Classical✓ Mathematical
◆ The PatternRegressing a series on its own past
An AR(p) model predicts each value as a weighted sum of the previous p values plus noise. The weights (φ coefficients) capture how strongly the series depends on each lag. PACF cutoff at lag p identifies the order.
yt = c + φ1yt−1 + φ2yt−2 + … + φpyt−p + εt
Each φ coefficient tells you how much influence a past value has. Stationarity requires all roots of the characteristic polynomial to lie outside the unit circle.
// Interactive — AR(p) process simulation
φ10.70
Order (p)1
# Python — fit AR model from statsmodels.tsa.ar_model import AutoReg model = AutoReg(series, lags=3).fit() print(model.summary()) forecast = model.predict(start=len(series), end=len(series)+10)
Pattern bridge: AR models are time-series linear regression where the features are your own lagged values. In markets, the idea that past prices predict future prices is the foundation of momentum.