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BUG: Use the expected variance path in RiskMetrics2006 forecasts - #934
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The analytic forecast repeated the one-step forecast for every horizon, which is only exact for a single EWMA. Iterate the expected values of the EWMA components instead, so multi-step analytic forecasts agree with the simulation forecasts.
The expected components follow a linear recursion, so iterating them is the same as applying horizon-dependent weights to the final component values, which is the form used in the RiskMetrics 2006 papers.
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Supersedes #885 by @raashish1601 (his two commits are unchanged; this branch adds a merge of
current main). Thanks @raashish1601.
Multi-step analytic forecasts from
RiskMetrics2006repeated the one-step forecast at everyhorizon. The RM2006 (Zumbach) process is linear in its EWMA components, so the h-step
forecast is
w' A^(h-1) s(t+1)withA = diag(mu) + (1 - mu) w'; this computes it ashorizon-dependent weights on the component values. One-step forecasts are unchanged.
Checked independently (release audit): the forecasts match a matrix-power reference to
5.6e-16 and arch's simulation forecasts (100,000 paths) within 1.4 standard errors at every
horizon; on main the flat forecast was up to 24% too low at h=10 and 120 standard errors
from simulation. Fast suite on this branch: 8154 passed, 17 skipped, 0 failed.
Supersedes #885.
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