Modelling the Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalized ARCH Model

Summary

A 9-page paper introducing the Constant Conditional Correlation (CCC) GARCH model: $H_t = D_t \Gamma D_t$, where $D_t$ is diagonal with individual GARCH(1,1) standard deviations and $\Gamma$ is a time-invariant positive definite correlation matrix. Applied to five European currencies vs. the USD over the European Monetary System (EMS, 1979–1985) and pre-EMS (1973–1979) periods, finding significantly higher correlations under EMS — attributable partly to policy coordination and partly to a common USD factor.

Key Claims

Econometric Methodology

• CCC model: $H_t = D_t \Gamma D_t$ with $h_{it} = \omega_i + \alpha_{i1}\varepsilon_{i,t-1}^2 + \beta_{i1}h_{i,t-1}$ (univariate GARCH(1,1)) and $h_{ijt} = \rho_{ij}(h_{iit}h_{jjt})^{1/2}$. Positive definiteness holds iff each univariate variance is positive and $\Gamma \succ 0$.
• Likelihood concentration: $\hat\Gamma = T^{-1}\sum_t \tilde\varepsilon_t\tilde\varepsilon_t'$ where $\tilde\varepsilon_t = D_t^{-1}e_t$ are standardized residuals — concentrated out analytically. Reduces matrix inversions from $T$ (the general case) to 1.
• Seemingly unrelated regression (SUR) interpretation: The model is an SUR extension allowing conditional heteroskedasticity; the information matrix is not block diagonal between $D_t$ and $\Gamma$ parameters, so joint maximum likelihood (ML) is required for correct standard errors (Berndt-Hall-Hall-Hausman, BHHH, algorithm with numerical derivatives).
• Quasi-maximum likelihood (QML) robustness (Bollerslev and Wooldridge 1989): if conditional mean and variance are correctly specified but normality fails, QML is still consistent and asymptotically normal; robust sandwich standard errors (SE) required.

Empirical Results (5 EMS currencies, weekly)

• Data: DM (Deutsche Mark), FF (French franc), IL (Italian lira), SF (Swiss franc), BP (British pound) vs. USD; July 1973 – August 1985. EMS period: 333 obs; pre-EMS: 299 obs.
• Martingale property not rejected for any currency (Phillips-Perron unit root tests).
• GARCH(1,1) for each conditional variance passes specification tests; likelihood-ratio (LR) statistic for no ARCH = 117.028 ~ $\chi^2(10)$.
• LR test for all $\rho_{ij} = 0$: 1,911.078 ~ $\chi^2(10)$ — overwhelmingly rejected.
• All conditional correlations significantly higher in EMS period: e.g. DM–FF 0.607→0.932; DM–IL 0.425→0.886; BP–DM 0.443→0.674.
• Even non-EMS currencies (BP, SF) show higher EMS-period correlations — suggests common USD factor, not only EMS policy coordination.
• Unconditional variances higher in EMS period for all currencies (linked to Federal Reserve (Fed) operating procedure change, October 1979).
• LR test for parameter constancy across the full 12-year period: 388.110 ~ $\chi^2(30)$ — highly significant.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH — Constant Conditional Correlation (CCC) model; $H_t = D_t\Gamma D_t$; likelihood concentration; SUR extension; predecessor to dynamic conditional correlation (DCC) (Engle 2002)

Entities Mentioned

• Tim Bollerslev

Quotes

"Compared to the linear diagonal GARCH model estimated in Bollerslev, Engle and Wooldridge (1988), the latent factor ARCH model in Diebold and Nerlove (1989), or the factor GARCH model in Engle, Ng, and Rothschild (1990), the parameterization proposed here with time varying conditional covariances but constant conditional correlations represents a major reduction in terms of computational complexity."

My Take

The CCC model earns its place in the GARCH lineage as the first tractable multivariate GARCH: one $N\times N$ inversion per dataset (not $T$), $\Gamma$ concentrated analytically, positive definiteness guaranteed without the BEKK (Baba-Engle-Kraft-Kroner) outer-product trick. The EMS application is a clean empirical illustration — the pre/post period comparison (same five currencies, same model) is exactly right for demonstrating what the model delivers. The main conceptual limitation — constant correlations — was later addressed by DCC (Engle 2002), which Engle explicitly frames as an extension of this model.