Per-step variance forecasts s_{T+1} … s_{T+h}, analytic: the first step
uses the observed last residual, later steps its expectation
(E[e²·[e<0]] = s/2), so s_{T+k} = ω + P·s_{T+k−1}. EGARCH's is
egarchForecast.
GARCH-X adds δ·x_{T+k−1} to each step: x_T is the fit's last value;
exogenous gives x_{T+1} … x_{T+h−1} (a model's forecast of x, or a
scenario), and without it x is held at x_T.
Per-step variance forecasts s_{T+1} … s_{T+h}, analytic: the first step uses the observed last residual, later steps its expectation (E[e²·[e<0]] = s/2), so s_{T+k} = ω + P·s_{T+k−1}. EGARCH's is
egarchForecast.GARCH-X adds δ·x_{T+k−1} to each step: x_T is the fit's last value;
exogenousgives x_{T+1} … x_{T+h−1} (a model's forecast of x, or a scenario), and without it x is held at x_T.