Does GARCH(1,1) beat naive volatility forecasts out of sample?

One full turn of the volatility-research loop on ~20 years of S&P 500 daily returns: verify the stylized facts, fit GARCH(1,1), produce one-day-ahead variance forecasts on a 30% holdout with frozen parameters, and score them against two naive benchmarks with a proxy-robust loss.

TL;DR

Using S&P 500 daily log returns from 2005, with a 70/30 chronological split and 1,629 test days, we find that GARCH(1,1)-N gives a solid reduction of 24.6% QLIKE and 25.7% MSE against unconditional volatility, and a modest reduction of 5.5% QLIKE and 16.8% MSE against a 21-day rolling variance random walk. These are one-day-ahead variance forecasts scored against squared returns, with parameters frozen at their training estimates. We are still missing the error bars that would support the apparent positive verdict.

Our test window opens on 7 February 2020, so training is entirely pre-COVID, which heavily penalizes the constant baseline. We fit both a normal and a Student-t innovation distribution and found the Gaussian slightly better (1.5375 vs 1.5436 QLIKE).

Summary of findings

• QLIKE over the test period: GARCH-N 1.54, random walk 1.63, unconditional 2.04
• Improvement against unconditional 24.6%, against the random walk 5.5%
• Fitted persistence α+β = 0.9773, shock half-life ≈ 30 trading days
• MSE gives the same ordering as QLIKE, so the ranking is not an artifact of the loss we chose

The question, and why it matters

The sign of tomorrow's return is nearly unpredictable, but its magnitude is strongly predictable. That is what makes a volatility forecast worth building at all. Volatility is an input to position sizing, to risk limits and VaR, and to option pricing, so the question is whether a standard conditional-variance model earns its complexity against forecasts anyone can produce in one line.

Data

We use the S&P 500 index (^GSPC) daily close from 2005 to 3 August 2026, obtained via yfinance, which gives 5,428 daily log returns. We request split- and dividend-adjusted prices, which protects the pipeline if it is ever pointed at a stock or an ETF, where unadjusted closes carry a fake return on every ex-dividend day and a fake crash at every split. For a price index such as ^GSPC the adjustment is a no-op, since it pays no dividends and does not split.

We work in trading time, using the series as it comes, and do not reindex to calendar days, which would inject artificial zero returns across weekends and holidays. Prices are cached under data/, which is not committed.

What the data demand of a model

• Daily returns are uncorrelated, but they are not independent. The dependence survives in the higher moments.

• Volatility clustering. High-volatility days are followed by high-volatility days, and the ACF of |r| stays positive out to 100 lags, decaying far more slowly than the exponential a GARCH(1,1) implies.

  

• Fat tails. Fitting a Student-t to the returns gives ν̂ = 2.4 degrees of freedom. Aggregating to weekly or monthly returns thins the tails, though slowly, as the central limit theorem grinds against both the dependence and the tail weight.

  

• Asymmetry. The sign of today's return affects tomorrow's volatility: negative returns are followed by higher volatility than positive returns of the same size. A symmetric model cannot reproduce this.

Model and evaluation design

We fit GARCH(1,1) with Gaussian and Student-t innovations by maximum likelihood on the first 70% of the sample. The chronological split puts the last training day at 6 February 2020 and opens the test window on 7 February 2020.

Over the remaining 30% we produce one-step-ahead variance forecasts with the parameters frozen at their training estimates, with no refitting. The benchmarks are a 21-day random-walk variance and the unconditional training variance. The proxy is the squared daily return.

QLIKE is the primary loss and MSE the secondary one. Patton (2011) shows that under a noisy proxy many intuitive losses systematically misrank models, and characterizes the class of losses whose rankings survive the noise; MSE and QLIKE are its two most-used members. There are no zero-return days in the test window, so no observation had to be dropped or clipped before taking the logarithm in QLIKE.

Results

We report one-step-ahead variance forecasts over the 1,629 test days (2020-02-07 to 2026-08-03), in daily variance units of %² per day.

Forecast	QLIKE	MSE
GARCH(1,1)-N	1.537518	34.937572
GARCH(1,1)-t	1.543563	35.575560
Random Walk	1.627217	41.970833
Unconditional	2.038582	46.998799

GARCH(1,1)-N ranks first on both losses. The gap to the unconditional benchmark is large and the gap to the random walk is small, which is the expected shape at a one-day horizon: daily volatility is persistent enough that "tomorrow looks like the recent past" is genuinely hard to beat.

Plotting the 21-day rolling volatility against the GARCH(1,1)-N conditional volatility shows the difference in responsiveness between a boxcar filter with equal weights and an exponentially weighted one whose decay is estimated from the data.

Limitations, and what I would do next

• The leverage effect measured in notebook 01 is unmodelled, since GARCH(1,1) is sign-blind. The model has to be extended to account for this.
• The ACF of |r| decays more slowly than any exponential (fig1), which points to HAR-RV on true realized volatility.
• A single split with frozen parameters should become walk-forward re-estimation.
• The QLIKE gap carries no significance statement. These should be computed to support the apparent positive verdict.
• We are comparing against the squared returns (a noisy proxy), not the true volatility itself. We could repeat the analysis with other proxies.

Reproduce it

Requires Python 3.14 or later.

uv sync                 # creates .venv and installs everything, incl. dev tools
uv run pytest           # the suite specifies the library functions
uv run jupyter lab      # run notebooks/01, then notebooks/02

Conventions

• Log returns throughout. Notebook 02 works in %²-per-day variance units.
• rolling_vol at t includes t, so it is an estimate at t. rw_variance_forecast at t excludes t, so it is a forecast for t. The two are one day apart on purpose.
• Chronological splits only. Nothing computed at time t may touch data dated after t.
• QLIKE primary, MSE secondary, no other losses.

How this was built

The scaffold of this repository was AI-generated under the constraints written down in CLAUDE.md: the test suite as an executable spec, the notebook skeletons, and a concept guide kept outside the repo. Every line of analysis, in src/garch_lab/ and in both notebooks, is mine. The point of the arrangement was to be able to defend the result rather than to arrive at it quickly.

References

Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1(2). The stylized-facts catalogue.

Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business 36. Volatility clustering, named and shown.

Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4). ARCH.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31. GARCH(1,1).

Patton, A. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160(1). Which losses rank models correctly when the target is a noisy proxy.

Diebold, F. and Mariano, R. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics 13(3). The forecast-comparison test this study still owes.

Sheppard, K. The arch package. https://arch.readthedocs.io