How to determine the joint distribution of two time series with serial dependence? (Application of Copula))

Question

I am going to determine the joint distribution of two time series. Each time series have serial dependence. How can I apply copula in this condition?

Thanks in advance for any helps.

Answer 1

You can model the development of the bivariate time series over time specifying e.g. the dynamics of the conditional mean vector and the conditional variance matrix and then assuming i.i.d. standardized innovations from a bivariate distribution which is modeled using a copula. If

• $x_t$ is a bivariate time series of interest,
• $\varepsilon_t$ is a bivariate series of (nonstandardized) innovations and
• $z_t$ is a bivariate time series of standardized innovation,

you would get something like this:

\begin{aligned} x_t &= \mu_t+\varepsilon_t, \\ \mu_t &= \text{some model of the conditional mean, e.g. VAR or VARMA}, \\ \varepsilon_t &= \Sigma_t^{1/2} z_t, \\ \Sigma_t &= \text{some model of the conditional variance, e.g. some sort of multivariate GARCH}, \\ z_t &\sim i.i.D(\xi). \end{aligned}

Here,

• $\mu_t$ is a bivariate vector of conditional mean,
• $\Sigma_t$ is a $2\times 2$ matrix of conditional variance of $\varepsilon_t$ and
• $D$ is some bivariate density parameterized by $\xi$ ($\xi$ could be a constant or a vector).

You would model $D(\xi)$ using copula.

More generally, given a multivariate time series, you would model some of the parameters of its joint distribution at time $t$ as time-varying and the remainder as constant over time. When adjusted for the time variation, the joint distribution would be constant and could thus be modeled using a copula. In the example above, adjusting for time variation means obtaining $z_t$ (which has a constant joint distribution) from $x_t$ (which has a time-varying joint distribution) via modelling the evolution of parameters $\mu_t$ and $\Sigma_t$.

Perhaps my formulation of ARMA-GARCH model here can be helpful. (It addresses a univariate case, but that is not important for the point I am making here.) It shows explicitly how the parameters of the distribution are evolving over time: there are equations for $\mu_t$ and $\sigma_t^2$. The equation for $\sigma_t^2$ is commonplace in representations of GARCH models, so no big deal there. But the equation for $\mu_t$ is not all that common in the time series literature on ARMA models.