Villani (2005) Bayesian Inference of General Linear Restrictions on the Cointegration Space

Summary

This paper develops Bayesian tests of linear restrictions on the cointegration space — the space spanned by the cointegration vectors $\beta$ in a cointegrated VAR. Assessing how much empirical support the data give to theory-implied long-run equilibria (restrictions on $\beta$) is often the centerpiece of a cointegration analysis, and classical likelihood-ratio tests are awkward when several competing "null" hypotheses are in play and only asymptotic distributions are available. Building on the finite-sample Bayesian procedures of Villani (2000) and Strachan–van Dijk (2003), which use a uniform prior on the cointegration space as the key ingredient, Villani extends the approach to the empirically important case where the restrictions differ across the individual cointegration vectors. He proposes prior distributions, devises tailored posterior-simulation algorithms, shows (on U.S. consumers'-expenditure data) how the results depend on the prior — especially the prior on the adjustment coefficients $\alpha$ — and finds in a simulation study that the Bayesian approach performs remarkably well against the LR test (with and without Bartlett correction) and information criteria. (Sveriges Riksbank Working Paper Series No. 189, September 2005.)

Key Claims

• The object of inference is the cointegration space. In the error-correction model $\Delta x_t = \alpha\beta' x_{t-1} + \sum_{i=1}^{k-1}\Gamma_i\Delta x_{t-i} + \Phi d_t + \varepsilon_t$, the columns of $\beta$ ($p\times r$) are the cointegration vectors and $\alpha$ the adjustment coefficients; the span of $\beta$ (identified only up to a linear transformation) is the "cointegration space," and restrictions from economic theory constrain that space.
• Bayesian model comparison over restrictions. Rather than a classical LR test, formulate a prior for each restricted model and compute the posterior probability over the set of restrictions/models. This sidesteps the difficulties of multiple nulls and small-sample-invalid asymptotics (Gredenhoff–Jacobson 2001), at the cost of needing a coherent prior on a curved parameter space.
• The uniform-prior-on-the-space device. Because $\beta$ is identified only up to rotation, priors are placed on the cointegration space (a point on the Grassmann manifold) — a uniform prior there (Villani 2000; Strachan–van Dijk 2003) gives a well-defined, invariant reference for calculating posterior probabilities of restrictions.
• Main contribution — vector-specific restrictions. Earlier work restricted all cointegration vectors the same way; Villani extends the finite-sample Bayesian machinery to different linear restrictions on different individual cointegration vectors, the common situation in applied work, with new priors and posterior simulators.
• The prior on $\alpha$ matters. Inference on the cointegration restrictions is shown to depend crucially on the prior on the adjustment coefficients $\alpha$; nuisance parameters (short-run dynamics $\Gamma$, deterministic terms, error covariance $\Omega$) can take improper priors since they are common across the hypothesized spaces, but $\alpha$ cannot be treated so cavalierly.
• Conditioning. The analysis proceeds conditional on a pre-specified lag length and a known cointegration rank $r$; it can be extended to joint inference on restrictions and rank (Strachan–van Dijk 2003) or paired with separate rank-posterior procedures (Kleibergen–Paap; Villani 2005a).
• Performance. A simulation study finds the Bayesian approach compares favorably to the Johansen LR test (with/without Bartlett correction) and standard information criteria for testing restrictions on $\beta$.

Concepts Introduced or Extended

• Cointegration
• Common Stochastic Trends
• Bayesian VAR

Entities Mentioned

• Mattias Villani

Quotes

"The current paper extends this approach to the empirically important case with different restrictions on the individual cointegration vectors. Prior distributions are proposed and posterior simulation algorithms are developed."

"A simulation study shows that the Bayesian approach performs remarkably well in comparison to other more established methods for testing restrictions on the cointegration vectors."

My Take

This paper is a careful piece of the Bayesian-cointegration program's hardest problem: putting a coherent, invariant prior on the cointegration space (a Grassmann manifold, since $\beta$ is only identified up to rotation) so that restrictions implied by economic theory can be scored by posterior probability rather than accepted/rejected by a fragile asymptotic LR test. Its practically important move is allowing different restrictions on different cointegration vectors — exactly what applied demand-system or term-structure work needs — and its most useful warning is that the answer is sensitive to the prior on the adjustment coefficients $\alpha$, not just the prior on $\beta$. It belongs beside Johansen's classical rank/restriction tests as the finite-sample Bayesian alternative, and it connects the Bayesian VAR literature (Villani's steady-state and Sims–Zha priors) to the geometry of reduced-rank regression. Note this is the Riksbank working-paper version and is distinct from Villani's other 2005 cointegration paper, "Bayesian Reference Analysis of Cointegration."