Phillips (1991) Optimal Inference in Cointegrated Systems

Summary

Phillips (1991) establishes when the Functional Central Limit Theorem (FCLT) framework yields a locally asymptotically mixed normal (LAMN) likelihood — enabling $\chi^2$ inference and Cramér-Rao efficiency — versus a locally Gaussian functional (LGF) likelihood that produces nonstandard distributions and nuisance-parameter contamination. The key finding is that the triangular Error Correction Model (ECM) parameterization (where the cointegrating vector $B$ appears linearly) delivers LAMN, while an unrestricted Vector AutoRegression (VAR) in levels (which implicitly estimates unit roots) produces LGF. The paper proves that the full-system Maximum Likelihood (ML) estimator of $B$ attains the mixed-normal information bound and is median unbiased, whereas Ordinary Least Squares (OLS) suffers from simultaneous-equations bias and single-equation ML fails unless strict exogeneity holds.

Key Claims

• Triangular ECM representation. The system (1) $y_{1t} = By_{2t} + u_{1t}$; (2) $\Delta y_{2t} = u_{2t}$; (3) $\Delta y_t = -EAy_{t-1} + v_t$ separates the unit-root components explicitly so the cointegrating vector $B$ ($n_1 \times n_2$) appears linearly in equation (1). This linearity is the structural feature that yields LAMN.

• LAMN vs. LGF dichotomy. An ECM that eliminates unit roots from its parameterization produces a LAMN likelihood $\to$ chi-squared test statistics $\to$ efficient ML. An unrestricted VAR in levels implicitly estimates unit roots alongside all other parameters $\to$ LGF likelihood $\to$ nonstandard, nuisance-parameter-dependent distributions $\to$ no optimal inference theory.

• Theorem 1 (independent and identically distributed (iid) errors): mixed-normal Maximum Likelihood Estimator (MLE). The full-system MLE satisfies

  $$T(\hat{B}-B) \Rightarrow \left(\int_0^1 dS_{1\cdot2}\,S_2'\right)\left(\int_0^1 S_2 S_2'\right)^{-1}$$

  which is conditionally $N(0,\,\Omega_{11\cdot2}\otimes G)$ given $G=\int_0^1 S_2 S_2' dt$ — a mixed normal (MN) distribution. The MLE is symmetrically distributed, median unbiased, and asymptotically efficient. Wald, Likelihood Ratio (LR), and Lagrange Multiplier (LM) tests all converge to $\chi^2_q$.

• OLS simultaneous-equations bias.

  $$T(\hat{B}^*-B) \sim \left(A\int_0^1 dS\,S_2' + \Sigma_{12}\right)\left(\int_0^1 S_2 S_2'\right)^{-1}$$

  The additional $\Sigma_{12}$ term (cross-equation innovation covariance) induces bias when $u_{1t}$ and $u_{2t}$ are correlated. OLS estimators of the cointegrating vector are not median unbiased and suffer from simultaneous-equations bias.

• Theorem 2 (Full Information Maximum Likelihood (FIML) in simultaneous system). The full-information ML in a simultaneous-equations ECM satisfies $T(\tilde{B}-B) \Rightarrow \left(A\int_0^1 dS S_2'\right)\left(\int_0^1 S_2 S_2'\right)^{-1}$ — equals the single-equation MLE iff $\Sigma_{12}=0$; otherwise contains a "unit root" component that places it outside LAMN.

• Theorem 1' (general linear process errors). The mixed-normal limit holds with $\Omega = 2\pi f(0)$ (the spectral density at frequency zero — the long-run covariance matrix) replacing the short-run $\Omega$.

• Theorem 1'' (variation dependence). Limit theory continues to apply even when the nuisance parameters $\theta$ (characterising short-run dynamics) and $B$ are not variation independent.

• Single-equation ECM fails. The partial MLE of $B$ from a single equation is efficient iff $\Sigma_{12}=0$ (strict exogeneity of $y_{2t}$). This condition fails in typical macro applications where all variables are jointly determined.

• Transient dynamics not needed. Only a consistent estimator of the long-run covariance $\hat\Omega = 2\pi\hat{f}(0)$ is required — not joint ML estimation of the transient dynamics. This underpins the Phillips-Hansen (1990) Fully Modified OLS (FM-OLS) approach.

• Standard tests valid under LAMN. Because the triangular ECM is LAMN, Wald, LR, and LM test statistics all have asymptotic $\chi^2$ distributions, enabling routine inference on cointegrating vectors.

Concepts Introduced or Extended

• Cointegration
• Spurious Regression
• Vector Autoregression

Entities Mentioned

• Peter C. B. Phillips

Quotes

"OLS estimators of the cointegrating vector do not have median unbiased limit distributions and they suffer from a simultaneous equations bias."

"The limit distribution of the ML estimator is shown to be mixed normal and, hence, chi-squared tests are valid."

"It is not necessary to estimate the complete system by maximum likelihood… a consistent estimate of the long run covariance matrix is all that is needed."

My Take

This paper provides the theoretical foundation for why Johansen's MLE dominates OLS for cointegrating vectors: not merely a statistical improvement but a qualitative change in the nature of the limit theory. The LAMN/LGF distinction explains why running a levels VAR and interpreting Wald tests at face value is invalid when rank conditions fail, while a properly specified ECM restores standard inference. The "transient dynamics not needed" result is practically powerful: it justifies two-step procedures (FM-OLS, Dynamic OLS (DOLS)) that only require a consistent long-run covariance estimate, avoiding costly full-system ML. The limitation is that strict exogeneity ($\Sigma_{12}=0$) is rarely plausible in macro applications — making single-equation ECM the exception rather than the rule and full-system Johansen ML the necessary default.