Volatility Forecasting Benchmarks: GARCH, HAR, and ML on Equity Indices

1. Target Definition: Realised Volatility

Before you can compare GARCH, HAR, and ML on a level field, you have to agree on what they are predicting. Volatility is never observed directly — it has to be estimated from returns — so the first decision in any benchmark is the target. For an index with daily returns r_t, define the realised volatility target as:

σ_{t+1}^{real} = √(Σ_{i=0}^{h-1} r_{t+1+i}^2)

for horizon h (often 1 or 5 days). Alternatively, use high-frequency-based realised measures where available.

2. Model Classes

2.1 GARCH Models

Standard GARCH(1,1):

r_t = σ_t ε_t, ε_t ~ N(0,1)
σ_t² = ω + α r_{t-1}² + β σ_{t-1}²

Variants: EGARCH, GJR-GARCH (asymmetry), long-memory variants, different error distributions.

2.2 HAR Model

HAR-RV model represents volatility as:

RV_{t+1} = β₀ + β₁ RV_t^{(d)} + β₂ RV_t^{(w)} + β₃ RV_t^{(m)} + ε_t

where RV_t^{(d)} is daily, RV_t^{(w)} is weekly average, RV_t^{(m)} is monthly average. This captures multi-horizon volatility dynamics in a simple linear framework.

2.3 ML Models

Input features may include:

lagged realised vol, RV d/w/m
lagged returns
volatility-of-volatility measures
macro proxies, VIX, term structure of implied vol (if available)

Models: Ridge / ElasticNet, Random Forest, Gradient Boosted Trees (XGBoost/LightGBM), MLP / small LSTM on volatility features.

Targets: next-day or next-5-day realised volatility (or log-vol to stabilise).

3. Benchmark Design

To avoid typical pitfalls:

Strict time-based splits
Rolling or expanding re-estimation (walk-forward)
Non-overlapping evaluation windows where feasible
Multi-criteria evaluation: RMSE / MAE on volatility, R² on log-vol, accuracy of volatility buckets, performance of strategies that use the forecasts

4. From Vol Forecast to Trading Performance

One useful research question: "Does a better volatility forecast translate into better risk-adjusted returns when used for position sizing?"

Example: vol targeting strategy w_t = τ / σ̂_t where τ is target risk, σ̂_t is forecast vol.

We can compare performance using GARCH-based vol, HAR vol, and ML vol. This closes the loop from forecasting metrics to trading metrics.

5. How This Maps to volarixs

A benchmark like this is really just a set of experiments run on a common footing, and that is the shape of an experiment in volarixs. You define the target as a realised volatility series, pick datasets and a feature set, choose the model class, and set a target horizon and time window:

the contenders map onto selectable model classes — Volatility for GARCH-type models, Time Series for HAR-style lag structures, and Regression / Tree & Boosted / Neural Networks for the ML side
each model is fit walk-forward over rolling windows, so the time-based splits this article argues for are how training runs by default rather than something you bolt on
every run is stored with its model, datasets, targets, status, and train/test R² — a durable record of each contender on the same target

That shared, per-run record is the foundation a benchmark is built on: with each model held to the same target, horizon, and window, the forecasts become genuinely comparable. The wider evaluation this article describes — error metrics beyond R², calibration, and closing the loop into vol-targeting strategy performance — is the lens volarixs is built around, layered on top of that stored history rather than a separate one-click report.

Volatility

GARCH

HAR

Benchmarks

Forecasting