Lütkepohl (1999) Vector Autoregressions

Summary

A thorough but compact survey of vector autoregression (VAR) methodology as of 1999. Covers the full chain from model representation through estimation, specification, and applications (forecasting, Granger causality, impulse responses, variance decomposition, policy analysis), with special attention to the complications introduced by integrated and cointegrated variables. Notable for precise treatment of the five deterministic-trend cases in Johansen rank tests, the Dolado-Lütkepohl sufficient condition for $\chi^2$ causality tests, and bootstrap limitations for impulse response function (IRF) confidence intervals (CIs).

Key Claims

The VAR(p) is stable iff $\det(I_K - A_1z - \cdots - A_pz^p) \neq 0$ for $|z| \leq 1$ ; a unit root ( $z=1$ ) allows for I(1) and cointegrated variables without changing the reduced-form OLS estimator.
OLS on the levels VAR equals generalized least squares (GLS) (Zellner 1962) and is asymptotically normal, but the covariance matrix is singular when I(1) variables are present — making standard Wald tests for restrictions involving all lag matrices non- $\chi^2$ .
Johansen vector error-correction model (VECM): $\hat\beta$ converges at rate $T$ (superconsistent), $\hat\alpha$ at rate $\sqrt{T}$ ; only the cointegration space is identified without additional restrictions.
Lag-order selection: AIC (Akaike information criterion) overestimates with positive probability; HQ (Hannan-Quinn) and SC (Schwarz criterion) are consistent. Finite-sample inequality $\hat{p}(\text{SC}) \leq \hat{p}(\text{HQ}) \leq \hat{p}(\text{AIC})$ holds for all $T > 16$ .
Johansen rank tests depend critically on the deterministic-trend specification: five distinct cases exist (no constant; restricted constant; restricted trend; unrestricted trend; unrestricted linear trend in cointegration space). Prior trend-adjustment (Saikkonen-Lütkepohl) yields higher local power.
Granger noncausality in $K > 2$ variable VARs: testing $\alpha_{21,i}=0$ is not the same as Granger noncausality — indirect causal chains persist through intermediary variables. True noncausality involves nonlinear restrictions (Dufour-Renault 1998).
Dolado-Lütkepohl (1996) sufficient condition: if at least one complete coefficient matrix $A_i$ is unrestricted under $H_0$ , the Wald test has standard $\chi^2$ asymptotics — adding one redundant lag suffices.
In cointegrated systems, $n-r$ structural shocks have permanent effects (rank of $\Psi(1)$ equals $n-r$ ); Blanchard-Quah long-run zero restrictions impose this directly.
Forecast mean squared error (MSE) for I(1) variables is unbounded as horizon $h \to \infty$ ; forecast MSE for cointegrating relations $\beta'y_t$ is bounded at all horizons.
Bootstrap CIs for IRFs can fail when asymptotic variance is zero (degenerate case) — bootstrap does not fix a fundamental identification zero-variance problem.

Concepts Introduced or Extended

Lag-Order Selection Criteria (AIC, HQ, SC) — introduced here; formulas and consistency properties
Cointegration — extended: five deterministic-trend cases for rank tests; Saikkonen-Lütkepohl prior trend-adjustment
Granger Causality — extended: higher-dimensional indirect causality; Dolado-Lütkepohl sufficient condition
Impulse Response Function — extended: permanent vs. transitory; Blanchard-Quah long-run restrictions; bootstrap CI limitations
Vector Autoregression — extended: stability condition; exogeneity taxonomy; two-stage structural estimation

Entities Mentioned

Quotes

"Restrictions are imposed to a large extent by statistical tools rather than by prior believes based on uncertain theoretical considerations."

"The concept of Granger-causality can also be investigated in the framework of the VECM. [...] in a bivariate situation the cointegrating rank $r$ can only be 0, 1 or 2, where $r = 1$ is the only case which may involve genuine cointegration."

"For this procedure [Dolado-Lütkepohl lag augmentation] to work it is neither necessary to know the cointegration properties of the system nor the order of integration of the variables."

My Take

Excellent 1999 reference for the frequentist underpinnings of VAR analysis. The section on Johansen rank tests (Table 1) is the clearest taxonomy of deterministic-trend cases I have seen — explicitly maps each assumption on $(\mu_0, \mu_1)$ to the recommended test and its reference. The Dolado-Lütkepohl sufficient condition clarifies in one sentence what Toda-Phillips requires three theorems to establish. The treatment of lag selection criteria is unusually explicit (formulas and finite-sample inequality). Major gap: no Bayesian content — this is the frequentist reference that the Bayesian papers in this wiki implicitly presuppose.