparisons must be made over the same time period across models.