Amisano-Serati (1999) Forecasting Cointegrated Series with BVAR Models

Summary

Amisano and Serati (1999) examine how to incorporate cointegrating relationships into Bayesian VAR (BVAR) models for forecasting. The central contribution is the observation that assigning diffuse priors to the error correction model (ECM) factor loadings $\alpha$ — as in the standard Bayesian ECM (BECM) literature — is harmful: since factor loading estimates are only $O_p(T^{-1/2})$ (not super-consistent), flat priors over-weight the ECM correction terms relative to short-run dynamics. The proposed fix — an informative Minnesota-style prior on $\alpha$ — yields the best forecasting performance across all horizons in an application to Italian gross domestic product (GDP), consumption, and investment.

Key Claims

Factor loadings $\hat\alpha$ converge at rate $O_p(T^{-1/2})$ , not the super-consistent rate of $\hat\beta$ ( $O_p(T^{-1})$ ). In small samples, flat priors on $\alpha$ trust unreliable estimates too much, causing the model to over-correct toward long-run equilibrium and under-weight short-run dynamics.
The IP-BECM model (informative prior on $\alpha$ , with $\lambda_{ij} \approx 0.08$ – $0.70$ ) wins in 13 out of 15 comparison points (5 horizons $\times$ 3 equations) against three competing specifications.
The FP-BECM model (flat prior on $\alpha$ , LeSage 1990 style) only outperforms the standard Minnesota BVAR at long horizons (12+ steps); at short horizons the Minnesota BVAR without cointegration is competitive — confirming that flat-prior ECM over-emphasizes long-run correction.
The AB-BVAR (Alvarez-Ballabriga 1995 stochastic restriction on $A_1$ from maximum likelihood (ML) $\hat\Pi$ ) is weakest at short horizons and never best in the ranking.
Low time variation in all models ( $\tau_i \approx 10^{-6}$ ) slightly but consistently beats fixed-coefficient versions.
A uni-equational optimization criterion (Theil's U equation-by-equation) is conceptually unappealing in a multivariate setting; multivariate criteria such as log-determinant of forecast error covariance are available but have poor small-sample properties.

Concepts Introduced or Extended

Cointegration — Bayesian ECM (BECM) approach; informative vs. flat priors on factor loadings; IP-BECM vs. FP-BECM vs. AB-BVAR comparison
Vector Autoregression — Bayesian VAR forecasting of cointegrated series

Entities Mentioned

Quotes

"a flat prior on the ECM terms combined with an informative prior on the lagged endogenous variables coefficients gives too much importance to the long-run properties with respect to the short-run dynamics."

"the gains in imposing long-run constraints with informative priors are evident over all forecasting horizons."

My Take

The paper's diagnostic — flat priors on $\alpha$ cause the ECM terms to dominate — is theoretically clean: $\alpha$ and the $\Gamma$ lags both converge at the same $\sqrt{T}$ rate, so there is no reason to treat them asymmetrically. The fix (informative prior on $\alpha$ with equation-specific hyperparameters $\lambda_{ij}$ ) is practical and the empirical results are convincing. The limitation is that $\lambda_{ij}$ must be tuned by grid search, adding a hyperparameter layer that is not always transparent. The paper also does not explore multivariate forecast evaluation metrics, which is a genuine gap for cointegrated systems where the forecast error structure is inherently multivariate.