Forecast Combination and Averaging

You have three different models forecasting next month's realized volatility — an ARCH-family model, a HAR-RV model, and a simple historical-average model — and each has its own track record of errors, none of them perfect. The obvious instinct is to run a horse race, find whichever model had the lowest historical error, and use only that one going forward. It's a reasonable instinct, but it throws away information: if the three models make different kinds of mistakes — one overreacts to recent spikes, another lags behind regime changes, a third smooths too much — then averaging their forecasts together can cancel out each model's particular blind spot, producing a combined forecast more accurate than any single model alone, even the "best" one picked by backtest.

An analogy: asking three friends to guess the jar's jelly beans

The classic "guess the number of jelly beans in a jar" demonstration: individual guesses are often wildly off, but the average of many people's independent guesses is consistently closer to the true count than most individual guesses, sometimes closer than nearly all of them. This works because each person's error has a different source and direction — one overestimates because the jar looks tall, another underestimates because the beans look packed — and averaging cancels out the idiosyncratic parts of each error while reinforcing whatever true signal everyone partially picked up on. Forecast combination is this exact phenomenon applied to models instead of people: as long as different models' errors aren't perfectly correlated, blending them cancels out some of each one's individual noise.

The mechanics, one symbol at a time

The simplest combination is an equal-weighted average of $K$ individual forecasts:

$$\hat{y}_t^{combo} = \frac{1}{K}\sum_{k=1}^{K} \hat{y}_{k,t},$$

where $\hat{y}_{k,t}$ is model $k$'s forecast for time $t$. In plain English: just average every model's prediction with equal weight — no fitting required, remarkably hard to beat in practice. A more refined version weights each model by its historical inverse error variance, giving more accurate models more say:

$$\hat{y}_t^{combo} = \sum_{k=1}^{K} w_k\, \hat{y}_{k,t}, \qquad w_k = \frac{1/\sigma_k^2}{\sum_{j=1}^{K} 1/\sigma_j^2},$$

where $\sigma_k^2$ is model $k$'s historical forecast error variance. In plain English: a model with a smaller historical error gets a proportionally larger weight in the blend, and the weights are constructed to sum to one so the combination stays on the same scale as the individual forecasts.

Worked example 1: equal-weighted average

Three volatility models forecast next month's realized vol at 18%, 22%, and 25%. The simple average combination:

$$\hat{y}^{combo} = \frac{18 + 22 + 25}{3} = \frac{65}{3} \approx 21.7\%.$$

Suppose the actual realized vol turns out to be 20%. The individual model errors are $|18-20|=2$, $|22-20|=2$, $|25-20|=5$ — the combination's error is $|21.7-20| = 1.7$, smaller than even the best individual model's error of 2. This isn't guaranteed every single period, but it's a common pattern: the combination smooths out whichever single model happened to overshoot or undershoot most.

Worked example 2: inverse-variance weighting

Suppose historical error variances for the three models are $\sigma_1^2 = 4$, $\sigma_2^2 = 9$, $\sigma_3^2 = 25$ (model 1 has been most reliable, model 3 least). Inverse variances: $0.25, 0.111, 0.04$, summing to $0.401$. Weights: $w_1 = 0.25/0.401 \approx 0.623$, $w_2 = 0.111/0.401 \approx 0.277$, $w_3 = 0.04/0.401 \approx 0.100$. Applying these to the same forecasts (18%, 22%, 25%):

$$\hat{y}^{combo} = 0.623 \times 18 + 0.277 \times 22 + 0.100 \times 25 = 11.2 + 6.1 + 2.5 = 19.8\%.$$

This weighted combination (19.8%) lands even closer to the actual 20% than the equal-weighted version (21.7%), because it correctly down-weighted the historically less reliable model 3, whose 25% forecast pulled the equal-weighted average further off target.

What this means in practice

Forecast combination is standard practice anywhere a desk maintains multiple competing models rather than a single blessed one — volatility forecasting, macro nowcasting, or blending analyst estimates. It is the natural next step after evaluating individual model accuracy (see Forecast Evaluation and Accuracy Metrics) and after checking with the Diebold-Mariano test (see The Diebold-Mariano Test) whether any one model is truly dominant — if none is, combining is usually the more robust choice.