Conditional Forecasting

Definition

Conditional forecasting is a vector autoregression (VAR)-based procedure for generating forecast paths that satisfy user-specified constraints on the future values of some variables, while allowing unconstrained variables to adjust freely. The constrained forecast is defined as the minimum mean-squared-error (MSE) perturbation of the unconditional forecast errors consistent with the imposed constraints (Doan-Litterman-Sims 1984; Waggoner-Zha 1999).

Key Ideas

Hard constraints: force some variables to follow a prescribed future path (e.g., hold the fed funds rate at its current level for 8 quarters).
Soft constraints: impose conditions only in expectation, via a tilted prior on forecast residuals.
Minimum-MSE residuals: among all combinations of forecast errors satisfying a constraint, choose the one closest to zero in mean-squared-error terms — this minimizes the distortion relative to the unconditional forecast.
Staggered data release: the most common practical motivation; some variables (e.g., quarterly gross domestic product (GDP)) are unavailable at the forecast origin, but their partially-complete values can be conditioned on to use all available monthly information.

How It Works

Waggoner-Zha (1999) Procedure

Let $\hat{e}_{T+1}, \ldots, \hat{e}_{T+H}$ be the stacked $nH$ -vector of one-step-ahead reduced-form residuals over the forecast horizon. An affine constraint takes the form:

$R \hat{e} = r$

where $R$ is a $q \times nH$ selection matrix ( $q$ constraints) and $r$ is a $q$ -vector of target values. For example, constraining the first variable to follow a known path for the first three months sets $R$ to the corresponding indicator rows and $r$ to the target values.

The minimum-norm residuals satisfying the constraint are:

$\hat{e}^* = R'(RR')^{-1}r$

These are the shortest (minimum-MSE) perturbation of the unconditional forecast consistent with $R\hat{e}^* = r$ . The implied conditional forecast path is obtained by feeding $\hat{e}^*$ through the VAR recursion.

Error Bands

Because the constraint removes $q$ degrees of freedom, the covariance matrix of the constrained forecast errors is rank-deficient:

$\text{Var}(\hat{e}^*) = \bigl(I - R'(RR')^{-1}R\bigr)$

Drawing from this reduced-rank distribution requires extracting only the non-zero eigencomponents. Robertson and Tallman (1999) note that this singular covariance means standard multivariate normal simulation must be replaced by a procedure that respects the constraint.

Application: Staggered Data Release (Robertson-Tallman 1999)

In a 6-variable monthly VAR including quarterly GDP distributed via Chow-Lin, the latest GDP observation is typically 1–3 months old at the forecast date. The VAR is estimated on a balanced panel ending at month $T$ . At the end of January, for example, the fed funds rate and commodity prices for January are available but M2, Consumer Price Index (CPI), unemployment, and GDP for January are not yet released. Setting $r$ to the known January values and $R$ to the corresponding selection rows conditions the VAR on these real-time observations and produces improved forecasts of the unknown series.

Relative Entropy Reweighting (Robertson-Tallman-Whiteman 2002)

An alternative to hard conditioning is to reweight the unconstrained predictive distribution to satisfy external moment constraints while minimising the Kullback-Leibler information criterion (KLIC). Let $\{w_s\}_{s=1}^S$ be weights on the $S$ Bayesian VAR (BVAR) sample paths, and let $\bar{\pi}$ be the implied reweighted distribution. The KLIC-minimising weights are:

$w_s^* = \frac{\exp(\lambda' f(y_s^*))}{S^{-1}\sum_{s'} \exp(\lambda' f(y_{s'}^*))}$

where $f(y_s^*)$ is the vector of constraint functions evaluated at path $s$ and $\lambda$ are Lagrange multipliers chosen so that $\sum_s w_s^* f(y_s^*) = c$ (the target moments). The KLIC at the optimum measures the informational distance between the constrained and unconstrained distributions:

$\mathrm{KLIC}(\bar{\pi} \| \pi_0) = \sum_s w_s^* \log(S w_s^*)$

In the Cogley-Morozov-Sargent (CMS) (2003) UK fan chart application:

Target constraint (inflation path pinned to official target): KLIC = 0.124
Monetary Policy Committee (MPC) mean constraint (match MPC modal forecast): KLIC = 0.502
MPC mean + variance constraint (match both mean and spread): KLIC = 0.675

Higher KLIC indicates that the external constraint is harder to reconcile with the model's predictive distribution — it requires more reweighting and therefore represents a stronger departure from the model.

This method generalises the minimum-MSE conditioning of Waggoner-Zha (1999) to soft, moment-based constraints and applies naturally to non-Gaussian predictive distributions from time-varying parameter VARs (TVP-VARs) and other nonlinear models where closed-form conditioning is unavailable.

Why It Matters

Policy analysis: produces "what if the central bank holds rates steady?" paths — the standard scenario exercise at central banks worldwide.
Real-time forecasting: handles staggered data release without requiring a balanced panel at the cost of re-estimating the model each period.
Fan chart construction: generates a coherent distribution over future paths that respects the conditioning information; relative entropy reweighting allows fan charts from TVP-VAR models to incorporate official forecast constraints.
Data vintages: allows the VAR coefficients to remain fixed while updating the conditional information set as new monthly releases arrive.

Open Questions

Whether minimum-MSE conditioning is the right loss function for policy analysis, as opposed to structural (shock-based) conditioning (Leeper-Zha 2003), which attributes the constrained path to specific structural shocks.
Extension to nonlinear or time-varying parameter models where the forecast distribution is non-Gaussian and the minimum-MSE solution lacks a closed form; relative entropy reweighting (Robertson-Tallman-Whiteman 2002) addresses this for moment constraints but the KLIC optimisation must be solved numerically.