Bayesian Fan Charts for U.K. Inflation: Forecasting and Sources of Uncertainty in an Evolving Monetary System

Summary

Cogley, Morozov, and Sargent (2003) fit a time-varying-parameter vector autoregression (TVP-VAR) with multivariate stochastic volatility to UK data (RPIX [Retail Prices Index excluding mortgage interest] inflation, output gap, nominal 3-month rate; 1957.Q1–2002.Q4) and use the estimated model to construct Bayesian fan charts for inflation. The paper decomposes forecast uncertainty into three sources — future shocks, end-of-sample parameter uncertainty, and future parameter drift — and applies relative entropy reweighting (Robertson-Tallman-Whiteman 2002) to tilt fan charts to satisfy external moment constraints from official forecasts.

Key Claims

• The drifting-coefficient model captures the full UK inflation cycle: core trend swept 3% → 13% → 2.2%, persistence evolved from white noise (1960s) to a dominant low-frequency 8-year peak (1975) to high-frequency dominance after 1992, and innovation variance fell 75–80%.
• Multivariate stochastic volatility $R_t = B^{-1}H_tB'^{-1}$ — diagonal $H_t$ with geometric random walk log-variances, lower-triangular $B$ fixed over time — captures time-varying conditional heteroskedasticity without tying error covariance dynamics to the VAR coefficient dynamics.
• The Metropolis-within-Gibbs sampler uses 5 blocks: $\theta_T$ via Carter-Kohn forward-filter backward-sampling (FFBS); $Q$ (coefficient drift covariance) via inverse-Wishart; $\beta$ (lower-triangular $B$ columns) via independent Normal seemingly-unrelated-regression (SUR) draws; $\sigma$ (log-variance scale parameters) via inverse-gamma; $\{h_{it}\}$ (log-variances) via single-move Metropolis with log-normal proposal (Jacquier-Polson-Rossi, JPR, 1994 style). The stability constraint on $\theta_t$ is imposed by rejection sampling.
• Forecast uncertainty decomposes into three sources: (1) uncertainty about future shocks (dominant), (2) end-of-sample parameter estimation uncertainty (moderate), (3) uncertainty about future parameter drift (smaller but non-negligible at longer horizons).
• Relative entropy reweighting (Kullback-Leibler information criterion, KLIC, minimisation over Bayesian VAR (BVAR) sample paths; Robertson-Tallman-Whiteman 2002) allows imposing external constraints on the predictive distribution: target constraint achieves KLIC = 0.124; Monetary Policy Committee (MPC) mean constraint KLIC = 0.502; MPC mean + variance constraint KLIC = 0.675 — each additional constraint requires progressively more distributional distortion.

Concepts Introduced or Extended

• Time-Varying Parameter VAR
• Stochastic Volatility
• Conditional Forecasting
• Gibbs Sampler

Entities Mentioned

• Timothy Cogley
• Thomas Sargent

Quotes

"We decompose the uncertainty in the fan chart into three sources: uncertainty about future shocks, about end-of-sample parameters, and about future parameter drift."

"The relative entropy approach of Robertson, Tallman, and Whiteman (2002) provides a principled way to impose external constraints on the predictive distribution while minimizing the distortion measured by the Kullback-Leibler information criterion."

My Take

The paper's main methodological advance is the multivariate stochastic volatility specification — separating time-varying conditional variances from TVP-VAR coefficient dynamics allows each component to evolve at its own pace, directly addressing Sims's (2001) critique of Cogley-Sargent (2001) for ignoring heteroskedasticity. The relative entropy reweighting is a flexible bridge between model-based fan charts and judgemental forecasts. The UK retrospective is valuable: the 75–80% decline in innovation variance is striking and suggests the Great Moderation arrived in the UK before Thatcher's 1979 regime change began to take effect. A limitation is that the stability constraint on $\theta_t$ is handled by rejection sampling rather than a proper prior, which may introduce implicit distributional assumptions that are hard to characterise.