13.7 COPULAS

13.7 COPULAS

Copulas provide an alternative representation to multivariate distribution functions or densities in a way that may be helpful in generating forecasts. Consider the case where $y_{t+1}=(y_{1t+1},\dots ,y_{nt+1})$ has a multivariate distribution function P .

By Sklar’s theorem (Sklar, 1959), we can decompose P into its individual univariate marginal distributions, $P_1(y_{1t+1}),\dots ,P_n(y_{nt+1})$ , and an n-dimensional copula, C , capturing the joint distribution of the marginal distributions, $P(y_{t+1})=$ $C(P_1(y_{1t+1}),\dots ,P_n(y_{nt+1}))$ . While P and C both have n-dimensional inputs, the key distinction is that the arguments in the copula function have uniform marginals. Each of the $P_i(y_i)$ functions are probability integral transforms (see chapter 18) and so will have uniform marginals.

The copula is simply a multivariate cumulative density function with uniform (0,1) marginals. Intuitively, it contains all the dependence information that is not captured in the marginal distributions of the individual variables. An advantage of copulas is that they are invariant under increasing and continuous transformations of the marginals. Hence if $(y_{1t+1},\dots ,y_{nt+1}{)}^{\prime }$ has copula C and $G_1,\dots ,G_n$ are increasing continuous functions, $(G_1(y_{1t+1}),\dots ,G_n(y_{nt+1}){)}^{\prime }$ also has copula $C;$ see Embrechts, McNeil, and Straumann (2002).

Multivariate ARCH models such as (13.52) capture dependencies across variables through their correlations. This suffices in the context of elliptical distributions which include the Gaussian family. Outside such restrictive families of densities, correlation will not, however, suffice to capture dependencies across variables. In particular, whereas independence of two random variables implies that they are uncorrelated (zero linear correlation), zero correlation does not in general imply independence. For example, $\mathrm{if}y\sim \mathrm{N}(0,1)$ , y and $y^2$ are uncorrelated but clearly not independent of each other.

Most relevant for forecasting situations with conditional dynamics in the distribution of the variables is the copula representation conditional on a set of variables $Z_t.$ Assume that $y_{t+1}|Z_t\sim P_{t+1|t}(y_{t+1})$ with marginal distributions $P_{it+1|t}\left(y_{it+1}\right)$ . Then the conditional copula representation takes the form (see Patton, 2006)

$$P_{t+1|t}(y_{t+1})=C_{t+1|t}(P_{1t+1|t}(y_{1t+1}),\dots ,P_{nt+1|t}(y_{nt+1})).$$

From a forecasting perspective, there are potential benefits admitted from the decomposition in (13.57) of the joint distribution into the marginal distributions and the copula. For example, the marginal distributions may be fairly easy to estimate using the large literature on estimation of univariate distributions. A two-stage estimation approach could first estimate the marginal distributions and then use $u_{it+1|t}=P_{it+1|t}(y_{it+1})$ as input into the conditional copula model. One option here is to estimate the marginal distributions nonparametrically and fit a parametric model to the copula. This approach is appealing given that the marginal distributions are univariate and hence more easily modeled using nonparametric methods than the n-dimensional copula function. The copula approach can also be applied to multivariate density modeling. Multiplying the univariate conditional density estimates, $p_{it+1|t}\left(y_{it+1}\right)$ , by the copula estimates yields the multivariate joint conditional density, $p_{t+1|t}\left(y_{t+1}\right)$

$$p_{t+1|t}(y_{t+1})=c_{t+1|t}(u_{1t+1},\dots ,u_{n_t+1})\times p_{1t+1|t}(y_{1t+1})\times \cdot \cdot \cdot \times p_{nt+1|t}(y_{nt+1}).$$

Decompositions such as these may be helpful particularly in cases where the focus is on modeling dependencies for high-dimensional vectors of variables, i.e., when n is very large.

Another advantage from the copula representation in (13.57) is that it affords great flexibility in pairing well-known marginal distributions (e.g., a conditional Gaussian and a skewed-t distribution for two asset returns) with an altogether different copula such as the Gumbel or logistic copula which takes the form

$$\begin{array}{rl}& C^{\mathrm{Gu}}(u_{1t+1},\dots ,u_{nt+1}|\beta )\\ & =\exp \left[-{\left\{(-\log (u_{1t+1}){)}^{1/\beta }+\cdots +(-\log (u_{nt+1}){)}^{1/\beta }\right\}}^{\beta }\right],\end{array}$$

where $0<\beta \le 1$

Patton (2013) reviews empirical applications of copula models. Most studies of copulas have been to financial data with numerous applications in areas such as risk management, pricing of derivatives, and portfolio selection. The jury is still out on whether, in practice, advantages such as those listed above translate into better multivariate forecasts being produced by copula models than if the forecasts were based on conventional multivariate distributions. Of course, given the oneto-one correspondence between the copula and multivariate distribution function representations, ultimately which approach is most useful may be a question of convenience, i.e., which approach lends itself best to estimation and implementation. For example, the bivariate Gaussian copula takes the form

$$C_{\rho }(y_1,y_2)={\int }_{-\infty }^{{\Phi }^{-1}(y_1)}{\int }_{-\infty }^{{\Phi }^{-1}(y_2)}\frac{1}{2\pi (1-{\rho }^2{)}^{1/2}}\exp \left(\frac{-(s^2-2\rho st+t^2)}{2(1-{\rho }^2)}\right)dsdt,\text{\hspace{1em}(13.59)}$$

and so depends on only a single correlation parameter, $\rho .$ Here  is the univariate c.d.f. of a standard Gaussian random variable. If the marginal distributions are also Gaussian, there is no obvious advantage from using the copula representation in (13.59) rather than the bivariate normal distribution function.