Toda-Phillips (1993) Vector Autoregressions and Causality

Summary

Toda and Phillips (1993) develop a complete asymptotic theory for Wald tests of Granger causality in both levels Vector Autoregression (VAR) systems estimated by Ordinary Least Squares (OLS) and Johansen-type Error Correction Models (ECM) estimated by Maximum Likelihood (ML), allowing for $I(1)$ variables and cointegration. The main finding is that standard chi-squared asymptotics fail in general — either because the limiting distribution depends on nuisance parameters (levels VAR) or because nonlinear restrictions on the cointegrating matrix and loading matrix produce non-standard limits (ECM). Sufficient rank conditions under which chi-squared asymptotics are restored are derived, and ECM-based sequential tests are shown to be preferable to levels VAR tests.

Key Claims

• In levels VAR (OLS), the Wald statistic $F_{WD}$ for noncausality is $\chi^2_{n_1 n_3 k}$ only if $\text{rank}(A_3) = n_3$ ("sufficient cointegration"), where $A_3$ is the subblock of the cointegrating matrix corresponding to the potentially causal variables. Otherwise the limit distribution mixes chi-squared and nonstandard components depending on nuisance parameters.
• If the system has no cointegration, $F_{WD}$ converges to a nonstandard but nuisance-parameter-free limit whose critical values can be tabulated.
• OLS estimators of cointegrating vectors suffer from simultaneous-equations bias; it is therefore impossible to verify the rank condition for chi-squared validity from a levels VAR alone — the test has no valid statistical basis in the general case.
• In the ECM (Johansen ML), $F_{ML} \xrightarrow{d} \chi^2_{n_1 n_3 k}$ if $\text{rank}(A_3) = n_3$ or $\text{rank}(\Gamma_1) = n_1$, where $\Gamma_1$ is the first $n_1$ rows of the loading matrix. When both are rank-deficient the limit involves a nonlinear function of chi-squared variates.
• The Wald statistic in the ECM decomposes as $F_{ML} = F_{ML,(1)} + F_{ML,(2)}$ where $F_{ML,(1)} \sim \chi^2_{n_1 n_3(k-1)+n_1 g}$ (short-run dynamics) and $F_{ML,(2)} \sim \chi^2_{n_1(n_3-g)}$ (long-run cointegrating restrictions), with $g = \text{rank}(A_3)$.
• Simulation evidence (companion paper, Toda-Phillips 1991b) shows the sequential ECM test performs well for $n = 3$–$4$ and $T > 100$.

Concepts Introduced or Extended

• Granger Causality — rank conditions for chi-squared validity; failure modes in levels VAR and ECM; sequential testing procedure
• Cointegration — Toda-Phillips rank conditions on $A_3$ and $\Gamma_1$; nuisance parameter problem in causality tests

Entities Mentioned

• Hiro Y. Toda
• Peter C. B. Phillips

Quotes

"We show that without explicit information on the number of unit roots in the system and the rank of certain submatrices in the cointegration space it is impossible to determine the appropriate limit theory in advance."

"We conclude that causality tests based on OLS estimation in levels VAR's are not to be recommended in general."

"Johansen-type ECM's do offer a sound basis for empirical testing of the rank of the cointegration space and the rank of key submatrices that influence the asymptotics."

My Take

The paper is theoretically definitive but practically sobering: it shows that essentially the most commonly performed test in applied VAR work (Granger causality in levels) is generically invalid. The ECM-based alternative requires a sequential procedure (test rank, then test causality) that is rarely carried out in applied work. The companion paper (Toda-Yamamoto 1995, published separately) proposes a simpler lag-augmentation approach that sidesteps the rank conditions entirely at the cost of some power — that procedure became more widely adopted in practice. This 1993 paper is the theoretical foundation.