1. Introduction
The heavy-tailed distribution is graphically thicker in the tails and sharper in the peaks than the normal distribution. Intuitively, this means that the probability of extreme values is greater for these data than for normally distributed data. The heavy-tailed phenomenon has been encountered empirically in various fields: physics, earth sciences, economics and political science, etc. Periodicity is a wave-like or oscillatory movement around a long-term trend presented in a time series. It is well-known that cyclicality caused by business and economic activities is usually different from trend movements, not in a single direction of continuous movement, but in alternating fluctuations between ups and downs. When these components, trend and cyclicality, do not evolve independently, traditional differencing filters may not be suitable (see for example, Franses and Paap [
1], Bell et al. [
2], Aliat and Hamdi [
3]). Periodic time series models are used to model periodically stationary time series. Periodic vector autoregressive (PVAR) models extend the classical vector autoregressive (VAR) models by allowing the parameters to vary with the cyclicality.
For fixed v and predetermined value T, the random vector represents the realization during the vth season, with , at year . The PVAR model order at season v is given by , whereas , represent the PVAR model coefficients during season v. The error process corresponds to a periodic white noise, with and , where is a zero vector of order d and is a unit matrix of order d.
This paper seeks to establish a PVAR model to simulate the heavy-tailed time series. Let random vectors be from a stationary stochastic process , where the transpose (indicated by ⊤) of a row vector is a column vector.
We consider first-order PVAR model,
where
,
T is the period of
,
denotes the latent heavy-tailed innovation, and
A is the transition matrix. In this paper, we assume that the entries of the heavy-tailed innovation
obey a power-law distribution with
and
, then the second moment is finite and the third moment is infinite. From the Equation (
2) we know that, given
k, the vector
follows a first-order vector autoregressive (VAR). We call the first-order PVAR process stable if all eigenvalues of transition matrix
A have modulus less than 1, the condition is equivalent to
for all
(see for example,
Section 2.1 in Lütkepohl [
4]). The transition matrix characterizes the temporal dependence for sequence data, and plays a fundamental role in forecasting. Moreover, the zero and nonzero entries of the transition matrix is often closely related to Granger causality. This manuscript focuses on estimating the transition matrix of high-dimensional heavy-tailed PVAR processes.
PVAR models have been extensively studied under the Gaussian assumption, the Gaussian PVAR models assume that the latent innovations are independent identity distribution Gaussian random vectors. Under this model, there are two kinds of methods to estimate the transition matrix under high dimensional setting, one is the Lasso-based estimation procedures, see [
5,
6,
7,
8], and the other is Dantzig-selector-type estimators, see [
9,
10,
11,
12]. Under the non-Gaussian VAR process, Qiu et al. [
13] proposed a quantile-based dantzig-selector-type estimator of the transition matrix for elliptical VAR process. Wong et al. [
14] provided an alternative proof of the consistency of the Lasso for sparse non-Gaussian VAR models. Maleki et al. [
15] extended the multivariate setting of autoregressive process, by considering the multivariate scale mixture of skew-normal distributions for VAR innovations.
The statistical second-order information contained in the data is usually expressed by the variance and covariance, most of the literature dealing with time series measure dependence using the variance and covariance. To investigate the validity of the variance estimates, we need the presence of the fourth order moment of the random variables. For the heavy-tailed nature of financial data, the third-order moments of random variables are usually non-existent. Schechtman and Yitzhaki [
16] proposed the concept of Gini covariance, which has been used widely to measure the dependence of heavy-tailed distributions. Let
H be the joint distribution of the random variables
X and
Y with the marginal distribution functions
and
, respectively. The standard Gini covariance is defined as
assuming the random variables with only a finite first moment. The Gini covariance has an advantage when analyzing bivariate data defined by both variate values and ranks of the values. The representation of Gini covariance
indicates that it has mixed properties of the variable
X and the rank of the variable
Y, and thus complements the usual covariance and rank covariance [
16,
17,
18]. In terms of balance between effciency and robustness, Gini covariance plays an important role in measuring association for variables from heavy-tailed distributions [
19].
The Yule–Walker equations arise naturally in the problem of linear prediction of any zero-mean weakly stationary process based on a finite number of contiguous observations. The Yule–Walker equations provide a straightforward connection between the autoregressive model parameters and the covariance function of the process. In this paper, relaxing the strong assumption of the existence of higher order moments of the regressors, we use a non-parametric method to estimate the Gini covariance matrix, establish the Gini–Yule–Walker equation to estimate the sparse transition matrix of stationary PVAR processes. The estimator falls into the category of Dantzig-selector-type estimators. With existence of a finite second moment, we investigate the asymptotic behavior of the estimator in high dimensions.
The paper is orginazed as follows: In
Section 2, we establish the Gini–Yule–Walker equation and estimate the sample Gini covariance matrix. In
Section 3, we derive the convergence rate of transfer matrix estimation. In
Section 4, we discuss the characterization and estimation of Granger causality under the heavy-tailed PVAR model. In
Section 5, both synthetic and real data are used to demonstrate the empirical performance of the proposed methodology.
2. Model
In this section, the notations are set. Then, we establish the Gini–Yule–Walker equation, obtain simple non-parametric estimators for Gini covariance matrix and investigate the convergence rate of the sample Gini covariance matrix.
2.1. Notation
Let be a d-dimensional real vector, and be a matrix. For , we define the vector norm of v as , and the vector norm of v as . Let the matrix norm of M as , the matrix norm , and be two random vectors.
2.2. Gini–Yule–Walker Equation
In this paper, we model the time series vector by a stationary PVAR process under the existence of second moment. For each , follow a lag-one VAR process, with independent of , and .
We define
this VAR process may be performed by concatenating the equation systems to analyze the following equation,
where
, and
the ⊗ is a Kronecker product operator.
Since the matrix
is a block diagonal matrix, the estimation problem can be decomposed into
d independent sub-problems. Let us consider the
i-th equation of the system.
the above equation system can be considered as a multiple regression
this equation system can be abbreviated as
with
samples to estimate the
i-th line of transition matrix
A.
Let
be the distribution of
, we assume the independence between
and
, for
. Then we get the Gini covariance matrix equation issue from Equation (
5),
From the above equation, we obtain the so called Gini–Yule–Walker Equation
where
,
. The entries of
are given by
, and the entries of
are given by
.
2.3. Sample Gini Covariance Matrix
We use a
statistic method to estimate the Gini covariance matrix
and
. From Equation (
6), the elements of the covariance matrix
and
can be divided into two categories: Gini covariance
and Gini mean difference
,
with
.
For the Gini covariance
, we have sample space
The
i-th ordered variable of
is expressed by
and the associated variable of
(matched with
) is expressed by
, which is the concomitant of the
i-th order statistic. In this set-up, in the context of non-parametric estimation of a regression function, Yang [
20] proposed a statistic of the form
where
is a bounded smooth function,
is a real valued function of
and
is the empirical distribution function corresponding to
. The Gini covariance defined in Equation (
3) can be rewritten as
Choosing
and
from Equation (
7), we obtain an estimator of
as
For the Gini mean difference
, we have sample space
. Let
and
be two independent random variables with distribution function
, the Gini mean difference
can be expressed as
The estimator of
based on U-statistics is given by
where
. After some simplification, we obtain
where
is the
i -th order statistic based on the sample space
.
2.4. Convergence Rates of the Estimator and
In this subsection, with the truncation method, we use the Bernstein’s inequality to investigate the convergence rates of the estimator
and
. From Equations (
8) and (
9), we define
, and
, where
is the variance of the variable
.
For analysis, we require the following three assumptions on the time series and the size of variables :
Assumption A1. From Equation (2), suppose that the entries of the heavy-tailed innovation obey a power-law distribution with and , then the second moment is finite and the third moment is infinite. Assumption A2. Suppose that , , c is a finite constant, for .
Assumption A3. Suppose , for
Lemma 1. Let be a stationary PVAR process from Equation (2), and be a sequence of observations from . Suppose that Assumptions (A1)–(A2) are satisfied. Then, for T and n large enough, with probability no smaller than we have Proof. Assuming that
m and
are constants greater than 0, and
Then,
is a bounded random variable and satisfies the property of independent identical distribution, it follows from the Bernstein’s inequality that
where
.
Let
and
, we have
As and , assuming and , then .
With similar proof methods, we obtain that
This completes the proof. □
From Equations (
5) and (
6), we define the sample estimation of Gini covariance matrix
G as
and the sample estimation of Gini covariance matrix
as
,
.
Next, we investigate the convergence rates of the estimator and under the norm.
Lemma 2. Let be a stationary PVAR process from Equation (2), and be a sequence of observations from . Suppose that Assumptions (A1)–(A3) are satisfied. Then, for T and n large enough, with probability no smaller than we have Proof. Let
, based on the Lemma 1, we have
As and , assuming , and , then .
Next, we study the convergence of sample Gini covariance matrix .
Let
, based on the Lemma 1,
Now, for the off-diagonal entries, we have
Combining Equations (
15) and (
16), we obtain
This completes the proof. □
4. Granger Causality
In this section, the practical example is conducted to verify the effectiveness of the proposed methods, moreover, the characterization and estimation of Granger causality under the heavy-tailed PVAR model are discussed. Firstly, we give the definition of Granger causality.
Definition 1.(Granger [22]) Let be a stationary process, where . For Granger causes if and only if there exists a measurable set A such that for all where is the subvector obtained by removing from
For a Gaussian VAR process
we have that
Granger causes
if and only if the
entry of the transition matrix is non-zero [
4]. For the heavy-tailed PVAR process, let
be a stationary PVAR process from Equation (
2), we define
In the next theorem, we show that a similar property holds for the heavy-tailed PVAR process.
Theorem 2. Let be a stationary PVAR process from Equation (2). Suppose that Assumptions (A1)–(A3) are satisfied, and for any Then, for we have - 1.
If then Granger causes .
- 2.
If we further assume that is independent of for any we have that Granger causes if and only if .
Proof. In order to prove Issue
we only need to prove that
doesn’t Granger cause
implies
Suppose for some
we have
for any measurable set
A. The above equation implies that conditioning on
,
is independent of
Hence, we have
Plugging
into the above equation, we have
The second term on the right hand side is
since given
is constant. Since
and
are independent for any
, we have
for any
. Using Theorem 2.18 in Fang et al. [
23], we have the third term is also
Thus, we have
and hence
This proves Issue 1.
Given Issue
to prove Issue
it remains to prove that
implies that
doesn’t Granger cause
Since
we have
Here
p is the conditional probability density function. The last equation is because
is independent of
and the fact that
is constant given
Hence, we have
and thus
□
Remark 1. The assumption that requires that cannot be perfectly predictable from the past or from the other observed random variables at time t. Otherwise, we can simply remove from the process since predicting is trivial. Assuming that is independent of for any the Granger causality relations among the processes is characterized by the non-zero entries of To estimate the Granger causality relations, we define wherefor some threshold parameter γ. To evaluate the consistency between and A regarding sparsity pattern, we define function . For a matrix M, define . The next theorem gives the rate of γ such that recovers the sparsity pattern of A with high probability. Theorem 3. Let be a stationary PVAR process from Equation (2). Suppose that Assumptions 1–3 are satisfied. The transition matrix , if we setthen, with probability no smaller than , we have provided that Proof. The proof is a consequence of Theorem 1. In detail, if
by Equation (
25), we have
By Theorem 1, with probability no smaller than
, we have
. Thus, we have
with probability no smaller than
. By the definition of
A, we have
.
If
by Equation (
25), we have
Using Theorem 1, we have
with probability no smaller than
, By the definition of
we have
.
If , using Theorem 1, we have with probability no smaller than , since 0. By the definition of we have .
□