QLIKE vs Squared Error for Volatility Forecasts: Noisy Proxies

A volatility model forecasts how spread out a future return may be, not whether the return will be positive or negative. To evaluate it, first define the target: for example, tomorrow’s conditional return variance, measured in squared basis points. A forecast of standard deviation, a forecast of variance, and an intraday realized-variance estimate are related quantities, but they are not interchangeable without a stated transformation and horizon.

This question is separate from how a model generates its forecast. The GARCH guide explains one way to update conditional variance from past returns. This guide starts after two or more models have made forecasts and asks how to score them against an imperfect measure of the variance that later occurred.

“Best forecast” has no single meaning until the target, evaluation sample, and loss function are fixed. A model may rank first under one loss and second under another because each loss assigns a different cost to forecast misses

Let vₜ be the unobserved conditional variance of the target return, hₜ a forecast made using information available before that return, and xₜ an observed proxy used for evaluation. A common proxy is the squared return, which is conditionally unbiased for variance only when the conditional return mean is zero. If the conditional mean is nonzero, the raw squared return estimates the conditional second moment, not the conditional variance; a proxy can instead be formed from returns after removing the conditional mean. Realized variance, built from intraday returns, can provide a more informative measure of variation over a session, but it is still a measurement of an interval and depends on the sampling and price-data rules.

Demeaning does not automatically remove proxy bias when the conditional mean is estimated. If mₜ = E[rₜ | Fₜ₋₁], hₜ = Var(rₜ | Fₜ₋₁), and an estimated mean m̂ₜ is fixed using information available at the forecast origin, then E[(rₜ − m̂ₜ)² | Fₜ₋₁] = hₜ + (mₜ − m̂ₜ)². Any forecastable mean-estimation error therefore adds a nonnegative term to the proxy. State how the mean is estimated and whether the proxy targets variance or the second moment before invoking a conditional-unbiasedness result.

Match xₜ and hₜ to the same asset, return definition, time window, horizon, and units. If xₜ is measured in bp², hₜ must also be in bp²; the squared-error score will then be in bp⁴. A daily close-to-close forecast should not be scored against a realized measure that omits an overnight interval unless that is the intended target.

Retain the proxy values and both forecasts for the same evaluation origins. If one model misses a date, define a common sample or a justified missing-data rule in advance. Comparing averages over different dates can mix model performance with differences in the market conditions each model faced

For a nonnegative variance proxy xₜ and a positive variance forecast hₜ, squared-error loss is

L_MSE(xₜ, hₜ) = (xₜ − hₜ)²

The QLIKE loss is

L_QLIKE(xₜ, hₜ) = log(hₜ) + xₜ / hₜ

Lower values score better. QLIKE is often written in an equivalent ratio form, xₜ/hₜ − log(xₜ/hₜ) − 1, when xₜ is strictly positive. The two formulas differ by a term that depends on the observed proxy but not on the forecast, so they give the same ranking on a common sample. The log(hₜ) form also remains defined when xₜ is zero, provided hₜ is positive. Patton gives these variance-loss definitions and studies their behavior with imperfect proxies (Patton, 2011).

QLIKE corresponds to a Gaussian quasi-likelihood score for variance forecasts (Hansen and Lunde, 2005). That connection does not establish that returns are Gaussian, nor does a low score verify the model’s distributional assumptions. Squared error instead measures the absolute gap between the proxy and forecast, squared. The question is which scoring rule matches the comparison goal and proxy assumptions

Squared error is in squared variance units and grows quickly when the proxy and forecast are far apart. A few very large proxy values can therefore dominate the average. That behavior can be appropriate when large absolute variance misses deserve much more weight, but it makes the result sensitive to extreme observations.

Rescaling the variance unit multiplies xₜ and hₜ by the same positive constant c and adds log(c) to log(hₜ) + xₜ/hₜ. That same constant applies to every forecast scored on the common proxy sample, so the ranking is unchanged. The ratio xₜ/hₜ is scale-free, while the absolute log(hₜ) term depends on the chosen unit. When a forecast is much too small relative to a large proxy, the ratio term rises sharply; overpredicting also costs more as log(hₜ) grows, but at a slower rate for comparable proportional errors. QLIKE is not an outlier-resistant loss: an extreme proxy still contributes through xₜ/hₜ.

Neither score wins for every decision. Squared error puts more weight on large absolute misses in variance units, while QLIKE has a different relative-error profile. Patton’s proxy-robustness result is not a claim that these losses rank forecasts identically or that either is immune to outliers

Suppose the observed variance proxies are x = [1, 4] bp². Model A forecasts hᴬ = [1, 1] bp², while Model B forecasts hᴮ = [4, 2] bp². The numbers are hypothetical and chosen to show how the loss can change the ordering.

MSE favors Model A: its average score is 4.5 bp⁴ versus 6.5 bp⁴ for Model B. QLIKE favors Model B: its average score is about 2.165 versus 2.5 for A. These two rows establish only the arithmetic. They are too few to estimate uncertainty, prove a forecast is better, or say anything about a trading strategy <!-- learn:illustration -->

A volatility proxy can be noisy and still be conditionally unbiased. Suppose E[xₜ | Fₜ₋₁] = vₜ, where Fₜ₋₁ is the information available when the forecast is made. For a forecast hₜ formed from that information, the expected squared-error loss is

E[(xₜ − hₜ)² | Fₜ₋₁] = (vₜ − hₜ)² + Var(xₜ | Fₜ₋₁)

The variance term does not depend on which forecast is being scored. It raises the expected MSE for both forecasts by the same amount, so it cancels when comparing their expected losses. For QLIKE, conditional unbiasedness gives

E[log(hₜ) + xₜ/hₜ | Fₜ₋₁] = log(hₜ) + vₜ/hₜ

That is the score that would be obtained using the latent variance in place of the proxy. Under these conditions, each loss preserves the expected ranking of competing variance forecasts relative to the latent target. Patton (2011) calls this proxy-robust ranking; the term concerns noise in expected rankings, not resistance to outliers.

The result assumes the same conditional information set, a conditionally unbiased proxy, positive forecasts for QLIKE, and finite expected losses. It does not say that MSE and QLIKE will rank the same models against each other. In the two-period example, each loss is calculated from the same noisy observations, but the two losses assign different importance to the forecast errors.

For two forecasts hᴬₜ and hᴮₜ fixed at the same origin, the expected MSE difference is E[L_MSE(xₜ,hᴬₜ) − L_MSE(xₜ,hᴮₜ) | Fₜ₋₁] = (vₜ − hᴬₜ)² − (vₜ − hᴮₜ)². Under the same proxy condition, the expected QLIKE difference is log(hᴬₜ) + vₜ/hᴬₜ − log(hᴮₜ) − vₜ/hᴮₜ. These expressions show what proxy robustness removes: the additional noise in the proxy's expected score. They remain different forecast comparisons because MSE and QLIKE impose different costs on errors.

Expected ranking invariance is not a promise that every realized sample gives the latent-variance ranking. Proxy noise adds variation to observed losses and can reduce the power of a forecast comparison. A short evaluation period may rank models differently just because a few proxy observations happened to be unusually high or low, even when conditional unbiasedness holds on average.

The guarantee also fails if the proxy is conditionally biased for the stated target. High-frequency realized variance can be affected by bid–ask bounce, stale or asynchronous quotes, sampling frequency, and omitted overnight returns. A proxy aimed at intraday integrated variance may not be an unbiased measure of a close-to-close conditional variance. The realized-volatility calculation guide discusses construction choices, and the two-scale realized-volatility guide explains one estimator designed to address some high-frequency microstructure noise. Hansen and Lunde (2006) document how market microstructure noise can affect realized-variance measurement.

“Robust” has a narrow meaning here: when the proxy is conditionally unbiased, the expected ranking is preserved. It does not mean the sample score ignores outliers, that any realized-volatility measure is unbiased, or that the comparison is protected from a target mismatch. State what the proxy measures and which assumptions connect it to vₜ

Once the score is chosen, form a paired loss difference on each common forecast origin. With dₜ = L_A,ₜ − L_B, a positive average favors B because B has the lower loss. A Diebold–Mariano test evaluates whether the expected difference is zero using the time series of these paired differences; it does not choose between QLIKE and MSE or establish which target should matter. The Diebold–Mariano guide explains the broader inference framework.

If forecasts overlap across multi-period horizons, or the score differences are otherwise serially correlated, use a dependence-aware standard error such as an appropriate HAC estimate. Newey and West (1987) provide a positive semi-definite HAC covariance estimator. State the horizon, evaluation origins, loss-difference sign, covariance method, kernel and bandwidth where relevant. A small p-value from a test does not undo a mismatched or biased proxy

Use a chronological out-of-sample design and save the predictions that each model would actually have made at each origin. Fix the target, proxy construction, horizon, unit, common sample, and loss before inspecting which model scores better. Record estimation windows, re-estimation rules, missing observations, and whether the evaluation period was repeatedly used to tune models. If you compare both MSE and QLIKE and report only the winner, disclose that choice rather than presenting it as a single pre-specified test.

Report each model’s mean loss, the paired difference, sample size, proxy definition, and inference method. QLIKE requires strictly positive hₜ; define any forecast floor before the evaluation because a floor can alter scores. Check sensitivity to defensible proxy and sampling choices, while making clear that trying many choices also creates a selection problem.

A lower volatility loss measures performance under the chosen score and proxy. It does not show that a predictor causes volatility, that returns are normally distributed, or that a strategy using the forecast will make money after spreads, fees, slippage, financing, and risk constraints. Evaluate forecast skill and trading outcomes as separate claims