# Multiplicative Error Models: 20 years on

Fabrizio Cipollini, Giampiero M. Gallo

###### Abstract

Several phenomena are available representing market activity: volumes, number
of trades, durations between trades or quotes, volatility – however measured –
all share the feature to be represented as positive valued time series. When
modeled, persistence in their behavior and reaction to new information
suggested to adopt an autoregressive–type framework. The Multiplicative Error
Model ($\mathsf{MEM}$) is borne of an extension of the popular GARCH approach
for modeling and forecasting conditional volatility of asset returns. It is
obtained by multiplicatively combining the conditional expectation of a
process (deterministically dependent upon an information set at a previous
time period) with a random disturbance representing unpredictable news:
$\mathsf{MEM}$s have proved to parsimoniously achieve their task of producing
good performing forecasts. In this paper we discuss various aspects of model
specification and inference both for the univariate and the multivariate case.
The applications are illustrative examples of how the presence of a slow
moving low–frequency component can improve the properties of the estimated
models.

##  1 Introduction

Twenty years have gone by since the Engle (2002) paper in which Rob Engle
provided a framework of generalization to the ARCH (Engle, 1982) and GARCH
(Bollerslev, 1986) modeling of the conditional variance of financial returns.
Following the work in Engle and Russell (1998), and the overall remarks in
Engle (2000), it was recognized that the availability of ultra–high frequency
data had opened the venue to widening the range of the analysis of market
activity. New financial time series became readily available: Andersen and
Bollerslev (1998) started a stream of literature addressing the properties of
measuring volatility from intradaily data and using it as an alternative
target to the squared returns (a very noisy measure of return variance); for a
review of various realized volatility measures, see Andersen et al. (2006);
for the care to be exerted in working with ultra–high frequency data, see
Brownlees and Gallo (2006). While in the GARCH literature measurement and
forecasting were steps performed simultaneously, the development of volatility
measures left the question open as of what model would be the most appropriate
to derive forecasts. Volatility being non–negatively valued and persistent
lends these measures to be represented in some autoregressive fashion; as
expressed by market trading activity volatility is accompanied by other
indicators which exhibit the stylized facts of volatility clustering: next to
realized volatility and durations, we can mention the volume, range, number of
trades, bid–ask spread. In Engle (2002), a new class of models was suggested
for non–negative valued processes, called Multiplicative Error Models (MEM),
obtained by multiplicatively combining the conditional expectation of a
process (deterministically dependent upon an information set at a previous
time period) with a random disturbance representing unpredictable news. This
approach is different in spirit from the linear and additive modeling
suggested by Corsi (2009) in his Heterogeneous Autoregressive (HAR) model for
realized variances, which has the merit of recognizing that long–memory
features, often cited in the profile of realized volatility especially, can be
mimicked upon suitable aggregation of the lagged variances (see also Andersen
et al., 2007, for a HAR with jumps).

In the past 20 years, the class of MEM models has grown in size, at times
applying and extending concepts that were originated in the GARCH literature,
also as a consequence of the fact that a GARCH model is a MEM for squared
returns. Within the $\mathsf{ACD}$, from the already mentioned seminal
contribution of Engle and Russell (1998), further developments for the
financial durations are documented in Hautsch (2004); Härdle et al. (2015)
provide a local adaptive approach to $\mathsf{ACD}$’s, using $\mathsf{MEM}$
and $\mathsf{ACD}$ as synonyms; Perera et al. (2016) provide a testing
framework for $\mathsf{ACD}$ parametric specifications. A $\mathsf{MEM}$ for
the daily range (Parkinson, 1980; Garman and Klass, 1980) is called
Conditional Autoregressive Range (CARR) model by Chou (2005) and a review of
this approach is contained in Chou et al. (2015).

Initial applications of the $\mathsf{MEM}$ were favored by the fact that GARCH
routines in standard software could expediently be used to estimate the new
model, by imposing that the ”dependent variable” were the square root of the
actual variable of interest, that the mean equation was forced to zero, and
that the innovation distribution was set to the Gaussian. This estimation
practice was later justified in a more general context by Engle and Gallo
(2006), who formally proved the result for a Gamma distribution for the
multiplicative innovation, and showed that the first order conditions for the
ML estimation of the parameters of the conditional expectation in a
$\mathsf{MEM}$ do not depend upon the unknown scale parameter of the Gamma,
establishing QMLE properties for the estimator. Independently developed, the
Generalized Autoregressive Score approach of Creal et al. (2013) and the
Dynamic Conditional Score approach by Harvey (2013) both address a wider class
of models which comprise the $\mathsf{MEM}$ as a special case under a
parametric assumption of a Gamma distribution for the error term.

For the case in which the series may contain zeros (as with absolute returns
or volume at very high frequency with thinly traded assets), Hautsch et al.
(2014) suggest a way to augment the $\mathsf{MEM}$ model to accommodate a mass
of the density at zero. In the case of intradaily processes, the
$\mathsf{MEM}$ gets expressed by several multiplicative components as in
Brownlees et al. (2011) who specify a model for volumes whose conditional
expectation is the result of the product of a daily component, a periodic
intradaily component (essentially capturing time of day) and a non periodic
intradaily component. Cipollini et al. (2021) address the issue of measurement
error in the realized volatility measures and show that the additive models
used by Bollerslev et al. (2016) in the univariate context which make use of
realized quarticity can be conveniently supplemented by $\mathsf{MEM}$s which,
thanks to their multiplicative structure, better capture the heteroskedastic
nature of measurement errors.

Brownlees and Gallo (2010) adopt a $\mathsf{MEM}$ to model several versions of
realized volatility in a risk management context where the VaR is used as a
basis to evaluate forecast performance. While they find that realized kernel
volatility has the most desirable prediction properties, they also establish
that the daily range is a good substitute when intradaily data are not
available.

Engle and Gallo (2006) contains an application of a trivariate $\mathsf{MEM}$
on three measures of volatility, estimated equation–by–equation, allowing for
lagged dynamic interdependence (a triangular structure with contemporaneous
dependence is contained in Manganelli, 2005). The same structure is adopted by
Engle et al. (2012) in the context of daily range for several East Asian
markets, where they test for market interdependence and spillovers in the
Asian crisis of 1997. A full–fledged theory for vector $\mathsf{MEM}$, i.e.
$\mathsf{vMEM}$, is suggested by Cipollini et al. (2006) in the context of
copula functions (later developed in Cipollini et al., 2017) and of
Generalized Method of Moments (later developed in Cipollini et al., 2013).
Diagnostics checking for $\mathsf{MEM}$ is provided by Koul et al. (2012) and
by Perera and Silvapulle (2017) and for $\mathsf{vMEM}$ by (Ng et al., 2016).
An automated procedure for $\mathsf{vMEM}$ specification selection is proposed
in Cipollini and Gallo (2010). Another development involving a trivariate
intradaily market activity process is suggested by Hautsch (2008) and a
dynamic conditional correlation model with $\mathsf{MEM}$ is developed by
Bodnar and Hautsch (2016).

The feature of a positive valued financial time series showing a slow–moving
low frequency component may be addressed adapting the component GARCH by Engle
and Lee (1999): the dynamics may depend on the sum of two components, one
forced to be persistent and another capturing more short-term dynamics. In the
$\mathsf{MEM}$ context, this is suggested by Brownlees et al. (2012) who
present an application on univariate volatilities of several stocks. Cipollini
and Gallo (2019) extend the idea to the multivariate context, by keeping the
additivity of the components in a $\mathsf{vMEM}$, but forcing the slow moving
volatility to be common to several Euro area market indices.

Gallo and Otranto (2015) investigate the features of a low–frequency component
in a $\mathsf{MEM}$ in the context of Markov Switching ($\mathsf{MS-AMEM}$)
and Smooth Transition ($\mathsf{ST-AMEM}$) models, suggesting the concept of
changing average volatility levels and comparing the behavior of the
$\mathsf{MS-AMEM}$ and of the $\mathsf{ST-AMEM}$ to various versions of the
$\mathsf{HAR}$ model by Corsi (2009) in forecasting. Among these models, there
is a $\mathsf{HAR}$ specification with jumps: the presence of jumps is
addressed within the $\mathsf{MEM}$ approach by Caporin et al. (2017).

This paper retraces the logic behind a $\mathsf{MEM}$, starting from some
stylized facts, and focuses on the presence of a component which captures
slow–moving behavior is of specific interest in our contribution;
specifically, we address the possibility of having components combining
multiplicatively, both in the univariate (extending Veredas et al., 2007, for
an ACD) and in the multivariate (extending Barigozzi et al., 2014) cases, the
latter with a common low–frequency component. We discuss model specification
and inference issues providing details for the Generalized Method of Moments
method. The applications are performed in reference to several volatility
market indices (US, Europe and East Asia), providing evidence of the relevance
of the low–frequency component which varies by market. The multivariate case
considers a trivariate system in which dynamic interdependencies in the
short–run components of absolute returns, realized kernel volatility and
option–based implied volatility indices are assessed with and without a common
low–frequency component. Being a GMM-based strategy, estimation bypasses the
specification of a parametric distribution for the error term. Yet, we provide
evidence of how this error behaves as an estimation residual, should some
applications for volatility–at–risk (in the sense of Caporin et al., 2017) be
needed.

The structure of the paper is as follows: in Section 2 we propose a
modification of the standard specification of a $\mathsf{MEM}$ (called
$\mathsf{SpMEM}$). In Section 3 we discuss inferential properties within a GMM
estimation strategy. In the univariate case, an empirical application is
performed on series of realized volatility from several markets (Section 4).
For the multivariate case, (Section 5) we extend the $\mathsf{SpvMEM}$ model
by Barigozzi et al. (2014) to allow for dynamic interdependence across
variables, for which the GMM properties of the estimation strategy followed
are presented in Section 6. The applications in the multivariate case relate
to a trivariate model with absolute returns, realized volatility and implied
volatility from option–based indices (Section 7). Concluding remarks follow.

##  2 The Model

In the univariate case, the general form of a Multiplicative Error Model
($\mathsf{MEM}$) can be specified as

$x_{t}=\underbrace{\mu\;\tau_{t}\;\xi_{t}}_{\mu_{t}}\varepsilon_{t},$ (1)

i.e., designed in a way that $\mu$ is introduced as the unconditional
expectation of $x_{t}$ (assumed mean stationary), around which there are three
components combining multiplicatively:

  * •

$\tau_{t}$ is a low–frequency component capturing the slow-moving secular
dynamics in the process (when constant and equal to $1$, we have the base
$\mathsf{MEM}$);

  * •

$\xi_{t}$ is a short–run component, parameterized in a way similar to a GARCH
process;

  * •

$\varepsilon_{t}$ is a conditionally unpredictable homoskedastic component.

Upon imposition of a unit expectation on the $\varepsilon_{t}$ term, the
expectation of $x_{t}$ conditional on the information set $\mathcal{I}_{t-1}$
is thus $\mu_{t}=\mu\tau_{t}\xi_{t}$; moreover, for identification purposes,
$E(\tau_{t})=E(\xi_{t})=1$, which is achieved via suitable normalizations.

To motivate this specification, let us refer to Figure 1 of a typical positive
valued financial time series, namely the realized kernel volatility measured
on the Dow Jones 30 Industrial Index.111The data span the period August 2nd,
2013 – April 13th, 2021 and the series is the annualized square root of the
realized kernel variance taken from the OMI Realized Library by Heber et al.
(2009). For future reference, and motivation for the subsequent analysis, we
have reported what we obtain as the sample estimate of the unconditional mean
$\mu$ (the horizontal red line); relative to it, the series exhibits a
slow–moving feature which is captured by the $\tau_{t}$ term (the brown line
depicts the estimated product $\mu\tau_{t}$); in turn, around it, we notice
the high–frequency, familiar persistent pattern deriving by the well-known
property of volatility clustering reproduced by $\xi_{t}$ (the blue line
depicts the estimated product $\mu_{t}=\mu\tau_{t}\xi_{t}$); finally, the
estimated error component $\varepsilon_{t}$ would be derived as the ratio
between the grey and the blue line.

As per the first component $\tau_{t}$, should one decide to do away with it
(equivalent to take $\tau_{t}\equiv 1\;\forall t$), the outcome would be the
base $\mathsf{MEM}$, as suggested by Engle (2002), with expectation targeting
$\mu$; to model $\tau_{t}$, one can resort to some smooth function of time (as
in, e.g, a spline or in the smooth transition framework of Amado and
Teräsvirta, 2017), or, even, a Markov Switching approach as in Gallo and
Otranto (2015) based on the concept of average level of volatility within each
regime. In what follows, also in view of the multivariate specification below,
we adopt a non parametric specification derived from Veredas et al. (2007) in
the context of the ACD model, where the only requirement is that $\tau_{t}$ is
a sufficiently smooth function of time: we call it
semi–nonparametric-$\mathsf{MEM}$ or $\mathsf{SpMEM}$. For practical purposes,
at the estimation stage, it takes the form:

$\widehat{\tau}_{t}=\frac{\displaystyle\sum_{s=1}^{T}x_{t}^{(\tau)}K\left(\frac{z_{t}-z_{s}}{h}\right)}{\displaystyle\sum_{s=1}^{T}K\left(\frac{z_{t}-z_{s}}{h}\right)}\\\
$ (2)

where

$x_{t}^{(\tau)}=\frac{x_{t}}{\mu\xi_{t}},$

$\mu$ and $\xi_{t}$ denote here estimates of the corresponding components,
$z_{t}=t/T$, $K(\cdot)$ is a kernel function and $h$ is the bandwidth. Further
details on model estimation are in Section 3.

The second component $\xi_{t}$ captures the short–run dynamics with an
autoregressive dependence on past observations of $x_{t}$ and past
$\xi_{t}$’s, as mentioned, in a GARCH structure of the type

$\xi_{t}=\left[1-\left(\beta_{1}+\alpha_{1}+\frac{\gamma_{1}}{2}\right)\right]+\beta_{1}\xi_{t-1}+\alpha_{1}x_{t-1}^{(\xi)}+\gamma_{1}x_{t-1}^{(\xi-)},$
(3)

where

$x_{t}^{(\xi)}=\frac{x_{t}}{\mu\tau_{t}}\qquad{}x_{t}^{(\xi-)}=x_{t}^{(\xi)}{D}^{-}_{t},$

${D}^{-}_{t}=I(r_{t}<0)$ and $r_{t}$ is the return. An equivalent way to
express the $\xi_{t}$ dynamics is

$\xi_{t}=\left(1-\beta_{1}^{*}\right)+\beta_{1}^{*}\xi_{t-1}+\alpha_{1}v_{t-1}+\gamma_{1}v_{t-1}^{(-)},$
(4)

where

$v_{t}=x^{(\xi)}_{t}-\xi_{t}\qquad{}v_{t}^{(-)}=x^{(\xi-)}_{t}-\xi_{t}/2$

are zero-mean innovations and
$\beta_{1}^{*}=\beta_{1}+\alpha_{1}+\gamma_{1}/2$ is the persistence
parameter.222The equivalence between (3) and (4) may be no longer valid in
case additional lags are added to some of the terms.

The third component $\varepsilon_{t}$ is a unit mean, homoskedastic r.v. with
non-negative support, in symbols

$\varepsilon_{t}\overset{\operatorname{i.i.d.}}{\sim}d^{+}(1,\sigma^{2}).$ (5)

In the univariate case, we will refer to r.v.’s with such characteristics,
namely the Gamma, the Log–Normal, the Beta′ and the Log-Logistic described in
detail in the Appendix.

Figure 1: DJI Realized kernel volatility $x_{t}$ (in grey); its unconditional
mean $\mu$ (horizontal red line); its low–frequency component $\mu\tau_{t}$
(in brown); its conditional expectation $\mu_{t}=\mu\tau_{t}\xi_{t}$ (in
blue). Sample Aug. 2, 2013–Apr. 13, 2021.

##  3 Inference Issues

Following Veredas et al. (2007) and Barigozzi et al. (2014), inference of the
$\mathsf{SpMEM}$ is obtained using the following strategy. $\mu$ is estimated
once for all by the sample average of the $\\{x_{t}\\}$ data; regarding the
other components, after initialization of all $\xi_{t}$’s to $1$, estimation
is done by iterating between these two steps, until convergence:

  1. 1.

$\tau_{t}$ is estimated using the kernel estimator detailed in Section 2 on
the $\\{x_{t}^{(\tau)}=x_{t}/(\mu\xi_{t})\\}$ current values;

  2. 2.

$\xi_{t}$ parameters and $\sigma^{2}$ are estimated on the
$\\{x_{t}^{(\xi)}=x_{t}/(\mu\tau_{t})\\}$ current values using the approach
detailed below.

Clearly, in case the model is pure $\mathsf{MEM}$ (i.e. $\tau_{t}\equiv 1\
\forall t$), only the second step is needed and inference is obtained by
estimating the $\xi_{t}$ parameters on the $\\{x_{t}^{(\xi)}=x_{t}/\mu\\}$
values.

To describe inference on the $\xi_{t}$ parameters and $\sigma^{2}$, we refer
to the framework presented in Brownlees et al. (2012, Section 9.2.2). Assuming
that $\xi_{t}$ is correctly specified and indicating with $\bm{\theta}$ the
vector of parameters entering it, two estimation strategies are illustrated:
Generalized Method of Moments (GMM), and Quasi Maximum Likelihood (QML).

###  3.1 Generalized Method of Moments Inference

GMM estimation can be conveniently used to estimate the parameters of interest
without an explicit choice of the error term distribution of Equation (5). Let

$\varepsilon_{t}=\frac{x^{(\xi)}_{t}}{\xi_{t}},$ (6)

where $x^{(\xi)}_{t}=x_{t}/(\mu\tau_{t})$. Under model assumptions,
$\varepsilon_{t}-1$ is a conditionally homoskedastic martingale difference,
with conditional expectation zero and conditional variance $\sigma^{2}$.
Following Brownlees et al. (2012, Section 9.2.2.2), the _efficient_ GMM
estimator of $\bm{\theta}$, say $\widehat{\bm{\theta}}_{GMM}$, solves the
criterion equation

$\sum_{t=1}^{T}(\varepsilon_{t}-1)\bm{a}_{t}=\bm{0},$ (7)

and has asymptotic variance matrix

$\displaystyle\operatorname{Avar}(\widehat{\bm{\theta}}_{GMM})$
$\displaystyle=\sigma^{2}\bm{A}^{-1},$ (8)

where

$\bm{a}_{t}=\frac{1}{\xi_{t}}\nabla_{\bm{\theta}}\xi_{t}$ (9)

and

$\bm{A}=\lim_{T\rightarrow\infty}\left[T^{-1}\sum_{t=1}^{T}E\left(\bm{a}_{t}\bm{a}_{t}^{\prime}\right)\right].$

As a consequence, a consistent estimator of the asymptotic variance matrix is

$\widehat{\operatorname{Avar}}(\widehat{\bm{\theta}}_{GMM})=\widehat{\sigma}^{2}\widehat{\bm{A}}^{-1},$

where

$\widehat{\sigma}^{2}=T^{-1}\sum_{t=1}^{T}\left(\widehat{\varepsilon}_{t}-1\right)^{2}$

is a Method of Moments estimator of $\sigma^{2}$,

$\widehat{\bm{A}}=T^{-1}\sum_{t=1}^{T}\widehat{\bm{a}}_{t}\widehat{\bm{a}}_{t}^{\prime},$

$\widehat{\varepsilon}_{t}$ and $\widehat{\bm{a}}_{t}$ correspond to (6) and
(9), respectively, evaluated at $\widehat{\bm{\theta}}_{GMM}$.

From a practical point of view, then, the procedure becomes fully feasible in
our context upon substituting the unknown $\mu$ and $\tau_{t}$ in
$x_{t}^{(\xi)}$ by consistent estimates $\widehat{\mu}$ and
$\widehat{\tau}_{t}$.

###  3.2 Quasi Maximum Likelihood Inference

Following Engle and Gallo (2006), an alternative approach is Quasi Maximum
Likelihood (QML): this is obtained assuming $\varepsilon_{t}$ as Gamma
distributed and then estimating $\bm{\theta}$ by Maximum Likelihood.

More specifically, if (5) is specified as
$\varepsilon_{t}|\mathcal{F}_{t-1}\sim Gamma\left(\phi,\phi\right)$ (so as
$E\left(\varepsilon_{t}|\mathcal{F}_{t-1}\right)=1$ and
$V\left(\varepsilon_{t}|\mathcal{F}_{t-1}\right)=\sigma^{2}=1/\phi$) the log-
likelihood function is

$l_{T}=\sum_{t=1}^{T}\left[\phi\ln\phi-\ln\Gamma(\phi)+\phi\ln\varepsilon_{t}-\phi\varepsilon_{t}-\ln
x^{(\xi)}_{t}\right].$

This equation shows that its maximization in $\bm{\theta}$ can be done
maximizing the quasi-log-likelihood

$\sum_{t=1}^{T}\left(\ln\varepsilon_{t}-\varepsilon_{t}\right),$

that does not depend on $\phi$. As a consequence, the first order condition
for $\bm{\theta}$ is given exactly by the GMM condition (7). Notice that this
condition can be rewritten as

$\sum_{t=1}^{T}\frac{\partial\xi_{t}}{\partial\bm{\theta}}\frac{x^{(\xi)}_{t}-\xi_{t}}{\xi_{t}^{2}}=\bm{0},$
(10)

whose LHS, in case $\xi_{t}$ is correctly specified, has a zero expectation
even in case $\varepsilon_{t}$ is not Gamma-distributed.

##  4 The Univariate Case: Applications

In the more general multivariate case, we will use series for the
open–to–close absolute returns (rescaled by $\sqrt{\pi/2}$ so that their mean
is expressed in volatility units), for the realized kernel volatility
(obtained as the percentage annualized square root of the realized variance
with the Parzen kernel) on market indices, both taken from the OMI Realized
Library (Heber et al., 2009), and for the option–based implied volatility
indices (built for several market indices like the VIX Whaley, 2009), taken
from the site Investing.com333at the URL https://www.investing.com. In Table
A1 in the Appendix, we report the symbol we will use and the description of
the market indices and volatility indices, as well as the start and end of the
available sample period. In what follows, we will use the acronyms arVol,
rkVol, impVol, respectively, for these series.

In what follows, we limit ourselves to the univariate case, showing the
results on just the realized kernel volatility.

Table 1: $\mathsf{MEM}$: Estimation summary table.

| DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---
| est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat
$\beta^{*}_{1}$ | 0.9459 | 97.70 | 0.9140 | 77.03 | 0.9750 | 281.41 | 0.9787 | 304.29 | 0.9763 | 288.26 | 0.9587 | 124.64 | 0.9798 | 253.40 | 0.9650 | 109.32 | 0.9674 | 187.69
$\alpha_{1}$ | 0.2721 | 11.04 | 0.3726 | 14.93 | 0.1621 | 14.07 | 0.1416 | 13.66 | 0.1808 | 15.85 | 0.1273 | 6.39 | 0.1602 | 15.46 | 0.1859 | 9.44 | 0.1797 | 14.05
$\gamma_{1}$ | 0.1328 | 8.80 | 0.0684 | 4.92 | 0.1238 | 16.21 | 0.0962 | 14.23 | 0.0865 | 13.61 | 0.1273 | 10.05 | 0.0170 | 2.58 | 0.0023 | 0.18 | 0.0818 | 8.95
$\sigma$ | 0.3977 |  | 0.3568 |  | 0.3941 |  | 0.3854 |  | 0.3374 |  | 0.4453 |  | 0.3756 |  | 0.4220 |  | 0.4581 |
$R^{2}$ | 0.6758 |  | 0.6372 |  | 0.6576 |  | 0.5791 |  | 0.6738 |  | 0.5114 |  | 0.5160 |  | 0.4771 |  | 0.4761 |
$LB(5)$ | 0.1820 |  | 0.0039 |  | 0.0046 |  | 0.0071 |  | 0.0169 |  | 0.5719 |  | 0.0012 |  | 0.0373 |  | 0.0000 |
$LB(10)$ | 0.0028 |  | 0.0031 |  | 0.0009 |  | 0.0436 |  | 0.0554 |  | 0.8704 |  | 0.0006 |  | 0.1938 |  | 0.0000 |
$LB(15)$ | 0.0000 |  | 0.0027 |  | 0.0013 |  | 0.0015 |  | 0.0043 |  | 0.5858 |  | 0.0078 |  | 0.4742 |  | 0.0001 |
$LB(20)$ | 0.0001 |  | 0.0012 |  | 0.0049 |  | 0.0007 |  | 0.0075 |  | 0.7550 |  | 0.0111 |  | 0.7305 |  | 0.0005 |

Table 2: $\mathsf{SpMEM}$: Estimation summary table.

| DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---
| est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat
$\beta^{*}_{1}$ | 0.8643 | 52.25 | 0.7974 | 38.59 | 0.8837 | 102.35 | 0.8798 | 80.13 | 0.8822 | 87.58 | 0.8972 | 67.31 | 0.8005 | 30.14 | 0.8749 | 32.24 | 0.8140 | 45.40
$\alpha_{1}$ | 0.2243 | 8.98 | 0.3363 | 13.09 | 0.0996 | 8.39 | 0.1041 | 9.14 | 0.1393 | 11.41 | 0.0626 | 3.34 | 0.1354 | 10.00 | 0.1560 | 7.02 | 0.1520 | 10.32
$\gamma_{1}$ | 0.1518 | 9.96 | 0.0876 | 6.25 | 0.1587 | 19.85 | 0.1134 | 15.04 | 0.1012 | 14.79 | 0.1498 | 11.67 | 0.0289 | 3.60 | 0.0106 | 0.79 | 0.1075 | 10.31
$\sigma$ | 0.3888 |  | 0.3461 |  | 0.3811 |  | 0.3778 |  | 0.3310 |  | 0.4360 |  | 0.3620 |  | 0.4186 |  | 0.4409 |
$R^{2}$ | 0.6842 |  | 0.6493 |  | 0.6724 |  | 0.5973 |  | 0.6870 |  | 0.5275 |  | 0.5410 |  | 0.4948 |  | 0.4978 |
$LB(5)$ | 0.4788 |  | 0.1884 |  | 0.0028 |  | 0.0466 |  | 0.0932 |  | 0.5518 |  | 0.0752 |  | 0.0232 |  | 0.2124 |
$LB(10)$ | 0.0127 |  | 0.0999 |  | 0.0035 |  | 0.2043 |  | 0.2755 |  | 0.8269 |  | 0.4399 |  | 0.1625 |  | 0.0927 |
$LB(15)$ | 0.0005 |  | 0.0818 |  | 0.0031 |  | 0.0472 |  | 0.0843 |  | 0.5525 |  | 0.7020 |  | 0.4415 |  | 0.3036 |
$LB(20)$ | 0.0021 |  | 0.1647 |  | 0.0059 |  | 0.0063 |  | 0.0748 |  | 0.6230 |  | 0.4299 |  | 0.6789 |  | 0.5765 |

We report the coefficient estimation results in Table 2 for the $\mathsf{MEM}$
and in Table 2 for the $\mathsf{SpMEM}$. In both cases we adopt a $(2,1)$
specification and the Tables contain the value of the unconditional mean
$\mu$, the persistence coefficient $\beta_{1}^{*}$ defined above, followed by
$\alpha_{1}$, $\alpha_{2}$ and $\gamma_{1}$. The coefficient $\sigma^{2}$ is
the variance of the estimated residuals, while $R^{2}$ is the squared
correlation coefficient between the observed realized volatility and the
estimated conditional mean. In the lower portion of the tables, we also report
the p-values of the Ljung–Box test statistics at various lags ($5$, $10$,
$15$, and $20$) with degrees of freedom net of the number of the parameters
(following the logic of Lütkepohl, 2005).

In reference to Table 2, we notice that the lowest persistence is had for the
DJI and the IXIC, the other indices being well above $0.96$. Correspondingly,
the $\alpha_{1}$ parameter is higher for the markets with less persistence,
while for the other markets it is comprised between $0.15$ and $0.21$. With
just one exception (KS11), the $\gamma_{1}$ has the right sign and is
significant. The measures of fit are comprised in a range between $0.47$ and
$0.67$. With the exception of STOXX50E and KS11, the p–values of the Ljung–Box
statistics signal the presence of undetected dynamics from this model.

In the presence of the slow–moving component $\tau_{t}$, the results of Table
2 for the $\mathsf{SpMEM}$ show results that are different in two substantial
directions: the first is the drastic drop in $\beta_{1}^{*}$ which now is well
below $0.9$ for most markets, the second is the generalized improvement of the
Ljung–Box p–values, showing that the low–frequency component is actually
capable of capturing relevant dynamics.

Figure 5 shows for all market the estimated pattern of the various components,
estimated in the $\mathsf{SpMEM}$ specification. In it, we reproduce the
overall unconditional mean $\mu$, as a flat line; the thicker red curve is the
estimated $\mu\tau_{t}$ and the product of the three components,
$\mu\tau_{t}\xi_{t}$, i.e. the estimated conditional expectation, is
reproduced in blue. The observed series is in the background in a shade of
grey. For most series, the low frequency component absorbs a substantial
portion of the volatility behavior.

Figure 2: $\mathsf{SpMEM}$ on rkVol. The overall unconditional mean $\mu$ is a
flat line; the thicker red curve is $\mu\tau_{t}$ (with $h=6$ months) and the
conditional expectation $\mu\tau_{t}\xi_{t}$ is in blue. The observed series
is in the background in a shade of grey. For the sample periods, see Table A1.

(a) DJI

(b) FCHI

(c) GDAXI

(d) HSI

(e) IXIC

(f) KS11

(g) N225

(h) SPX

(i) STOXX50E

Just to provide further evidence on the features of the $\mathsf{SpMEM}$ model
relative to the $\mathsf{MEM}$, we report the behavior of the Autocorrelation
Function (ACF), as well as the fitted distribution of the multiplicative
residuals with superimposed some known target distributions (Gamma,
Log–Normal, Beta′, and Log–Logistic, cf. the description in the Appendix),
calibrated on the basis of the estimated residual variance.

The evidence, shown in the various panels of Figure 3, indicates that the
ACF’s do not offer any systematic pattern: there is a low number of
significant individual autocorrelations and, when that happens, there is a
prevalence of negative over positive values. As per the fitting of
distributions, the first pattern of the varioous panels in Figure 4 is that
the Gamma seems to systematically miss the peak and hence it overfits the
sides of the distribution; by contrast, a generally remarkable performance is
had by the Log–Logistic density, which, for all markets manages to reproduce
adequately the behavior of the realized kernel volatility residuals. Also we
note the almost perfect overlap between the Log–Normal and the Beta′, with an
intermediate behavior relative to the other two.

Figure 3: Residual correlogram for the rkVol estimated with $\mathsf{SpMEM}$.
For the sample periods by market, cf. Table A1.

(a) DJI

(b) IXIC

(c) SPX

(d) FCHI

(e) GDAXI

(f) STOXX50E

(g) HSI

(h) KS11

(i) N225

Figure 4: Histograms of residuals for the rkVol estimated with
$\mathsf{SpMEM}$. For the sample periods by market, cf. Table A1.

(a) DJI

(b) IXIC

(c) SPX

(d) FCHI

(e) GDAXI

(f) STOXX50E

(g) HSI

(h) KS11

(i) N225

More formally, the distribution fit of these $\mathsf{SpMEM}$ residuals can be
tested with two popular test statistics, the Anderson–Darling (AD) and Cramèr-
von Mises (CvM) tests, as done in Table 8 reporting the p–values of the tests,
using the Gamma, Log–Normal (Log–N), Beta′, Log–Logistic (Log–L) as target
densities. While the Gamma density is never supported by the data, the
performance of the Log–Logistic is very satisfactory, especially with the CvM
test which shows a more conservative behavior, failing to reject the reference
null hypothesis in 6 out of 9 cases at 5%, with an additional case (GDAXI)
added at 1%. The evidence is weaker, but still strong with the AD test. As
noted graphically, the performance of the Log–Normal and Beta′ densities is
somewhat complementary to the Log–L, with p–values higher than 1% in
correspondence to 5 cases out of 9 for both tests. In view of the GMM
estimation procedure suggested, this goodness of distributional fit tests help
in guiding the ex post choice of a parametric distribution, in cases in which
volatility at risk (in the sense of Caporin et al., 2017) evaluation is of
interest (for volatility of volatility concerns, see Corsi et al., 2008).

Table 3: Realized kernel volatility. P–values of the Anderson–Darling (AD) and
Cramèr-von Mises (CvM) tests for distribution fitting on $\mathsf{SpMEM}$
residuals. Target densities: Gamma, Log–Normal (Log–N), Beta′, Log–Logistic
(Log–L).

Test | Distr. | DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---|---
AD | Gamma | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-N | 0.0764 | 0.0270 | 0.1278 | 0.0081 | 0.0005 | 0.0000 | 0.0000 | 0.0185 | 0.0000
Beta′ | 0.1055 | 0.0369 | 0.2194 | 0.0184 | 0.0011 | 0.0000 | 0.0000 | 0.0325 | 0.0000
Log-L | 0.0496 | 0.0705 | 0.0000 | 0.0000 | 0.0218 | 0.4204 | 0.1194 | 0.0513 | 0.0524
CvM | Gamma | 0.0004 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0008 | 0.0000
Log-N | 0.1578 | 0.0482 | 0.1650 | 0.0237 | 0.0034 | 0.0001 | 0.0002 | 0.0430 | 0.0000
Beta′ | 0.2082 | 0.0628 | 0.2486 | 0.0450 | 0.0059 | 0.0002 | 0.0005 | 0.0699 | 0.0000
Log-L | 0.0708 | 0.0883 | 0.0001 | 0.0007 | 0.0340 | 0.7570 | 0.1034 | 0.0833 | 0.0580

##  5 The $\mathsf{SpvMEM}$

In view of the fact that financial time series tend to show a similar slow
moving pattern (both across measures of market activity and different
markets), we extend the model representation of Equation (1) to a
$K-$dimensional time series with non-negative components $\\{\bm{x}_{t}\\}$ as

$\bm{x}_{t}=\underbrace{\tau_{t}\;\bm{\mu}\odot\bm{\xi}_{t}}_{\bm{\mu}_{t}}\odot\bm{\varepsilon}_{t},$

where $\odot$ denotes the element–by–element product, and

  * •

$\tau_{t}$ is a scalar slow-moving component satisfying
$E\left(\tau_{t}\right)=1$, a common synthesis of low–frequency features of
the $K$ series, possessing a smooth behavior, similarly to what was done in
Section 2, in the univariate case. Also in this case, when $\tau_{t}=1,\
\forall t$, we are in the base $\mathsf{vMEM}$ of Cipollini et al. (2013). The
semi–nonparametric vector MEM $\mathsf{SpvMEM}$ was derived in Barigozzi et
al. (2014), extending the Veredas et al. (2007) approach illustrated in
Section 3 to the multivariate case: here it will be extended in ways to be
seen in Section 6;

  * •

$\bm{\mu}$ is the $K\times 1$ vector of the unconditional means;

  * •

$\bm{\xi}_{t}$ is a $K\times 1$ vector of interconnected short-run components
satisfying $E\left(\bm{\xi}_{t}\right)=\mathds{1}$. More specifically, the
$K$-dimensional counterpart of Equation (3) is

$\bm{\xi}_{t}=\left[\mathds{1}-\left(\bm{\beta}_{1}+\bm{\alpha}_{1}+\frac{\bm{\gamma}_{1}}{2}\right)\mathds{1}\right]+\bm{\beta}_{1}\bm{\xi}_{t-1}+\bm{\alpha}_{1}\bm{x}_{t-1}^{(\xi)}+\bm{\gamma}_{1}\bm{x}_{t-1}^{(\xi-)},$
(11)

where

$x_{t,i}^{(\xi)}=\frac{x_{t,i}}{\mu_{i}\tau_{t}}\qquad{}x_{t,i}^{(\xi-)}=x_{t,i}^{(\xi)}\mathbbm{1}^{-}_{t}\qquad{}i=1,\ldots,K$

and $\bm{\beta}_{1}$, $\bm{\alpha}_{1}$ and $\bm{\gamma}_{1}$ are in general
$K\times K$ matrices. In the base formulation, $\bm{\beta}_{1}$ and
$\bm{\gamma}_{1}$ are diagonal, but $\bm{\alpha}_{1}$ is allowed to be full in
order to capture interdependency structures. Paralleling (4), an equivalent
specification is

$\bm{\xi}_{t}=(\mathds{1}-\bm{\beta}_{1}^{*}\mathds{1})+\bm{\beta}_{1}^{*}\bm{\xi}_{t-1}+\bm{\alpha}_{1}\bm{v}_{t-1}+\bm{\gamma}_{1}\bm{v}^{(-)}_{t-1},$
(12)

where

$v_{t,i}=x^{(\xi)}_{t,i}-\xi_{t,i},\qquad{}v_{t,i}^{(-)}=x^{(\xi-)}_{t,i}-\frac{\xi_{t,i}}{2},\qquad{}i=1,\ldots,K$

and $\bm{\beta}_{1}^{*}=\bm{\beta}_{1}+\bm{\alpha}_{1}+\bm{\gamma}_{1}/2$;

  * •

$\bm{\varepsilon}_{t}$ is a $K\times 1$ conditionally homoskedastic unit mean
error term,

$\bm{\varepsilon}_{t}\overset{\operatorname{i.i.d.}}{\sim}d^{+}(\mathds{1},\bm{\Sigma});$

the actual form of the multivariate distribution $d^{+}$ is left unspecified,
with the proviso that it is not straighforward to find a suitable parametric
specification. As a matter of fact, as pointed out by Cipollini et al. (2017),
one could resort to a copula function–based solution (more details provided
below) or a multivariate Log–Normal, later used also in Cattivelli and Gallo
(2020).

Mirroring the univariate case, under the above conditions, the conditional
expectation of $\bm{x}_{t}$ is
$\bm{\mu}_{t}=\tau_{t}\bm{\mu}\odot\bm{\xi}_{t}$.

In the empirical application in the multivariate context below, for each
market described in Table A1, we use the three measures of volatility observed
at a daily frequency introduced before: the absolute return, the realized
kernel volatility, and a synthetic index derived from implied volatilities in
option prices. For illustration purposes, one can refer to Figure 5, where the
three series are reproduced for the Dow Jones Index: in each panel we have the
observed series $i$ in light grey, its unconditional mean $\mu_{i}$ as a flat
line, the product of $\tau_{t}\mu_{i}$ as a smooth line and then the full
conditional expectation $\tau_{t}\mu_{i}\xi_{t,i}$ in blue. The reported
estimation follows the $\mathsf{SpvMEM}$ specification.

Figure 5:  $\mathsf{SpvMEM}$ estimated on DJI absolute returns (arVol – panel
(a)), realized kernel volatility (rkVol – panel(b)), and implied volatility
(impVol – panel(c)). The overall unconditional mean $\bm{\mu}$ is a flat line;
the thicker red curve is the estimated (with $h=3$ months) $\bm{\mu}\tau_{t}$
and the estimated conditional expectation $\tau_{t}\bm{\mu}\bm{\xi}_{t}$ is
reproduced in blue. The observed series is in the background in a shade of
grey. For sample periods refer to Table A1.

(a) arVol.

(b) rkVol.

(c) impVol.

##  6 Model Inference

The starting point for the estimation procedure is to get $\bm{\mu}$ as the
average centroid of the $\\{\bm{x}_{t}\\}$ series. Then, after initialization
of all $\bm{\xi}_{t}$’s to $\mathds{1}$, the following two steps are
alternated and iterated until convergence:

  1. 1.

$\tau_{t}$ is estimated using the kernel estimator

$\widehat{\tau}_{t}=\frac{\displaystyle\sum_{s=1}^{T}\bm{x}^{(\tau)}_{t}K\left(\frac{z_{t}-z_{s}}{h}\right)}{\displaystyle\sum_{s=1}^{T}K\left(\frac{z_{t}-z_{s}}{h}\right)}$
(13)

where

$\bm{x}^{(\tau)}_{t}=\sum_{j=1}^{K}\frac{x_{j,t}}{\mu_{j}\xi_{j,t}}\frac{\sigma_{j}^{-2}}{\sum_{j=1}^{k}\sigma_{j}^{-2}},$

$\bm{\mu}$, $\bm{\xi}_{t}$ and $\sigma_{j}^{2}=\bm{\Sigma}_{j,j}$ denote
current estimates of the corresponding parameters. Notice that estimation of
$\tau_{t}$ is based on all $K$ elements of $\bm{x}_{t}$, with higher
contributions, the lower the individual residual’s variance;

  2. 2.

the parameters in $\bm{\xi}_{t}$ and in $\bm{\Sigma}$ are estimated from the
current values ${x_{t,i}^{(\xi)}=x_{t,i}/(\mu_{i}\tau_{t})}$, using the
approach detailed below.

Clearly, when $\tau_{t}\equiv 1\;\forall t$, the model is the base vMEM
introduced by Cipollini et al. (2006), so that only the second step above is
needed, and inference is obtained by estimating the $\bm{\xi}_{t}$ parameters
on the $\\{x^{(\xi)}_{t,i}=x_{t,i}/\mu_{i}\\}$ values.

To define inference on the $\bm{\xi}_{t}$ parameters and $\bm{\Sigma}$
adopting a parametric specification of the distribution $d^{+}$ of the error
term in a multivariate context has, as a matter of fact, several problems:

  1. 1.

Multivariate distributions deﬁned on the non-negative orthant are often not
sufﬁciently ﬂexible: see, for example, the discussion on the multivariate
Gamma in Cipollini et al. (2006)). The only reasonable choice seems to be the
multivariate log-Normal which, on the other hand, does not carry over to QML
characteristics similarly to what was seen for the Gamma in the univariate
case (as discussed in Section 3.2).

  2. 2.

Considering the one-to-one correspondence between the cumulative distribution
function of an absolutely continuous $K-$dimensional random variable and its
copula representation, resorting to copulas bypasses some issues: some
proposals are in Cipollini et al. (2017), where marginals for the components
are linked together using Gaussian or Student-T copulas. However, there is no
guidance for choices of a specific copula function except for convenience.
Moreover, using copulas in a vMEM context makes inference considerably
complex.

  3. 3.

It is not said that all components of the error term share the same marginal
distribution. Even though, in principle, one may be able to select the
appropriate pdf one by one, this would require a lengthy model tuning (we will
provide some evidence on our three volatility series below).

  4. 4.

As a ﬁnal remark, the dynamics of the conditional mean is almost always the
main focus of the analysis, so that a full ex ante speciﬁcation of the
distribution of $\bm{\varepsilon}_{t}$ may be of secondary importance.

To by–pass these specific issues, we advocate basing the inference procedure
within a GMM context (Cipollini et al., 2013; Cipollini and Gallo, 2019).
Hence, assuming that $\bm{\xi}_{t}$ is correctly specified and indicating with
$\bm{\theta}$ the vector of parameters entering it, we let

$\varepsilon_{t,i}=\frac{x^{(\xi)}_{t,i}}{\xi_{t,i}},$ (14)

where $x^{(\xi)}_{t,i}=x_{t,i}/(\mu_{i}\tau_{t})$. Under model assumptions,
$\bm{\varepsilon}_{t}-\mathds{1}$ is a conditionally homoskedastic martingale
difference, with conditional expectation the zero vector and conditional
variance matrix $\bm{\Sigma}$. Then, the _efficient_ GMM estimator of
$\bm{\theta}$, say $\widehat{\bm{\theta}}_{GMM}$, solves the criterion
equation

$\sum_{t=1}^{T}\bm{A}_{t}(\bm{\varepsilon}_{t}-\mathds{1})=\bm{0},$ (15)

and has asymptotic variance matrix

$\displaystyle\operatorname{Avar}(\widehat{\bm{\theta}}_{GMM})$
$\displaystyle=\left(\lim_{T\rightarrow\infty}\left[T^{-1}\sum_{t=1}^{T}E\left(\bm{A}_{t}\bm{\Sigma}\bm{A}_{t}^{\prime}\right)\right]\right)^{-1},$
(16)

where

$\bm{A}_{t}=\frac{\partial\bm{\xi}_{t}^{\prime}}{\partial\bm{\theta}}\operatorname{diag}(\bm{\xi}_{t})^{-1}\bm{\Sigma}^{-1}.$
(17)

As a consequence, a consistent estimator of the asymptotic variance matrix is

$\widehat{\operatorname{Avar}}(\widehat{\bm{\theta}}_{GMM})=\left(T^{-1}\sum_{t=1}^{T}\widehat{\bm{A}}_{t}\widehat{\bm{\Sigma}}\widehat{\bm{A}}_{t}^{\prime}\right)^{-1},$

where

$\widehat{\bm{\Sigma}}=T^{-1}\sum_{t=1}^{T}\left(\widehat{\bm{\varepsilon}}_{t}-\mathds{1}\right)\left(\widehat{\bm{\varepsilon}}_{t}-\mathds{1}\right)^{\prime}$

is the Method of Moments estimator of $\bm{\Sigma}$,
$\widehat{\bm{\varepsilon}}_{t}$ and $\widehat{\bm{A}}_{t}$ correspond to (14)
and (17), respectively, evaluated at $\widehat{\bm{\theta}}_{GMM}$. As with
the univariate case, the feasible GMM estimator is obtained upon replacement
of $\tau_{t}\mu_{i}$ with the system estimated counterparts.

###  6.1 Forecasting

Although, we will not provide empirical evidence of these models in
forecasting (for which we refer to the original contributions, notably QQQ),
forecasts from the $\mathsf{SpvMEM}$ can be obtained via a general expression
of the conditional expectation at time $t+h$ given the information available
at time $t$ which exploits the fact that forecasts of the components
$\tau_{t+h|t}$ and $\bm{\xi}_{t+h|t}$ can be obtained separately:

$E\left(\bm{x}_{t+h}|\mathcal{I}_{t}\right)=\bm{\mu}_{t+h|t}=\tau_{t+h|t}\;\bm{\mu}\odot\bm{\xi}_{t+h|t}.$

Regarding the slow-moving component, in view of the way it is estimated and
its high persistence, it is reasonable to take $\tau_{t+h|t}\equiv\tau_{t}$
(at least for small $h$; cf. Cattivelli and Gallo, 2020). As per the short-run
component, Equation (11) implies

$\bm{\xi}_{t+h|t}=\left\\{\begin{array}[]{ll}\displaystyle\left(\mathds{1}-\bm{\beta}_{1}^{*}\mathds{1}\right)+\bm{\beta}_{1}^{*}\bm{\xi}_{t}+\bm{\alpha}_{1}\bm{x}_{t}^{(\xi)}+\bm{\gamma}_{1}\bm{x}_{t}^{(\xi-)}&\text{if
}h=1\\\\[5.69054pt]
\displaystyle\left(\mathds{1}-\bm{\beta}_{1}^{*}\mathds{1}\right)+\bm{\beta}_{1}^{*}\bm{\xi}_{t+h-1|t}&\text{if
}h>1.\end{array}\right.$

Other refinements in the formulas for $\bm{\xi}_{t+h|t}$ over horizons $h>1$
are needed when further lags occur in any of the components of the RHS of
Equation (11).

##  7 The Multivariate Case: Applications

As mentioned at various stages so far, in order to illustrate the behavior of
the $\mathsf{vMEM}$ in the multivariate case, we refer to three series for
each market, absolute returns ($i=1$), realized kernel volatility ($i=2$) and
implied volatility ($i=3$). The general behavior of the series was presented,
in the case of the Dow Jones index in Figure 5. Graphs for the other markets
reproduce the same features by variable and are left for supplementary
material.

In Table 4, we report the sample means of the series across markets, to
provide some reference about the overall level $\bm{\mu}$ around which the
estimated components will move. We can notice that, in a generalized manner,
the absolute returns (which include zeros) have mean values similar to those
of the realized volatility, and that average implied volatilities are higher.

Table 4: Sample averages of the three volatility indicators. For sample
periods refer to Table A1.

| DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---
arVol | 11.38 | 12.64 | 14.80 | 15.94 | 16.84 | 14.38 | 13.97 | 10.12 | 14.97
rkVol | 10.53 | 11.61 | 13.01 | 15.38 | 16.26 | 13.73 | 13.10 | 9.34 | 13.52
impVol | 16.80 | 19.84 | 19.98 | 22.37 | 22.02 | 19.96 | 22.70 | 16.06 | 24.78

The first set of results relates to the base $\mathsf{vMEM}$ in the absence of
the low–frequency component, where we have extended Equation (11) to
accommodate a $(1,1)$ structure with diagonal $\bm{\gamma}_{1}$ and
$\bm{\beta}_{1}$ and $\bm{\alpha}_{1}$ is a full matrix: estimated parameters
are presented, by market, in Table 5, where, in view of the choices of
diagonality just mentioned, we report just the diagonal elements of
$\bm{\beta}_{1}^{*}$, namely,
$\beta_{i,i,1}^{*}=\beta_{i,i,1}+\alpha_{i,i,1}+\gamma_{i,i,1}/2$.

Table 5: $\mathsf{vMEM}$: Estimation summary table.

| DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---
| est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat
$\beta^{*}_{1,1,1}$ | 0.9558 | 142.28 | 0.9182 | 92.66 | 0.9639 | 280.12 | 0.9587 | 245.30 | 0.9638 | 245.42 | 0.9560 | 156.19 | 0.9722 | 267.54 | 0.9210 | 66.23 | 0.9567 | 198.87
$\alpha_{1,1,1}$ | -0.0022 | -0.20 | -0.0096 | -0.83 | 0.0001 | 0.01 | -0.0149 | -2.47 | -0.0069 | -1.19 | -0.0359 | -4.15 | 0.0071 | 1.32 | 0.0164 | 1.19 | 0.0140 | 1.78
$\alpha_{1,2,1}$ | 0.1425 | 5.20 | 0.1923 | 5.46 | 0.0813 | 5.44 | 0.1022 | 7.15 | 0.1244 | 8.56 | 0.0603 | 3.30 | 0.0381 | 3.25 | 0.0249 | 0.83 | 0.0946 | 5.81
$\alpha_{1,3,1}$ | 1.2051 | 11.33 | 0.9998 | 9.05 | 1.3461 | 18.62 | 0.5885 | 24.27 | 1.1816 | 14.58 | 1.0537 | 9.43 | 0.6913 | 11.90 | 1.1183 | 7.98 | 0.6517 | 11.46
$\gamma_{1,1,1}$ | 0.0229 | 2.05 | 0.0364 | 2.93 | 0.0170 | 3.07 | 0.0432 | 6.84 | 0.0130 | 2.44 | 0.0432 | 4.34 | 0.0062 | 1.47 | 0.0014 | 0.13 | 0.0190 | 3.06
$\beta^{*}_{2,2,1}$ | 0.9603 | 165.01 | 0.9242 | 110.20 | 0.9679 | 319.61 | 0.9628 | 293.96 | 0.9677 | 298.06 | 0.9570 | 162.04 | 0.9711 | 272.17 | 0.9418 | 109.71 | 0.9569 | 212.34
$\alpha_{2,1,1}$ | 0.0156 | 2.19 | 0.0203 | 3.09 | 0.0161 | 4.27 | 0.0112 | 2.88 | 0.0117 | 3.17 | -0.0025 | -0.43 | 0.0155 | 4.54 | 0.0272 | 3.90 | 0.0269 | 5.14
$\alpha_{2,2,1}$ | 0.1046 | 5.75 | 0.1424 | 7.00 | 0.0463 | 4.83 | 0.0812 | 8.42 | 0.0963 | 9.78 | 0.0364 | 2.60 | 0.0388 | 4.71 | 0.0241 | 1.46 | 0.0779 | 6.81
$\alpha_{2,3,1}$ | 0.9404 | 15.66 | 0.8860 | 15.42 | 1.0202 | 28.47 | 0.5022 | 20.64 | 0.9797 | 22.35 | 0.9395 | 15.12 | 0.6211 | 19.00 | 0.7539 | 11.98 | 0.5609 | 15.89
$\gamma_{2,2,1}$ | 0.0200 | 2.24 | 0.0045 | 0.50 | 0.0089 | 2.22 | 0.0404 | 8.53 | 0.0132 | 3.14 | 0.0331 | 4.14 | 0.0038 | 1.01 | 0.0012 | 0.17 | 0.0146 | 2.93
$\beta^{*}_{3,3,1}$ | 0.9773 | 269.96 | 0.9653 | 229.42 | 0.9809 | 519.88 | 0.9692 | 362.80 | 0.9805 | 446.38 | 0.9713 | 219.26 | 0.9822 | 408.62 | 0.9665 | 190.15 | 0.9762 | 384.95
$\alpha_{3,1,1}$ | 0.0051 | 1.45 | 0.0033 | 1.03 | 0.0023 | 1.13 | 0.0064 | 3.15 | 0.0034 | 2.28 | 0.0045 | 1.37 | 0.0031 | 1.65 | -0.0001 | -0.03 | -0.0008 | -0.32
$\alpha_{3,2,1}$ | 0.0125 | 1.42 | 0.0003 | 0.03 | 0.0032 | 0.60 | 0.0166 | 3.58 | 0.0047 | 1.15 | 0.0006 | 0.08 | -0.0086 | -1.78 | 0.0020 | 0.21 | 0.0153 | 2.82
$\alpha_{3,3,1}$ | 0.9013 | 30.92 | 0.9197 | 32.75 | 0.9293 | 51.10 | 0.8507 | 75.81 | 0.9946 | 56.25 | 0.9499 | 33.47 | 0.9769 | 58.58 | 0.9493 | 35.04 | 0.7995 | 48.02
$\gamma_{3,3,1}$ | -0.0066 | -2.05 | -0.0065 | -2.17 | -0.0055 | -2.79 | 0.0042 | 2.10 | -0.0045 | -2.65 | -0.0034 | -0.98 | -0.0039 | -2.54 | -0.0076 | -2.48 | -0.0015 | -0.82
$\sigma_{1}$ | 0.8608 |  | 0.8595 |  | 0.8426 |  | 0.8245 |  | 0.8421 |  | 0.8673 |  | 0.8189 |  | 0.8252 |  | 0.8611 |
$\sigma_{2}$ | 0.3744 |  | 0.3331 |  | 0.3683 |  | 0.3774 |  | 0.3245 |  | 0.4077 |  | 0.3507 |  | 0.3870 |  | 0.4368 |
$\sigma_{3}$ | 0.0719 |  | 0.0720 |  | 0.0740 |  | 0.0728 |  | 0.0575 |  | 0.0734 |  | 0.0557 |  | 0.0694 |  | 0.0684 |
$\rho_{1,2}$ | 0.5042 |  | 0.4546 |  | 0.5382 |  | 0.5352 |  | 0.5016 |  | 0.5627 |  | 0.5697 |  | 0.5851 |  | 0.6490 |
$\rho_{1,3}$ | 0.3187 |  | 0.3205 |  | 0.3153 |  | 0.2299 |  | 0.2401 |  | 0.2859 |  | 0.2492 |  | 0.2738 |  | 0.3331 |
$\rho_{2,3}$ | 0.4319 |  | 0.5300 |  | 0.3952 |  | 0.3004 |  | 0.3587 |  | 0.3542 |  | 0.3128 |  | 0.3344 |  | 0.4111 |
$R^{2}(1)$ | 0.3602 |  | 0.2372 |  | 0.3380 |  | 0.2346 |  | 0.2613 |  | 0.2150 |  | 0.2251 |  | 0.2633 |  | 0.2408 |
$R^{2}(2)$ | 0.7087 |  | 0.6729 |  | 0.6969 |  | 0.5983 |  | 0.6927 |  | 0.5505 |  | 0.5711 |  | 0.5441 |  | 0.5008 |
$R^{2}(3)$ | 0.9596 |  | 0.9432 |  | 0.9590 |  | 0.9597 |  | 0.9718 |  | 0.9425 |  | 0.9718 |  | 0.9543 |  | 0.9547 |
$LB(5)$ | 0.0040 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.0014 |  | 0.0000 |  | 0.0019 |  | 0.0000 |
$LB(10)$ | 0.0028 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.1022 |  | 0.0000 |  | 0.0158 |  | 0.0000 |
$LB(15)$ | 0.0246 |  | 0.0003 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.3265 |  | 0.0000 |  | 0.0121 |  | 0.0000 |
$LB(20)$ | 0.0750 |  | 0.0003 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.3307 |  | 0.0000 |  | 0.0524 |  | 0.0000 |

The first element to comment on are the parameters on the diagonal of
$\bm{\beta}_{1}^{*}$: the degree of persistence is very high, and it increases
moving from the absolute returns to the realized volatility, to the implied
volatility which has the highest, in excess of $0.97$.

As far as the relevance of the off-diagonal elements of $\bm{\alpha}_{1}$, we
can point to the generalized significance of the parameters that feed into the
absolute returns from the other measures (the $\alpha_{1,j,1}$’s, $j=2,3$, the
only exception is for KS11 for realized volatility); the same is true for the
$\alpha_{2,j,1}$’s, $j=1,3$, with an exception for the coefficient from lagged
absolute returns for STOXX50E; by contrast, the coefficients
$\alpha_{3,j,1}$’s, $j=1,2$ are generally not significant with some rare
exceptions (both for FCHI and the coefficient from lagged realized volatility
for N225). These results point to the autonomous, so–to–speak, dynamics of the
implied volatility, but also (and foremost) to a full interdependence in what
concerns the other two measures. As a matter of fact, if one were to extend
the logic of the HEAVY model by Shephard and Sheppard (2010) to this
framework, the question would be whether absolute returns can be considered
irrelevant in their own dynamics (while keeping the lagged dependence on the
other measures) and also in the dynamics of the other two measures. We address
this question formally by jointly testing the significance of the four
parameters $\alpha_{1,1,1}$, $\gamma_{1,1,1}$, $\alpha_{2,1,1}$,
$\alpha_{3,1,1}$, obtaining the results in the first panel of Table 6. For the
$\mathsf{vMEM}$ these implied ‘HEAVY’ restrictions are strongly rejected, with
the DJI providing significant evidence at 5% but not at 1%.

Table 6: Joint hypothesis testing on the ‘HEAVY’ zero restriction of
$\alpha_{1,1,1}$, $\gamma_{1,1,1}$, $\alpha_{2,1,1}$, $\alpha_{3,1,1}$,
corresponding to the irrelevance of absolute returns in their own dynamics and
in the dynamics of the other two series. Wald statistics and p–values reported
for $\mathsf{vMEM}$ and for $\mathsf{SpvMEM}$.

Model |  | DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---|---
$\mathsf{vMEM}$ | statistic | 12.3104 | 25.1204 | 51.7463 | 61.3916 | 39.4824 | 31.1643 | 28.7689 | 23.0047 | 42.0453
p-value | 0.0064 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
$\mathsf{SpvMEM}$ | statistic | 16.1555 | 32.5538 | 83.4393 | 57.5994 | 76.6602 | 46.0709 | 56.6530 | 31.1979 | 87.8567
p-value | 0.0011 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000

The evidence on asymmetric effects is also mixed: for the absolute returns
almost all parameters are significant (except HSI and KS11); for the realized
volatility we have substantially the same results as in the univariate case
(the only insignificant cases are IXIC, HSI and KS11); finally, for the
implied volatility only one (FCHI) is significantly positive, while 6 of 9 are
significantly negative, 2 are not significant.

The overall behavior of the estimated components can be graphically
illustrated in Figure 6 with reference to the DJI as an example. In it, we
have reported the unconditional means $\mu_{i}$ as a flat line, and the
conditional expectations of the three volatility measures, keeping the
original series in the background with a shade of grey.

Figure 6:  $\mathsf{vMEM}$ estimated on DJI absolute returns (arVol – panel
(a)), realized kernel volatility (rkVol – panel(b)), and implied volatility
(impVol – panel(c)). The individual unconditional mean ${\mu}_{i}$ is a flat
line; the estimated conditional expectation $\mu_{i}{\xi}_{i,t}$ is reproduced
in blue. The observed series is in the background in a shade of grey. For
sample periods, refer to Table A1.

(a) arVol.

(b) rkVol.

(c) impVol.

The estimated covariance matrix for the residuals $\bm{\Sigma}$ is represented
using standard deviations $\sigma_{i}\equiv{\Sigma}_{i,i}$, ($i=1,2,3$), which
shows a larger uncertainty surrounding absolute returns, with realized
volatility in the middle position and a very low value for the implied
volatilities. Off-diagonal, the derived estimated correlations $\rho_{i,j}$
($i>j$) show a higher correlation between the residuals of the absolute
returns and those of the realized volatility, followed by the correlations
between realized volatility and implied volatilities and last, but in a range
between $0.24$ and $0.33$ for the remaining pair.

As in the univariate case, we report under the label $R^{2}$, the squared
correlation coefficient between the observed values and the estimated
conditional expectations for each variable. As one would expect, the fit is
lower for absolute returns ($R^{2}(1)$ generally in the $0.20$’s), for the
realized volatility the performance is higher, but somewhat mixed ($R^{2}(2)$
between $0.50$ and $0.71$); finally for the implied volatility we get a very
close fit ($R^{2}(3)$ in excess of $0.94$).

The autocorrelation results show some generalized presence of residual
dynamics with very low p-values of the Ljung–Box statistics.

The second set of results relates to the $\mathsf{SpvMEM}$ model, presented in
Section 5 the estimation results of which are reported in Table 7 with the
same choice of specification before: $\bm{\gamma}_{1}$ and $\bm{\beta}_{1}$
are both diagonal matrices and $\bm{\alpha}_{1}$ is a full matrix.

By and large, the most striking result is the generalized reduction in the
persistence parameters $\bm{\beta}^{*}$, in some cases even a drastic one,
especially for arVol and rkVol which go below $0.9$. This is to be expected,
since the $\tau_{t}$ term in the $\mathsf{SpvMEM}$ characterizes part of the
persistence in the series as a low–frequency component.

While the analysis is carried out by market on the three volatility measures,
it is nevertheless interesting to compare the behavior of the estimated
$\tau_{t}$ across markets, as done in Figure 7, where we have reproduced the
various series, with a color reference for ease of recognition of the
geographic area the market is in, black for the US, red for Europe and blue
for East Asia (we also superimpose vertical bars for the start of the sample
for the corresponding ticker). The graphical appraisal allows us to pinpoint
that this common volatility component has a strikingly similar behavior, but
there is also a great deal of different behavior across markets with bursts of
volatility affecting some areas at different rates. For example, commonality
in the reaction is seen when the COVID-19 health emergency erupted and when
the Great Financial Crisis had its worse episode in September 2008; a similar
common episode coincides with 9/11. By the same token, the acceleration of the
consequences of the dot com bubble burst were felt especially in Europe in the
second half of 2002 (with the DAX and CAC40 indices hitting historical lows),
the Euro area sovereign debt crisis spread out to all areas, with a sharp and
progressive reduction in market fears induced by the July 2012 Whatever it
takes speech by Mario Draghi. These results point out to a warning that
attempts to model a single common low–frequency component across these market
volatilities by an $\mathsf{SpvMEM}$ would probably be frustrated by the
presence of market–specific, slow–moving idiosyncratic behavior.

Table 7: $\mathsf{SpvMEM}$: Estimation summary table.

| DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---
| est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat | est | zstat
$\beta^{*}_{1,1,1}$ | 0.9324 | 96.43 | 0.8424 | 44.84 | 0.9368 | 167.80 | 0.8474 | 51.25 | 0.9106 | 103.13 | 0.9307 | 95.30 | 0.8734 | 41.86 | 0.8353 | 23.35 | 0.8717 | 58.33
$\alpha_{1,1,1}$ | -0.0065 | -0.49 | -0.0270 | -1.86 | -0.0214 | -2.99 | -0.0419 | -4.15 | -0.0350 | -4.62 | -0.0537 | -5.15 | -0.0238 | -2.59 | 0.0288 | 1.54 | -0.0082 | -0.77
$\alpha_{1,2,1}$ | 0.1795 | 5.51 | 0.2340 | 5.31 | 0.1102 | 6.24 | 0.1587 | 6.68 | 0.1527 | 7.48 | 0.0856 | 3.66 | 0.0432 | 2.15 | -0.0190 | -0.48 | 0.1277 | 5.35
$\alpha_{1,3,1}$ | 1.0742 | 9.22 | 0.7873 | 5.96 | 1.0448 | 13.70 | 0.5222 | 13.17 | 0.9163 | 9.91 | 0.6964 | 6.53 | 0.5428 | 6.38 | 0.9552 | 5.92 | 0.3938 | 5.55
$\gamma_{1,1,1}$ | 0.0369 | 2.93 | 0.0612 | 3.56 | 0.0465 | 6.54 | 0.0675 | 6.47 | 0.0419 | 5.39 | 0.0699 | 5.81 | 0.0324 | 3.75 | 0.0009 | 0.05 | 0.0549 | 5.54
$\beta^{*}_{2,2,1}$ | 0.9369 | 114.88 | 0.8315 | 55.57 | 0.9409 | 200.29 | 0.8652 | 84.52 | 0.9134 | 137.19 | 0.9239 | 99.50 | 0.8484 | 63.42 | 0.8711 | 48.37 | 0.8602 | 77.88
$\alpha_{2,1,1}$ | 0.0179 | 2.30 | 0.0249 | 3.36 | 0.0157 | 3.76 | 0.0102 | 2.06 | 0.0132 | 3.11 | -0.0061 | -0.92 | 0.0222 | 4.59 | 0.0412 | 4.96 | 0.0402 | 6.18
$\alpha_{2,2,1}$ | 0.1243 | 6.09 | 0.1798 | 7.52 | 0.0576 | 5.30 | 0.0907 | 7.21 | 0.1165 | 9.55 | 0.0437 | 2.62 | 0.0582 | 4.59 | 0.0072 | 0.36 | 0.0955 | 6.30
$\alpha_{2,3,1}$ | 0.8819 | 13.43 | 0.8018 | 12.20 | 0.9443 | 24.00 | 0.4381 | 13.61 | 0.8066 | 16.18 | 0.7747 | 11.80 | 0.5685 | 12.85 | 0.6636 | 9.53 | 0.3878 | 8.90
$\gamma_{2,2,1}$ | 0.0310 | 2.94 | 0.0155 | 1.38 | 0.0229 | 4.17 | 0.0653 | 10.04 | 0.0327 | 5.74 | 0.0546 | 5.60 | 0.0143 | 2.33 | 0.0031 | 0.34 | 0.0500 | 6.61
$\beta^{*}_{3,3,1}$ | 0.9662 | 161.30 | 0.9046 | 100.94 | 0.9754 | 308.19 | 0.8758 | 127.73 | 0.9457 | 194.48 | 0.9461 | 122.43 | 0.9190 | 147.93 | 0.9344 | 95.21 | 0.9111 | 145.11
$\alpha_{3,1,1}$ | 0.0048 | 1.38 | 0.0033 | 1.11 | 0.0016 | 0.79 | 0.0055 | 2.88 | 0.0036 | 2.43 | 0.0038 | 1.16 | 0.0033 | 1.77 | -0.0008 | -0.21 | -0.0008 | -0.32
$\alpha_{3,2,1}$ | 0.0115 | 1.32 | 0.0028 | 0.30 | 0.0007 | 0.14 | 0.0101 | 2.30 | 0.0019 | 0.48 | -0.0013 | -0.18 | -0.0124 | -2.63 | -0.0032 | -0.35 | 0.0160 | 2.99
$\alpha_{3,3,1}$ | 0.8783 | 29.88 | 0.8564 | 30.88 | 0.9173 | 49.83 | 0.8177 | 71.21 | 0.9613 | 54.36 | 0.9267 | 32.41 | 0.9298 | 56.29 | 0.9187 | 34.09 | 0.7496 | 45.20
$\gamma_{3,3,1}$ | -0.0046 | -1.37 | -0.0040 | -1.34 | -0.0067 | -3.12 | 0.0047 | 2.45 | -0.0048 | -2.76 | -0.0044 | -1.21 | -0.0037 | -2.39 | -0.0079 | -2.53 | 0.0002 | 0.13
$\sigma_{1}$ | 0.8631 |  | 0.8511 |  | 0.8450 |  | 0.8304 |  | 0.8423 |  | 0.8699 |  | 0.8194 |  | 0.8304 |  | 0.8644 |
$\sigma_{2}$ | 0.3748 |  | 0.3306 |  | 0.3682 |  | 0.3745 |  | 0.3254 |  | 0.4139 |  | 0.3494 |  | 0.3873 |  | 0.4344 |
$\sigma_{3}$ | 0.0710 |  | 0.0684 |  | 0.0734 |  | 0.0683 |  | 0.0565 |  | 0.0726 |  | 0.0542 |  | 0.0682 |  | 0.0664 |
$\rho_{1,2}$ | 0.5034 |  | 0.4499 |  | 0.5343 |  | 0.5238 |  | 0.4987 |  | 0.5707 |  | 0.5653 |  | 0.5844 |  | 0.6477 |
$\rho_{1,3}$ | 0.3123 |  | 0.3149 |  | 0.3048 |  | 0.1939 |  | 0.2316 |  | 0.2675 |  | 0.2398 |  | 0.2601 |  | 0.3196 |
$\rho_{2,3}$ | 0.4231 |  | 0.5311 |  | 0.3810 |  | 0.2678 |  | 0.3438 |  | 0.3341 |  | 0.2938 |  | 0.3205 |  | 0.3895 |
$R^{2}(1)$ | 0.3610 |  | 0.2427 |  | 0.3457 |  | 0.2413 |  | 0.2676 |  | 0.2191 |  | 0.2269 |  | 0.2526 |  | 0.2437 |
$R^{2}(2)$ | 0.7135 |  | 0.6761 |  | 0.7019 |  | 0.6096 |  | 0.6986 |  | 0.5537 |  | 0.5758 |  | 0.5500 |  | 0.5099 |
$R^{2}(3)$ | 0.9608 |  | 0.9477 |  | 0.9599 |  | 0.9622 |  | 0.9728 |  | 0.9441 |  | 0.9733 |  | 0.9559 |  | 0.9570 |
$LB(5)$ | 0.0115 |  | 0.0003 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.0033 |  | 0.0000 |  | 0.0105 |  | 0.0000 |
$LB(10)$ | 0.0051 |  | 0.0006 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.1062 |  | 0.0000 |  | 0.0590 |  | 0.0000 |
$LB(15)$ | 0.0100 |  | 0.0002 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.2393 |  | 0.0000 |  | 0.0736 |  | 0.0003 |
$LB(20)$ | 0.0391 |  | 0.0013 |  | 0.0000 |  | 0.0000 |  | 0.0000 |  | 0.2361 |  | 0.0000 |  | 0.2317 |  | 0.0009 |

Figure 7: Estimated slow moving component $\tau_{t}$ for all tickers. Vertical
bars mark the start of the sample by ticker. Colors represent geographical
areas; black for the US, red for Europe and blue for East Asia.  Figure 8:
$\mathsf{SpvMEM}$ on DJI ($h=3$ months).

(a) Arvol, acf

(b) Arvol, hist

(c) rkVol, acf

(d) rkVol, hist

(e) impVol, acf

(f) impVol, hist

Table 8: Absolute returns (arVol), Realized kernel volatility (rkVol) and
Implied volatility (impVol). P–values of the Anderson–Darling (AD) and Cramèr-
von Mises (CvM) tests for distribution fitting on $\mathsf{SpMEM}$ residuals.
Target densities: Gamma, Log–Normal (Log–N), Beta′, Log–Logistic (Log–L).

Ind. | Test | Distr. | DJI | IXIC | SPX | FCHI | GDAXI | STOXX50E | HSI | KS11 | N225
---|---|---|---|---|---|---|---|---|---|---|---
arVol | AD | Gamma | 0.0000 | 0.0835 | 0.0000 | 0.0000 | 0.0000 | 0.0117 | 0.0000 | 0.0000 | 0.0001
Log-N | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Beta′ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-L | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
CvM | Gamma | 0.0077 | 0.1368 | 0.0000 | 0.0000 | 0.0009 | 0.0120 | 0.0001 | 0.0356 | 0.0196
Log-N | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Beta′ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-L | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
rkVol | AD | Gamma | 0.0003 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0043 | 0.0000
Log-N | 0.3403 | 0.1577 | 0.4096 | 0.0113 | 0.0038 | 0.0001 | 0.0052 | 0.1816 | 0.0000
Beta′ | 0.4102 | 0.1964 | 0.6147 | 0.0248 | 0.0074 | 0.0003 | 0.0106 | 0.2603 | 0.0000
Log-L | 0.0123 | 0.0204 | 0.0000 | 0.0000 | 0.0132 | 0.4805 | 0.0066 | 0.0066 | 0.0240
CvM | Gamma | 0.0033 | 0.0010 | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0170 | 0.0000
Log-N | 0.4764 | 0.2425 | 0.3540 | 0.0315 | 0.0137 | 0.0010 | 0.0204 | 0.2419 | 0.0000
Beta′ | 0.5704 | 0.2878 | 0.5225 | 0.0583 | 0.0235 | 0.0022 | 0.0353 | 0.3227 | 0.0001
Log-L | 0.0260 | 0.0286 | 0.0000 | 0.0007 | 0.0243 | 0.6014 | 0.0099 | 0.0218 | 0.0267
impVol | AD | Gamma | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-N | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Beta′ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-L | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
CvM | Gamma | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-N | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Beta′ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Log-L | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000

##  8 Concluding Remarks

In reviewing theoretical and empirical aspects of the Multiplicative Error
Model, the paper has two main sections, devoted to the univariate,
respectively, multivariate versions of the $\mathsf{MEM}$. Its contribution
can be synthesized as follows: we have proposed a modification of the standard
specification of a $\mathsf{MEM}$ to highlight the composition of the
conditional expectation of a positive variable of interest, in a way inspired
by Veredas et al. (2007) ($\mathsf{SpMEM}$). From a theoretical point of view
we have discussed how GMM applies to this model and the properties of the
estimator, showing an equivalence with the Quasi Maximum Likelihood estimator
under the Gamma assumption. In this respect, we have proceeded to an original
set of applications in the univariate case on series of realized volatility
from several markets, presenting estimation results both in tables and in the
graphical decomposition achieved among the various components. Moreover, we
have discussed the properties of estimated residuals graphically (correlogram
and histograms), and more formally with distribution fitting tests. In the
multivariate context, we have introduced the extension to the
$\mathsf{SpvMEM}$model by Barigozzi et al. (2014) to accommodate
interdependence in the multivariate framework, showing, again, our preference
for the GMM estimation strategy. The applications in the multivariate case
relate to a trivariate model with absolute returns, realized volatility and
implied volatility from option–based indices. Estimation results both for the
$\mathsf{vMEM}$ and the $\mathsf{SpvMEM}$are complemented by some tests for
HEAVY–type dynamics (Shephard and Sheppard, 2010) which show that the impact
of absolute returns on realized volatility is relevant. Once again, attention
is paid to the properties of estimated residuals, showing how substantially
different their profile is in terms of underlying distributions, to the point
that while a Gamma often fits the behavior for absolute returns, and a Beta′
or a Log–Normal that of realized volatility, none of the distributions
considered is suitable for implied volatility.

This paper marks 20 years of developments within the area of the
Multiplicative Error Models, starting from the seminal paper of Engle (2002).
Overall, the conceptual advantages of using a $\mathsf{MEM}$ lie in the fact
that positive processes can be modeled directly without a log–transformation,
and that GMM inference is widely available avoiding the need for a parametric
specification of the error distribution. Residual diagnostics typically
improves relative to non–multiplicative specifications, especially in what
concerns autocorrelation. In forecasting, the various versions of the
$\mathsf{MEM}$ have proved successful, especially in reproducing the dynamic
behavior of realized volatility (Cipollini et al., 2021, cf., for example),
but also of intradaily volumes (Brownlees et al., 2011) for Volume Weighted
Average Price (VWAP) applications (for volumes and number of trades, see
Cipollini et al., 2017).

When the number of variables grows, the $\mathsf{vMEM}$ is somewhat affected
by a curse of dimensionality problem. In the context of a factor approach, one
should explore the possibility that several variables (e.g. realized
volatilities on a large number of assets) share some common component beyond
the single nonparametric one suggested by Barigozzi et al. (2014). On the
other hand, the matrix $\bm{\alpha}_{1}$ which we envisaged to be full both in
theory and in the empirical applications may contain some zeros, and
estimating them affects the precision of the estimates. A first attempt was
made by Cattivelli and Gallo (2020) with an adaptive Lasso approach which
imposes some penalization on making parameters different from zero at the
estimation stage. While promising, there are further refinements in the
penalized ML approach, and Lasso is not the only tool available among
regularization techniques (cf. ridge–type and/or elastic-net): many of these
techniques were successfully introduced in the Vector Autoregression (for
recent overviews of the approaches and of the results obtained, cf. Nicholson
et al., 2017, 2020), and, given the similarity of the models could be explored
in the context of the $\mathsf{vMEM}$ world.

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Appendices

##  Appendix A Data Details

Table A1: Data reference for realized kernel volatility and volatility
indices.

Symbol | OMI Realized Library (arVol, rkVol) | Investing.com (impVol) | Start | End
---|---|---|---|---
DJI | Dow Jones Industrial Average | VXD (DJIA Volatility) | Aug 2, 2013 | Apr 13, 2021
IXIC | Nasdaq 100 | VXN (CBOE NASDAQ100 Volatility) | Oct 6, 2011 | Apr 13, 2021
SPX | S&P 500 | VIX (CBOE Volatility index) | Jan 3, 2000 | Apr 13, 2021
FCHI | CAC 40 | VCAC (CAC40 VIX) | Jan 4, 2000 | Dec 31, 2020
GDAXI | DAX | V1XI (DAX New Volatility) | May 11, 2001 | Apr 13, 2021
STOXX50E | EURO STOXX 50 | V2TX (STOXX 50 Volatility VSTOXX) | Dec 28, 2012 | Apr 13, 2021
HSI | HANG SENG | VHSI (HSI Volatility) | Jan 3, 2001 | Apr 13, 2021
KS11 | Korea Composite Stock Price Index (KOSPI) | KSVKOSPI (KOSPI Volatility) | Aug 6, 2013 | Apr 13, 2021
N225 | Nikkei 225 | JNIV (Nikkei Volatility) | Nov 13, 2000 | Apr 13, 2021

##  Appendix B Univariate Distributions of the Error Term

One of the maintained assumptions of the $\mathsf{MEM}$ (Section 2) is that
the multiplicative error term $\varepsilon_{t}$ has non-negative support and
unit mean. In this appendix we list four possible parametric specifications of
the $\varepsilon_{t}$ distribution, reporting the corresponding probability
density functions and the parameter constraints needed so as the mean and the
variance are one and $\sigma^{2}$, respectively. In the application, density
estimates are retrieved from the GMM estimate of the $\sigma^{2}$.

###  B.1 Gamma

$f_{\varepsilon}(\varepsilon)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\varepsilon^{\alpha-1}\exp(-\beta\varepsilon)$

where $\beta=\alpha=\sigma^{-2}$.

###  B.2 Log-Normal

$f_{\varepsilon}(\varepsilon)=\frac{1}{\varepsilon\sqrt{2\pi
V}}\exp\left(-\frac{1}{2}\frac{\left(\ln\varepsilon-m\right)^{2}}{V}\right)$

where $V=\ln\left(\sigma^{2}+1\right)$ and $m=-V/2$.

###  B.3 Beta Prime

$f_{\varepsilon}(\varepsilon)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\varepsilon^{\alpha-1}\left(1+\varepsilon\right)^{-(\alpha+\beta)}$

where $\beta=2+2\sigma^{-2}$ and $\alpha=\beta-1$.

###  B.4 Log-Logistic

$f_{\varepsilon}(\varepsilon)=\frac{\beta}{\alpha}\left(\frac{\varepsilon}{\alpha}\right)^{\beta-1}\left[1+\left(\frac{\varepsilon}{\alpha}\right)^{\beta}\right]^{-2}$

where $\beta$ satisfies $\sigma^{2}=\tan(\pi/\beta)/(2\pi/\beta)-1$ and
$\alpha=\sin(\pi/\beta)/(\pi/\beta)$.