autoregressiv

Autoregressiv, or autoregressive in English, describes a class of time series models in which the current value is a linear function of a finite number of its own past values plus a random shock. The term is widely used in statistics, econometrics, and signal processing to capture persistence and temporal dependence in observed data.

An AR(p) model expresses this idea as

X_t = c + φ1 X_{t-1} + φ2 X_{t-2} + ... + φp X_{t-p} + ε_t,

where ε_t is a white-noise term with E[ε_t] = 0 and Var(ε_t) = σ^2. The parameters to estimate are c and φ1, ..., φp. The order p determines how many lags influence the present value.

Stationarity is a central consideration. For an AR(p) process to be covariance stationary, the roots of the characteristic polynomial 1 − φ1 z − φ2 z^2 − ... − φp z^p = 0 must lie outside the unit circle (|z| > 1). If this condition is not met, the series may be nonstationary, potentially requiring differencing or other transformations before modeling.

Estimation and use: With a fixed p, the model can be estimated by ordinary least squares using observed X_t and its past values, or by maximum likelihood methods. Model order selection commonly relies on information criteria such as AIC or BIC. Autoregressive models are fundamental components of ARMA and ARIMA models and serve for forecasting, modeling residual correlation, and serving as benchmarks in time-series analysis. They can be extended to seasonal forms (SAR, SARIMA) and to include exogenous inputs (ARIMAX).