1. Introduction
In this paper, we show that a class of uniformitypreserving transformations for uniform random variables can facilitate the application of copula modelling to time series exhibiting the serial dependence characteristics that are typical of volatile financial return data. Our main aims are twofold: to establish the fundamental properties of vtransforms and show that they are a natural fit to the volatility modelling problem; to develop a class of processes using the implied copula process of a Gaussian ARMA model that can serve as an archetype for copula models using vtransforms. Although the existing literature on volatility modelling in econometrics is vast, the models we propose have some attractive features. In particular, as copulabased models, they allow the separation of marginal and serial dependence behaviour in the construction and estimation of models.
A distinction is commonly made between genuine stochastic volatility models, as investigated by
Taylor (
1994) and
Andersen (
1994), and GARCHtype models as developed in a long series of papers by
Engle (
1982),
Bollerslev (
1986),
Ding et al. (
1993),
Glosten et al. (
1993) and
Bollerslev et al. (
1994), among others. In the former an unobservable process describes the volatility at any time point while in the latter volatility is modelled as a function of observable information describing the past behaviour of the process; see also the review articles by
Shephard (
1996) and
Andersen and Benzoni (
2009). The generalized autoregressive score (GAS) models of
Creal et al. (
2013) generalize the observationdriven approach of GARCH models by using the score function of the conditional density to model time variation in key parameters of the time series model. The models of this paper have more in common with the observationdriven approach of GARCH and GAS but have some important differences.
In GARCHtype models, the marginal distribution of a stationary process is inextricably linked to the dynamics of the process as well as the conditional or innovation distribution; in most cases, it has no simple closed form. For example, the standard GARCH mechanism serves to create powerlaw behaviour in the marginal distribution, even when the innovations come from a lightertailed distribution such as Gaussian (
Mikosch and Stărică 2000). While such models work well for many return series, they may not be sufficiently flexible to describe all possible combinations of marginal and serial dependence behaviour encountered in applications. In the empirical example of this paper, which relates to logreturns on the Bitcoin price series, the data appear to favour a marginal distribution with subexponential tails that are lighter than power tails and this cannot be well captured by standard GARCH models. Moreover, in contrast to much of the GARCH literature, the models we propose make no assumptions about the existence of secondorder moments and could also be applied to very heavytailed situations where variancebased methods fail.
Let
${X}_{1},\dots ,{X}_{n}$ be a time series of financial returns sampled at (say) daily frequency and assume that these are modelled by a strictly stationary stochastic process
$\left({X}_{t}\right)$ with marginal distribution function (cdf)
${F}_{X}$. To match the stylized facts of financial return data described, for example, by
Campbell et al. (
1997) and
Cont (
2001), it is generally agreed that
$\left({X}_{t}\right)$ should have limited serial correlation, but the squared or absolute processes
$\left({X}_{t}^{2}\right)$ and
$\left(\right{X}_{t}\left\right)$ should have significant and persistent positive serial correlation to describe the effects of volatility clustering.
In this paper, we refer to transformed series like $\left(\right{X}_{t}\left\right)$, in which volatility is revealed through serial correlation, as volatility proxy series. More generally, a volatility proxy series $\left(T\left({X}_{t}\right)\right)$ is obtained by applying a transformation $T:\mathbb{R}\mapsto \mathbb{R}$ which (i) depends on a change point ${\mu}_{T}$ that may be zero, (ii) is increasing in ${X}_{t}{\mu}_{T}$ for ${X}_{t}\ge {\mu}_{T}$ and (iii) is increasing in ${\mu}_{T}{X}_{t}$ for ${X}_{t}\le {\mu}_{T}$.
Our approach in this paper is to model the probabilityintegral transform (PIT) series $\left({V}_{t}\right)$ of a volatility proxy series. This is defined by ${V}_{t}={F}_{T\left(X\right)}\left(T\left({X}_{t}\right)\right)$ for all t, where ${F}_{T\left(X\right)}$ denotes the cdf of $T\left({X}_{t}\right)$. If $\left({U}_{t}\right)$ is the PIT series of the original process $\left({X}_{t}\right)$, defined by ${U}_{t}={F}_{X}\left({X}_{t}\right)$ for all t, then a vtransform is a function describing the relationship between the terms of $\left({V}_{t}\right)$ and the terms of $\left({U}_{t}\right)$. Equivalently, a vtransform describes the relationship between quantiles of the distribution of ${X}_{t}$ and the distribution of the volatility proxy $T\left({X}_{t}\right)$. Alternatively, it characterizes the dependence structure or copula of the pair of variables $({X}_{t},T\left({X}_{t}\right))$. In this paper, we show how to derive flexible, parametric families of vtransforms for practical modelling purposes.
To gain insight into the typical form of a vtransform, let
${x}_{1},\dots ,{x}_{n}$ represent the realized data values and let
${u}_{1},\dots ,{u}_{n}$ and
${v}_{1},\dots ,{v}_{n}$ be the samples obtained by applying the transformations
${v}_{t}={F}_{n}^{\left(\rightX\left\right)}\left(\right{x}_{t}\left\right)$ and
${u}_{t}={F}_{n}^{\left(X\right)}\left({x}_{t}\right)$, where
${F}_{n}^{\left(X\right)}\left(x\right)=\frac{1}{n+1}{\sum}_{t=1}^{n}{I}_{\{{x}_{t}\le x\}}$ and
${F}_{n}^{\left(\rightX\left\right)}\left(x\right)=\frac{1}{n+1}{\sum}_{t=1}^{n}{I}_{\left\{\right{x}_{t}\le x\}}$ denote scaled versions of the empirical distribution functions of the
${x}_{t}$ and
${x}_{t}$ samples, respectively. The graph of
$({u}_{t},{v}_{t})$ gives an empirical estimate of the vtransform for the random variables
$({X}_{t},{X}_{t}\left\right)$. In the lefthand plot of
Figure 1 we show the relationship for a sample of
$n=1043$ daily logreturns of the Bitcoin price series for the years 2016–2019. Note how the empirical vtransform takes the form of a slightly asymmetric ‘V’.
The righthand plot of
Figure 1 shows the sample autocorrelation function (acf) of the data given by
${z}_{t}={\Phi}^{1}\left({v}_{t}\right)$ where
$\Phi $ is the standard normal cdf. This reveals a persistent pattern of positive serial correlation which can be modelled by the implied ARMA copula. This pattern is not evident in the acf of the raw
${x}_{t}$ data in the centre plot.
To construct a volatility model for $\left({X}_{t}\right)$ using vtransforms, we need to specify a process for $\left({V}_{t}\right)$. In principle, any model for a series of serially dependent uniform variables can be applied to $\left({V}_{t}\right)$. In this paper, we illustrate concepts using the Gaussian copula model implied by the standard ARMA dependence structure. This model is particularly tractable and allows us to derive model properties and fit models to data relatively easily.
There is a large literature on copula models for time series; see, for example, the review papers by
Patton (
2012) and
Fan and Patton (
2014). While the main focus of this literature has been on crosssectional dependencies between series, there is a growing literature on models of serial dependence. Firstorder Markov copula models have been investigated by
Chen and Fan (
2006),
Chen et al. (
2009) and
Domma et al. (
2009) while higherorder Markov copula models using Dvines are applied by
Smith et al. (
2010). These models are based on the paircopula apporoach developed in
Joe (
1996),
Bedford and Cooke (
2001,
2002) and
Aas et al. (
2009). However, the standard bivariate copulas that enter these models are not generally effective at describing the typical serial dependencies created by stochastic volatility, as observed by
LoaizaMaya et al. (
2018).
The paper is structured as follows. In
Section 2, we provide motivation for the paper by constructing a symmetric model using the simplest example of a vtransform. The general theory of vtransforms is developed in
Section 3 and is used to construct the class of VTARMA processes and analyse their properties in
Section 4.
Section 5 treats estimation and statistical inference for VTARMA processes and provides an example of their application to the Bitcoin return data;
Section 6 presents the conclusions. Proofs may be found in the
Appendix A.
2. A Motivating Model
Given a probability space $(\Omega ,\mathcal{F},\mathbb{P})$, we construct a symmetric, strictly stationary process ${\left({X}_{t}\right)}_{t\in \mathbb{N}\backslash \left\{0\right\}}$ such that, under the even transformation $T\left(x\right)=\leftx\right$, the serial dependence in the volatility proxy series $\left(T\left({X}_{t}\right)\right)$ is of ARMA type. We assume that the marginal cdf ${F}_{X}$ of $\left({X}_{t}\right)$ is absolutely continuous and the density ${f}_{X}$ satisfies ${f}_{X}\left(x\right)={f}_{X}(x)$ for all $x>0$. Since ${F}_{X}$ and ${F}_{\leftX\right}$ are both continuous, the properties of the probabilityintegral (PIT) transform imply that the series $\left({U}_{t}\right)$ and $\left({V}_{t}\right)$ given by ${U}_{t}={F}_{X}\left({X}_{t}\right)$ and ${V}_{t}={F}_{\leftX\right}\left(\right{X}_{t}\left\right)$ both have standard uniform marginal distributions. Henceforth, we refer to $\left({V}_{t}\right)$ as the volatility PIT process and $\left({U}_{t}\right)$ as the series PIT process.
Any other volatility proxy series that can be obtained by a continuous and strictly increasing transformation of the terms of $\left(\right{X}_{t}\left\right)$, such as $\left({X}_{t}^{2}\right)$, yields exactly the same volatility PIT process. For example, if ${\tilde{V}}_{t}={F}_{{X}^{2}}\left({X}_{t}^{2}\right)$, then it follows from the fact that ${F}_{{X}^{2}}\left(x\right)={F}_{\leftX\right}\left(\sqrt[+]{x}\right)$ for $x\ge 0$ that ${\tilde{V}}_{t}={F}_{{X}^{2}}\left({X}_{t}^{2}\right)={F}_{\leftX\right}\left(\right{X}_{t}\left\right)={V}_{t}$. In this sense, we can think of classes of equivalent volatility proxies, such as $\left(\right{X}_{t}\left\right)$, $\left({X}_{t}^{2}\right)$, $(exp{X}_{t}\left\right)$ and $(ln(1+{X}_{t}\left\right))$. In fact, $\left({V}_{t}\right)$ is itself an equivalent volatility proxy to $\left(\right{X}_{t}\left\right)$ since ${F}_{\leftX\right}$ is a continuous and strictly increasing transformation.
The symmetry of
${f}_{X}$ implies that
${F}_{\leftX\right}\left(x\right)=2{F}_{X}\left(x\right)1=12{F}_{X}(x)$ for
$x\ge 0$. Hence, we find that
which implies that the relationship between the volatility PIT process
$\left({V}_{t}\right)$ and the series PIT process
$\left({U}_{t}\right)$ is given by
where
$\mathcal{V}\left(u\right)=2u1$ is a perfectly symmetric vshaped function that maps values of
${U}_{t}$ close to 0 or 1 to values of
${V}_{t}$ close to 1, and values close to
$0.5$ to values close to 0.
$\mathcal{V}$ is the canonical example of a vtransform. It is related to the socalled tentmap transformation
$\mathcal{T}\left(u\right)=2min(u,1u)$ by
$\mathcal{V}\left(u\right)=1\mathcal{T}\left(u\right)$.
Given
$\left({V}_{t}\right)$, let the process
$\left({Z}_{t}\right)$ be defined by setting
${Z}_{t}={\Phi}^{1}\left({V}_{t}\right)$ so that we have the following chain of transformations:
We refer to
$\left({Z}_{t}\right)$ as a
normalized volatility proxy series. Our aim is to construct a process
$\left({X}_{t}\right)$ such that, under the chain of transformations in (
2), we obtain a Gaussian ARMA process
$\left({Z}_{t}\right)$ with mean zero and variance one. We do this by working back through the chain.
The transformation $\mathcal{V}$ is not an injection and, for any ${V}_{t}>0$, there are two possible inverse values, $\frac{1}{2}}(1{V}_{t})$ and $\frac{1}{2}}(1+{V}_{t})$. However, by randomly choosing between these values, we can ‘stochastically invert’ $\mathcal{V}$ to construct a random variable ${U}_{t}$ such that $\mathcal{V}\left({U}_{t}\right)={V}_{t}$, This is summarized in Lemma 1, which is a special case of a more general result in Proposition 4.
Lemma 1. Let V be a standard uniform variable. If $V=0$, set $U={\textstyle \frac{1}{2}}$. Otherwise, let $U={\textstyle \frac{1}{2}}(1V)$ with probability 0.5 and $U={\textstyle \frac{1}{2}}(1+V)$ with probability 0.5. Then, U is uniformly distributed and $\mathcal{V}\left(U\right)=V$.
This simple result suggests Algorithm 1 for constructing a process
$\left({X}_{t}\right)$ with symmetric marginal density
${f}_{X}$ such that the corresponding normalized volatility proxy process
$\left({Z}_{t}\right)$ under the absolute value transformation (or continuous and strictly increasing functions thereof) is an ARMA process. We describe the resulting model as a VTARMA process.
Algorithm 1: 
Generate $\left({Z}_{t}\right)$ as a causal and invertible Gaussian ARMA process of order $(p,q)$ with mean zero and variance one. Form the volatility PIT process $\left({V}_{t}\right)$ where ${V}_{t}=\Phi \left({Z}_{t}\right)$ for all t. Generate a process of iid Bernoulli variables $\left({Y}_{t}\right)$ such that $\mathbb{P}({Y}_{t}=1)=0.5$. Form the PIT process $\left({U}_{t}\right)$ using the transformation ${U}_{t}=0.5{(1{V}_{t})}^{{I}_{\{{Y}_{t}=0\}}}{(1+{V}_{t})}^{{I}_{\{{Y}_{t}=1\}}}$. Form the process $\left({X}_{t}\right)$ by setting ${X}_{t}={F}_{X}^{1}\left({U}_{t}\right)$.

It is important to state that the use of the Gaussian process $\left({Z}_{t}\right)$ as the fundamental building block of the VTARMA process in Algorithm 1 has no effect on the marginal distribution of $\left({X}_{t}\right)$, which is ${F}_{X}$ as specified in the final step of the algorithm. The process $\left({Z}_{t}\right)$ is exploited only for its serial dependence structure, which is described by a family of finitedimensional Gaussian copulas; this dependence structure is applied to the volatility proxy process.
Figure 2 shows a symmetric VTARMA(1,1) process with ARMA parameters
${\alpha}_{1}=0.95$ and
${\beta}_{1}=0.85$; such a model often works well for financial return data. Some intuition for this observation can be gained from the fact that the popular GARCH(1,1) model is known to have the structure of an ARMA(1,1) model for the squared data process; see, for example,
McNeil et al. (
2015) (
Section 4.2) for more details.
3. VTransforms
To generalize the class of vtransforms, we admit two forms of asymmetry in the construction described in
Section 2: we allow the density
${f}_{X}$ to be skewed; we introduce an asymmetric volatility proxy.
Definition 1 (Volatility proxy transformation and profile)
. Let ${T}_{1}$ and ${T}_{2}$ be strictly increasing, continuous and differentiable functions on ${\mathbb{R}}^{+}=[0,\infty )$ such that ${T}_{1}\left(0\right)={T}_{2}\left(0\right)$. Let ${\mu}_{T}\in \mathbb{R}$. Any transformation $T:\mathbb{R}\to \mathbb{R}$ of the form is a volatility proxy transformation. The parameter ${\mu}_{T}$ is the change point of T and the associated function ${g}_{T}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, ${g}_{T}\left(x\right)={T}_{2}^{1}\circ {T}_{1}\left(x\right)$ is the profile function of T. By introducing
${\mu}_{T}$, we allow for the possibility that the natural change point may not be identical to zero. By introducing different functions
${T}_{1}$ and
${T}_{2}$ for returns on either side of the change point, we allow the possibility that one or other may contribute more to the volatility proxy. This has a similar economic motivation to the
leverage effects in GARCH models (
Ding et al. 1993); falls in equity prices increase a firm’s leverage and increase the volatility of the share price.
Clearly, the profile function of a volatility proxy transformation is a strictly increasing, continuous and differentiable function on ${\mathbb{R}}^{+}$ such that ${g}_{T}\left(x\right)=0$. In conjunction with ${\mu}_{T}$, the profile contains all the information about T that is relevant for constructing vtransforms. In the case of a volatility proxy transformation that is symmetric about ${\mu}_{T}$, the profile satisfies ${g}_{T}\left(x\right)=x$.
The following result shows how vtransforms
$V=\mathcal{V}\left(U\right)$ can be obtained by considering different continuous distributions
${F}_{X}$ and different volatility proxy transformations
T of type (
3).
Proposition 1. Let X be a random variable with absolutely continuous and strictly increasing cdf ${F}_{X}$ on $\mathbb{R}$ and let T be a volatility proxy transformation. Let $U={F}_{X}\left(X\right)$ and $V={F}_{T\left(X\right)}\left(T\left(X\right)\right)$. Then, $V=\mathcal{V}\left(U\right)$ where The result implies that any two volatility proxy transformations
T and
$\tilde{T}$ which have the same change point
${\mu}_{T}$ and profile function
${g}_{T}$ belong to an equivalence class with respect to the resulting vtransform. This generalizes the idea that
$T\left(x\right)=\leftx\right$ and
$T\left(x\right)={x}^{2}$ give the same vtransform in the symmetric case of
Section 2. Note also that the volatility proxy transformations
${T}^{\left(V\right)}$ and
${T}^{\left(Z\right)}$ defined by
are in the same equivalence class as
T since they share the same change point and profile function.
Definition 2 (vtransform and fulcrum)
. Any transformation $\mathcal{V}$ that can be obtained from Equation (4) by choosing an absolutely continuous and strictly increasing cdf ${F}_{X}$ on $\mathbb{R}$ and a volatility proxy transformation T is a vtransform. The value $\delta ={F}_{X}\left({\mu}_{T}\right)$ is the fulcrum of the vtransform. 3.1. A Flexible Parametric Family
In this section, we derive a family of vtransforms using construction (
4) by taking a tractable asymmetric model for
${F}_{X}$ using the construction proposed by
Fernández and Steel (
1998) and by setting
${\mu}_{T}=0$ and
${g}_{T}\left(x\right)=k{x}^{\xi}$ for
$k>0$ and
$\xi >0$. This profile function contains the identity profile
${g}_{T}\left(x\right)=x$ (corresponding to the symmetric volatility proxy transformation) as a special case, but allows cases where negative or positive returns contribute more to the volatility proxy. The choices we make may at first sight seem rather arbitrary, but the resulting family can in fact assume many of the shapes that are permissible for vtransforms, as we will argue.
Let
${f}_{0}$ be a density that is symmetric about the origin and let
$\gamma >0$ be a scalar parameter. Fernandez and Steel suggested the model
This model is often used to obtain skewed normal and skewed Student distributions for use as innovation distributions in econometric models. A model with $\gamma >1$ is skewed to the right while a model with $\gamma <1$ is skewed to the left, as might be expected for asset returns. We consider the particular case of a Laplace or double exponential distribution ${f}_{0}\left(x\right)=0.5exp(x\left\right)$ which leads to particularly tractable expressions.
Proposition 2. Let ${F}_{X}(x;\gamma )$ be the cdf of the density (6) when ${f}_{0}\left(x\right)=0.5exp(x\left\right)$. Set ${\mu}_{T}=0$ and let ${g}_{T}\left(x\right)=k{x}^{\xi}$ for $k,\xi >0$. The vtransform (4) is given bywhere $\delta ={F}_{X}\left(0\right)={(1+{\gamma}^{2})}^{1}\in (0,1)$ and $\kappa =k/{\gamma}^{\xi +1}>0$. It is remarkable that (
7) is a uniformitypreserving transformation. If we set
$\xi =1$ and
$\kappa =1$, we get
which obviously includes the symmetric model
${\mathcal{V}}_{0.5}\left(u\right)=2u1$. The vtransform
${\mathcal{V}}_{\delta}\left(u\right)$ in (
8) is a very convenient special case, and we refer to it as the
linear vtransform.
In
Figure 3, we show the vtransform
${\mathcal{V}}_{\delta ,\kappa ,\xi}$ when
$\delta =0.55$,
$\kappa =1.4$ and
$\xi =0.65$. We will use this particular vtransform to illustrate further properties of vtransforms and find a characterization.
3.2. Characterizing vTransforms
It is easily verified that any vtransform obtained from (
4) consists of two arms or branches, described by continuous and strictly monotonic functions; the left arm is decreasing and the right arm increasing. See
Figure 3 for an illustration. At the fulcrum
$\delta $, we have
$\mathcal{V}\left(\delta \right)=0$. Every point
$u\in [0,1]\backslash \left\{\delta \right\}$ has a
dual point ${u}^{*}$ on the opposite side of the fulcrum such that
$\mathcal{V}\left({u}^{*}\right)=\mathcal{V}\left(u\right)$. Dual points can be interpreted as the quantile probability levels of the distribution of
X that give rise to the same level of volatility.
We collect these properties together in the following lemma and add one further important property that we refer to as the
square property of a vtransform; this property places constraints on the shape that vtransforms can take and is illustrated in
Figure 3.
Lemma 2. A vtransform is a mapping $\mathcal{V}:[0,1]\to [0,1]$ with the following properties:
 1.
$\mathcal{V}\left(0\right)=\mathcal{V}\left(1\right)=1$;
 2.
There exists a point δ known as the fulcrum such that $0<\delta <1$ and $\mathcal{V}\left(\delta \right)=0$;
 3.
$\mathcal{V}$ is continuous;
 4.
$\mathcal{V}$ is strictly decreasing on $[0,\delta ]$ and strictly increasing on $[\delta ,1]$;
 5.
Every point $u\in [0,1]\backslash \left\{\delta \right\}$ has a dual point ${u}^{*}$ on the opposite side of the fulcrum satisfying $\mathcal{V}\left(u\right)=\mathcal{V}\left({u}^{*}\right)$ and ${u}^{*}u=\mathcal{V}\left(u\right)$ (square property).
It is instructive to see why the square property must hold. Consider
Figure 3 and fix a point
$u\in [0,1]\backslash \left\{\delta \right\}$ with
$\mathcal{V}\left(u\right)=v$. Let
$U\sim U(0,1)$ and let
$V=\mathcal{V}\left(U\right)$. The events
$\left\{V\le v\right\}$ and
$\left\{min(u,{u}^{*})\le U\le max(u,{u}^{*})\right\}$ are the same and hence the uniformity of
V under a vtransform implies that
The properties in Lemma 2 could be taken as the basis of an alternative definition of a vtransform. In view of (
9), it is clear that any mapping
$\mathcal{V}$ that has these properties is a uniformitypreserving transformation. We can characterize the mappings
$\mathcal{V}$ that have these properties as follows.
Theorem 1. A mapping $\mathcal{V}:[0,1]\to [0,1]$ has the properties listed in Lemma 2 if and only if it takes the formwhere Ψ is a continuous and strictly increasing distribution function on $[0,1]$. Our arguments so far show that every vtransform must have the form (
10). It remains to verify that every uniformitypreserving transformation of the form (
10) can be obtained from construction (
4), and this is the purpose of the final result of this section. This allows us to view Definition 2, Lemma 2, and the characterization (
10) as three equivalent approaches to the definition of vtransforms.
Proposition 3. Let $\mathcal{V}$ be a uniformitypreserving transformation of the form (10) and ${F}_{X}$ a continuous distribution function. Then, $\mathcal{V}$ can be obtained from construction (4) using any volatility proxy transformation with change point ${\mu}_{T}={F}_{X}^{1}\left(\delta \right)$ and profile Henceforth, we can view (
10) as the general equation of a vtransform. Distribution functions
$\Psi $ on
$[0,1]$ can be thought of as
generators of vtransforms. Comparing (
10) with (
7), we see that our parametric family
${\mathcal{V}}_{\delta ,\kappa ,\xi}$ is generated by
$\Psi \left(x\right)=exp(\kappa ({(lnx)}^{\xi}))$. This is a 2parameter distribution whose density can assume many different shapes on the unit interval including increasing, decreasing, unimodal, and bathtubshaped forms. In this respect, it is quite similar to the beta distribution which would yield an alternative family of vtransforms. The uniform distribution function
$\Psi \left(x\right)=x$ gives the family of linear vtransforms
${\mathcal{V}}_{\delta}$.
In applications, we construct models starting from the building blocks of a tractable vtransform
$\mathcal{V}$ such as (
7) and a distribution
${F}_{X}$; from these, we can always infer an implied profile function
${g}_{T}$ using (
11). The alternative approach of starting from
${g}_{T}$ and
${F}_{X}$ and constructing
$\mathcal{V}$ via (
4) is also possible but can lead to vtransforms that are cumbersome and computationally expensive to evaluate if
${F}_{X}$ and its inverse do not have simple closed forms.
3.3. VTransforms and Copulas
If two uniform random variables are linked by the vtransform $V=\mathcal{V}\left(U\right)$, then the joint distribution function of $(U,V)$ is a special kind of copula. In this section, we derive the form of the copula, which facilitates the construction of stochastic processes using vtransforms.
To state the main result, we use the notation ${\mathcal{V}}^{1}$ and ${\mathcal{V}}^{\prime}$ for the the inverse function and the gradient function of a vtransform $\mathcal{V}$. Although there is no unique inverse ${\mathcal{V}}^{1}\left(v\right)$ (except when $v=0$), the fact that the two branches of a vtransform mutually determine each other allows us to define ${\mathcal{V}}^{1}\left(v\right)$ to be the inverse of the left branch of the vtransform given by ${\mathcal{V}}^{1}:[0,1]\to [0,\delta ],\phantom{\rule{0.277778em}{0ex}}{\mathcal{V}}^{1}\left(v\right)=inf\{u:\mathcal{V}\left(u\right)=v\}$. The gradient ${\mathcal{V}}^{\prime}\left(u\right)$ is defined for all points $u\in [0,1]\backslash \left\{\delta \right\}$, and we adopt the convention that ${\mathcal{V}}^{\prime}\left(\delta \right)$ is the left derivative as $u\to \delta $.
Theorem 2. Let V and U be random variables related by the vtransform $V=\mathcal{V}\left(U\right)$.
 1.
The joint distribution function of $(U,V)$ is given by the copula  2.
Conditional on $V=v$, the distribution of U is given bywhere  3.
$\mathbb{E}\left(\Delta \left(V\right)\right)=\delta $.
Remark 1. In the case of the symmetric vtransform $\mathcal{V}\left(u\right)=12u$, the copula in (12) takes the form $C(u,v)=max(min(u+\frac{v}{2}\frac{1}{2},v),0)$. We note that this copula is related to a special case of the tent map copula family ${C}_{\theta}^{\mathcal{T}}$ in Rémillard (2013) by $C(u,v)=u{C}_{1}^{\mathcal{T}}(u,1v)$. For the linear vtransform family, the conditional probability
$\Delta \left(v\right)$ in (
14) satisfies
$\Delta \left(v\right)=\delta $. This implies that the value of
V contains no information about whether
U is likely to be below or above the fulcrum; the probability is always the same regardless of
V. In general, this is not the case and the value of
V does contain information about whether
U is large or small.
Part (2) of Theorem 2 is the key to stochastically inverting a vtransform in the general case. Based on this result, we define the concept of stochastic inversion of a vtransform. We refer to the function $\Delta $ as the conditional down probability of $\mathcal{V}$.
Definition 3 (Stochastic inversion function of a vtransform)
. Let $\mathcal{V}$ be a vtransform with conditional down probability Δ. The twoplace function ${\mathcal{V}}^{1}:[0,1]\times [0,1]\to [0,1]$ defined byis the stochastic inversion function of $\mathcal{V}$. The following proposition, which generalizes Lemma 1, allows us to construct general asymmetric processes that generalize the process of Algorithm 1.
Proposition 4. Let V and W be iid $U(0,1)$ variables and let $\mathcal{V}$ be a vtransform with stochastic inversion function $\mathcal{V}$. If $U={\mathcal{V}}^{1}(V,W)$, then $\mathcal{V}\left(U\right)=V$ and $U\sim U(0,1)$.
In
Section 4, we apply vtransforms and their stochastic inverses to the terms of time series models. To understand the effect this has on the serial dependencies between random variables, we need to consider multivariate componentwise vtransforms of random vectors with uniform marginal distributions and these can also be represented in terms of copulas. We now give a result which forms the basis for the analysis of serial dependence properties. The first part of the result shows the relationship between copula densities under componentwise vtransforms. The second part shows the relationship under the componentwise stochastic inversion of a vtransform; in this case, we assume that the stochastic inversion of each term takes place independently given
$\mathit{V}$ so that all serial dependence comes from
$\mathit{V}$.
Theorem 3. Let $\mathcal{V}$ be a vtransform and let $\mathit{U}={({U}_{1},\dots ,{U}_{d})}^{\prime}$ and $\mathit{V}={({V}_{1},\dots ,{V}_{d})}^{\prime}$ be vectors of uniform random variables with copula densities ${c}_{\mathit{U}}$ and ${c}_{\mathit{V}}$, respectively.
 1.
If $\mathit{V}={(\mathcal{V}\left({U}_{1}\right),\dots ,\mathcal{V}\left({U}_{d}\right))}^{\prime}$, thenwhere ${u}_{i1}={\mathcal{V}}^{1}\left({v}_{i}\right)$ and ${u}_{i2}={\mathcal{V}}^{1}\left({v}_{i}\right)+{v}_{i}$ for all $i\in \{1,\dots ,d\}$.  2.
If $\mathit{U}={({\mathcal{V}}^{1}({V}_{1},{W}_{1}),\dots ,{\mathcal{V}}^{1}({V}_{d},{W}_{d}))}^{\prime}$ where ${W}_{1},\dots ,{W}_{d}$ are iid uniform random variables that are also independent of ${V}_{1},\dots ,{V}_{d}$, then
4. VTARMA Copula Models
In this section, we study some properties of the class of time series models obtained by the following algorithm, which generalizes Algorithm 1. The models obtained are described as VTARMA processes since they are stationary time series constructed using the fundamental building blocks of a vtransform $\mathcal{V}$ and an ARMA process.
We can add any marginal behaviour in the final step, and this allows for an infinitely rich choice. We can, for instance, even impose an infinitevariance or an infinitemean distribution, such as the Cauchy distribution, and still obtain a strictly stationary process for $\left({X}_{t}\right)$. We make the following definitions.
Definition 4 (VTARMA and VTARMA copula process). Any stochastic process $\left({X}_{t}\right)$ that can be generated using Algorithm 2 by choosing an underlying ARMA process with mean zero and variance one, a vtransform $\mathcal{V}$, and a continuous distribution function ${F}_{X}$ is a VTARMA process. The process $\left({U}_{t}\right)$ obtained at the penultimate step of the algorithm is a VTARMA copula process.
Algorithm 2: 
Generate $\left({Z}_{t}\right)$ as a causal and invertible Gaussian ARMA process of order $(p,q)$ with mean zero and variance one. Form the volatility PIT process $\left({V}_{t}\right)$ where ${V}_{t}=\Phi \left({Z}_{t}\right)$ for all t. Generate iid $U(0,1)$ random variables $\left({W}_{t}\right)$. Form the series PIT process $\left({U}_{t}\right)$ by taking the stochastic inverses ${U}_{t}={\mathcal{V}}^{1}({V}_{t},{W}_{t})$. Form the process $\left({X}_{t}\right)$ by setting ${X}_{t}={F}_{X}^{1}\left({U}_{t}\right)$ for some continuous cdf ${F}_{X}$.

Figure 4 gives an example of a simulated process using Algorithm 2 and the vtransform
${\mathcal{V}}_{\delta ,\kappa ,\xi}$ in (
7) with
$\kappa =0.9$ and MA parameter
$\xi =1.1$. The marginal distribution is a heavytailed skewed Student distribution of type (
6) with degreesoffreedom
$\nu =3$ and skewness
$\gamma =0.8$, which gives rise to more large negative returns than large positive returns. The underlying time series model is an ARMA(1,1) model with AR parameter
$\alpha =0.95$ and MA parameter
$\beta =0.85$. See the caption of the figure for full details of parameters.
In the remainder of this section, we concentrate on the properties of VTARMA copula processes $\left({U}_{t}\right)$ from which related properties of VTARMA processes $\left({X}_{t}\right)$ may be easily inferred.
4.1. Stationary Distribution
The VTARMA copula process $\left({U}_{t}\right)$ of Definition 4 is a strictly stationary process since the joint distribution of $({U}_{{t}_{1}},\dots ,{U}_{{t}_{k}})$ for any set of indices ${t}_{1}<\cdots <{t}_{k}$ is invariant under time shifts. This property follows easily from the strict stationarity of the underlying ARMA process $\left({Z}_{t}\right)$ according to the following result, which uses Theorem 3.
Proposition 5. Let $\left({U}_{t}\right)$ follow a VTARMA copula process with vtransform $\mathcal{V}$ and an underlying ARMA(p,q) structure with autocorrelation function $\rho \left(k\right)$. The random vector $({U}_{{t}_{1}},\dots ,{U}_{{t}_{k}})$ for $k\in \mathbb{N}$ has joint density ${c}_{P({t}_{1},\dots ,{t}_{k})}^{\mathrm{Ga}}(\mathcal{V}\left({u}_{1}\right),\dots ,\mathcal{V}\left({u}_{k}\right))$, where ${c}_{P({t}_{1},\dots ,{t}_{k})}^{\mathrm{Ga}}$ denotes the density of the Gaussian copula ${C}_{P({t}_{1},\dots ,{t}_{k})}^{\mathrm{Ga}}$ and $P({t}_{1},\cdots ,{t}_{k})$ is a correlation matrix with $(i,j)$ element given by $\rho \left(\right{t}_{j}{t}_{i}\left\right)$.
An expression for the joint density facilitates the calculation of a number of dependence measures for the bivariate marginal distribution of
$({U}_{t},{U}_{t+k})$. In the bivariate case, the correlation matrix of the underlying Gaussian copula
${C}_{P(t,t+k)}^{\mathrm{Ga}}$ contains a single offdiagonal value
$\rho \left(k\right)$ and we simply write
${C}_{\rho \left(k\right)}^{\mathrm{Ga}}$. The Pearson correlation of
$({U}_{t},{U}_{t+k})$ is given by
This value is also the value of the Spearman rank correlation
${\rho}_{S}({X}_{t},{X}_{t+k})$ for a VTARMA process
$\left({X}_{t}\right)$ with copula process
$\left({U}_{t}\right)$ (since the Spearman’s rank correlation of a pair of continuous random variables is the Pearson correlation of their copula). The calculation of (
18) typically requires numerical integration. However, in the special case of the linear vtransform
${\mathcal{V}}_{\delta}$ in (
8), we can get a simpler expression as shown in the following result.
Proposition 6. Let $\left({U}_{t}\right)$ be a VTARMA copula process satisfying the assumptions of Proposition 5 with linear vtransform ${\mathcal{V}}_{\delta}$. Let $\left({Z}_{t}\right)$ denote the underlying Gaussian ARMA process. Then, For the symmetric vtransform
${\mathcal{V}}_{0.5}$, Equation (
19) obviously yields a correlation of zero so that, in this case, the VTARMA copula process
$\left({U}_{t}\right)$ is a white noise with an autocorrelation function that is zero, except at lag zero. However, even a very asymmetric model with
$\delta =0.4$ or
$\delta =0.6$ gives
$\rho ({U}_{t},{U}_{t+k})=0.04{\rho}_{S}({Z}_{t},{Z}_{t+k})$ so that serial correlations tend to be very weak.
When we add a marginal distribution, the resulting process
$\left({X}_{t}\right)$ has a different autocorrelation function to
$\left({U}_{t}\right)$, but the same rank autocorrelation function. The symmetric model of
Section 2 is a white noise process. General asymmetric processes
$\left({X}_{t}\right)$ are not perfect white noise processes but have only very weak serial correlation.
4.2. Conditional Distribution
To derive the conditional distribution of a VTARMA copula process, we use the vector notation ${\mathit{U}}_{t}={({U}_{1},\dots ,{U}_{t})}^{\prime}$ and ${\mathit{Z}}_{t}={({Z}_{1},\dots ,{Z}_{t})}^{\prime}$ to denote the history of processes up to time point t and ${\mathit{u}}_{t}$ and ${\mathit{z}}_{t}$ for realizations. These vectors are related by the componentwise transformation ${\mathit{Z}}_{t}={\Phi}^{1}\left(\mathcal{V}\left({\mathit{U}}_{t}\right)\right)$. We assume that all processes have a time index set given by $t\in \{1,2,\dots \}$.
Proposition 7. For $t>1$, the conditional density ${f}_{{U}_{t}\mid {\mathit{U}}_{t1}}(u\mid {\mathit{u}}_{t1})$ is given bywhere ${\mu}_{t}=\mathbb{E}({Z}_{t}\mid {\mathit{Z}}_{t1}={\Phi}^{1}\left(\mathcal{V}\left({\mathit{u}}_{t1}\right)\right))$ and ${\sigma}_{\u03f5}$ is the standard deviation of the innovation process for the ARMA model followed by $\left({Z}_{t}\right)$. When
$\left({Z}_{t}\right)$ is iid white noise
${\mu}_{t}=0$,
${\sigma}_{\u03f5}=1$ and (
20) reduce to the uniform density
${f}_{{U}_{t}\mid {\mathit{U}}_{t1}}(u\mid {\mathit{u}}_{t1})=1$ as expected. In the case of the firstorder Markov AR(1) model
${Z}_{t}={\alpha}_{1}{Z}_{t1}+{\u03f5}_{t}$, the conditional mean of
${Z}_{t}$ is
${\mu}_{t}={\alpha}_{1}{\Phi}^{1}\left(\mathcal{V}\left({u}_{t1}\right)\right)$ and
${\sigma}_{\u03f5}^{2}=1{\alpha}_{1}^{2}$. The conditional density (
20) can be easily shown to simplify to
${f}_{{U}_{t}\mid {U}_{t1}}(u\mid {u}_{t1})={c}_{{\alpha}_{1}}^{\mathrm{Ga}}\left(\mathcal{V}\left(u\right),\mathcal{V}\left({u}_{t1}\right)\right)$ where
${c}_{{\alpha}_{1}}^{\mathrm{Ga}}\left(\mathcal{V}\left({u}_{1}\right),\mathcal{V}\left({u}_{2}\right)\right)$ denotes the copula density derived in Proposition 5. In this special case, the VTARMA model falls within the class of firstorder Markov copula models considered by
Chen and Fan (
2006), although the copula is new.
If we add a marginal distribution
${F}_{X}$ to the VTARMA copula model to obtain a model for
$\left({X}_{t}\right)$ and use similar notational conventions as above, the resulting VTARMA model has conditional density
with
${f}_{{U}_{t}\mid {\mathit{U}}_{t1}}$ as in (
20). An interesting property of the VTARMA process is that the conditional density (
21) can have a pronounced bimodality for values of
${\mu}_{t}$ in excess of zero that is in high volatility situations where the conditional mean of
${Z}_{t}$ is higher than the marginal mean value of zero; in low volatility situations, the conditional density appears more concentrated around zero. This phenomenon is illustrated in
Figure 4. The bimodality in high volatility situations makes sense: in such cases, it is likely that the next return will be large in absolute value and relatively less likely that it will be close to zero.
The conditional distribution function of
$\left({X}_{t}\right)$ is
${F}_{{X}_{t}\mid {\mathit{X}}_{t1}}(x\mid {\mathit{x}}_{t1})={F}_{{U}_{t}\mid {\mathit{U}}_{t1}}({F}_{X}\left(x\right)\mid {F}_{X}\left({\mathit{x}}_{t1}\right))$ and hence the
$\psi $quantile
${x}_{\psi ,t}$ of
${F}_{{X}_{t}\mid {\mathit{X}}_{t1}}$ can be obtained by solving
For $\psi <0.5$, the negative of this value is often referred to as the conditional $(1\psi )$VaR (valueatrisk) at time t in financial applications.
5. Statistical Inference
In the copula approach to dependence modelling, the copula is the object of central interest and marginal distributions are often of secondary importance. A number of different approaches to estimation are found in the literature. As before, let ${x}_{1},\dots ,{x}_{n}$ represent realizations of variables ${X}_{1},\dots ,{X}_{n}$ from the time series process $\left({X}_{t}\right)$.
The semiparametric approach developed by
Genest et al. (
1995) is very widely used in copula inference and has been applied by
Chen and Fan (
2006) to firstorder Markov copula models in the time series context. In this approach, the marginal distribution
${F}_{X}$ is first estimated nonparametrically using the scaled empirical distribution function
${F}_{n}^{\left(X\right)}$ (see definition in
Section 1) and the data are transformed onto the
$(0,1)$ scale. This has the effect of creating pseudocopula data
${u}_{t}=\mathrm{rank}\left({x}_{t}\right)/(n+1)$ where
$\mathrm{rank}\left({x}_{t}\right)$ denotes the rank of
${x}_{t}$ within the sample. The copula is fitted to the pseudocopula data by maximum likelihood (ML).
As an alternative, the inferencefunctionsformargins (IFM) approach of
Joe (
2015) could be applied. This is also a twostep method although in this case a parametric model
${\widehat{F}}_{X}$ is estimated under an iid assumption in the first step and the copula is fitted to the data
${u}_{t}={\widehat{F}}_{X}\left({x}_{t}\right)$ in the second step.
The approach we adopt for our empirical example is to first use the semiparametric approach to determine a reasonable copula process, then to estimate marginal parameters under an iid assumption, and finally to estimate all parameters jointly using the parameter estimates from the previous steps as starting values.
We concentrate on the mechanics of deriving maximum likelihood estimates (MLEs). The problem of establishing the asymptotic properties of the MLEs in our setting is a difficult one. It is similar to, but appears to be more technically challenging than, the problem of showing consistency and efficiency of MLEs for a BoxCoxtransformed Gaussian ARMA process, as discussed in
Terasaka and Hosoya (
2007). We are also working with a componentwise transformed ARMA process, although, in our case, the transformation
$\left({X}_{t}\right)\to \left({Z}_{t}\right)$ is via the nonlinear, nonincreasing volatility proxy transformation
${T}^{\left(Z\right)}\left(x\right)$ in (
5), which is not differentiable at the change point
${\mu}_{T}$. We have, however, run extensive simulations which suggests good behaviour of the MLEs in large samples.
5.1. Maximum Likelihood Estimation of the VTARMA Copula Process
We first consider the estimation of the VTARMA copula process for a sample of data
${u}_{1},\dots ,{u}_{n}$. Let
${\mathit{\theta}}^{\left(V\right)}$ and
${\mathit{\theta}}^{\left(A\right)}$ denote the parameters of the vtransform and ARMA model, respectively. It follows from Theorem 3 (part 2) and Proposition 5 that the loglikelihood for the sample
${u}_{1},\dots ,{u}_{n}$ is simply the log density of the Gaussian copula under componentwise inverse vtransformation. This is given by
where the first term
${L}^{*}$ is the loglikelihood for an ARMA model with a standard N(0,1) marginal distribution. Both terms in the loglikelihood (
23) are relatively straightforward to evaluate.
The evaluation of the ARMA likelihood ${L}^{*}({\mathit{\theta}}^{\left(A\right)}\mid {z}_{1},\dots ,{z}_{n})$ for parameters ${\mathit{\theta}}^{\left(A\right)}$ and data ${z}_{1},\dots ,{z}_{n}$ can be accomplished using the Kalman filter. However, it is important to note that the assumption that the data ${z}_{1},\dots ,{z}_{n}$ are standard normal requires a bespoke implementation of the Kalman filter, since standard software always treats the error variance ${\sigma}_{\u03f5}^{2}$ as a free parameter in the ARMA model. In our case, we need to constrain ${\sigma}_{\u03f5}^{2}$ to be a function of the ARMA parameters so that $var\left({Z}_{t}\right)=1$. For example, in the case of an ARMA(1,1) model with AR parameter ${\alpha}_{1}$ and MA parameter ${\beta}_{1}$, this means that ${\sigma}_{\u03f5}^{2}={\sigma}_{\u03f5}^{2}({\alpha}_{1},{\beta}_{1})=(1{\alpha}_{1}^{2})/(1+2{\alpha}_{1}{\beta}_{1}+{\beta}_{1}^{2})$. The constraint on ${\sigma}_{\u03f5}^{2}$ must be incorporated into the statespace representation of the ARMA model.
Model validation tests for the VTARMA copula can be based on residuals
where
${z}_{t}$ denotes the implied realization of the normalized volatility proxy variable and where an estimate
${\widehat{\mu}}_{t}$ of the conditional mean
${\mu}_{t}=\mathbb{E}({Z}_{t}\mid {\mathit{Z}}_{t1}={\mathit{z}}_{t})$ may be obtained as an output of the Kalman filter. The residuals should behave like an iid sample from a normal distribution.
Using the estimated model, it is also possible to implement a likelihoodratio (LR) test for the presence of stochastic volatility in the data. Under the null hypothesis that
${\mathit{\theta}}^{\left(A\right)}=\mathbf{0}$, the loglikelihood (
23) is identically equal to zero. Thus, the size of the maximized loglikelihood
$L({\widehat{\mathit{\theta}}}^{\left(V\right)},{\widehat{\mathit{\theta}}}^{\left(A\right)}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}{u}_{1},\dots ,{u}_{n})$ provides a measure of the evidence for the presence of stochastic volatility.
5.2. Adding a Marginal Model
If
${F}_{X}$ and
${f}_{X}$ denote the cdf and density of the marginal model and the parameters are denoted
${\mathit{\theta}}^{\left(M\right)}$, then the full loglikelihood for the data
${x}_{1},\dots ,{x}_{n}$ is simply
where the first term is the loglikelihood for a sample of iid data from the marginal distribution
${F}_{X}$ and the second term is (
23).
When a marginal model is added, we can recover the implied form of the volatility proxy transformation using Proposition 3. If
$\widehat{\delta}$ is the estimated fulcrum parameter of the vtransform, then the estimated change point is
${\widehat{\mu}}_{T}={F}_{X}^{1}(\widehat{\delta};{\widehat{\mathit{\theta}}}^{\left(M\right)})$ and the implied profile function is
Note that is is possible to force the change point to be zero in a joint estimation of marginal model and copula by imposing the constraint ${F}_{X}(0;{\mathit{\theta}}^{\left(M\right)})=\delta $ on the fulcrum and marginal parameters during the optimization. However, in our experience, superior fits are obtained when these parameters are unconstrained.
5.3. Example
We analyse
$n=1043$ daily logreturns for the Bitcoin price series for the period 2016–2019; values are multiplied by 100. We first apply the semiparametric approach of
Genest et al. (
1995) using the loglikelihood (
23) which yields the results in
Table 1. Different models are referred to by VT(
n)ARMA(
p,
q), where
$(p,q)$ refers to the ARMA model and
n indexes the vtransform: 1 is the linear vtransform
${\mathcal{V}}_{\delta}$ in (
8); 3 is the threeparameter transform
${\mathcal{V}}_{\delta ,\kappa ,\xi}$ in (
7); 2 is the twoparameter vtransform given by
${\mathcal{V}}_{\delta ,\kappa}:={\mathcal{V}}_{\delta ,\kappa ,1}$. In unreported analyses, we also tried the threeparameter family based on the beta distribution, but this had negligible effect on the results.
The column marked L gives the value of the maximized loglikelihood. All values are large and positive showing strong evidence of stochastic volatility in all cases. The model VT(1)ARMA(1,0) is a firstorder Markov model with linear vtransform. The fit of this model is noticeably poorer than the others suggesting that Markov models are insufficient to capture the persistence of stochastic volatility in the data. The column marked SW contains the pvalue for a Shapiro–Wilks test of normality applied to the residuals from the VTARMA copula model; the result is nonsignificant in all cases.
According to the AIC values, the VT(2)ARMA(1,1) is the best model. We experimented with higher order ARMA processes, but this did not lead to further significant improvements.
Figure 5 provides a visual of the fit of this model. The pictures in the panels show the QQplot of the residuals against normal, acf plots of the residuals and squared residuals and the estimated conditional mean process
$\left({\widehat{\mu}}_{t}\right)$, which can be taken as an indicator of high and low volatility periods. The residuals and absolute residuals show very little evidence of serial correlation and the QQplot is relatively linear, suggesting that the ARMA filter has been successful in explaining much of the serial dependence structure of the normalized volatility proxy process.
We now add various marginal distributions to the VT(2)ARMA(1,1) copula model and estimate all parameters of the model jointly. We have experimented with a number of locationscale families including Studentt, Laplace (double exponential), and a doubleWeibull family which generalizes the Laplace distribution and is constructed by taking backtoback Weibull distributions. Estimation results are presented for these three distributions in
Table 2. All three marginal distributions are symmetric around their location parameters
$\mu $, and no improvement is obtained by adding skewness using the construction of
Fernández and Steel (
1998) described in
Section 3.1; in fact, the Bitcoin returns in this time period show a remarkable degree of symmetry. In the table, the shape and scale parameters of the distributions are denoted
$\eta $ and
$\sigma $, respectively; in the case of Student, an infinitevariance distribution with degreeoffreedom parameter
$\eta =1.94$ is fitted, but this model is inferior to the models with Laplace and doubleWeibull margins; the latter is the favoured model on the basis of AIC values.
Figure 6 shows some aspects of the joint fit for the fully parametric VT(2)ARMA(1,1) model with doubleWeibull margin. A QQplot of the data against the fitted marginal distribution confirms that the doubleWeibull is a good marginal model for these data. Although this distribution is subexponential (heaviertailed than exponential), its tails do not follow a power law and it is in the maximum domain of attraction of the Gumbel distribution (see, for example,
McNeil et al. 2015, Chapter 5).
Using (
26), the implied volatility proxy profile function
${\widehat{g}}_{T}$ can be constructed and is found to lie just below the line
$y=x$ as shown in the upperright panel. The change point is estimated to be
${\widehat{\mu}}_{T}=0.06$. We can also estimate an implied volatility proxy transformation in the equivalence class defined by
${\widehat{g}}_{T}$ and
${\widehat{\mu}}_{T}$. We estimate the transformation
$T={T}^{\left(Z\right)}$ in (
5) by taking
$\widehat{T}\left(x\right)={\Phi}^{1}({\mathcal{V}}_{{\widehat{\mathit{\theta}}}^{\left(V\right)}}({F}_{X}(x;{\widehat{\mathit{\theta}}}^{\left(M\right)})))$. In the lowerleft panel of
Figure 6, we show the empirical vtransform formed from the data
$({x}_{t},\widehat{T}\left({x}_{t}\right))$ together with the fitted parametric vtransform
${\mathcal{V}}_{{\widehat{\mathit{\theta}}}^{\left(V\right)}}$. We recall from
Section 1 that the empirical vtransform is the plot
$({u}_{t},{v}_{t})$ where
${u}_{t}={F}_{n}^{\left(X\right)}\left({x}_{t}\right)$ and
${v}_{t}={F}_{n}^{\left(\widehat{T}\left(X\right)\right)}(\widehat{T}\left({x}_{t}\right))$. The empirical vtransform and the fitted parametric vtransform show a good degree of correspondence. The lowerright panel of
Figure 6 shows the volatility proxy transformation
$\widehat{T}\left(x\right)$ as a function of
x superimposed on the points
$({x}_{t},{\Phi}^{1}\left({v}_{t}\right))$. Using the curve, we can compare the effects of, for example, a logreturn (× 100) of −10 and a logreturn of 10. For the fitted model, these are 1.55 and 1.66 showing that the up movement is associated with slightly higher volatility.
As a comparison to the VTARMA model, we fit standard GARCH(1,1) models using Studentt and generalized error distributions for the innovations; these are standard choices available in the popular
rugarch package in R. The generalized error distribution (GED) contains normal and Laplace as special cases as well as a model that has a similar tail behaviour to Weibull; note, however, that, by the theory of
Mikosch and Stărică (
2000), the tails of the marginal distribution of the GARCH decay according to a power law in both cases. The results in
Table 3 show that the VT(2)ARMA(1,1) models with Laplace and doubleWeibull marginal distributions outperform both GARCH models in terms of AIC values.
Figure 7 shows the insample 95% conditional valueatrisk (VaR) estimate based on the VT(2)ARMA(1,1) model which has been calculated using (
22). For comparison, a dashed line shows the corresponding estimate for the GARCH(1,1) model with GED innovations.
Finally, we carry out an outofsample comparison of conditional VaR estimates using the same two models. In this analysis, the models are estimated daily throughout the 2016–2019 period using a 1000day moving data window and onestepahead VaR forecasts are calculated. The VTARMA model gives 47 exceptions of the 95% VaR and 11 exceptions of the 99% VaR, compared with expected numbers of 52 and 10 for a 1043 day sample, while the GARCH model leads to 57 and 12 exceptions; both models pass binomial tests for these exception counts. In a followup paper (
Bladt and McNeil 2020), we conduct more extensive outofsample backtests for models using vtransforms and copula processes and show that they rival and often outperform forecast models from the extended GARCH family.
6. Conclusions
This paper has proposed a new approach to volatile financial time series in which vtransforms are used to describe the relationship between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable. We have characterized vtransforms mathematically and shown that the stochastic inverse of a vtransform may be used to construct stationary models for return series where arbitrary marginal distributions may be coupled with dynamic copula models for the serial dependence in the volatility proxy.
The construction was illustrated using the serial dependence model implied by a Gaussian ARMA process. The resulting class of VTARMA processes is able to capture the important features of financial return series including nearzero serial correlation (white noise behaviour) and volatility clustering. Moreover, the models are relatively straightforward to estimate building on the classical maximumlikelihood estimation of an ARMA model using the Kalman filter. This can be accomplished in the stepwise manner that is typical in copula modelling or through joint modelling of the marginal and copula process. The resulting models yield insights into the way that volatility responds to returns of different magnitude and sign and can give estimates of unconditional and conditional quantiles (VaR) for practical risk measurement purposes.
There are many possible uses for VTARMA copula processes. Because we have complete control over the marginal distribution, they are very natural candidates for the innovation distribution in other time series models. For example, they could be applied to the innovations of an ARMA model to obtain ARMA models with VTARMA errors; this might be particularly appropriate for longer interval returns, such as weekly or monthly returns, where some serial dependence is likely to be present in the raw return data.
Clearly, we could use other copula processes for the volatility PIT process $\left({V}_{t}\right)$. The VTARMA copula process has some limitations: the radial symmetry of the underlying Gaussian copula means that the serial dependence between large values of the volatility proxy must mirror the serial dependence between small values; moreover, this copula does not admit tail dependence in either tail and it seems plausible that very large values of the volatility proxy might have a tendency to occur in succession.
To extend the class of models based on vtransforms, we can look for models for the volatility PIT process
$\left({V}_{t}\right)$ with higher dimensional marginal distributions given by asymmetric copulas with upper tail dependence. Firstorder Markov copula models as developed in
Chen and Fan (
2006) can give asymmetry and tail dependence, but they cannot model the dependencies at longer lags that we find in empirical data. Dvine copula models can model higherorder Markov dependencies and
Bladt and McNeil (
2020) show that this is a promising alternative specification for the volatility PIT process.