Comparing Copula VAR Models

VAR(1) Marginals

Each variable follows a VAR(1) process with innovations:

\[y_t = \mu + \Phi \, (y_{t-1} - \mu) + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \text{diag}(\sigma_{\varepsilon}^2))\]

The normal marginal densities provide the per-variable likelihoods in this special case. For non-normal margins, dcvar uses the same VAR recursion and Gaussian copula but swaps in the corresponding marginal likelihood and transform.

Gaussian Copula

The dependence between variables is captured by a Gaussian copula with time-varying correlation \(\rho_t\). The log-likelihood contribution of the copula at time \(t\) is:

\[\log c(u_{1t}, u_{2t}; \rho_t) = -\frac{1}{2}\log(1 - \rho_t^2) + \frac{\rho_t^2 \, (z_{1t}^2 + z_{2t}^2) - 2\rho_t \, z_{1t} z_{2t}}{2(1 - \rho_t^2)}\]

where \(z_{it} = \Phi^{-1}(u_{it})\) and \(u_{it} = \Phi(\varepsilon_{it} / \sigma_{\varepsilon,i})\).

How the Models Differ

• DC-VAR: The correlation evolves as a random walk on the Fisher-z scale: \(z_t = z_{t-1} + \omega_t\) with \(\omega_t \sim N(0, \sigma_\omega^2)\), and \(\rho_t = \tanh(z_t)\).

• HMM Copula: The correlation is state-dependent: \(\rho_t = \rho_{s_t}\), where \(s_t \in \{1, \ldots, K\}\) follows a Markov chain with transition matrix \(A\). State-specific correlations use an ordered constraint (\(\rho_1 < \rho_2 < \ldots\)) to prevent label switching.

• Constant Copula: \(\rho_t = \rho\) for all \(t\). A special case of both the DC-VAR (\(\sigma_\omega = 0\)) and HMM (\(K = 1\)).