We compare the Heterogeneous Autoregressive
(HAR) model with a novel Continuous Memory System (CMS) for forecasting
realized volatility. CMS employs 12 exponential moving averages with adaptive
decay rates modulated by learned, level-specific shock sensitivities through a
rank-1 gating mechanism. The response of each memory level to volatility shocks
is governed by an optimized sensitivity parameter that determines whether the
level accelerates or decelerates during turbulent periods. Using 1,234 daily
observations from February 2021 to January 2026, we estimate the model through
bounded constrained optimization and compare its performance with that of the
parsimonious HAR benchmark.
CMS
learns a surprisingly intuitive pattern in how different time horizons respond
when markets become turbulent. Short-term memory reacts aggressively to
volatility spikes, updating rapidly to capture sudden regime shifts. Medium- to
long-term memory behaves in the opposite way, slowing down sharply during
stress to preserve a stable baseline, with the strongest dampening occurring at
horizons of roughly 4 to 6 weeks (levels 10 and 11). This creates an asymmetric
response pattern: high reactivity at short horizons and strong stabilization at
medium to long horizons. Notably, the model discovers this structure
automatically from the data, without being explicitly designed to behave in
this way, and the resulting pattern aligns closely with financial intuition
about how different forecast horizons should weight past information during
volatile periods. The sole exception is the 60-day horizon (level 12), which
exhibits a large positive sensitivity. This may reflect distinct very long-term
dynamics, or it may be an overfitting artifact, so it should be interpreted
with caution.
Despite
its theoretical appeal, CMS underperforms in practice. Its out-of-sample
forecasting error is 30% higher than that of HAR, even though it fits the
training data extremely well. This is consistent with the classic problem of
overfitting, in which a model captures historical patterns too closely and then
fails to generalize well to new observations. The added complexity of CMS, with
25 tunable parameters versus HAR’s 4, appears to be a liability rather than an
asset in a limited-sample setting. The sophisticated level-specific shock
responses also provide almost no forecasting improvement, only 0.13%, over a
simpler uniform-gating specification, while also creating numerical instability
at very short horizons. Ultimately, HAR’s simplicity is a strength: fewer
parameters leave less room for overfitting, making the model more reliable out
of sample. At the same time, the response patterns learned by CMS, particularly
how different forecast horizons adjust to volatility shocks, provide useful
economic intuition that may help guide the design of better hybrid models in
future work.
