Autoregressions
Autoregressions are a class of statistical models for time series in which the current observation is expressed as a function of its own past values and a stochastic term. The most common form is the autoregressive model of order p, denoted AR(p). It specifies X_t = c + phi1 X_{t-1} + ... + phip X_{t-p} + epsilon_t, where epsilon_t is white noise and c is a constant. The model assumes stationarity, meaning its statistical properties do not change over time; this requires the roots of the characteristic polynomial 1 - phi1 z - ... - phip z^p = 0 to lie outside the unit circle.
Estimation and model selection: Parameters phi1 through phip can be estimated by ordinary least squares for
Relation to other models: Autoregressions are central to broader families such as ARIMA, which incorporate differencing
Applications and limitations: Autoregressions are widely used for forecasting in economics, finance, meteorology, and other domains