Modelling Interest Rates with a Cointegrated VAR-GARCH Model

Summary

Bauwens, Deprins, and Vandeuren (1998) fit a bivariate vector error-correction model (VECM) for the long and short interest rates $(R_t, r_t)'$ of five countries (Belgium, Germany, France, UK, USA) while allowing the innovation covariance matrix to follow a BEKK (Baba-Engle-Kraft-Kroner)-GARCH process. The key finding is that GARCH heteroscedasticity materially sharpens cointegration test power — UK and Germany appear cointegrated only once GARCH is incorporated in the likelihood. The unconditional covariance matrix does not exist in any country (multivariate integrated GARCH, IGARCH, behavior), and off-diagonal BEKK elements are required everywhere except the USA, indicating a common volatility factor in short and long rates.

Key Claims

• GARCH sharpens cointegration evidence. Standard Johansen tests (homoscedastic innovations) reject cointegration for UK and Germany. Estimating the VECM-BEKK jointly shifts this conclusion: both countries produce significant cointegrating ranks once the GARCH structure is accounted for in the likelihood.
• Multivariate IGARCH in all countries. The covariance-stationarity eigenvalue condition $\rho(A_1'\otimes A_1' + G_1'\otimes G_1') < 1$ fails for all five countries: the dominant eigenvalue exceeds unity, so the unconditional covariance matrix $\mathrm{Vech}(H) = (I - A_1' \otimes A_1' - G_1' \otimes G_1')^{-1}\mathrm{Vech}(C'C)$ does not exist. Short-run volatility forecasts remain valid, but long-run variance is infinite.
• Off-diagonal BEKK required. The restricted diagonal BEKK (where $A_1$ and $G_1$ are diagonal) is rejected at 5% for Belgium, Germany, France, and UK via a likelihood-ratio test. Only for the USA is diagonal BEKK adequate. This implies that shocks to one rate spill over into the conditional variance of the other — a common volatility factor is present.
• Present value motivation. Campbell and Shiller (1987) show that if long rates are present values of expected future short rates, the spread $R_t - \gamma r_t$ (with $\gamma = 1$ under the expectations hypothesis) must be stationary. The cointegrating vector is $(1, -\gamma)'$; $\gamma = 1$ is accepted for France, UK, and USA but rejected for Belgium and Germany.
• Asymmetric adjustment. In all countries the short rate $r_t$ bears the larger adjustment coefficient $\alpha$ (error-correction loading), indicating it corrects faster toward the long-run equilibrium than the long rate $R_t$. This is consistent with monetary policy targeting short-term rates.
• GARCH(1,1) suffices. Portmanteau tests on standardized squared residuals fail to reject white noise for all countries at lag orders beyond one GARCH term, justifying the parsimonious GARCH(1,1) (i.e., $p=q=1$ in the BEKK).
• Short prediction horizon. Out-of-sample conditional covariance forecasts are informative for approximately one to three months; beyond that horizon the GARCH estimates of $H_{t+h}$ converge rapidly toward the long-run target (which, under IGARCH, is infinite) and provide no predictive advantage over unconditional estimates.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH — BEKK multivariate GARCH; covariance stationarity eigenvalue condition; IGARCH; quasi-maximum likelihood (QML) estimation
• Cointegration — interaction of GARCH heteroscedasticity with cointegration testing; present value cointegrating vector; asymmetric adjustment

Entities Mentioned

• Luc Bauwens

Quotes

"The evidence of cointegration is stronger when the GARCH structure is taken into account." (paraphrase of main empirical finding, Section 4)

My Take

The paper makes a methodologically important point: ignoring volatility clustering inflates the residual variance estimate and therefore biases cointegration test statistics toward non-rejection of the null. The VECM-BEKK framework is theoretically sound but computationally demanding (QML on a nonlinear system with O(n²) GARCH parameters alongside the VECM coefficients). A limitation is that QML consistency relies on correct specification of the conditional mean (the VECM), and rank pre-testing with the homoscedastic Johansen procedure before BEKK-QML estimation introduces the usual pre-test bias. The IGARCH finding is also fragile — near-unit-root behavior in variance is difficult to distinguish in short samples from a stationary GARCH with high persistence, and the paper does not formally test for an exact unit root in variance.