Felix-Nunes (2003) Forecasting Euro Area Aggregates with Bayesian VAR and VECM Models

Summary

Félix and Nunes (2003) compare twelve forecasting models for euro area aggregates — from random walks and classical vector autoregressions (VARs) to Bayesian VARs (BVARs) and Bayesian error-correction models (ECMs) — over a pseudo out-of-sample evaluation period 1989:1–1997:4. The Minnesota prior is extended with a real/price block hyperparameterization and a separate tightness parameter $\Omega$ on error-correction factor loadings. The main finding is that BVAR in levels dominates all competitors with an average root mean squared error (RMSE) ratio of 0.731 relative to a random walk (RW) benchmark, and that assigning a flat (diffuse) prior to the factor loadings in a Bayesian error-correction model (BECM) built on Johansen's multiple cointegrating vectors is worse than a random walk (RMSE ratio 1.278).

Key Claims

• BVAR in levels is best overall (avg RMSE 0.731 vs. RW = 1.000) across six endogenous variables (gross domestic product (GDP), unemployment, consumer price index (CPI), wages, long rate, exchange rate) and horizons 1–12 quarters.
• All Bayesian models beat all non-Bayesian models. Every model without shrinkage (unrestricted VAR, vector error-correction model (VECM), autoregression (AR), RW) has an average RMSE ratio ≥ 1.000.
• Flat prior on factor loadings is dangerous when Johansen's trace test finds multiple cointegrating vectors. BECM(J)-FP (4 cointegrating vectors, flat prior on $\alpha$) achieves only 1.278 — 28% worse than a random walk. The unrestricted factor loadings absorb noise from all four estimated equilibrium relations.
• Informative prior on factor loadings (IP) partially rescues the BECM. BECM(EG)-IP (Engle-Granger single cointegrating vector, finite $\Omega$) achieves 0.754; BECM(J)-IP achieves 0.805.
• BECM(EG)-IP is best for real GDP at long horizons (> 12 quarters), outperforming even the BVAR. Long-run cointegration constraints add genuine information at extended horizons.
• Integration order: P (private consumption deflator), W (nominal wage rate), and PW (external GDP deflator) are $I(2)$ (integrated of order two); all other variables are $I(1)$. $I(2)$ variables enter in first differences within levels models.
• Johansen cointegration: trace test finds 5–6 cointegrating relationships; authors use only 4 after the 5th estimated vector displays non-stationary behavior in-sample.
• Hyperparameterization: $\lambda_1$ (own-lag tightness, real variables), $\lambda_2$ (own-lag tightness, price variables), $\theta_1$–$\theta_2$ (same-block/cross-block cross-variable), $\theta_3$–$\theta_6$ (exogenous variables), $\Omega$ (prior variance on factor loadings; $\Omega=0$ suppresses ECM, $\Omega\to\infty$ recovers flat prior). Hyperparameters chosen by grid search on in-sample fit.

Concepts Introduced or Extended

• Minnesota Prior — real/price block extension with separate $\lambda_1$, $\lambda_2$ and loading tightness $\Omega$
• Cointegration — empirical demonstration of flat-prior pathology with multiple Johansen vectors; informative-prior fix

Entities Mentioned

• Ricardo Mourinho Félix — Banco de Portugal
• Luís C. Nunes — Universidade Nova de Lisboa

Quotes

"The results suggest that BVAR models clearly outperform the other models, including the BECM models, for most variables and forecast horizons."

"When using a flat prior for the factor loadings, the BECM models estimated with the Johansen cointegrating vectors perform clearly worse than the other models."

My Take

The paper's most useful contribution is the empirical demonstration that combining a Johansen cointegration test (which tends to find many cointegrating vectors in finite samples) with a diffuse prior on the factor loadings creates a badly overfit model — one that would have been better replaced by a pure random walk. The fix — an informative prior on $\alpha$ — is theoretically well motivated ($\alpha$ estimates converge at the slow $O_p(T^{-1/2})$ rate) and empirically effective. The result resonates with Amisano-Serati (1999), which showed the same phenomenon for Italian data. The BVAR's dominance at average horizons partly reflects the fact that the hyperparameters are tuned in-sample; the model that fits best in-sample tends to forecast well in the short run. For researchers choosing between BVAR and BECM in practice, the implication is clear: if you use Johansen-based cointegrating vectors, impose an informative prior on the factor loadings, or fall back to the Engle-Granger single-vector approach.