Does Anything Beat a Naive Forecast? Machine Learning, GARCH, and Equity Volatility

Motivation & Research Questions

Machine learning keeps reopening Hansen & Lunde's (2005) question — "does anything beat a GARCH(1,1)?" — but most horse races stop at one loss function and skip the naive benchmark. This project asks three questions of eleven volatility forecasters — including a naive persistence floor — at once:

RQ1 — Accuracy. Do ML/hybrid models significantly beat econometric benchmarks and a model that does no fitting at all, once corrected for testing eleven models at once, under more than one loss function?
RQ2 — Economic value. Does that accuracy translate into better risk-adjusted portfolio returns, net of costs?
RQ3 — Resilience. Does any edge survive across regimes, sub-periods, and cost/exposure assumptions?

Data & Design

18 U.S. equities (3 indices + 15 large-caps), daily 2010–2025. Expanding-window walk-forward: trained through Dec 2017, 96 monthly refits tested Jan 2018–Dec 2025, forecasting 1/5/20-day realized volatility (21-day rolling std of log returns, annualized).

Econometric GARCH(1,1) · EGARCH(1,1) · GJR-GARCH(1,1) · HAR-RV · HAR-X

Naive baseline Persistence (tomorrow = today, no fitting)

Machine learning Random Forest · XGBoost · LSTM

Hybrid GARCH-RF · GARCH-XGBoost

Portfolio Strategies

Four monthly-rebalanced strategies translate a forecast into a position, each testing a different economic mechanism:

Strategy	Mechanism
Vol-Timing	SPY/bond, target 15% vol
Risk-Parity	15-stock, inverse-vol weights
Regime-Based	Discrete equity/bond by regime
VRP	Harvests implied–forecast spread

$5/trade transaction cost on every strategy; all metrics from log returns, equity compounded as exp(cumsum(r)).

Result 1 — Nothing Beats a Naive Floor (It Depends on the Loss Function)

0 / 11

Models beating Persistence at 1-day, squared-error

3 / 11

Survive 90% MCS: HAR-X, HAR-RV, Persistence

2 / 11

Survive 90% MCS under QLIKE instead: XGBoost, GARCH-XGBoost

Under squared-error loss, a trivial "tomorrow's volatility equals today's" forecast is essentially unbeatable at the 1-day horizon: every GARCH-family, tree-based, LSTM, and hybrid model is significantly worse than it (Diebold-Mariano, all p<0.001). Switch to QLIKE — a loss the literature treats as more appropriate for volatility forecasts — and the ranking inverts: XGBoost and GARCH-XGBoost become the survivors instead. Which model "wins" depends entirely on which question is asked.

Heatmap of 1-day RMSE by model (rows) and asset (columns); HAR-X, HAR-RV, and Persistence rows are lightest/best across nearly every column, Random Forest is not the best model on any single asset

Fig. 11. 1-day-ahead RMSE, 11 models × 18 assets. The three persistence-adjacent models (HAR-X, HAR-RV, Persistence — top rows) are lightest — lowest error — almost everywhere; Random Forest is the best model on zero individual assets.

But this 1-day edge evaporates by the month: the same three models that lead at 1-day are among the worst performers once benchmarked against implied volatility at the 21-day horizon — a forecast built to track tomorrow's value carries little information about a month ahead.

Result 2 — Mechanism Beats Model

Sharpe ratio, sorted by strategy (full sample; benchmarks: Buy&Hold SPY = 0.53, Equal-Weight 15-Stock = 0.64):

Strategy	Best	Worst
VRP	2.96	1.20
Regime-Based	0.71	0.58
Vol-Timing	0.51	0.44
Risk-Parity	0.22	0.21

Strategies needing accurate point forecasts (Vol-Timing, Risk-Parity) lose to passive benchmarks under every one of the eleven models. Strategies resting on a structural mechanism (Regime-Based, VRP) beat their benchmarks under every model — and best-in-class for the strongest strategy (VRP) is GJR-GARCH(1,1), a plain econometric model.

Result 3 — Sharpe Ratio by Model and Strategy

Grouped bar charts of Sharpe ratio by model, faceted by strategy, colored by model family (econometric blue, ML orange, hybrid violet, naive gray)

Fig. 9. Sharpe ratio by model (colored by family: econometric / ML / hybrid / naive) faceted by strategy. No single model or family wins every panel — and VRP, the strategy with by far the largest payoff, is won by a plain econometric model at negligible compute cost.

Cost vs. Benefit

~24 hrs

LSTM walk-forward compute

LSTM's 86,864 seconds (~24 hours) of walk-forward compute buys a +0.824 Sharpe gain over GARCH(1,1) in its best strategy (VRP). GJR-GARCH(1,1) — fit in just 3.6 seconds, over 24,000× cheaper — buys +0.833, the best economic gain of any model in the study, with HAR-X (+0.730) and HAR-RV (+0.660) close behind at essentially zero marginal compute (a single OLS fit). Measured per unit of compute, the cheapest models win outright.

Robustness Checks

Cross-asset: HAR-X has the lowest RMSE on 11/18 assets, Persistence on 5/18, HAR-RV on 2/18 — Random Forest wins on zero individual assets.
Cost/exposure sensitivity: sweeping VRP's transaction cost ($0–$50/trade) and exposure cap (0.5–3x) for the two highest-Sharpe VRP combinations, GJR-GARCH(1,1)'s edge over LSTM widens as costs rise (2.8% Sharpe decline vs. LSTM's 6.3%) — the cost-benefit case for the cheap model is not an artifact of the paper's $5/trade assumption.
Sub-period stability: zero Sharpe sign flips, 11 models × 2 strategies × pre/post-2022 halves.

Conclusion

No model beats a naive "tomorrow equals today" forecast under squared-error loss — and which model does "win" statistically flips entirely once QLIKE is used instead. Statistical accuracy at 1-day is nearly irrelevant to portfolio performance: strategies that need it (Vol-Timing, Risk-Parity) lose to passive benchmarks under every model, while strategies resting on a structural mechanism (Regime-Based, VRP) win under every model — and the largest economic payoff is captured by an econometric model costing a rounding error of compute next to the LSTM's 24-hour walk-forward. Match model complexity to the forecasting horizon and the strategy's mechanism, and never report an accuracy claim without checking it against a naive floor and more than one loss function.

Selected References

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.

Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2), 174–196.

Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13(3), 253–263.

Fleming, J., Kirby, C., & Ostdiek, B. (2001–2003). The economic value of volatility timing. Journal of Finance / Journal of Financial Economics.

Gu, S., Kelly, B., & Xiu, D. (2020). Empirical asset pricing via machine learning. Review of Financial Studies, 33(5), 2223–2273.

Hansen, P. R., & Lunde, A. (2005). Does anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20(7), 873–889.

Hansen, P. R., Lunde, A., & Nason, J. M. (2011). The model confidence set. Econometrica, 79(2), 453–497.

Moreira, A., & Muir, T. (2017). Volatility-managed portfolios. Journal of Finance, 72(4), 1611–1644.

Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, 160(1), 246–256.