What is the difference between autoregressive (AR) and moving average (MA) models?

Autoregressive (AR) and moving average (MA) models are foundational in time series analysis, but they capture different aspects of temporal patterns. AR models predict future values based on a linear combination of past observations. For example, an AR(1) model uses the immediately preceding value: ( y_t = c + \phi_1 y_{t-1} + \epsilon_t ), where ( \phi_1 ) is the coefficient and ( \epsilon_t ) is noise. In contrast, MA models forecast using past forecast errors rather than past data points. An MA(1) model is defined as ( y_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} ), where ( \theta_1 ) weights the prior error. The key distinction is their input: AR relies on lagged values of the series, while MA incorporates lagged errors.

These models address different types of dependencies. AR models excel at capturing trends or inertia in data, such as temperature readings where today’s value depends on yesterday’s. For instance, if a stock price tends to drift upward after a rise, an AR model could capture this momentum. MA models, however, are better suited for sudden, short-lived shocks. Imagine a manufacturing process where a machine error disrupts output for one day; an MA model would account for the residual impact of that error in subsequent predictions. MA’s focus on errors makes it effective for scenarios where recent anomalies influence future values, but their effects fade quickly.

Practically, AR models require the time series to be stationary (mean and variance stable over time), often achieved through differencing. MA models, by design, handle noise more directly and are inherently stationary. Developers often combine both into ARMA models to leverage their strengths: AR captures long-term trends, while MA addresses transient noise. For example, predicting monthly sales might use AR(2) to model seasonal patterns and MA(1) to account for temporary supply chain disruptions. Understanding these differences helps in selecting the right tool—AR for persistent trends, MA for abrupt, short-term fluctuations—or blending them for complex datasets.