Dolado-Lütkepohl Granger Causality Test

Both augment a levels VAR by d_max extra lags to handle integration, but they differ in which parameters receive the Wald restriction. Toda-Yamamoto applies a standard Wald test to the first p lag coefficients in the same way; the two procedures are often described as equivalent in implementation. The DL paper provides a formal asymptotic justification within a cointegrated VAR framework, while Toda-Yamamoto's proof uses a different technical route. In practice, the two methods yield identical test statistics when applied consistently.

Set d_max to the highest plausible integration order — typically one for most macroeconomic and financial series. Over-estimating d_max is conservative: you lose degrees of freedom but retain valid chi-squared inference. Under-estimating d_max, however, can reintroduce non-standard asymptotics, so erring on the side of a larger d_max is generally advisable in ambiguous cases.

The test is designed for time-series VARs and works best with moderate system dimensions and sufficient sample length relative to the total number of parameters (K times (p + d_max)). In large systems the degrees-of-freedom cost of extra lags becomes substantial. Panel extensions exist in the literature but require additional assumptions about cross-sectional dependence and are not part of the original Dolado-Lütkepohl framework.