Determining AR Model Order Using Information Criteria: AIC, CAT and Other Methods

In time series analysis, the autoregressive (AR) model is a widely used forecasting method, where selecting the appropriate model order (p-value) constitutes a critical modeling step. Information criteria provide quantitative basis for this selection, primarily including metrics such as AIC (Akaike Information Criterion) and CAT (Criterion Autoregressive Transfer).

The AIC criterion, grounded in information theory, selects the optimal order by balancing model goodness-of-fit against complexity. Its calculation formula is AIC = 2k - 2ln(L), where k represents the number of parameters and L denotes the likelihood function value. Smaller AIC values indicate better model performance. Implementation note: When computing AIC programmatically, ensure proper maximum likelihood estimation for the AR parameters. The ar() function in MATLAB or fit() methods in Python's statsmodels library automatically calculate AIC values for different orders.

The CAT criterion is specifically designed for AR models, selecting the order by minimizing prediction error variance. Compared to AIC, CAT imposes stricter penalties on model complexity, making it particularly suitable for small sample sizes. Algorithmically, CAT involves computing the residual variance for different orders while applying progressive complexity adjustments.

In practical applications, the following steps are typically employed for order determination: - Preset a maximum candidate order P - Calculate AIC/CAT values for AR models at each order - Select the order minimizing the criterion value as optimal solution - Validate the selection using auxiliary tools like PACF plots Code implementation typically involves looping through orders 1 to P, fitting AR models, and extracting information criteria values for comparison.

Notably, different criteria may yield varying results. AIC emphasizes prediction accuracy, BIC (Bayesian Information Criterion) favors simpler models, while CAT focuses specifically on AR characteristics. Recommendation: Compare results from multiple criteria simultaneously and make final judgments based on practical business requirements. Many statistical packages provide automated functions like aic() or cat() that return criterion values directly from fitted model objects.