Time Series Forecasting: ARIMA and Seasonal Models
Apply ARIMA and seasonal time series models for actuarial forecasting on Exam MAS-I.
ARIMA Models
ARIMA(p,d,q) combines autoregressive (AR), differencing (I), and moving average (MA) components. The AR(p) model: Y_t = phi_1*Y_{t-1} + ... + phi_p*Y_{t-p} + epsilon_t. The MA(q) model: Y_t = epsilon_t + theta_1*epsilon_{t-1} + ... + theta_q*epsilon_{t-q}. Differencing (order d) makes non-stationary series stationary. Model identification uses the autocorrelation function (ACF) and partial autocorrelation function (PACF): AR(p) has PACF cutting off at lag p, MA(q) has ACF cutting off at lag q, and ARMA shows both tailing off.
Seasonal Models
Seasonal ARIMA (SARIMA) adds seasonal AR and MA terms: ARIMA(p,d,q)(P,D,Q)_s where s is the seasonal period. Seasonal differencing removes periodic patterns. For monthly data with annual seasonality, s=12. The model captures both short-term autocorrelation and seasonal patterns. Box-Jenkins methodology involves identification (ACF/PACF plots), estimation (MLE), and diagnostic checking (residual analysis, Ljung-Box test). Exam MAS-I tests model identification from ACF/PACF plots, parameter estimation, forecasting, and forecast interval construction.