Time Series Analysis: AR, MA, and ARMA Models for Exam SRM

Autoregressive (AR) Models

An AR(p) model expresses the current value as a linear combination of past values plus white noise: Y_t = phi_1*Y_{t-1} + phi_2*Y_{t-2} + ... + phi_p*Y_{t-p} + epsilon_t, where epsilon_t ~ WN(0, sigma^2). For AR(1): Y_t = phi*Y_{t-1} + epsilon_t. The process is stationary when |phi| < 1, with mean 0, variance sigma^2/(1-phi^2), and autocorrelation function (ACF) rho(k) = phi^k, which decays exponentially.

The partial autocorrelation function (PACF) of an AR(p) process cuts off at lag p, while the ACF tails off. This pattern helps identify the order of an AR model from data.

Moving Average (MA) Models

An MA(q) model expresses the current value as a linear combination of current and past error terms: Y_t = epsilon_t + theta_1*epsilon_{t-1} + ... + theta_q*epsilon_{t-q}. For MA(1): Y_t = epsilon_t + theta*epsilon_{t-1}. An MA process is always stationary. The ACF of MA(q) cuts off at lag q, while the PACF tails off. This is the opposite pattern from AR models.

ARMA Models

An ARMA(p, q) model combines AR and MA components: Y_t = phi_1*Y_{t-1} + ... + phi_p*Y_{t-p} + epsilon_t + theta_1*epsilon_{t-1} + ... + theta_q*epsilon_{t-q}. Both the ACF and PACF tail off for ARMA models, making order identification harder. In practice, examine the ACF and PACF patterns, fit candidate models, and compare using AIC or BIC. For Exam SRM, be able to identify AR, MA, and ARMA models from ACF/PACF patterns, compute forecasts, and understand stationarity conditions.