Forecast Combination

Definition

Forecast combination (forecast pooling) aggregates multiple individual forecasts into a single combined forecast. The Bayesian approach due to Winkler (1981) derives optimal combination weights from the estimated covariance matrix of forecast errors, accounting simultaneously for relative precision and cross-forecast error correlation.

Key Ideas

Combining $k$ forecasts with correlated errors generally outperforms all individual forecasts, even if one method dominates in isolation — the correlations provide extra information not exploited by any single model.
Optimal Bayesian weights depend on $\Sigma^{-1}$ , not just diagonal precision: the inverse covariance down-weights forecasters whose errors move together.
Weights sum to unity but need not be non-negative; negative weights arise naturally when forecast errors are positively correlated across methods.
When at least one source cannot be represented by a probability model (e.g., judgment-based forecasts), standard Bayesian model averaging via posterior model probabilities is infeasible; Winkler's approach remains applicable.
An Inverted-Wishart prior on $\Sigma$ stabilizes weights when only a short history of realized forecast errors is available — critical in real-time central bank settings.

How It Works

Setup

Suppose $k$ forecasting methods are available over $T$ evaluation periods. Let $\hat{x}_{jt}$ be method $j$ 's forecast of $x_t$ and $e_{jt} = \hat{x}_{jt} - x_t$ the corresponding error. Assume:

$e_t = (e_{1t},\ldots,e_{kt})' \sim \mathcal{N}_k(0,\Sigma)$

with errors independent across time. The combined forecast is the posterior mean of $x_t$ :

$E(x_t \mid \hat{x}_t) = \sum_{j=1}^k w_{jt}\,\hat{x}_{jt}$

Optimal Weights (Winkler 1981)

With a uniform prior on $x_t$ and posterior mean $\tilde\Sigma_t$ of $\Sigma$ :

$w_t' = (w_{1t},\ldots,w_{kt}) = \frac{u'\,\tilde\Sigma_t^{-1}}{u'\,\tilde\Sigma_t^{-1}\,u}$

where $u = (1,\ldots,1)'$ . These weights are optimal in the sense of minimizing the posterior expected squared loss for $x_t$ .

Bayesian Estimation of $\Sigma$

With a conjugate Inverted-Wishart prior $\Sigma \sim \mathrm{IW}(\Sigma_0, v)$ , the posterior mean is a precision-weighted blend:

$\tilde\Sigma_t = E(\Sigma \mid e_1,\ldots,e_t) = \frac{v}{t+v}\,\Sigma_0 + \frac{t}{t+v}\,\hat\Sigma_t$

where $\hat\Sigma_t$ is the standard unbiased sample covariance. As $t \to \infty$ , $\tilde\Sigma_t \to \hat\Sigma_t$ ; for small $t$ , the prior $\Sigma_0$ dominates and stabilizes the weights.

Prior specification (Adolfson et al. 2005 application with 3 forecasters, 1–8 quarter horizons):

$\Sigma_0 = \sigma_0^2 \begin{pmatrix} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \rho & 1 \end{pmatrix}$

$\rho$ increases linearly from 0.5 (horizon 1) to 0.92 (horizon 8) — at long horizons all methods converge to their steady states, inflating correlation
$\sigma_0^2$ increases from 0.25 to 1.0 across horizons — longer-horizon errors are larger
$v = 50$ degrees of freedom; gives a 95% prior interval for $\rho$ of $(0.40, 0.75)$ at horizon 1

Assumptions and Robustness

Unbiasedness: The derivation assumes $E(e_{jt}) = 0$ . Biased forecasters (e.g., Dynamic Stochastic General Equilibrium (DSGE) models systematically over-predicting interest rates) distort weights toward underweighting the biased forecaster. A bias-robust alternative weights by inverse mean squared error (MSE):

$w_{jt} \propto \frac{1}{\mathrm{MSE}_{jt}}$

This ignores cross-correlations but incorporates any systematic bias. In Adolfson et al. (2005) both weighting schemes produce similar results for Consumer Price Index (CPI) inflation.

Independence across time: Strictly, the IW prior assumes independent and identically distributed (i.i.d.) errors over $t$ . At horizons beyond one quarter, forecast errors overlap and are autocorrelated. The Winkler weights remain usable but lose Bayesian optimality; a Newey-West adjustment to $\hat\Sigma_t$ is an alternative but tends to be unstable at small $t$ .

Why It Matters

In Adolfson et al. (2005), the combined forecast (Riksbank + Bayesian VAR (BVAR) + DSGE) achieves the lowest root-mean-square error (RMSE) at all 1–8 quarter horizons for both CPI inflation and the short-term interest rate.
Provides a principled, operationally simple way to merge model-based and judgment-based forecasts — the combination literature's key challenge at central banks where official forecasts lack an explicit likelihood.
The IW prior solves a practical problem: combination weights start equal and converge to data-driven values as the error history grows, rather than being set to equal weights ad hoc throughout.

Open Questions

How to handle the case where $\Sigma$ is time-varying — the prior is static, but actual correlations may shift.
Optimal choice of $v$ when the true $\Sigma$ is unknown; sensitivity to this choice is not systematically studied in the literature.
Extension to high-dimensional combination (many models) requires sparse estimation of $\Sigma^{-1}$ .