$\beta$-mixing and moments properties of a non-stationary copula-based Markov process

TL;DR

This paper establishes conditions for a non-stationary copula-based Markov process to be $eta$-mixing and introduces a convolution-based Gaussian Markov process that generalizes random walks with dependent increments.

Contribution

It provides new conditions for $eta$-mixing in non-stationary copula-based Markov processes and introduces a generalized Gaussian Markov process with dependent increments.

Findings

Conditions for $eta$-mixing are derived.

A new convolution-based Gaussian Markov process is introduced.

The process generalizes standard random walks with dependent increments.

Abstract

This paper provides conditions under which a non-stationary copula-based Markov process is $\beta$-mixing. We introduce, as a particular case, a convolution-based gaussian Markov process which generalizes the standard random walk allowing the increments to be dependent.

Equations

$$C_{s,t}(u,v)=C_{s,r}\ast C_{r,t}(u,v)={\int }_0^1\frac{\partial }{\partial w}{C}_{s,r}(u,w)\frac{\partial }{\partial w}{C}_{r,t}(w,v)dw,\forall s<r<t.$$

$$C_{t,t+k}(u,v)=C_{t,t+k-1}\ast C_{t+k-1,t+k}(u,v)=C_{t,t+1}\ast C_{t+1,t+2}\ast \cdots \ast C_{t+k-1,t+k}(u,v).$$

$$\tilde{\beta }({\mathcal{F}}_{-\infty }^t,{\mathcal{F}}_{t+k}^{+\infty })=\underset{\{A_i\},\{B_j\}}{\sup }\frac{1}{2}\sum _{i=1}^I\sum _{j=1}^J|\mathbb{P}(A_i\cap B_j)-\mathbb{P}(A_i)\mathbb{P}(B_j)|,$$

$${\beta }_k=\underset{t\in \mathbb{Z}}{\sup }\tilde{\beta }({\mathcal{F}}_{-\infty }^t,{\mathcal{F}}_{t+k}^{+\infty }).$$

$$\underset{f,g}{\sup }\left|{\int }_0^1{\int }_0^{1}f(x)g(x)C(dx,dy)\right|$$

$$\hat{\eta }=\underset{t\in \mathbb{Z}}{\sup }{\eta }_t<1,$$

$$\tilde{\beta }({\mathcal{F}}_{-\infty }^t,{\mathcal{F}}_{t,t+k}^{+\infty })$$

$$=\frac{1}{2}\parallel C_{t,t+k}(F_t(x),F_{t+k}(y))-F_t(x)F_{t+k}(y){\parallel }_{TV}\le$$

$$\le \frac{1}{2}\parallel C_{t,t+k}(u,v)-uv){\parallel }_{TV}.$$

$$\tilde{\beta }({\mathcal{F}}_{-\infty }^t,{\mathcal{F}}_{t+k}^{+\infty })\le \frac{1}{2}\parallel c_{t,t+k}(u,v)-1{\parallel }_{{\mathbb{L}}^1}\le \frac{1}{2}\parallel c_{t,t+k}(u,v)-1{\parallel }_{{\mathbb{L}}^2}$$

$${\beta }_k\le \frac{1}{2}\underset{t\in \mathbb{Z}}{\sup }\parallel c_{t,t+k}(u,v)-1{\parallel }_{{\mathbb{L}}^2}.$$

$$c_{t,t+1}(u,v)=1+\sum _{i=1}^{+\infty }{\lambda }_{i,t}{\varphi }_i(u){\varphi }_i(v),$$

$$\underset{i\ge 1}{\max }{\lambda }_{i,t}={\eta }_t.$$

$$c_{t,t+k}(u,v)=1+\sum _{i=1}^{+\infty }\left(\prod _{j=0}^{k-1}{\lambda }_{i,t+j}\right){\varphi }_i(u){\varphi }_i(v).$$

$$\parallel c_{t,t+k}(u,v)-1{\parallel }_{{\mathbb{L}}^2}$$

$$={\left[\sum _{i=1}^{+\infty }{\lambda }_{i,t}^2\left(\prod _{j=1}^{k-1}{\lambda }_{i,t+j}^2\right)\right]}^{1/2}\le {\left[\sum _{i=1}^{+\infty }{\lambda }_{i,t}^2\left(\prod _{j=1}^{k-1}{\eta }_{t+j}^2\right)\right]}^{1/2}\le$$

$$\le {\hat{\eta }}^{k-1}{\left[\sum _{i=1}^{+\infty }{\lambda }_{i,t}^2\right]}^{1/2}={\hat{\eta }}^{k-1}\parallel c_{t,t+1}(u,v)-1{\parallel }_{{\mathbb{L}}^2}.$$

$${\beta }_k\le \frac{1}{2}{\hat{\eta }}^{k-1}\underset{t\in \mathbb{Z}}{\sup }\parallel c_{t,t+1}(u,v)-1{\parallel }_{{\mathbb{L}}^2}$$

$$Y_t=Y_{t-1}+{\xi }_t,Y_0=0,$$

$$F_t(y_t)=(F_{t-1}\stackrel{C}{\ast }{H}_t)(y_t)={\int }_0^1{D}_{1}C(w,H_t(y_t-F_{t-1}^{-1}(w)))dw,t\ge 2$$

$$C_{t-1,t}(u,v)={\int }_0^u{D}_{1}C(w,H_t(F_t^{-1}(v)-F_{t-1}^{-1}(w)))dw,t\ge 2$$

$$Y_t\sim N(0,V_t^2),$$

$$V_t^2=Var(Y_t)=V_1^2+(t-1){\sigma }_{\xi }^2+2\rho {\sigma }_{\xi }\sum _{i=1}^{t-1}{V}_{t-i},t\ge 2,$$

$${\tau }_{t,t+1}=\frac{V_t+\rho {\sigma }_{\xi }}{V_{t+1}},t\ge 2$$

$$\underset{t\rightarrow+\infty}{\lim}V_{t}=\left\{\begin{array}[]{ll}-\frac{\sigma_{\varepsilon}}{2\rho},&\hbox{ if \ $\rho\in(-1,0)$}\\ +\infty,&\hbox{otherwise.}\\ \end{array}\right.$$

$${\tau }_{t,t+k}=\prod _{s=0}^{k-1}\frac{V_{t+s}+\rho {\sigma }_{\xi }}{V_{t+s+1}}.$$

$$\frac{V_{t+s}+\rho {\sigma }_{\xi }}{V_{t+s+1}}\to \frac{-\frac{{\sigma }_{\xi }}{2\rho }+\rho {\sigma }_{\xi }^2}{-\frac{{\sigma }_{\xi }}{2\rho }}=1-2{\rho }^2,$$

$$\mathbb{E}[{\xi }_t{\xi }_{t+k}]$$

$$=\mathbb{E}[Y_t{Y}_{t+k}]-\mathbb{E}[Y_t{Y}_{t+k-1}]-\mathbb{E}[Y_{t-1}{Y}_{t+k}]+\mathbb{E}[Y_{t-1}{Y}_{t+k-1}]=$$

$$={\tau }_{t,t+k}{V}_t{V}_{t+k}-{\tau }_{t,t+k-1}{V}_t{V}_{t+k-1}-{\tau }_{t-1,t+k}{V}_{t-1}{V}_{t+k}+{\tau }_{t-1,t+k-1}{V}_{t-1}{V}_{t+k-1}.$$

$$\mathbb{E}[{\xi }_t{\xi }_{t+k}]\to$$

$$=-{\rho }^2{\sigma }_{\xi }^2(1-2{\rho }^2{)}^{k-1},ast\to +\infty .$$

$$corr({\xi }_t{\xi }_{t+k})\to -{\rho }^2(1-2{\rho }^2{)}^{k-1}.$$

$$|{\tau }_{t,t+1}|<1,\forall t.$$

Full text

JEL classification: C22,C10

Mathematics Subject Classification (2010): 62M10, 62H20

Keywords: Markov process, copula, $\beta$-mixing, gaussian process.

1 Introduction

In this paper we analyze the temporal dependence properties satisfied by a discrete times non-stationary Markov process. Temporal dependence is relevant since it permits to verify of how well theoretical models explain temporal persistency observed in financial data. Moreover, it is also a useful tool to establish large sample properties of estimators for dynamic models. In particular, in this paper we analyze the $\beta$-mixing property and we give sufficient conditions that ensure this property be satisfied.

In the copula approach to univariate time series modelling, the finite dimensional distributions are generate by copulas. Darsow et al. (1992) provide necessary and sufficient conditions for a copula-based time series to be a Markov process. Recent literature on this topic has mainly focused on the stationary case. Chen and Fan (2006) introduce a copula-based strictly stationary first order Markov process generated from $(G_{0}(\cdot),C(\cdot,\cdot,\alpha_{0}))$ where $G_{0}(\cdot)$ is the invariant distribution of $Y_{t}$ and $C(\cdot,\cdot,\alpha_{0})$ is the parametric copula for $(Y_{t-1},Y_{t})$. The authors show that the $\beta$-mixing temporal dependence measure is purely determined by the properties of copulas and present sufficient conditions to ensure that the process $(Y_{t})_{t}$ based on gaussian and EFGM copulas are geometric $\beta$-mixing. Beare (2010) shows that all Markov models generated via symmetric copulas with positive and square integrable densities are geometric $\beta$-mixing. Many commonly used bivariate copulas without tail dependence such as gaussian, EFGM and Frank copulas satisfy this condition. Chen et al. (2009) show that Clayton, Gumbel and Student’s $t$ copula based Markov models are geometrically ergodic which is a stronger condition than the geometric $\beta$-mixing.

In this paper we focus on Markov processes where some dependence between each state variable and increment is allowed and modeled through a time-invariant copula. In particular, we introduce a gaussian Markov process, which is non-stationary and generalizes the classical gaussian random walk, and we study related moments properties and provide conditions under wich the process is $\beta$-mixing.

The paper is organized as follows. Section 2 presents a general result on the $\beta$-mixing properties satisfied by non-stationary Markov processes. Section 3 restricts the study to the gaussian case. Section 4 concludes.

2 Copula-based Markov processes and $\beta$-mixing properties

Throughout the paper ${\bf Y}=(Y_{t})_{t\in\mathbb{Z}}$ is a discrete time Markov process. Thanks to the seminal paper of Darsow et al. (1992), the markovianity of a stochastic process can be characterized through a specific requirement that the copulas, representing the dependence structure of the finite dimensional distributions induced by the stochastic process (for a detailed discussion on copulas see Nelsen (2006), Joe (1997), Cherubini et al. (2012) and Durante and Sempi (2015)), must satisfy. In particular, in Darsow et al (1992) it is proved that the Chapman-Kolmogorov equations for transition probabilities are equivalent to the requirement that, if $C_{i,j}$ is the copula associated to the vector $(Y_{i},Y_{j})$, then

[TABLE]

As a consequence, since ${\bf Y}$ is a discrete times Markov process, if we assume that the set of bivariate copulas $C_{t,t+1}$ (representing the dependence structure of the stochastic process at two adjacent times) is given for $t\in\mathbb{Z}$ and $k>0$, then necessarily (we remind that the $\ast$-operator is associative)

[TABLE]

Notice that, in the stationary case considered in Beare (2010), $C_{t,t+1}=C$ for all $t\in\mathbb{Z}$, therefore all bivariate copulas $C_{t,t+k}$ are functions of the copula $C$ and of the lag $k$ and not of the time $t$. In this paper we extend the study to the more general non-stationary case. In particular we analyze the temporal dependence problem with a special attention to mixing properties.

The notion of $\beta$-mixing was introduced by Volkonskii and Rozanov (1959 and 1961) and was attribute there to Kolmogorov. Given a (not necessarily stationary) sequence of random variables ${\bf Y}=(Y_{t})_{t\in\mathbb{Z}}$, let $\mathcal{F}_{t}^{l}$ be the $\sigma$-field $\mathcal{F}_{t}^{l}=\sigma(Y_{t},t\leq t\leq l)$ with $-\infty\leq t\leq l\leq+\infty$ and let

[TABLE]

where the second supremum is taken over all finite partitions $\{A_{1},...A_{I}\}$ and $\{B_{1},...B_{J}\}$ of $\Omega$ such that $A_{i}\in\mathcal{F}_{-\infty}^{t}$ for each $i$ and $B_{j}\in\mathcal{F}_{t+k}^{\infty}$ for each $j$. Define the following dependence coefficient

[TABLE]

We say that the sequence $(Y_{t})_{t\in\mathbb{Z}}$ is $\beta-$mixing (or absolutely regular) if $\beta_{k}\rightarrow 0$ as $k\rightarrow+\infty$.

In next Theorem we give conditions on the set of copulas $C_{t,t+1}$, $t\in\mathbb{Z}$ in order to guarantee that the Markov resulting process is $\beta$-mixing. These conditions are based on specific requirements on the maximal correlation coefficients of the copulas $C_{t,t+1}$. We remind that the maximal correlation $\eta$ of a copula $C$ is given by

[TABLE]

where $f,g\in L^{2}([0,1])$, $\int_{0}^{1}f(x)dx=\int_{0}^{1}g(x)dx=0$ and $\int_{0}^{1}f^{2}(x)dx=\int_{0}^{1}g^{2}(x)dx=1$ and we refer to Beare (2010) and Rényi (1959) for more details.

Theorem 2.1**.**

Let ${\bf Y}=(Y_{t})_{t\in\mathbb{Z}}$ be a Markov process. Let $C_{t,t+1}$ be the copula associated to the vector $(Y_{t},Y_{t+1})$ for $t\in\mathbb{Z}$ that we assume to be absolutely continuous, with symmetric and square-integrable density $c_{t,t+1}$ so that $(c_{t,t+1})_{t\in\mathbb{Z}}$ is uniformly bounded in $L^{2}([0,1])$.

If the maximal correlation coefficients $\eta_{t}$ of $C_{t,t+1}$ satisfy

[TABLE]

then ${\bf Y}$ is $\beta$-mixing.

Proof.

The proof follows that of Theorem 3.1 in Beare (2010) who proves a similar result for stationary copula-based Markov processes. First of all, since the stochastic process is Markovian, (2) can be rewritten in terms of the cumulative distribution functions of $(Y_{t},Y_{t+k})$, $Y_{t}$ and $Y_{t+k}$ ($F_{t,t+k}$, $F_{t}$ and $F_{t+k}$, respectively) and the total variation norm $\|\cdot\|_{TV}$ (see Bradley, 2007) and then, applying Sklar’s theorem, we can write

[TABLE]

From (1) it follows that all bivariate copulas of type $C_{t,t+k}$ for $t\in\mathbb{Z}$ and $k\geq 1$ are absolutely continuous: let us denote their density as $c_{t,t+k}$. Then

[TABLE]

and

[TABLE]

Since $c_{t,t+1}$ is a symmetric square-integrable joint density with uniform margins, it admits the following series expansion in terms of a complete orthonormal sequence $(\phi_{i})_{i\geq 1}$ in $\mathbb{L}^{2}[0,1]$,

[TABLE]

where the eigenvalues $(\lambda_{i,t})_{i}$ form a square-summable sequence of nonnegative real numbers: notice that, as proved in Lancaster(1958)

[TABLE]

Applying (1), we get

[TABLE]

Then, using (4) and (3), we get

[TABLE]

Therefore

[TABLE]

which, since $(c_{t,t+1})_{t\in\mathbb{Z}}$ is uniformly bounded in $L^{2}([0,1])$, tends to zero as $k\rightarrow+\infty$. ∎

3 A gaussian convolution-based Markov process

From now on, we assume that the Markov process ${\bf Y}$ is obtained through

[TABLE]

where $(\xi_{t})_{t\geq 1}$ is a sequence of identically distributed random variables such that $\xi_{t}$ is dependent on $Y_{t-1}$ for each $t$. The dependence structure is modelled by a time-invariant copula function $C$.

The process defined in (6) is not stationary. However, we can determine the distribution of $Y_{t}$ for each $t$ thanks to the $C$-convolution operator (denoted by $\stackrel{{\scriptstyle C}}{{\ast}}$), introduced in Cherubini et al. (2011) as a tool to recover the distribution of the sum of two dependent random variables. As shown in Cherubini et al. (2011, 2012 and 2015) the $C$-convolution technique may be used in the construction of dependent increments stochastic processes like (6). More precisely, if $F_{t-1}$ is the cumulative distribution function of $Y_{t-1}$ and $H_{t}$ that of $\xi_{t}$, we may recover the cumulative distribution function of $Y_{t}$ iterating the $C$-convolution for all $t$

[TABLE]

while, the copula associated to $(Y_{t-1},Y_{t})$ is

[TABLE]

where $D_{1}C(u,v)=\frac{\partial}{\partial u}C(u,v)$.

Equations (7) and (8) provide the ingredients to construct discrete times Markov processes according to Darsow et al. (1992).

Our model (6) is a sort of a modified version of a random walk process where the independence assumption for the innovations $(\xi_{t})_{t\geq 1}$ is no longer required: however, its weakness is that in most cases the distribution function cannot be expressed in closed form and it may be evaluated only numerically.

From now on we assume that innovations $(\xi_{t})_{t\geq 1}$ are gaussian identically distributed with zero mean and standard deviation $\sigma_{\xi}$ and that the copula between $Y_{t-1}$ and $\xi_{t}$ is a (stationary) gaussian copula with constant parameter $\rho$ for all $t$. This way, the distribution of $Y_{t}$ is gaussian for all $t$ and, more specifically, in section 4.3.1 of Cherubini et al. (2016) it is shown that

[TABLE]

where

[TABLE]

where $V_{1}^{2}=\sigma_{\xi}^{2}$ since by assumption $Y_{1}=\xi_{1}$. Moreover, the copula between $Y_{t}$ and $Y_{t+1}$ is gaussian with parameters

[TABLE]

since $\mathbb{E}[Y_{t}Y_{t+1}]=V_{t}^{2}+\rho V_{t}\sigma_{\xi}$.

The limiting behavior of the standard deviation $V_{t}$ has also been analyzed in Cherubini et al. (2016) where it is proved that

[TABLE]

Notice that only in case of negative correlation with the increments, the standard deviation of the levels does not explode: in the following we will restrict the analysis to the case $\rho\in(-1,0)$.

3.1 Moments and autocorrelation function

In this subsection we study the behavior of moments and autocorrelation functions of the process $(Y_{t})_{t\geq 1}$ when $t\rightarrow+\infty$. It is just the case to recall that in the standard random walk model the $k$-th order autocorrelation function of $(Y_{t})_{t\geq 1}$ tends to $1$ as $t\rightarrow+\infty$, for each lag $k$. In our more general setting, this is no longer true. The limit of the $k$-th order autocorrelation function of $(Y_{t})_{t\geq 1}$ is a function of $k$ and $\rho$ as the following proposition shows.

Proposition 3.1**.**

Let $\rho\in(-1,0)$. The $k$-th order autocorrelation function of $(Y_{t})_{t\geq 1}$ tends to $(1-2\rho^{2})^{k}$ for any $k\geq 1$ as $t\rightarrow+\infty$.

Proof.

As proved in section 4.3.1 in Cherubini et al. (2016), using the fact that the $\ast$-product of two gaussian copulas has a parameter given by the product of the parameters of the copulas involved in the $\ast$-product, we have that the copula between $Y_{t}$ and $Y_{t+k}$ is gaussian with parameter

[TABLE]

Therefore, since as $t\rightarrow+\infty$ and for any $s\geq 1$

[TABLE]

we easily get the result. ∎

On the other hand, the innovations $(\xi_{t})_{t\geq 1}$ are no longer serially independent as in the random walk case and the $k$-th order autocorrelation function approaches to a limit which again depends on $\rho$ and $k$.

Proposition 3.2**.**

Let $\rho\in(-1,0)$. The $k$-th order autocorrelation function of $(\xi_{t})_{t\geq 1}$ tends to $-\rho^{2}(1-2\rho^{2})^{k-1}$ for any $k\geq 1$ as $t\rightarrow+\infty$.

Proof.

We compute first the autocovariance of order $k$, with $k\geq 1$, $\mathbb{E}[\xi_{t}\xi_{t+k}]$. We have

[TABLE]

Since for any fixed $k\geq 1$, $\tau_{t,t+k}\rightarrow(1-2\rho^{2})^{k}$ and $V_{t}\rightarrow-\frac{\sigma_{\xi}}{2\rho}$ as $t\rightarrow+\infty$ we get

[TABLE]

Moreover, it is immediate to find the statement of the proposition since as $t\rightarrow+\infty$

[TABLE]

∎

3.2 $\beta$-mixing properties

In our gaussian framework $c_{t,t+1}$ is the density of a gaussian copula for which it is well known that the maximal correlation coefficient is equal to the absolute value of the simple correlation coefficient (see Lancaster, 1957). Therefore, according to the notation of Theorem 2.1, for each $t$, $\eta_{t}=|\tau_{t,t+1}|$. The following results, which is an application of Theorem 2.1, holds

Corollary 3.1**.**

The Markov process defined by (6) with $\xi_{t}\sim N(0,\sigma_{\xi})$ and $\rho\in(-1,0)$ is $\beta$-mixing.

Proof.

Firstly notice that

[TABLE]

In fact this is equivalent to $(V_{t}+\rho\sigma_{\xi})^{2}<V_{t+1}^{2}$ which is always verified since $\rho^{2}<1$ by assumption.

Thanks to (11), since $|1-2\rho^{2}|<1$, we have that $|\tau_{t,t+1}|=\eta_{t}$ is bounded by a constant smaller than 1, that is (3) is satisfied. Furthermore, it is not hard to prove that for any $t$

[TABLE]

Thus Theorem 2.1 applies.

∎

4 Concluding remarks

In this paper we provide conditions under which a non-stationary copula-based Markov process is $\beta$-mixing. Our results represent a generalization of those in Beare (2010), where the author considers the stationary case. Our analysis is focused on the particular case of a gaussian Markov process with dependent increments that represents a generalization of the standard gaussian random walk. In this particular non-stationary setting it is proved that the $k$-th order autocorrelation function of the process does not converge to $1$, as in the random walk case, but to a quantity that depends on the lag and the correlation between the state variable and the innovation, which is assumed to be time-invariant. Additionally, it is proved that the process satisfies the conditions required to be $\beta$-mixing.