In many applications, we want to explain a response series $Y_t$ using covariates while still accounting for autocorrelation. A standard approach is regression with ARMA errors:

$$ Y_t = \beta^T X_t + R_t $$

where the residual process $R_t$ follows an ARMA model:

$$ R_t = \phi_1 R_{t-1} + \cdots + \phi_p R_{t-p} + Z_t - \theta_1 Z_{t-1} - \cdots - \theta_q Z_{t-q} $$

with $Z_t$ as white noise.

Why This Matters

• Ordinary least squares assumes independent errors.
• Autocorrelated residuals lead to biased standard errors and misleading inference.
• Modeling the residuals as ARMA provides more reliable uncertainty estimates.

Practical Workflow

1. Fit a regression model for the deterministic part ($\beta^T X_t$).
2. Inspect residuals using ACF/PACF to identify ARMA structure.
3. Fit the combined model with ARMA errors using MLE or GLS.
4. Validate residuals to confirm they resemble white noise.

This approach blends explanatory modeling (regression) with time series dependence (ARMA), which is common in econometrics and forecasting.