Forecasting with Autoregressive (AR) Models

Predict future values using past observations (Autoregression), assuming linear dependence on previous time steps.

Procedure

1. Ensure Stationarity

   • Remove trend and seasonality via differencing or decomposition.
   • Test for stationarity with the ADF test.

2. Model Identification

   • Use PACF to determine AR order $p$.
   • Fit an AR($p$) model: $X_t=c+{\varphi }_1{X}_{t-1}+{\varphi }_2{X}_{t-2}+\cdots +{\varphi }_p{X}_{t-p}+{ϵ}_t$

3. Model Fitting & Diagnostics

   • Estimate coefficients via Maximum Likelihood or Least Squares.
   • Check residuals: should resemble white noise (no autocorrelation).

4. Forecasting

   • Generate forecasts ${\hat{X}}_{t+h}$ recursively using the fitted coefficients.
   • For differenced data, reconstruct forecasts by reversing the differencing operation.

5. Interpretation

   • AR models describe persistence or memory in data.
   • A slow decay in ACF or significant PACF lags indicates how many past values drive predictions.