Autoregressive (AR) is a term commonly used in time series analysis and statistics to describe a model where a variable is regressed against its own past values. In other words, an autoregressive model is a mathematical representation that uses the relationship between an observation and several past observations (lags) of the same variable.

The notation for an autoregressive model of order \(p\) is often denoted as AR(p), where \(p\) represents the number of lags included in the model.

The general form of an AR(p) model is:

\[ Y_t = \phi_0 + \phi_1 \cdot Y_{t-1} + \phi_2 \cdot Y_{t-2} + \ldots + \phi_p \cdot Y_{t-p} + \varepsilon_t \]

– \( Y_t \) is the value of the time series variable at time \(t\),
– \( \phi_0 \) is a constant term,
– \( \phi_1, \phi_2, \ldots, \phi_p \) are the autoregressive coefficients,
– \( Y_{t-1}, Y_{t-2}, \ldots, Y_{t-p} \) are the lagged values of the variable,
– \( \varepsilon_t \) is the error term at time \(t\), representing the unobserved factors affecting the variable.

Key points about autoregressive models:

1. **Order \(p\):** The order of the autoregressive model (\(p\)) indicates how many past values of the variable are used in the regression. For example, AR(1) uses only the immediate past value, AR(2) uses two past values, and so on.

2. **Stationarity:** The stability and stationarity of the time series data are important considerations when using autoregressive models. Non-stationary data may lead to spurious results.

3. **Parameter Estimation:** The autoregressive coefficients (\( \phi_1, \phi_2, \ldots, \phi_p \)) are estimated from the data using statistical methods.

4. **Forecasting:** Autoregressive models are often used for time series forecasting. Forecasted values are generated based on past observations and the estimated coefficients.

Autoregressive models are part of a broader class of models used in time series analysis, including moving average (MA) models and autoregressive integrated moving average (ARIMA) models. These models are widely used in fields such as finance, economics, signal processing, and meteorology for analyzing and forecasting time-dependent data.