Warne (1997) Inference in Cointegrated VAR Systems

Summary

A short technical note (4 pages) extending the Sims-Stock-Watson (1990) Wald statistic distribution theory in Vector Autoregression (VAR) systems with unit roots from linear to nonlinear restrictions. The central result is that cross-equation restrictions derived from rational expectations (RE) models — called nonlinear cross-equation (NCE) restrictions — generically produce a nonstandard limiting distribution because they constrain the row space of the long-run impact matrix $A(1)$. Correct critical values require simulating the nonstandard limit using the cointegration rank $r$, the canonical decomposition matrix $D$, the moving-average matrix $C$, and the error covariance $\Omega$; crucially, the NCE restrictions themselves supply a lower bound for $r$, making the simulation approach tractable. An empirical illustration decisively rejects the expectations hypothesis for U.S. term structure data (1952–1987). A small Monte Carlo study finds the Wald statistic is somewhat oversized in small samples.

Key Claims

• In VARs with unit roots, the Wald statistic for restrictions on the VAR coefficients $\beta$ converges to a functional of multivariate Brownian motion — not $\chi^2$ — when restrictions involve the non-stationary canonical regressors. Warne (1997) extends this result from linear to arbitrary smooth nonlinear restrictions (Proposition 1).
• NCE restrictions from RE models constrain the row space of $A(1) = I - \sum_i A_i$ (the long-run impact matrix). Whenever $r < n$, the partial Jacobian $P_1(\delta)$ does not have full row rank, so the Wald statistic $W$ has a nonstandard limiting distribution.
• The NCE restriction structure provides a lower bound for the cointegration rank $r$: the rows of $N_0^\perp$ (induced by the model matrices $N_i$) must be cointegrating vectors, so $\text{rank}(N_0^\perp) \leq r$. This bound guides selection of $r$ for the simulation of nonstandard critical values.
• Only when $r = n-1$ and $\mu \neq 0$ (non-zero drift) does $P_1(\delta)$ retain full row rank and the Wald statistic recover a standard $\chi^2$ limit.
• Empirical application — expectations hypothesis (EH): Using Campbell-Shiller (1991) U.S. term structure data (one- and three-month bond yields, January 1952–February 1987, $T \approx 422$), the Wald statistic $W = 59.48$ ($q = 8$ restrictions). The 99th percentile critical values are 32.00 ($\chi^2$), 33.86 (simulated nonstandard asymptotic), and 37.58 (empirical bootstrap). The EH is decisively rejected at all levels.
• Small-sample oversize: Monte Carlo evidence shows that the 5% nominal simulated asymptotic critical value corresponds to a roughly 10% empirical rejection rate at $T = 422 - p$, suggesting practitioners should use low nominal significance levels (1–2%) when applying this approach.

Concepts Introduced or Extended

• Cointegration — Establishes that RE cross-equation restrictions imply a lower bound for the cointegration rank r, which is needed to simulate the correct asymptotic critical region for the Wald test.
• Granger Causality — Proposition 1 is the nonlinear generalization of the Toda-Phillips (1993) result: standard $\chi^2$ asymptotics fail generically in cointegrated systems under NCE restrictions, just as they fail for linear Granger causality tests when rank conditions are not met.
• Term Structure of Interest Rates — Empirical illustration: expectations hypothesis strongly rejected for U.S. 1- and 3-month bond yields.

Entities Mentioned

• Anders Warne
• Christopher Sims — Sims-Stock-Watson (1990) is the framework being extended
• Mark W. Watson — co-author of Sims-Stock-Watson (1990)

Quotes

"The objective of this note is to extend the theory in Sims et al. to nonlinear restrictions ... We show that such restrictions typically imply a nonstandard limiting distribution and that they provide a lower bound for the number of cointegration relations as well as the complete cointegration space for that bound."

"A Monte Carlo study indicates that the Wald statistic is somewhat oversized in small samples, suggesting the use of low nominal levels in practice."

My Take

A focused, technically precise note that fills a genuine gap: Sims-Stock-Watson (1990) addressed linear restrictions; this paper handles the empirically important case of RE cross-equation restrictions. The key contribution is the lower-bound result for r, which makes the nonstandard simulation approach practical — without it, the entire nuisance parameter space would need to be searched. The EH application confirms Campbell-Shiller (1991) and primarily serves to validate the method.

The main limitation is brevity: at four pages the paper omits all proofs and relies heavily on Sims-Stock-Watson (SSW) (1990) infrastructure. The companion working paper Warne (1993) is the fuller version. The later Warne (2006) moved the entire framework to a Bayesian setting, making the frequentist simulation approach less central in practice — but the asymptotic results here remain the theoretical foundation for understanding why naive $\chi^2$ tests fail in cointegrated systems under RE restrictions.