A copula approach for dependence modeling in multivariate nonparametric time series
This paper is concerned with modeling the dependence structure of two (or more) time-series in the presence of a (possible multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which we estimate nonparametrically, and the copula of the innovations, which represents the dependency structure. We consider a nonparametric as well as a semi-parametric estimator based on the estimated residuals. We show that under suitable assumptions these copula estimators are asymptotically equivalent to estimators that would be based on the unobserved innovations. The theoretical results are illustrated by simulations and a real data example.
Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
Department of Probability and Statistics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
March 7, 2018
Keywords and phrases: Asymptotic representation; CHARN model; empirical copula process; goodness-of-fit testing; nonparametric AR-ARCH model; nonparametric SCOMDY model; weak convergence.
March 7, 2018
Modeling the dependency of observed time series can be of utmost importance for applications, e. g. in risk management (for instance to model the dependence between several exchange rates). We will take the approach to model dependent nonparametric AR-ARCH time series
where the covariate may include past values of the process, (), or other exogenous variables. Further the innovations , , are assumed to be independent and identically distributed random vectors and is independent of the past and present covariates , . For identifiability we assume , (), such that the functions and represent the conditional mean and volatility function in the th time series. Such models are also called multivariate nonparametric CHARN (conditional heteroscedastic autoregressive nonlinear) models and have gained much attention over the last decades, see Fan and Yao (2005) and Gao (2007) for extensive overviews.
Note that due to the structure of the model and Sklar’s theorem (see e.g., Nelsen, 2006), for () one has
where () denote the marginal distributions of the innovations and their copula. Thus the joint conditional distribution of the observations, given the covariate, is completely specified by the individual conditional mean and variance functions, the marginal distributions of the innovations, and their copula. The copula describes the dependence structure of the time series, conditional on the covariates, after removing influences of the conditional means and variances as well as marginal distributions.
We will model the conditional mean and variance function nonparametrically like Härdle et al. (1998), among others. Semiparametric estimation, e. g. with additive structure for and multiplicative structure for as in Yang et al. (1999) can be considered as well and all presented results remain valid under appropriate changes for the estimators and assumptions. Further we will model the marginal distributions of the innovations nonparametrically, whereas we will take two different approaches to estimate the copula : nonparametrically and parametrically. As the innovations are not observable, both estimators will be based on estimated residuals. We will show that the asymptotic distribution is not affected by the necessary pre-estimation of the mean and variance functions. This remarkable result is intrinsic for copula estimation, while the asymptotic distribution of empirical distribution functions is typically influenced by pre-estimation of mean and variance functions. Moreover, comparison of the nonparametric and parametric copula estimator gives us the possibility to test goodness-of-fit of a parametric class of copulas.
Our approach extends the following parametric and semiparametric approaches in time series contexts. Chen and Fan (2006) introduced SCOMDY (semiparametric copula-based multivariate dynamic) models, which are very similar to the model considered here. However, the conditional mean and variance functions are modeled parametrically, while the marginal distributions of innovations are estimated nonparametrically and a parametric copula model is applied to model the dependence. See also Kim et al. (2007) for similar methods for some parametric time series models including nonlinear GARCH models, Rémillard et al. (2012), Kim et al. (2008) and the review by Patton (2012). Chan et al. (2009) give a goodness-of fit test for the innovation copula in the GARCH context. Further, in an i.i.d. setting Gijbels et al. (2015) show that in nonparametric location-scale models the asymptotic distribution of the empirical copula is not influenced by pre-estimation of the mean and variance function. This results was further generalized by Portier and Segers (2015) to a completely nonparametric model for the marginals.
The remainder of the paper is organized as follows. In Section 2 we define the estimators and state some regularity assumptions. In Subsection 2.1 we show weak convergence of the copula process, while in Subsection 2.2 we show asymptotic normality of a parameter estimator when considering a parametric class of copulas. Subsection 2.3 is devoted to goodness-of-fit testing. In Section 3 we present simulation results and in Section 4 a real data example. All proofs are given in the Appendix.
2. Main results
For the ease of presentation we will focus on the case of two time series, i. e. , but all results can be extended to general in an obvious manner. Suppose we have observed for a section of the stationary stochastic process that satisfies
where is a -dimensional covariate and the innovations are independent identically distributed random vectors. Further is independent of the past and present covariates , , and , . If the marginal distribution functions and of the innovations are continuous, then the copula function of the innovations is unique and can be expressed as
As the innovations are unobserved, the inference about the copula function is based on the estimated residuals
where and are the estimates of the unknown functions and . In what follows we will consider the local polynomial estimators of order ; see Fan and Gijbels (1996) or Masry (1996), among others. I.e. is for a given defined as , the component of with multi-index , where is the solution to the minimization problem
Here denotes the set of multi-indices with and . Further
with being a kernel function and the smoothing parameter.
Further is estimated as
where is obtained in the same way as but with replaced with .
For any function defined on , interval in , define for , ,
where and is the Euclidean norm on . Denote by the set of -times differentiable functions on , such that . Denote by the subset of of the functions that satisfy .
In what follows we are going to prove that under appropriate regularity assumptions using the estimated residuals (3) instead of the (true) unobserved innovations affects neither the asymptotic distribution of the empirical copula estimator nor the parametric estimator of a copula.
2.1. Empirical copula estimation
Mimicking the copula function can be estimated nonparametrically as
is the estimate of the joint distribution function and
the corresponding marginal empirical cumulative distribution functions. Here we make use of a weight function and put . For some real positive sequence we set .
Now let be the ‘oracle’ estimator based on the unobserved innovations, i.e.
where is the estimator of based on the unobserved innovations and the corresponding marginal empirical cumulative distribution functions.
The process is strictly stationary and absolutely regular (-mixing) with the mixing coefficient that satisfies with
where is specified in assumption below.
The second-order partial derivatives , and of the joint cumulative distribution function , with , satisfy
Further the innovation density satisfies
The observations () have density that is bounded and differentiable with bounded uniformly continuous first order partial derivatives. We assume the existence of some and some positive sequence such that for and ,
For some , for , , the functions and are bounded and there are some , such that for all ,
where denotes the joint density of and is bounded (for ).
Let, for and for each , and be elements of for some and a sequence for some . Further, and for some positive sequence that fulfils for some .
There exists a sequence such that , where , . Further, for from assumption and for some ,
for all , and
with from assumption and from assumption .
is a symmetric ()-times continuously differentiable probability density function supported on .
Using assumption requires that
where and stand for the first and second order partial derivatives of the copula function.
Thus provided that for some
then we need that the functions are of order .
Parts of our assumptions are reproduced from Hansen (2008) because we apply his results about uniform rates of convergence for kernel estimators several times in our proofs. Note that in his Theorem 2 we set .
Suppose that assumptions , , , , , , and are satisfied. Then
Note that Theorem 1 together with the weak convergence of (see e.g., Proposition 3.1 of Segers, 2012) implies that that process weakly converges in the space of bounded functions to a centred Gaussian process , which can be written as
where is a Brownian bridge on with covariance function
Nevertheless in applications we recommend to use rather instead of as the sample size.
2.2. Semiparametric copula estimation
The copula describes the dependency between the two time series of interest, given the covariate. For applications modeling this dependency structure parametrically is advantageous because a parametric model often gives easier access to interpretations. Goodness-of-fit testing will be considered in the next section.
Suppose that the joint distribution of is given by the copula function , where is an unknown parameter that belongs to a parametric space . In copula settings we are often interested in semi-parametric estimation of the parameter , i.e. estimation of without making any parametric assumption on the marginal distributions and . The methods of semi-parametric estimation for i.i.d. settings are summarized in Tsukahara (2005). The question of interest is what happens if we use the estimated residuals (3) instead of the unobserved innovations . Generally speaking, thanks to Theorem 1 the answer is that using instead of does not change the asymptotic distribution provided that the parameter of interest can be written as a Hadamard differentiable functional of a copula.
Method-of-Moments using rank correlation
This method is in a general way described for instance in Embrechts et al. (2005, Section 5.5.1). To illustrate the application of Theorem 1 for this method consider that the parameter is one-dimensional. Then the method of the inversion of Kendall’s tau is a very popular method of estimating the unknown parameter. For this method the estimator of is given by
is the theoretical Kendall’s tau and is an estimate of Kendall’s tau. In our settings the Kendall’s tau would be computed from the estimated residuals for which . By Theorem 1 and Hadamard differentiability of Kendall’s tau proved in Veraverbeke et al. (2011, Lemma 1), the estimators of Kendall’s tau based on or on are asymptotically equivalent. Thus provided that one gets that
Analogously one can show that working with residuals has asymptotically negligible effects also for the method of moments introduced in Brahimi and Necir (2012).
Minimum distance estimation
Here one can follow for instance Tsukahara (2005, Section 3.2). Note that thanks to Theorem 1 the proof of Theorem 3 of Tsukahara (2005) does not change when is replaced with . Thus provided assumptions (B.1)-(B.5) of Tsukahara (2005) are satisfied with , then the estimator defined as
is asymptotically normal and satisfies
M-estimator, rank approximate Z-estimators
To define a general -estimator let us introduce
that can be viewed as estimates of the unobserved . Note that the multiplier is introduced in order to have both of the coordinates of the vector bounded away from zero and one. The -estimator of the parameter is now defined as
where is a given loss function. This class of estimators includes among others the pseudo-maximum likelihood estimators (), for which , with being the copula density function.
Note that the estimator is usually searched for as a solution to the estimating equations
In what follows we give general assumptions under which there exists a consistent root () of the estimating equations (10) that is asymptotically equivalent to the consistent root () of the ‘oracle’ estimating equations given by
are the standard pseudo-observations calculated from the unobserved innovations and their marginal empirical distribution functions .
Unfortunately, these general assumptions exclude some useful models (e.g. pseudo-maximum likelihood estimator in the Clayton family of copulas) for which the function viewed as a function of is unbounded. The reason is that for empirical distribution functions calculated from estimated residuals we lack some of the sophisticated results for an empirical distribution function calculated from (true) innovations . For such copula families one can use for instance the Method-of-Moments using rank correlation (see Section 2.2.1) to stay on the safe side. Nevertheless the simulation study in Section 3 suggests that the pseudo-maximum likelihood estimation can be used also for the Clayton copula (and probably also for other families of copulas with non-zero tail dependence) provided that the dependence is not very strong.
In what follows let stand for the true value of the parameter and for an open neighbourhood of .
is a unique minimizer of the function and is an inner point of .
There exists such that for each the functions and are for each continuous in and of uniformly bounded Hardy-Kraus variation (see e.g., Berghaus et al., 2017).
There exists and a function such that for each
Each element of the (matrix) function is a continuous function and the matrix is positively definite.
The proof of Theorem 2 is given in Appendix B. Note that the asymptotic distribution of the estimator coincides with the distribution given in Section 4 of Genest et al. (1995) that corresponds to the consistent root of the estimating equations (11). Thus using the residuals instead of the true innovations has asymptotically negligible effect on the (first-order) asymptotic properties. In fact, it can be even shown that both and have the same asymptotic representations and thus
2.3. Goodness-of-fit testing
When modeling multivariate data using copulas parametrically one needs to choose a suitable family of copulas. When choosing the copula family tests of goodness-of-fit are often a useful tool. Thus we are interested in testing , where is a given parametric family of copulas.
Many testing methods have been proposed (see e.g. Genest et al., 2009; Kojadinovic and Holmes, 2009, and the references therein). The most standard ones are based on the comparison of nonparametric and parametric estimators of a copula. For instance the Cramér-von Mises statistics is given by
where is an estimate of the unknown parameter . As the asymptotic distributions of and are the same as the asymptotic distribution of and we suggest that the significance of the test statistic can be assessed in the same way as in i.i.d. settings. Thus one can use for instance the parametric bootstrap by simply generating independent and identically distributed observations from the copula function . The test statistic is then simply recalculated from this observations in the same way as if we directly observed the innovations. The only difference is that instead of generating observations we recommend to generate only observations.
Similar remarks hold when testing other hypotheses about the copula such as symmetry, for instance. Note that testing provides a test for conditional independence of the two time series, given the covariate.
3. Simulation study
A small Monte Carlo study was conducted in order to compare the semiparametric estimators based on the residuals with the ‘oracle’ estimators based on (unobserved) innovations. The inversion of Kendall’s tau (IK) method and the maximum pseudo-likelihood (MPL) method were considered for the following five copula families: Clayton, Frank, Gumbel, normal, and Student with 4 degrees of freedom. The values of the parameters are chosen so that they correspond to the Kendall’s tau , and . The data were simulated from the following four models:
where the innovations , , follow marginally the standard normal distribution, and is an exogenous variable following the ARMA model with being i.i.d. from a standard normal distribution. The simulations were conducted also for innovations , with Student marginals with 5 degrees of freedom but the results are similar. For brevity of the paper we do not present them here.