4. Conditional forecasting using relative entropy

Besides the model-based conditional forecasts – where conditions are achieved through the model’s own structural shocks (the conditions argument to forecast; see the reduced-form VAR chapter, and, for perfect-foresight conditioning, the simulation-plan object) – RISE can impose conditions by reweighting an already-computed predictive distribution. This is entropic tilting (relative-entropy / exponential tilting), in the tradition of Robertson, Tallman and Whiteman, of Cogley, Morozov and Sargent, and of Krüger, Clark and Ravazzolo.

4.1. The idea

Start from a sample of forecast paths – \(N\) draws, for every variable, over the forecast horizon – carrying equal weights \(\pi_i = 1/N\). You have additional information you want to impose, expressed as a moment condition: some function \(g(\cdot)\) of a path should average to a target \(\bar c\). Entropic tilting replaces the equal weights by new weights \(\omega_i\) that are as close as possible – in Kullback-Leibler distance – to the originals, subject to that condition:

\[\min_{\omega}\;\sum_i \omega_i \log\frac{\omega_i}{\pi_i} \quad\text{s.t.}\quad \sum_i \omega_i g(x_i) = \bar c,\;\; \sum_i \omega_i = 1.\]

The solution is the exponential tilt \(\omega_i \propto \pi_i\,\exp\!\left(\gamma' g(x_i)\right)\), where the multiplier \(\gamma\) is chosen so the condition holds. The reweighted draws are then summarised as usual (means, quantiles, fan charts) – nothing about the model or its shocks changes; you have simply moved probability mass toward the paths consistent with your information.

4.2. When to use it

  • The condition is distributional – a target mean, a quantile, a variance – rather than an exact path value;

  • you want to bring in external information (a survey, a judgmental view, an off-model forecast) that is not naturally expressed through the model’s shocks;

  • you already have the predictive sample (it need not even come from a RISE model) and want a cheap reweighting rather than a fresh model solve.

The price is that the conditions hold only in expectation under the tilted measure, and there is no structural-shock decomposition of the result – if you need exact conditioning with a shock interpretation, use the model-based route.

4.3. The API

The high-level entry point is utils.entropic_tilting.tilt_scenario:

[omega, fkst_tilted] = utils.entropic_tilting.tilt_scenario(fkst, mfun, c, options);

  • fkst – a structure of simulated paths, one field per variable, each a T x N matrix of draws (the kind of object forecast / simulate return when given parameter draws);

  • mfun – a function handle; mfun(X) returns the moment(s) of interest from one path X (a variables-by-horizon array);

  • c – the target value(s) \(\bar c\) for those moments;

  • options – options passed through to the tilting solver (utils.entropic_tilting.core).

It returns omega (the N x 1 vector of tilted weights) and a forecast structure with the reweighted statistics; pass omega to the fan-chart helpers (or compute your own weighted quantiles) to display the tilted forecast alongside the untilted one.

4.4. A sketch

Suppose you want to tilt a VAR forecast distribution toward a path on which a Taylor-type policy-rule residual is zero on average – i.e. “the central bank broadly follows its rule over the forecast”:

% 1. an unconditional forecast distribution (parameter draws -> paths)
params = mest.estim_.sampler(2000);
fkst   = forecast(mest, db, '2024Q1', params, 12);     % fields: var -> T x N

% 2. the moment to condition on: the per-period rule residual
%    R{t} - rho*R{t-1} - (1-rho)*(dpai*PAI{t} + dy*YGAP{t})
rho = 0.8; dpai = 1.5; dy = 0.3;
mfun = @(X) rule_residual(X, rho, dpai, dy);            % returns a vector of residuals
c    = 0;                                               % target: zero on average

% 3. tilt
[omega, fkst_tilted] = utils.entropic_tilting.tilt_scenario(fkst, mfun, c);

% 4. fan charts, tilted vs untilted
plot_fanchart(fanchart(fkst.PAI,        [30 50 68 90]))
plot_fanchart(fanchart(fkst.PAI, [30 50 68 90], omega))   % weighted

4.5. Solver options

utils.entropic_tilting.tilt_scenario passes its options argument through to utils.entropic_tilting.core, which solves for the Lagrange multiplier \(\gamma\) with fsolve. The accepted fields:

Field

Meaning

Default

tol

fsolve TolFun (convergence on the moment residual)

1e-8

maxiter

fsolve MaxIter

500

lambda0

initial multiplier guess, k x 1 (where k is the number of moment conditions)

zeros(k, 1)

A non-default lambda0 is useful for sequential or rolling tilting where the previous solution is a warm-start for the next.

4.5.1. Companion helper

utils.entropic_tilting.weighted_quantile(x, omega, q) computes weighted quantiles of a sample x under weights omega. Use it to build the tilted summary statistics that feed the fan-chart helper when plot_fanchart is called with an explicit omega.

4.6. Worked example shipped with the toolbox

A runnable end-to-end example is in the toolbox repository at development/tests/models/var/panel/panel/tut13_conditional_forecast_entropic_tilting.m, with companion notes in entropic_tilting.pdf next to it. The script fits a small VAR, draws an unconditional forecast distribution, and tilts it toward a Taylor-rule residual of zero on average – exactly the pattern sketched above. The five steps in the script (%% housekeeping → %% conditional forecast distribution → fan charts without tilting → with tilting → side-by-side comparison) are a template to adapt for any model.

To impose other condition types (quantile targets, an external mean forecast, multiple moments at once), change only mfun and c:

  • Point mean target on variable Yj at horizon h: mfun = @(X) X(j, h); c = target_value;

  • Quantile target (e.g. “median of inflation over the next 8 quarters is 2%”): build mfun from an indicator \(\mathbf{1}\{X_{j,h} \le q\}\) and target the desired probability level for c (see the entropic-tilting references below for the derivation).

  • External forecast (e.g. a survey mean for Yj at horizons 1:H): stack mfun(X) = X(j, 1:H).' and set c to the vector of survey means.

4.7. References

The entropic-tilting / relative-entropy reweighting framework used here:

  • Robertson, J. C., Tallman, E. W., & Whiteman, C. H. (2005). “Forecasting using relative entropy.” Journal of Money, Credit and Banking, 37(3), 383-401.

  • Cogley, T., Morozov, S., & Sargent, T. J. (2005). “Bayesian fan charts for U.K. inflation: Forecasting and sources of uncertainty in an evolving monetary system.” Journal of Economic Dynamics and Control, 29(11), 1893-1925.

  • Krüger, F., Clark, T. E., & Ravazzolo, F. (2017). “Using entropic tilting to combine BVAR forecasts with external nowcasts.” Journal of Business & Economic Statistics, 35(3), 470-485.