Sims-Stock-Watson (1990) Inference in Linear Time Series Models with Some Unit Roots

Summary

Sims, Stock, and Watson (1990) develop a unified canonical-regressor framework for ordinary least squares (OLS) inference in vector autoregression (VAR) models that may contain unit roots, cointegration, and deterministic trends. They show that OLS is consistent for all parameters regardless of the integration structure, and provide a complete characterization of when Wald statistics have standard $\chi^2$ distributions versus nonstandard Wiener-process-based limits. As a corollary, the Engle-Granger two-step procedure is asymptotically redundant, and Bayesian inference on VAR parameters requires no special treatment at the unit root boundary.

Key Claims

• Consistency (Theorem 1): OLS on the levels VAR is consistent for all coefficient matrices regardless of how many unit roots or cointegrating relations are present; convergence rate is $T^{1/2}$ for stationary canonical regressors and $T$ for I(1) components.
• $\chi^2$ validity criterion (Theorem 2): An F-test (Wald statistic) has a standard $\chi^2_q$ limiting distribution if and only if the null restrictions can be written entirely as restrictions on coefficients of mean-zero stationary canonical regressors. Restrictions that also involve I(1), deterministic trend, or constant canonical regressors yield nonstandard limits involving functionals of multivariate Wiener processes.
• Canonical decomposition: Via the Jordan form of the companion matrix $A$, the system $Y_t = AY_{t-1} + G\Omega^{1/2}\eta_t$ with $k_1$ eigenvalues inside and $k - k_1$ eigenvalues on the unit circle decomposes through $Z_t = DY_t$ into four blocks: $Z^1_t$ (mean-zero stationary), $Z^2_t$ (constant), $Z^3_t$ (I(1) stochastic trends), $Z^4_t$ (deterministic trend).
• Engle-Granger two-step asymptotically redundant: When the levels VAR is estimated directly, the limiting distribution of coefficient estimators is identical to the case where the cointegrating vector is known a priori — no efficiency gain from the two-step pre-estimation of $\beta$.
• Dickey-Fuller as special case (Section 5): The unit-root F-test in a univariate AR($p$) is a special case of the general framework; the limit involves $\int W(t)\,dW(t)$ and $\int W(t)^2\,dt$, recovering the Dickey-Fuller nonstandard distribution.
• Lag-length tests always $\chi^2$ (Section 6, Proposition 6.1): In a VAR with possible unit roots, the F-test for whether the last lag matrix is zero has a standard $\chi^2$ distribution in all four cases (cointegrated/not $\times$ deterministic trend/not).
• Granger causality tests (Section 6): Standard $\chi^2$ asymptotics hold when the system is cointegrated and the excluded variable's coefficient loads on the stationary equilibrium error ($A_{12}(1)$ loads on $\zeta_{1t}$, Cases 3–4); nonstandard asymptotics obtain when the excluded variable's coefficient loads on the I(1) component (Cases 1–2, no cointegration).
• Bayesian invariance (Section 7): The Gaussian likelihood of the levels VAR has the same shape regardless of nonstationarity; posterior inference using any prior absolutely continuous with respect to Lebesgue measure proceeds exactly as in the stationary case.

Concepts Introduced or Extended

• Unit Root Inference
• Granger Causality
• Cointegration
• Vector Autoregression

Entities Mentioned

• Christopher Sims
• James H. Stock
• Mark W. Watson

Quotes

"It is not necessary to have consistent estimates of the cointegrating vector when making inferences about VAR parameters; the limiting distribution of the OLS estimator is the same whether or not the cointegrating vector is known."

"From a Bayesian point of view, inference about the coefficients in this model is straightforward. The Gaussian likelihood function has the same shape whether or not the model has unit roots."

My Take

This paper resolves a major source of applied confusion in the late 1980s: whether standard inference procedures remain valid in VARs with integrated variables. The $\chi^2$ validity criterion — restrictions must fall entirely on mean-zero stationary canonical regressors — is elegant and practically actionable, giving a decisive answer about when you can use standard F-tables without worrying about integration order. The Bayesian invariance result (Section 7) anticipates Sims-Uhlig (1991): the likelihood is Gaussian regardless of unit roots; nonstandard frequentist asymptotics are a property of the estimator's sampling distribution, not of the data-generating process. The Engle-Granger two-step redundancy is a significant negative finding that rationalized estimating VARs in levels rather than going through the cointegration pre-testing ritual. The trivariate system analysis in Section 6 provides four concrete empirical cases that have been widely used as a teaching device for the asymptotics of integrated systems.