††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Hiding neutrinoless double beta decay in the minimal seesaw mechanism

Takehiko Asaka  Department of Physics, Niigata University, Niigata 950-2181,
Japan  Hiroyuki Ishida  KEK Theory Center, IPNS, Tsukuba, Ibaraki 305-0801,
Japan  Kazuki Tanaka  Graduate School of Science and Technology, Niigata
University, Niigata 950-2181, Japan

###### Abstract

We present a possibility that the neutrinoless double beta decay can be hidden
in the minimal seesaw mechanism where the standard model is extended by two
right-handed neutrinos which have a hierarchical mass structure. In this
framework, the lepton number is violated due to the massive Majorana
neutrinos. Especially, we investigate the case that the heavier right-handed
neutrino is sufficiently heavy to decouple from the decay while the lighter
one is lighter enough than the typical Fermi-momentum scale of nuclei and
gives a sizable contribution to the decay. Under the specific condition on
mixing elements, the lighter right-handed neutrino can give a significant
destructive contribution which suppresses or even hides to the effective mass
of the neutrinoless double beta decay. In this case, the flavor structure of
the mixing element of the lighter right-handed neutrino with ordinary
neutrinos is predicted depending on the Majorana $CP$ violating phase of
active neutrinos.

††preprint: KEK-TH-2253

##  I Introduction

Neutrino oscillation experiments have been developed so far and have provided
many properties of three active neutrinos, including mass square differences,
mixing angles, and even $CP$ violating phase which has been getting to be
revealed recently. Unfortunately, all the fascinated neutrino oscillation
experiments can give no hint of whether neutrinos are Dirac or Majorana
fermions, which is one of the most interesting missing pieces of neutrinos.
One promising possibility to attack this issue is finding the phenomenon of
the neutrinoless double beta ($0\nu\beta\beta$) decay. (See as a theoretical
review, for instance Ref. Pas:2015eia .) It violates the lepton number by two
units and shows a clear signal for physics beyond the Standard Model (SM). The
decay can be mediated by massive Majorana neutrinos and the rate is
characterized by the so-called effective mass $m_{\rm eff}$ of Majorana
neutrinos, which has been constrained by various $0\nu\beta\beta$ decay
experiments until today Arnaboldi:2002te ; Umehara:2008ru ; Barabash:2010bd ;
Gando:2012zm ; Agostini:2013mzu ; Albert:2014awa ; Andringa:2015tza ;
Arnold:2015wpy ; Arnold:2016ezh ; KamLAND-Zen:2016pfg ; Elliott:2016ble ;
Arnold:2016qyg ; Arnold:2016bed ; Agostini:2017iyd ; Aalseth:2017btx ;
Albert:2017owj ; Alduino:2017ehq ; Agostini:2018tnm ; Azzolini:2018dyb ;
Arnold:2018tmo ; Adams:2019jhp ; Alvis:2019sil ; Agostini:2019hzm ;
Azzolini:2019tta ; Alenkov:2019jis ; Anton:2019wmi .

The seesaw mechanism Minkowski:1977sc ; Yanagida:1979as ; Yanagida:1980xy ;
Ramond:1979 ; GellMann:1980vs ; Glashow:1979 ; Mohapatra:1979ia  by
introducing right-handed neutrinos with Majorana masses is the one of the most
attractive scenarios for explaining the origin and the observed smallness of
neutrino masses. In this case the lepton number is violated by the Majorana
masses and the $0\nu\beta\beta$ decay is possible to occur. The effective mass
is expressed in terms of neutrino masses, mixing angles, and $CP$ violating
phases. In the effective SM with three massive neutrinos, there is a
possibility where the effective mass is highly suppressed or even vanishes by
tuning the lightest active neutrino mass (see, e.g., Pas:2015eia ).

When right-handed neutrinos are much lighter than the unification scale $\sim
10^{16}~{}{\rm GeV}$, or even lighter than the weak scale $\sim 100~{}{\rm
GeV}$, they can give a sizable contribution to $m_{\rm eff}$ in addition to
active neutrinos’ one. In such cases, the masses and mixing elements of right-
handed neutrinos must be chosen appropriately without conflicting with the
experimental limits on $m_{\rm eff}$. When all right-handed neutrinos are
lighter than a typical scale of Fermi momentum of a nucleus $\Lambda_{\beta}$
($\sim\mathcal{O}(100)$ MeV), the contributions of active neutrinos and
heavier ones exactly cancel out each other due to the intrinsic property of
the seesaw mechanism Blennow:2010th . Further, it is shown that, when right-
handed neutrinos are degenerate, $m_{\rm eff}$ becomes smaller than the one
solely from active neutrinos Asaka:2011pb . Although such right-handed
neutrinos are attractive to realize the baryogenesis via the oscillation
mechanism Akhmedov:1998qx ; Asaka:2005pn , there is no concrete reason to
constrain ourselves to keep the degeneracy in general.

In this paper, we present another possibility to suppress the $0\nu\beta\beta$
decay by hierarchical right-handed neutrinos. As the simplest example, we
consider the extended SM with two right-handed neutrinos where one is
sufficiently heavier than $\Lambda_{\beta}$ to decouple from the system while
the other is lighter than $\Lambda_{\beta}$ giving a destructive contribution
to the decay rate. It is shown that $m_{\rm eff}=0$ is possible due to the
exact cancellation of the contributions between active neutrinos and the
lighter right-handed neutrino if the mixing elements are chosen to be specific
values. We then discuss the impacts of this cancellation conditions,
especially, on the Majorana $CP$ violating phase and the mass hierarchy of
active neutrinos.

##  II Seesaw model with two right-handed neutrinos

We consider here the simplest extension of the SM to explain the observed
neutrino masses by adding two right-handed neutrinos#1#1#1 The extension to
the case with three right-handed neutrinos is straightforwardly possible.
However, since the number of parameters is increased, the impacts discussed
below would be blurred. This issue is beyond our scope. $\nu_{RI}$ ($I=1,2$)

$\displaystyle{\cal
L}_{\nu}=i\overline{\nu_{RI}}\gamma^{\mu}\partial_{\mu}\nu_{RI}-\left(F_{\alpha
I}\,\overline{\ell_{\alpha}}\,\Phi\,\nu_{RI}+\frac{M_{I}}{2}\,\overline{\nu_{RI}^{c}}\,\nu_{RI}+H.c.\right)\,,$
(1)

where $\Phi$ and $\ell_{\alpha}$ ($\alpha=e,\mu,\tau$) are the Higgs and
lepton doublets of the weak SU(2). Neutrino Yukawa coupling constants and
Majorana masses of right-handed neutrinos are denoted by $F_{\alpha I}$ and
$M_{I}$, respectively. Here and hereafter, we work in the basis where the
Yukawa coupling matrix of charged leptons and the Majorana mass matrix of
right-handed neutrinos are diagonal.

The electroweak symmetry breaking gives the neutrino masses of Dirac type
$[M_{D}]_{\alpha I}=F_{\alpha I}\langle\Phi\rangle$ in addition to the
Majorana type $M_{I}$. When $|[M_{D}]_{\alpha I}|\ll M_{I}$, the seesaw
mechanism for neutrino masses is realized. In addition to massive active
neutrinos $\nu_{i}$ ($i=1,2,3$) there are heavy neutrinos $N_{I}$ with masses
$M_{I}$ which almost correspond to right-handed neutrino states (we simply
call them as right-handed neutrinos from now on). These states take part in
weak gauge interactions through the mixing as

$\displaystyle\nu_{L\alpha}=U_{\alpha i}\,\nu_{i}+\Theta_{\alpha
I}N_{I}^{c}\,,$ (2)

where $U_{\alpha i}$ is the mixing matrix of active neutrinos Pontecorvo:1958
; Maki:1962mu  while the mixing elements of $N_{I}$ are given by
$\Theta_{\alpha I}=[M_{D}]_{\alpha I}\,M_{I}^{-1}$.

Based on the parametrization proposed by Casas and Ibarra Casas:2001sr ;
Abada:2006ea , the Yukawa couplings are written as

$\displaystyle
F=\frac{i}{{\langle\Phi\rangle}}\,U\,D_{\nu}^{1/2}\,\Omega\,D_{N}^{1/2}\,.$
(3)

Here $D_{\nu}=\mbox{diag}(m_{1},m_{2},m_{3})$ is the diagonal mass matrix of
active neutrinos. In the considering case, the lightest active neutrino is
massless, and then $m_{3}>m_{2}>m_{1}=0$ for the normal hierarchy (NH) case
and $m_{2}>m_{1}>m_{3}=0$ for the inverted hierarchy (IH) case.
$D_{N}=\mbox{diag}(M_{1},M_{2})$ is the mass matrix of right-handed neutrinos.
The mixing matrix of active neutrinos is expressed as

$\displaystyle U=\left(\begin{array}[]{c c
c}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-c_{23}s_{12}-s_{23}c_{12}s_{13}e^{i\delta}&c_{23}c_{12}-s_{23}s_{12}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{23}s_{12}-c_{23}c_{12}s_{13}e^{i\delta}&-s_{23}c_{12}-c_{23}s_{12}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)\times\mbox{diag}(1\,,~{}e^{i\eta}\,,~{}1)\,,$
(7)

with $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. $\delta$ and
$\eta$ are the Dirac and Majorana $CP$ violating phases, respectively. The
$3\times 2$ matrix $\Omega$ can be expressed as

$\displaystyle\Omega=\left\\{\begin{array}[]{l l}\left(\begin{array}[]{c
c}0&0\\\ c_{\omega}&-s_{\omega}\\\ \xi s_{\omega}&\xi
c_{\omega}\end{array}\right)&\hskip 5.69054pt\mbox{for the NH
case}\\\\[25.83325pt] \left(\begin{array}[]{c c}c_{\omega}&-s_{\omega}\\\ \xi
s_{\omega}&\xi c_{\omega}\\\ 0&0\end{array}\right)&\hskip 5.69054pt\mbox{for
the IH case}\end{array}\right.\,,$ (16)

where $s_{\omega}=\sin\omega$ and $c_{\omega}=\cos\omega$, respectively.
$\xi=\pm 1$ is sign parameter and $\omega$ is a complex parameter, i.e.,
$\omega=\omega_{r}+i\omega_{i}$. Further, we introduce

$\displaystyle X_{\omega}=\exp[\omega_{i}]\,,$ (17)

since it represents the overall strength of the Yukawa couplings (see, e.g.,
the discussion in Ref. Asaka:2011pb ). In practice, the Yukawa couplings scale
as $F\propto X_{\omega}$ or $X_{\omega}^{-1}$ for $X_{\omega}\gg 1$ or $\ll
1$.

Throughout this analysis, we choose the convention in which $\xi$ is selected
to be positive, and fix $\theta_{ij}$ and $\delta$ in the mixing matrix $U$ to
be the central values of the latest global fit of neutrino oscillation data
Esteban:2018azc ; nufit .

##  III Neutrinoless double beta decay

In the considering model, the effective mass in the 0$\nu\beta\beta$ decay is
given by

$\displaystyle m_{\rm eff}=m_{\rm eff}^{\nu}+m_{\rm eff}^{N}\,,$ (18)

where the contribution from active neutrinos is

$\displaystyle m_{\rm eff}=\sum_{i}\,U_{ei}^{2}\,m_{i}\,.$ (19)

Note that, since only two right-handed neutrinos are introduced, it is
impossible to cancel the effective mass from active neutrinos by tuning the
lightest neutrino mass. The contribution from right-handed neutrinos is

$\displaystyle m_{\rm
eff}^{N}=\sum_{I}f_{\beta}(M_{I})\,\Theta_{eI}^{2}\,M_{I}\,.$ (20)

The function $f_{\beta}$ represents the suppression by the propagator effect
of right-handed neutrinos, and we use the approximate formula

$\displaystyle
f_{\beta}(M_{I})=\frac{\Lambda_{\beta}^{2}}{\Lambda_{\beta}^{2}+M_{I}^{2}}\,,$
(21)

where $\Lambda_{\beta}$ is a typical scale of Fermi momentum of a nucleus
which is evaluated as a few hundred MeV varied depending on nucleus and
modelings Faessler:2014kka ; Hyvarinen:2015bda ; Menendez:2017fdf ;
Barea:2015kwa .

We consider the case where $M_{1}<\Lambda_{\beta}\ll M_{2}$ and take
$f_{\beta}(M_{1})=1$ and $f_{\beta}(M_{2})=0$ approximately. Thus, the
effective mass becomes independent of $\Lambda_{\beta}$. In this case the
effective neutrino mass is expressed as

$\displaystyle m_{\rm eff}=\left\\{\begin{array}[]{l
l}\left(s_{\omega}\,U_{e2}m_{2}^{1/2}-c_{\omega}U_{e3}m_{3}^{1/2}\right)^{2}&\hskip
5.69054pt\mbox{for the NH case}\\\\[17.22217pt]
\left(s_{\omega}\,U_{e1}m_{1}^{1/2}-c_{\omega}U_{e2}m_{2}^{1/2}\right)^{2}&\hskip
5.69054pt\mbox{for the IH case}\end{array}\right.\,.$ (24)

Importantly, we find out that the effective mass vanishes if the complex
parameter $\omega$ satisfies

$\displaystyle\tan\omega=\left\\{\begin{array}[]{l
l}\displaystyle\frac{U_{e3}m_{3}^{1/2}}{U_{e2}m_{2}^{1/2}}&~{}~{}~{}\mbox{for
the NH case}\\\\[17.22217pt]
\displaystyle\frac{U_{e2}m_{2}^{1/2}}{U_{e1}m_{1}^{1/2}}&~{}~{}~{}\mbox{for
the IH case}\end{array}\right.\,.$ (27)

Figure 1:  Required values of $\omega_{r}$ (left) and $X_{\omega}$ (right) for
the vanishing effective mass in the NH or IH case (red solid or blue dashed
line).

In Fig. 1 we show the real and imaginary parts of $\omega$ satisfying the
cancellation condition (27). It is found from Eq. (27) that the maximal value
of $X_{\omega}$ is achieved by the $CP$ violating phases $\delta+\eta=-\pi/2$
for the NH case while $\eta=\pi/2$ in the IH case. Notice that $X_{\omega}$
becomes unity (i.e., no imaginary part of $\omega$) when $\eta=-\delta$ in the
NH case and $\eta=0~{}(\pi)$ in the IH case, respectively.

##  IV Discussions and conclusions

It is, therefore, found that the contribution to $m_{\rm eff}$ from active
neutrinos can be obscured by the light right-handed neutrino when Eq. (27) is
fulfilled. In this case, we can determine the mixing elements of of $N_{1}$
for a given mass depending on the Majorana phase. This point is illustrated in
Fig. 2.

Figure 2:  Mixing elements $|\Theta_{\alpha 1}|^{2}$ for the vanishing
effective neutrino mass in the NH (left) and IH (right) cases. Here,
$M_{1}=10$ MeV for both cases. The solid (red), dashed (blue), and dot-dashed
(black) lines represent $\alpha=e$, $\mu$, and $\tau$, respectively.

Interestingly, the flavor structure of the mixing elements highly depends on
the values of Majorana phase $\eta$. This pattern of the mixing elements may
be tested by future direct search experiments of right-handed neutrinos. In
the NH case $|\Theta_{\mu 1}|^{2}\gg|\Theta_{e1}|^{2}$ and then the
experiments like the peak search in $K\to\mu+N_{1}$, for instance, would help
the observation. On the other hand, in the IH case $|\Theta_{e1}|^{2}$ or
$|\Theta_{\mu 1}|^{2}$ becomes dominant depending on $\eta$, and $K\to
e+N_{1}$ or $K\to\mu+N_{1}$ would be the golden channel for the discovery.
Since the relative sizes of the mixing elements are not so much identical we
can extract important information of the mass hierarchy and the Majorana phase
$\eta$ from the combination of $|\Theta_{e1}|^{2}$ and $|\Theta_{\mu 1}|^{2}$
under the situation of no $0\nu\beta\beta$ decay is observed.

When the effective mass vanishes, the lifetime of $N_{1}$ can be predicted by
$M_{1}$ and $\eta$. In the mass region of interest the possible decay channels
are $N_{1}\to\nu\nu\nu$ and $N_{1}\to\nu e^{+}e^{-}$ (when $M_{1}>2m_{e}$) and
we find the range of the lifetime is

$\displaystyle\tau\simeq\left\\{\begin{array}[]{l}(5\mathchar 45\relax
6)\times
10^{7}~{}\mbox{sec}\left(\displaystyle\frac{10~{}\mbox{MeV}}{M_{1}}\right)^{4}\hskip
14.22636pt\mbox{for the NH case}\\\\[17.22217pt] (0.9\mathchar 45\relax
2)\times
10^{7}~{}\mbox{sec}\left(\displaystyle\frac{10~{}\mbox{MeV}}{M_{1}}\right)^{4}\hskip
5.69054pt\mbox{for the IH case}\end{array}\right.\,.$ (30)

The suggested values of the lifetime are so long that $N_{1}$ decays after the
onset of the big bang nucleosynthesis (BBN) and would destroy the success of
the BBN and/or conflict with the observational data of the cosmic microwave
background radiation. One possibility to avoid this difficulty is the dilution
of the $N_{1}$ abundance by the late time entropy production. Such an
additional production may be realized by the decay of the heavier right-handed
neutrino $N_{2}$ Asaka:2006ek .

To summarize we have examined the neutrinoless double beta decay in the seesaw
mechanism by two right-handed neutrinos. The Majorana nature of active
neutrinos and right-handed neutrinos breaks the lepton number of the theory,
which may lead to the neutrinoless double beta decay. When the masses of
right-handed neutrinos, however, are lighter than or comparable to the scale
$\Lambda_{\beta}$, they can give a significant effect, and the effective mass
can vanish in some cases.

In this paper we have found a possibility when
$M_{1}\lesssim\Lambda_{\beta}\ll M_{2}$. It has been shown that $N_{1}$
contribution can obliterate the neutrinoless double beta decay even if active
neutrinos do contribute it. If this is the case, the unknown parameters
related to right-handed neutrinos appeared in $\Omega$ are highly restricted
and the mixing elements of $N_{1}$ can be determined by the Majorana phase and
the mass hierarchy of active neutrinos. Inversely speaking, we may obtain
important information of the missing piece of neutrino properties, namely the
Majorana phase and the mass hierarchy, from the relative sizes among the
mixing elements of right-handed neutrinos measured at the future terrestrial
experiments together with no neutrinoless double beta decay.

## Acknowledgments

The work of T.A. was partially supported by JSPS KAKENHI Grants No. 17K05410,
No. 18H03708, No. 19H05097, and No. 20H01898. The work of H.I. was supported
by JSPS KAKENHI Grant No. 18H03708.

## References

  * (1) H. Päs and W. Rodejohann, New J. Phys. 17 (2015) no.11, 115010 doi:10.1088/1367-2630/17/11/115010 [arXiv:1507.00170 [hep-ph]].
  * (2) C. Arnaboldi et al., Phys. Lett. B 557 (2003) 167 doi:10.1016/S0370-2693(03)00212-0 [hep-ex/0211071].
  * (3) S. Umehara et al., Phys. Rev. C 78 (2008) 058501 doi:10.1103/PhysRevC.78.058501 [arXiv:0810.4746 [nucl-ex]].
  * (4) A. S. Barabash et al. [NEMO Collaboration], Phys. Atom. Nucl. 74 (2011) 312 doi:10.1134/S1063778811020062 [arXiv:1002.2862 [nucl-ex]].
  * (5) A. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev. Lett. 110 (2013) no.6, 062502 doi:10.1103/PhysRevLett.110.062502 [arXiv:1211.3863 [hep-ex]].
  * (6) M. Agostini et al. [GERDA Collaboration], Phys. Rev. Lett. 111 (2013) no.12, 122503 doi:10.1103/PhysRevLett.111.122503 [arXiv:1307.4720 [nucl-ex]].
  * (7) J. B. Albert et al. [EXO-200 Collaboration], Nature 510 (2014) 229 doi:10.1038/nature13432 [arXiv:1402.6956 [nucl-ex]].
  * (8) S. Andringa et al. [SNO+ Collaboration], Adv. High Energy Phys. 2016 (2016) 6194250 doi:10.1155/2016/6194250 [arXiv:1508.05759 [physics.ins-det]].
  * (9) R. Arnold et al. [NEMO-3 Collaboration], Phys. Rev. D 92 (2015) no.7, 072011 doi:10.1103/PhysRevD.92.072011 [arXiv:1506.05825 [hep-ex]].
  * (10) R. Arnold et al. [NEMO-3 Collaboration], Phys. Rev. D 93 (2016) no.11, 112008 doi:10.1103/PhysRevD.93.112008 [arXiv:1604.01710 [hep-ex]].
  * (11) A. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev. Lett. 117 (2016) no.8, 082503 Addendum: [Phys. Rev. Lett. 117 (2016) no.10, 109903] doi:10.1103/PhysRevLett.117.109903, 10.1103/PhysRevLett.117.082503 [arXiv:1605.02889 [hep-ex]].
  * (12) S. R. Elliott et al., J. Phys. Conf. Ser. 888 (2017) no.1, 012035 doi:10.1088/1742-6596/888/1/012035 [arXiv:1610.01210 [nucl-ex]].
  * (13) R. Arnold et al. [NEMO-3 Collaboration], Phys. Rev. D 94 (2016) no.7, 072003 doi:10.1103/PhysRevD.94.072003 [arXiv:1606.08494 [hep-ex]].
  * (14) R. Arnold et al. [NEMO-3 Collaboration], Phys. Rev. D 95 (2017) no.1, 012007 doi:10.1103/PhysRevD.95.012007 [arXiv:1610.03226 [hep-ex]].
  * (15) M. Agostini et al., Nature 544 (2017) 47 doi:10.1038/nature21717 [arXiv:1703.00570 [nucl-ex]].
  * (16) C. E. Aalseth et al. [Majorana Collaboration], Phys. Rev. Lett. 120 (2018) no.13, 132502 doi:10.1103/PhysRevLett.120.132502 [arXiv:1710.11608 [nucl-ex]].
  * (17) J. B. Albert et al. [EXO Collaboration], Phys. Rev. Lett. 120 (2018) no.7, 072701 doi:10.1103/PhysRevLett.120.072701 [arXiv:1707.08707 [hep-ex]].
  * (18) C. Alduino et al. [CUORE Collaboration], Phys. Rev. Lett. 120 (2018) no.13, 132501 doi:10.1103/PhysRevLett.120.132501 [arXiv:1710.07988 [nucl-ex]].
  * (19) M. Agostini et al. [GERDA Collaboration], Phys. Rev. Lett. 120 (2018) no.13, 132503 doi:10.1103/PhysRevLett.120.132503 [arXiv:1803.11100 [nucl-ex]].
  * (20) O. Azzolini et al. [CUPID-0 Collaboration], Phys. Rev. Lett. 120 (2018) no.23, 232502 doi:10.1103/PhysRevLett.120.232502 [arXiv:1802.07791 [nucl-ex]].
  * (21) R. Arnold et al., Eur. Phys. J. C 78 (2018) no.10, 821 doi:10.1140/epjc/s10052-018-6295-x [arXiv:1806.05553 [hep-ex]].
  * (22) D. Q. Adams et al. [CUORE Collaboration], Phys. Rev. Lett. 124 (2020) no.12, 122501 doi:10.1103/PhysRevLett.124.122501 [arXiv:1912.10966 [nucl-ex]].
  * (23) S. I. Alvis et al. [Majorana Collaboration], Phys. Rev. C 100 (2019) no.2, 025501 doi:10.1103/PhysRevC.100.025501 [arXiv:1902.02299 [nucl-ex]].
  * (24) M. Agostini et al. [GERDA Collaboration], Science 365 (2019) 1445 doi:10.1126/science.aav8613 [arXiv:1909.02726 [hep-ex]].
  * (25) O. Azzolini et al. [CUPID Collaboration], Phys. Rev. Lett. 123 (2019) no.3, 032501 doi:10.1103/PhysRevLett.123.032501 [arXiv:1906.05001 [nucl-ex]].
  * (26) V. Alenkov et al., Eur. Phys. J. C 79 (2019) no.9, 791 doi:10.1140/epjc/s10052-019-7279-1 [arXiv:1903.09483 [hep-ex]].
  * (27) G. Anton et al. [EXO-200 Collaboration], Phys. Rev. Lett. 123 (2019) no.16, 161802 doi:10.1103/PhysRevLett.123.161802 [arXiv:1906.02723 [hep-ex]].
  * (28) P. Minkowski, Phys. Lett. 67B (1977) 421. doi:10.1016/0370-2693(77)90435-X
  * (29) T. Yanagida, Conf. Proc. C 7902131 (1979) 95.
  * (30) T. Yanagida, Prog. Theor. Phys. 64 (1980) 1103. doi:10.1143/PTP.64.1103
  * (31) P. Ramond, in Talk given at the Sanibel Symposium, Palm Coast, Fla., Feb. 25-Mar. 2, 1979, preprint CALT-68-709 (retroprinted as hep-ph/9809459).
  * (32) M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity, edited by.P. van Niewwenhuizen and D. Freedman (North Holland, Amsterdam, 1979) [arXiv:1306.4669 [hep-th]].
  * (33) S. L. Glashow, in Proc. of the Cargése Summer Institute on Quarks and Leptons, Cargése, July 9-29, 1979, eds. M. Lévy et. al, , (Plenum, 1980, New York), p707.
  * (34) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.
  * (35) M. Blennow, E. Fernandez-Martinez, J. Lopez-Pavon and J. Menendez, JHEP 07 (2010), 096 doi:10.1007/JHEP07(2010)096 [arXiv:1005.3240 [hep-ph]].
  * (36) T. Asaka, S. Eijima and H. Ishida, JHEP 1104 (2011) 011 doi:10.1007/JHEP04(2011)011 [arXiv:1101.1382 [hep-ph]].
  * (37) E. K. Akhmedov, V. A. Rubakov and A. Y. Smirnov, Phys. Rev. Lett. 81 (1998), 1359-1362 doi:10.1103/PhysRevLett.81.1359 [arXiv:hep-ph/9803255 [hep-ph]].
  * (38) T. Asaka and M. Shaposhnikov, Phys. Lett. B 620 (2005), 17-26 doi:10.1016/j.physletb.2005.06.020 [arXiv:hep-ph/0505013 [hep-ph]].
  * (39) Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870.
  * (40) B. Pontecorvo, Sov. Phys. JETP 7 (1958) 172.
  * (41) J. A. Casas and A. Ibarra, Nucl. Phys. B 618 (2001) 171 doi:10.1016/S0550-3213(01)00475-8 [hep-ph/0103065].
  * (42) A. Abada, S. Davidson, A. Ibarra, F.-X. Josse-Michaux, M. Losada and A. Riotto, JHEP 0609 (2006) 010 doi:10.1088/1126-6708/2006/09/010 [hep-ph/0605281].
  * (43) I. Esteban, M. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, JHEP 01 (2019), 106 doi:10.1007/JHEP01(2019)106 [arXiv:1811.05487 [hep-ph]].
  * (44) I. Esteban, M. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, “NuFiT 4.1: Three-neutrino fit based on data available in July 2019,” www.nu-fit.org.
  * (45) A. Faessler, M. González, S. Kovalenko and F. Šimkovic, Phys. Rev. D 90 (2014) no.9, 096010 doi:10.1103/PhysRevD.90.096010 [arXiv:1408.6077 [hep-ph]].
  * (46) J. Hyvärinen and J. Suhonen, Phys. Rev. C 91 (2015) no.2, 024613 doi:10.1103/PhysRevC.91.024613
  * (47) J. Barea, J. Kotila and F. Iachello, Phys. Rev. C 91 (2015) no.3, 034304 doi:10.1103/PhysRevC.91.034304 [arXiv:1506.08530 [nucl-th]].
  * (48) J. Menéndez, J. Phys. G 45 (2018) no.1, 014003 doi:10.1088/1361-6471/aa9bd4 [arXiv:1804.02105 [nucl-th]].
  * (49) T. Asaka, M. Shaposhnikov and A. Kusenko, Phys. Lett. B 638 (2006) 401 doi:10.1016/j.physletb.2006.05.067 [hep-ph/0602150].


# When is $F(p)$ the Laplace transform of a bounded $f(t)$?

Alexander G. Ramm
Department of Mathematics, Kansas State University,
Manhattan, KS 66506, USA
<EMAIL_ADDRESS>
http://www.math.ksu.edu/ $\sim$ ramm

###### Abstract

MSC: 44A10 Key words: Laplace transform; hyper-singular integrals;

Sufficient conditions are given for a function $F(p)$, analytic in Re$p>0$, to
be a Laplace transform of a function $f(t)$, such that $max_{t\geq
0}|f(t)|<\infty$, $f(0)=0$.

##  1 Introduction

There is a large literature on the Laplace transform, e.g., [2], [3]. Tables
of the Laplace transform of distributions are published, see, e.g., [1]. The
Laplace transform of a function $f(t)$, $f(t)=0$ for $t<0$, $|f(t)|\leq
Ce^{at}$ for some positive constants $C$ and $a$, is defined by the formula

$L(f):=F(p):=\int_{0}^{\infty}e^{-pt}f(t)dt,\quad p=s+i\eta,\,\,s>a.$ (1)

Under this assumption $F(p)$ is an analytic function of $p$ in the half-plane
$s>a$ and $|F(p)|\leq\frac{C}{s-a}$, $s=Re\,p$, as follows from the estimate

$|F(p)|=|\int_{0}^{\infty}e^{-pt}f(t)dt|\leq
C\int_{0}^{\infty}e^{-(s-a)t}dt=\frac{C}{s-a},\quad s>a.$ (2)

We assume that

$\lim_{|p|\to\infty,-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}}F(p)=0,$ (3)

where $\phi=arg\,p$ is the argument of $p$.

If $f\in L^{2}(0,\infty)$, then $a=0$. In Section 3 we discuss possible
generalizations.

We are interested in the sufficient conditions for $F(p)$ to be the Laplace
transform of a function $f(t)$, such that

$\sup_{t\geq 0}|f(t)|<\infty,\quad f(0)=0.$ (4)

These conditions are (2), (3) and

$|F(i\eta)|=O(|i\eta|^{-b}),\,\,b>1\text{ for }|\eta|\gg 1;\,\,F(p)\text{is
analytic for }s>0.$ (5)

The condition $F(i\eta)\in L^{2}(-\infty,\infty)$ follows from the assumption
$f(t)\in L^{2}(0,\infty)$ by the Parceval’s identity. This condition is not
necessary for $F(p)$ to be the Laplace transform of $f(t)$ satisfying
conditions (4).

Our result is formulated in the following theorem.

Theorem 1. If conditions (2), (3) and (5) hold, then $F(p)=L(f)$ and $f(t)$
satisfies conditions (4).

This result is new. It differs from the known results: usually some
assumptions are made on $f(t)$ and necessary conditions are derived for $F(p)$
to be the Laplace transform. In Theorem 1 assumptions are made on $F(p)$ and
sufficient conditions are given for $F(p)$ to be the Laplace transform of a
function $f(t)$ satisfying conditions (4) and vanishing for $t<0$.

##  2 Proof

Let

$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{it\eta}F(i\eta)d\eta.$ (6)

The integral (6) converges since

$b>1$. Therefore, $L(f)=F(p)$, $f(t)$ is a continuous function and
$\sup_{t\geq 0}|f(t)|<\infty$. This follows from formula (6) and the known
inversion formula for the Laplace transform, [3], namely:

$f(t)=\frac{1}{2\pi i}\int_{C_{\sigma}}e^{pt}F(p)dp,$ (7)

where $C_{\sigma}$, $\sigma>a$, is the straight line
$\sigma-i\eta,\sigma+i\eta$, $-\infty<\eta<\infty$. In our case $\sigma=0$,
$dp=id\eta$ and $C_{\sigma}$ is the straight line $-i\eta,i\eta$,
$-\infty<\eta<\infty$.

For convenience of the reader, let us give a version of a proof of the
inversion formula. Under our assumptions $f(t)$, defined in (6) is a
continuous function on $[0,\infty)$, uniformly bounded because the function
$F(i\eta)$ is absolutely integrable on the whole axis $-\infty<\eta<\infty$.
Consider

$\int_{0}^{\infty}f(t)e^{-qt}dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\eta
F(i\eta)\int_{0}^{\infty}e^{-(q-i\eta)t}dt=-\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}F(p)\frac{1}{p-q}dp,\,\,\,p=i\eta.$ (8)

Let $C_{n}$ be the closed contour consisting of $[in,-in]\cup L_{n}$, where
$L_{n}$ is the semi-circle $p=ne^{i\phi}$,
$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$. Since $b>1$, it follows that

$\lim_{n\to\infty}\int_{L_{n}}F(p)\frac{1}{p-q}dp=0.$ (9)

Therefore, the integral on the right side of (8) can be considered as the
integral over the closed contour $C_{n}=$ with $n\to\infty$. The minus sign in
(8) is used to get the closed contour passed counterclockwise. Consequently,
the integral in the right side of (8) is equal to $F(q)$ by the Cauchy formula
and the analyticity of $F(p)$ in the half-plane Re$p>0$. Thus,

$\int_{0}^{\infty}f(t)e^{-qt}dt=F(q).$ (10)

Therefore, the inversion formula for the Laplace transform is proved.

Let us prove that $f(0)=0$. It follows from (6) that

$f(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(i\eta)d\eta.$ (11)

If $N$ is sufficiently large for the estimate

$|F(p)|\leq c(1+|p|)^{-b},\quad|p|>N,\quad b>1,$ (12)

to hold, then

$f(0)=\frac{1}{2\pi}\int_{-N}^{N}F(i\eta)d\eta+o(1)\text{ as }N\to\infty.$
(13)

The function $f(t)$ is continuous and uniformly bounded on $[0,\infty)$.
Indeed, by the inversion formula (7) with $\sigma=0$, where the integral over
the straight line $\sigma=0$ absolutely converges if $b>1$, one has

$f(t)=\frac{1}{2\pi i}\int_{-iN}^{iN}e^{pt}F(p)dp+\frac{1}{2\pi
i}\int_{-i\infty}^{-iN}e^{pt}F(p)dp+\frac{1}{2\pi
i}\int_{iN}^{i\infty}e^{pt}F(p)dp.$ (14)

The first integral is a continuous function of $t$ because it is taken over a
compact set and $F(p)\in L^{1}(-iN,iN)$, the second and third integrals are
continuous functions of $t$ because $b>1$. At $t=0$ formula (14) reduces to
(13).

If we prove that

$I_{N}:=\frac{1}{2\pi}\int_{-N}^{N}F(i\eta)d\eta\to 0\text{ as }N\to\infty,$
(15)

then relations (13)-(15) imply that $f(0)=0$ and Theorem 1 is proved.

To prove (15), consider a closed contour $C_{N}$, consisting of $[-iN,iN]$ and
the semi-circle $L_{N}:=Ne^{i\phi}$, where
$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$. Since $F(p)$ is analytic in the
half-plane $s>0$, one has:

$\int_{C_{N}}F(p)dp=0.$ (16)

Since $\lim_{N\to\infty}\frac{1}{2\pi}\int_{L_{N}}F(p)dp=0$ because $b>1$, it
follows from (16) that equation (15) is valid.

Finally, the condition $f(t)=0$ for $t<0$ follows from the following argument.
If $t<0$ then the function $e^{-pt}F(p),\,\,t>0$ is analytic in Re$p>0$ and

$f(-t)=\lim_{N\to\infty}\frac{1}{2\pi i}\int_{C_{N}}e^{-pt}F(p)dp=0,\quad
t>0,$ (17)

because by the Cauchy theorem the integral over the closed contour $C_{N}$,
inside which the function $e^{-pt}F(p)$ is analytic, is equal to zero.

Theorem 1 is proved. $\Box$

The Jordan lemma in the following form, compare [3], pp. 412, 469, is useful:

Lemma. If a function $h(p)$ tends to zero uniformly with respect to the
argument $\phi$ of $p$, $-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$ on the
contour $L_{n}$ as $n\to\infty$ then for $t>0$ one has
$\lim_{n\to\infty}\int_{L_{n}}h(p)e^{pt}dp=0$.

##  3 Discussion

In this Section we discuss possible generalization and applications of Theorem
1. We may replace the assumption $F(i\eta)\in L^{2}(-\infty,\infty)$ by the
assumption

$F(i\eta)\in L^{p}(-iN,iN),\,\,p\geq 1,\quad\text{ \eqref{e3} holds. }$ (18)

The proof of Theorem 1 remains unchanged because the assumption $F(i\eta)\in
L^{p}(-iN,iN),\,\,p\geq 1$ implies that the first integral in (14) is a
continuous function of $t$ and assumption (12) guarantees that the second and
third integrals in (14) are continuous functions of $t$, which tend to zero as
$N\to\infty$.

In applications Theorem 1 is useful in the definition of hyper-singular
integrals, see [4]. To give an idea of this application, consider the integral
equation:

$q(t)+\frac{1}{\Gamma(\lambda)}\int_{0}^{t}(t-\tau)^{\lambda-1}q(\tau)d\tau=f(t).$
(19)

If Re$\lambda>0$, then the integral in (19) is defined classically, i.e., from
the classical point of view. If $\lambda<0$, then this integral is a hyper-
singular integral,it diverges classically. For Re$\lambda>0$ we take the
Laplace transform of (19) and get

$L(q)(1+p^{-\lambda})=L(f),\quad L(q)=\frac{L(f)}{1+p^{-\lambda}},$ (20)

where the following formula was used:
$L(t^{\lambda-1})=\Gamma(\lambda)p^{-\lambda}$. This formula is valid for all
$\lambda\in{\mathbb{C}}$ except for $\lambda=0,-1,-2,...$. The $L(q)$ in (20)
admits analytic continuation with respect to $\lambda$ from the region
Re$\lambda>0$ to the region Re$\lambda<0$, for example, to the point
$\lambda=-\frac{1}{4}$ which is of interest in the Navier-Stokes problem.
Theorem 1 is of use to prove that $\frac{L(f)}{1+p^{-\lambda}}$ for
$\lambda=-\frac{1}{4}$ is the Laplace transform of a function $q(t)$
satisfying conditions (4). This can be checked if $f(t)$ smooth and rapidly
decaying as $t\to\infty$, so that $|L(f)|\leq c(1+|p|)^{-1}$. In this case the
function $(1+p^{\frac{1}{4}})^{-1}$ is analytic in the half-plane Re$p\geq 0$
and is $O(\frac{1}{|p|^{1/4}})$ for $|p|\gg 1$, the function $L(f)$ is
analytic in the half-plane Re$p\geq 0$ and is $O(\frac{1}{|p|})$ for $|p|\gg
1$ on the imaginary axis of the complex plane $p=s+i\eta$. By Theorem 1, the
function $\frac{L(f)}{1+p^{\frac{1}{4}}}$ is the Laplace transform of a
function $q(t)$ satisfying (4). In this example $b=\frac{5}{4}>1$. We have
proved the following result.

Theorem 2. Assume $\lambda=-\frac{1}{4}$ and $f(t)$ be a smooth rapidly
decaying as $t\to\infty$. Then equation (19) has a unique solution $q(t)$
satisfying (4).

The kernel of equation (19) with $\lambda=-\frac{1}{4}$ is hyper-singular. The
integral in this equation with $\lambda=-\frac{1}{4}$ diverges classically.
Theorem 2 is of prime interest in a study of the Navier-Stokes problem, see
[4].

##  4 Conclusion

Sufficient conditions are given for a function $F(p)$, $p=s+i\eta$, analytic
in the half-plane $s>0$ to be the Laplace transform of a function $f(t)$,
$f(t)=0$ for $t<0$, $\sup_{t\geq 0}|f(t)<\infty$ and $f(0)=0$. This result is
useful in a study of the Navier-Stokes problem in ${\mathbb{R}}^{3}$ and in a
study of integral equations with hyper-singular kernels, see [4].

## References

  * [1] Yu. Brychkov, A. Prudnikov, Integral tranforms of generalized functions, Nauka, Moskow, 1977 (in Russian)
  * [2] J. Schiff, The Laplace transform, Springer, Berlin, 1999.
  * [3] M.Lavrentiev, B. Shabat, Methods of the theory of functions of complex variable, GIFML, Moscow, 1958. (in Russian)
  * [4] A.G.Ramm, The Navier-Stokes problem, Morgan & Claypool publishers, 2021.


# A Machine Learning Based Regulatory Risk Index for Cryptocurrencies

Xinwen Ni Financial support of the European Union’s Horizon 2020 research and
innovation program “FIN- TECH: A Financial supervision and Technology
compliance training programme” under the grant agreement No 825215 (Topic:
ICT-35-2018, Type of action: CSA), the European Cooperation in Science &
Technology COST Action grant CA19130 - Fintech and Artificial Intelligence in
Finance - Towards a transparent financial industry, the Deutsche
Forschungsgemeinschaft’s IRTG 1792 grant, the Yushan Scholar Program of Taiwan
and the Czech Science Foundation’s grant no. 19-28231X / CAS: XDA 23020303 are
greatly acknowledged. School of Business and Economics, Humboldt-Universität
zu Berlin, Berlin, Germany Wolfgang Karl Härdle  School of Business and
Economics, Humboldt-Universität zu Berlin, Berlin, Germany Sim Kee Boon
Institute for Financial Economics, Singapore Management University, Singapore
W.I.S.E. - Wang Yanan Institute for Studies in Economics, Xiamen University,
Fujian, China  Department of Probability and Mathematical Statistics, Faculty
of Mathematics and Physics, Charles University, Prague, Czech Republic
Department of Information Management and Finance, National JiaoTong
University, Taiwan  Taojun Xie  Asia Competitiveness Institute, Lee Kuan Yew
School of Public Policy National University of Singapore, Singapore

###### Abstract

Cryptocurrencies’ values often respond aggressively to major policy changes,
but none of the existing indices informs on the market risks associated with
regulatory changes. In this paper, we quantify the risks originating from new
regulations on FinTech and cryptocurrencies (CCs), and analyse their impact on
market dynamics. Specifically, a Cryptocurrency Regulatory Risk IndeX (CRRIX)
is constructed based on policy-related news coverage frequency. The unlabeled
news data are collected from the top online CC news platforms and further
classified using a Latent Dirichlet Allocation model and Hellinger distance.
Our results show that the machine-learning-based CRRIX successfully captures
major policy-changing moments. The movements for both the VCRIX, a market
volatility index, and the CRRIX are synchronous, meaning that the CRRIX could
be helpful for all participants in the cryptocurrency market. The algorithms
and Python code are available for research purposes on www.quantlet.de.

Keywords: Cryptocurrency, Regulatory Risk, Index, LDA, News Classification
JEL classification: C45, G11, G18

##  1 Introduction

Today, there are nearly 2,500 cryptocurrencies worth more than $252.5 trillion
trading in the market (Dolan, 2020). The original boom of cryptocurrencies
occurred in an unregulated environment. Even as news outlets and investors
paid closer attention to the market, regulators and international actors
remained largely distant from the action, and prices continued to soar
unabated. However, the situation changed since 2014.

Regulations are designed to protect the investors, to put a stop on money
laundering, or to prevent the fiat currency from being crowded out. Despite
these good wills, speculation and implementation of regulations have resulted
in volatile price movements in the cryptocurrency markets. Recent incidents,
including China’s ban on cryptocurrency exchanges and the rumours of Korea
doing the same, have caused major sell-offs and losses among investors. It is
therefore important to identify the extent to which new regulations and
speculations on them have affected the cryptocurrency markets. Ignoring this
source of risk, regulators could end up with self-destroying outcomes and
create thus systemic risk bias.

In this paper, we aim to quantify the risks originating from introducing
regulations on the Cryptocurrency (CC) markets and identify their impact on
the cryptocurrency investments. In order to measure the regulatory risk,
particularly the effects of regulations, some researchers considered event-
study methods (Binder, 1985; Buckland and Fraser, 2001; Binder, 1985; Binder,
1985; Schwert, 1981 and etc.). However, cryptocurrency market is young and so
different from other financial market that the previous regulatory event may
not appear again, such as a certain country ban the market. Therefore, we need
a measurement tool, which is able to represent the risk level, be comparative
and track the changes over time. An index matches all those requirement.

Indices have been applied to track the Cryptocurrency markets already. The CC
index developed in Trimborn and Härdle (2018), known as CRIX 111seen in
crix.berlin, or thecrix.de is a benchmark and tracks the price movements in
the CC markets on a daily basis. A volatility measure, VCRIX (Kim et al.,
2019), similar to VIX, is also presented. there to reflect the market’s
volatility However, even though the VCRIX shows jumps that are to some extent
attributable to political decisions, neither of these indices and other
indices, like e.g. CCI30 222The CCi30 is a rules-based index designed to
objectively measure the overall growth, daily and long-term movement of the
blockchain sector. It does so by tracking the 30 largest cryptocurrencies by
market capitalization, seen in http://cci30.com directly address regulatory
risks although it plays an important role for the future of CCs and in
addition might be created by text mining techniques from a sufficiently rich
corpus.

The here proposed Cryptocurrency Regulatory Risk Index (CRRIX) will be
constructed by evaluating regulation-related news articles. The indices
introduced in Baker et al. (2016) reveal economic policy uncertainty. Similar
to their indices, our index is also based on the policy-related news coverage
frequency. Unlike their algorithm, which involved a meticulous manual process
to label a pool of 12,000 articles, we pursue a Machine Learning (ML)
technique to classify policy-related news in our data. We use the support
vector machine (SVM), a widely used text classifier, as benchmark and apply
Latent Dirichlet Allocation (LDA) method to capture the distances between
articles and further classify the unannotated news according to their
similarity to policy-related news.

The Regulatory Risk Index (CC) will subsequently be used to analyse the
association between regulatory risk and market activities. The algorithms have
been programmed in Python. All numerical calculations are available for
research purposes on www.quantlet.de and also on Github in the organization
QuantLet , see Borke and Härdle (2018).

In the next Section 2, we discuss the background and research questions. The
Section 3 presents the data in detail and shows the basic statistics. The
Section 4 enters into the methodology and the Section 5 presents results from
LDA model and the time series of our regulatory risk index. Finally, Section 6
concludes.

##  2 Background and Research Questions

In this section, we first generally review the development of cryptocurrency.
Based on that, we display the picture of trends of regulatory dynamics in the
CC market. The research questions touch the identification of the regulatory
risk, the construction of the regulatory risk index, and the interaction of
the regulatory risk index with market variables.

###  2.1 Development of Cryptocurrency

The history of digital or programmable monies can be traced back to as far as
four decades ago, although the word “cryptocurrency” became popular only
recently. In the 1980s, the concept of E-cash was introduced by David Chaum in
a paper entitled “Blind Signatures for Untraceable Payments” (Fiorillo, 2018).
Subsequently, DigiCash, B-Money, and Bit Gold were proposed by Chaum, Wei Dai,
and Nick Szabo in the 1990s, respectively. Most of these digital monies did
not survive as they failed to address the practical issues of “double
spending” and “third-party trust”.

The milestone development in this field came in after the global financial
crisis. Nakamoto, in the seminal white paper, 2008, proposed the bitcoin. This
is a peer-to-peer electronic cash system implemented via the blockchain
technology, with the participation of a network of computer owners known as
miners. The blockchain technology, later known as the distributed ledger
technology, or DLT, ensures that transaction records are easy to be updated
but costly to be changed, avoiding the “double spending” issue. Miners, after
solving complex mathematical puzzles, are rewarded with a predetermined amount
of bitcoins. The amount of the reward can only be amended with the agreement
from majority of the miners. Such a mechanism avoids the “thrid-party trust”
problem in fiat monies, whose issuance depends on the central banks’ sole
discretion. Since its birth, the BTC model has defined the meaning of
“cryptocurrency”, which now typically refers to a decentralized digital
network that facilitates secured transactions using cryptographic methods.

Figure 1: Cryptocurrency timeline

Bitcoin has sparked a series of events since 2009. In Figure 1, we show a
timeline of a few major events in the last decade. Some events pertained to
innovations that aimed at improving the technology behind cryptocurrencies
(highlighted in blue). For example, competing CCs, also known as altcoins,
began to emerge in 2011. The events highlighted in yellow are the ones
involving CCs being used in legitimate real-life applications. A well-known
example was the two pizzas in 2010 that cost 10,000 bitcoins. Through this
series of events, the general public became aware of the strengths and
weaknesses of CCs. As good and bad news took turns to be reported, the price
of bitcoin, highlighted in gray, and that of the other CCs also experienced
volatile movements. Notably, events relating to losses of cryptocurrency
exchanges have been associated with the largest price movements. For instance,
as Mt. Gox went bankruptcy in 2014 after losing over 850,000 bitcoins, the
price of bitcoin fell from over $1,000 to the around $400. This event has been
foreseeable through textual analysis of BTC blog and other solid media
channels. Linton et al. (2017) and the chapter 3 of Härdle et al. (2017)
employ a technique similar to ours to evaluate discussions in social media. A
recent price movement was an upswing of 1300% in year 2017, followed by a fall
of more than half in May 2018.

###  2.2 Regulations of the Cryptocurrency Market

Among the features of cryptocurrencies, anonymity has been the most
controversial one. Users of cryptocurrencies like this feature for it is
difficult to trace one’s spending history, but regulators dislike it for the
exact same reason. We thus see an interesting interaction here. Users have
proposed numerous improvements to enhance anonymity. Zcash and Monero were
designed to facilitate anonymous transactions. At the same time, incidents
such as the Silk Road going live and terrorists using cryptocurrencies for
remittance got the regulators on the toes. Regulators, on the one hand,
insisted on know-your-customer (KYC) measures to trace any illegitimate
transactions, but on the other hand, prepared to launch their own
cryptocurrencies (Barrdear and Kumhof, 2016; George et al., 2020). Fighting
against illegitimate transactions became one of the first tasks for the
cryptocurrency regulators.

Trading activities at the exchanges was the next issue that regulators reacted
to. The anonymity feature and a lack of regulation at the cryptocurrency
exchanges cultivated illegal and unethical behaviors, such as money
laundering, pump-and-dump activities and scams. Ignorant users of the
cryptocurrency exchanges faced high risks while trading. Responding to these,
starting in 2011, the US Treasury Department’s Financial Crimes Enforcement
Network (FinCEN) began oversight of cryptocurrency exchanges, transmitters,
and administrators under the Bank Secrecy Act related to anti-money laundering
and combating the financing of terrorism (AML/CFT) (Lee and Deng, 2018). In
the same year, the United States Department of Homeland Security (DHS)
initiated first investigation relating to cryptocurrency. The number of cases
raised to over 200 in 2017 (seen in Figure 1).

However, there has been a dilemma in regulating the cryptocurrency space.
While it was important to protect retail investors and to prevent unlawful
transactions, new technology driving the development of cryptocurrencies
needed to be incubated until the ecosystem was matured. It was then imperative
for the regulators to move towards systematic governance. In Figure 2, we show
a timeline of countries’ publications of guidance on the cryptocurrency space.

Figure 2: Timeline for Cryptocurrency Guidance

In order to apply the existing law and regulations, the very first step for
the governors is to identify the cryptocurrency nature as means of holding,
transferring or investing “money”. Since the birth of cryptocurrency, the
debate, whether BTC, or generally cryptocurrencies, are currency or asset,
eolved with the growth of the market (Glaser et al., 2014, Baur et al., 2018).
Despite using “currency” as the name name, most countries that have permitted
cryptocurrencies view these monies as assets. In 2014, the US Internal Revenue
Service (IRS) stipulated that virtual currency is considered property for the
purpose of federal tax”. Purchases using cryptocurrencies as the media of
exchange are considered barter trades, or exchanges between properties and
services (Blandin et al., 2019). The Swiss Financial Market Supervisory
Authority (FINMA), in 2018, published guidelines for initial coin offerings
(ICOs). In those guidelines, tokens were categorised into three types based on
their economic functions: payment tokens, utility tokens and asset tokens
(Caytas, 2018).

Regulatory responses to cryptocurrencies vary from strict bans to government
adoption. Most of these regulations were designed to protect retail investors
or to ensure that cryptocurrencies are not used for illegal activities. The
Financial Action Task Force (FATF), recommended that regulations be
implemented to prevent the use of cryptocurrencies in money laundering and
terrorist finance (Gold and McBride, 2019). At the G20 Summit in 2018, the
FATF urged on all countries to take necessary preventive measures towards the
misuse of cryptocurrencies. In the European Union, by 2020, all member states
will implement AML/CFT333AML/CFT: Anti-money laundering / combating the
financing of terrorism. rules to cryptocurrency exchanges and wallet operators
(Houben and Snyers, 2018). On the contrary, some countries began to change
their attitude towards the adoption of DLT. In China, the use of BTC as
currency for financial institutions was prohibited in 2013 (Glaser et al.,
2014), but in Octobor 2019, the president of China announced that the country
will encourage enterprises to seize the opportunity in the up-and-coming
technology. Subsequently, the People’s Bank of China announced that it would
launch digital Yuan, commonly known as the central bank digital currency
(Zhong, 2019). Such support for DLT lead to a new round of debate globally.

This disparity in regulatory approaches creates interesting dynamics in the
cryptocurrency markets. As poited out earlier, good and bad news is likely to
induce different movements in the price of cryptocurrencies. As more
regulations are on their way, and because the cryptocurrency market is
globally unified, policy changes in one country or even the rumor about the
attitude adjustment of one government, would be widely discussed in the news
platform and bring turbulence to the whole market. Under these circumstances,
and in consistent with Auer and Claessens (2020), the construction of a
regulatory risk measurement for the whole market is necessary and the news
data, which includes information from different sources and countries, would
be helpful.

###  2.3 Research Questions

For the most cases, the regulation changes do not come out from nowhere. A
good example is the Fed’s rate cut (or hike). Before the Fed’s announcement of
changes, there are discussions and predictions about the Fed’s move in
newspapers. Numerous studies have attempted to examine whether the text mining
technology would contribute to the forecasting for financial markets
(Nassirtoussi et al., 2014), e.g. Ghiassi et al. (2013), Geva and Zahavi
(2014), Tu and Härdle (2018). But not many use textual data to analyze
financial regulatory risk. Gulen and Ion (2016), Baker et al. (2016) and Kang
and Ratti (2013) argue that news from newspapers could be a good indicator for
macro policy uncertainty.

As discussed before, indices have been introduced to trace the movement of CC
market, but none of them addresses regulatory risks which plays an important
role for the future of CCs. We try to, in this paper, quantify the risks
brought by introducing policies on the CC market and further discuss its
impact on the CC investment. A regulatory risk index for the CC market, which
could serve as a tool for passive investors, for the fund manager, and even
for policy-makers.

There are mainly three kinds of indices as to the data sources which were
applied to construct the index. First, and most commonly, some indices use
real data, e.g. VIX, S&P 500, and DAX, which employed real market price or
volume data. Second, some are based on a regular survey, e.g. IFO business
Climate Index and Purchasing Managers’ Index (PMI), which is relied on a
monthly survey of supply chain managers. Recently, the third source, news
data, or generally speaking the text data, becomes popular, e.g. Thomson
Reuters MarketPsych Indices 444seen in
https://www.refinitiv.com/en/products/world-news-data/, sentiment Indices,
which are standard input to trading desks and are provided by a variety of
news data channels. The majority of these indices are based on text mining
tools that establish a dictionary and then calculate in a bag of words
approach the frequency of key words with identifiable sentiment, like good,
bad etc. Such input for asset trading and risk management is a refined data
source that in many cases contains useful directional information. Baker et
al. (2016) introduced an index to reveal economic policy uncertainty based on
newspaper coverage frequency. They calculated the number of articles which
contained “economic” or ”economy”; “uncertain” or “uncertainty”; and one or
more of “congress”, “deficit”, “Federal Reserve”, “legislation”, “regulation”
or “White House” from 10 large newspapers. Their index successfully
represented movements in policy-related economic uncertainty and they also
found that policy uncertainty rise stock price volatility.

Unlike Baker et al.’s algorithm, which involved a meticulous manual process to
label a pool of 12,000 articles, we propose a machine learning based approach
to classify policy-related news in our data. With a sufficiently rich corpus,
it might be able to conduct the regulatory risk index by some NLP techniques,
such as topic modelling methods. We are also curious about whether the policy-
related uncertainty index could be helpful for the market participants.
Therefore, the research questions discussed in this paper are as follows:

######  Research Question 1

How to indentify the regulatory risk for Cryptocurrencies?

######  Research Question 2

How to construct an index of regulatory risk for Cryptocurrency market based
on news data?

######  Research Question 3

What is the impact of regulatory risk to the market?

##  3 Data

###  3.1 News from Top Cryptocurrency Online Platforms

Since CCs are frequently mentioned in newspapers only for the very recent
years, we choose to use news data from the top online cryptocurrency news
platform (Guides, 2018). The representative news platform Coindesk
555https://www.coindesk.com and Bitcoin Magazine
666https://bitcoinmagazine.com were considered in this paper, because both of
them are not only pioneers and leaders in the market but also they offer news
data which could trace back to the beginning of BTC’s boost.

Coindesk is a news website with a particular focus on Blockchain, Bitcoin and
Cryptocurrencies as a whole. The site launched in April 2013 and released
close to 25000 articles. The articles have been classified already in
categories: markets, technology, business, policy & regulation and people. The
aim of this paper is to introduce a policy uncertainty index by calculating
the frequency of policy-related news. The pre-classified news data from
Coindesk perfectly matches our demand and will be further applied as training
data for the ML models. The textual data from the source was collected via a
dynamic web scraper.

After checking for duplicates, eventually we keep 16,528 articles from 01
April 2013 to 18 July 2019. The data covers 76 months, 329 weeks and 2300
days. Out of the total over 16,000 articles, 2,468 are marked as policy-
related news. The data is available for further research at the Blockchain
Research Center (BRC) 777https://hu.berlin/BRC and on Quantlet.de.

Figure 3 represents the average number of news per week related to policy and
the average number of all news. The number of daily articles increased in 2014
and 2018 both in total and in regulation-related term. In those years, the
price of Bitcoin underwent volatile movements, declining by more than 70 % in
2014 in 2018. Before these massive corrections, the end of 2013 and 2017
marked periods of price discovery and all-time highs in USD valuation being
broken every other day. During the same periods, number of blockchain-related
news articles increased, indicating growing interests in blockchain and
distributed ledger technology. There is no doubt that, simultaneously, the
market attracted policy-makers’ strong attention as well.

Figure 3: Weekly Average Number of total News (in red) and of Policy-Related
News (in blue)

CRRIX

Bitcoin Magazine is another leading and pioneer platform supplying information
on the new market. It was founded in Feb 2012, one year earlier than Coindesk.
There are fewer news collected from Bitcoin Magazine, only 4,586, but it
covers longer period from 28 Feb 2012 to 18 July 2019. Similar to Coindesk,
the platform creates channels with different topics like “Policy and Laws”,
“Payment”, “Blockchain” and etc. But not all news have been classified. We
found that 2,976 articles didn’t belong to any category. Classification is
needed and machine learning methods will be employed. However, our final goal
is to apply our methods to multiple data sources, especially the other
financial news platforms, such as the NASDAQ news platform in which 887, 018
articles are available since Jan 2013. In order to test the performance of
applying our methods to other platforms, we manually marked each article from
Bitcoin Magazine with “policy-related” and “non-policy-related” to compare
with our ML classification results. 582 of 4,586 articles are related to
cryptocurrency market regulations.

###  3.2 CRIX and VCRIX

The CRIX and VCRIX is chosen to represent the value and the volatility of the
entire cryptocurrency market for the later analysis. The CRIX (CRyptocurrency
IndeX), created by Trimborn and Härdle (2018), closely tracks the entire
cryptocurrency market performance. Its construction is robust in the sense it
takes into account the dynamics of market structure, thus ensuring the
representativity and the tracking performance of the index. It follows that
constituents of CRIX change over time, depending on market conditions and the
relative dominance of CCs. The CRIX series begins from July 2014, and is
available through thecrix.de. Reallocation of the CRIX happens on a monthly
and quarterly basis. It adopts a liquidity rule when incorporating a certain
cryptocurrency into CRIX, and hence guarantees the trading of CRIX, which is
good for ETFs and traders. CRIX has been widely investigated in the pioneering
research on cryptocurrencies, including Hafner (2020), Klein et al. (2018),
Trimborn et al. (2018), and da Gama Silva et al. (2019).

Like VIX or VDAX, which provide a measure for implied volatility, VCRIX,
created by Kim et al. (2019), is a volatility index, able to grasp the risk
induced by the cryptocurrency market. This index accurately addresses the
market dynamics on the basis of CRIX and thus proved to be a proper basis for
option pricing. Similar to CRIX, the data of VCRIX is also be able to be
download from thecrix.de.

##  4 Methodology

Based on the rich text corpus, on can now enter the machine learning text
mining step to identify policy-related news from others. In this paper, the
classification problem is simply binary: policy-related or not. In the
literature, SVM is widely applied to solve such kind of binary problem.
However, this method didn’t performance well with imbalanced cases, whilst our
target is to classify the regulatory news (a very small subgroup) from all. In
our training data, the ratio is $1:7$. Indeed there are multiple ways to solve
the unbalanced data problem, such as oversampling or class-weighted SVM by
assigning higher misclassification penalties. But the pre-process of
oversampling or undersampling will change the distribution of labels and
further change the distribution of test data. Our index is constructed based
on regulatory news frequency and it is therefore sensitive to the distribution
of classes.

On the other hand, we could assume that when policy-related topics are
discussed, similar topics with their key words are used and their
distributions are close. Based on that assumption, the Latent Dirichlet
Allocation (LDA) method can be employed to analyze the topic distribution and
words distribution for the corpus and further identify the policy-related
articles according to the similarity calculation.

###  4.1 LDA Topic Modeling

The Latent Dirichlet Allocation (LDA) technique proposed by Blei et al.
(2003), is an unsupervised machine learning algorithm that learns the
unobserved topics of a corpus (individual news articles in this paper). This
technique is widely applied to establish the thematic structure of text and
other discrete data in the linguistic, information retrieval, biologic and
even engineering literatures (see Blei, 2012 for a review of topic modelling
and its application to various text collections).

The LDA technique is based on a generative statistical method to identify the
distribution of words that contribute to a topic, while simultaneously
constructing documents with different probabilities of topics, meaning that
each topic $z$ is annotated with a collections of the most probable words $w$,
and each document $d$ is annotated with a collections of the most probable
topics $z$. It is an unsupervised algorithm which requires no labeled texts
and learns these two latent (unobserved) distributions $p\left(w|z\right)$ and
$p\left(z|d\right)$ by acquiring model parameters that maximize the
probability of each word appearing in each document with the number of topics
$K$ as given.

Then, with Bayes theorem, the probability of observed word $w_{n}$ appearing
in a document $d_{m}$ is given by:

$\displaystyle p\left(d_{m},w_{n}\right)$ $\displaystyle=$ $\displaystyle
p\left(d_{m}\right)p\left(w_{n}|d_{m}\right)$ (1) $\displaystyle=$
$\displaystyle
p\left(d_{m}\right)\sum_{k=1}^{K}p\left(w_{n}|z_{k}\right)p\left(z_{k}|d_{m}\right)$
(2)

where $z_{k}$ is a latent variable indicating the $k$th topic from which the
words were drawn ($Z$ in Figure 4), $p\left(w_{n}|z_{k}\right)$ is a
distribution for each topic over the vocabulary ($\phi$ in Figure 4), and
$p\left(z_{k}|d_{m}\right)$ denotes the topic proportions for the $m$th
document (article in this paper) ($\theta$ in Figure 4). Intuitively, $\phi$
indicates which words weight more to a topic, while $\theta$ states the
importance of those topics to a document.

Figure 4: Graphic LDA Model

Both $\theta$ and $\phi$ follow the Dirichlet distribution with hyper-
parameter $\alpha$ and $\beta$ respectively. With higher $\alpha$, the topic
distribution per article turns to be more specific, while similarly, higher
$\beta$ leads to a more specific word distribution per topic. In general,
$\alpha$ links to the similarity of documents, meaning that a higher alpha
value implies that documents are embodied by more similar weights of each
topic. The same holds for $\beta$ but meaning that a higher beta value
indicates that topics contents more similar weights of each word. In the
Python package gensim, the symmetric or asymmetric hyper-parameters are
learned from data. The generative process of LDA is based on the following
joint distribution of the observed variables $w$ and the unobserved variables
$z$, $\theta$, $\phi$, $\alpha$ and $\beta$,

${\displaystyle
p({\boldsymbol{w}},{\boldsymbol{z}},{\boldsymbol{\theta}},{\boldsymbol{\phi}};\alpha,\beta)=\prod_{k=1}^{K}p(\phi_{k};\beta)\prod_{d=1}^{M}p(\theta_{d};\alpha)\prod_{n=1}^{N}p(z_{d,k}\mid\theta_{d})p(w_{d,n}\mid\phi_{z_{d,k}}),}$
(3)

###  4.2 Number of Topics for LDA

The standard LDA model proposed by Blei et al. (2003) has a significant
weakness that it requires pre-determination of the number of topics, meaning
that users should set the number of unobserved topics manually before applying
the method. The quality of LDA model is heavily depended on the choice of
topic number. Therefore, many believe that choosing the best value for the
topic numbers is more art than science (Azqueta-Gavaldón, 2017).

A common solution is to plug in a set of values and pick the optimal topic
number either based on some intrinsic criterion, such as the coherence of the
topics, or based on some extrinsic criterion, such as accuracy on a specific
task, e.g. paraphrase identification. There also exist other methods to help
with choosing the number of topics. For example, nonparametric Bayesian
models, e.g. Hierarchical Dirichlet process are employed to automatically
generate the number of topics (Teh et al., 2004). However, it is
computationally inefficient to apply such nonparametric models to LDA (Wallach
et al., 2009).

Coherence measures which are based on word co-occurrence are widely applied to
quantify the quality of topic models. The poor quality topics with the type of
“chained”, “intruded” and “random” could be detected using detected with
coherence measures (Mimno et al., 2011). Newman et al. (2010) proposed a
coherence measure which is comparable to the human rating of topics. Their
coherence measure ($C_{\mathrm{UCI}}$) takes the set of the top J words
($w_{1},...,w_{J}$) for a given topic and sum a confirmation measure over all
word pairs. The function is given as follows:

$C_{\mathrm{UCI}}=\frac{2}{J\cdot(J-1)}\sum_{i=1}^{J-1}\sum_{j=i+1}^{J}\log\frac{P\left(w_{i},w_{j}\right)+\epsilon}{P\left(w_{i}\right)\cdot
P\left(w_{j}\right)}$ (4)

where the probabilities are estimated on Wikipedia outperform which is used as
external reference corpus. Mimno et al. (2011) employ an asymmetrical
confirmation measure between top word pairs in the calculation of coherence
$C_{\mathrm{UMass}}$:

$C_{\mathrm{UMass}}=\frac{2}{J\cdot(J-1)}\sum_{i=2}^{J}\sum_{j=1}^{i-1}\log\frac{P\left(w_{i},w_{j}\right)+\epsilon}{P\left(w_{j}\right)}$
(5)

Unlike $C_{\mathrm{UCI}}$, the probabilities in function 5 are estimated based
on the original corpus with applied to trained topic models. Röder et al.
(2015) build a coherence framework and report a measure ($C_{v}$) with the
best performance. Different from $C_{\mathrm{UMass}}$ and $C_{\mathrm{UCI}}$,
$C_{v}$ defines the confirmation using normalized point-wise mutual
information (NPMI) for the $j-th$ element of the context vector
$\overrightarrow{v_{i}}$ of word $w_{i}$:

$v_{ij}=\mathrm{NPMI}\left(w_{i},w_{j}\right)^{\gamma}=\left(\frac{\log\frac{P\left(w_{i},w_{j}\right)+\epsilon}{P\left(w_{i}\right)\cdot
P\left(w_{j}\right)}}{-\log\left(P\left(w_{i},w_{j}\right)+\epsilon\right)}\right)^{\gamma}$
(6)

where $\gamma$ denotes the weight for NPMI. In this paper, we use coherence
value $C_{v}$ as the criteria to select model.

###  4.3 LDA-Based Similarity Measurement and Classification

Semantic similarity problems can be classified according to different levels
of granularity, specifically ranging from word-to-word to sentence-to-sentence
to document-to-document similarities (Niraula et al., 2013). In this paper,
our task is to analyze document-to-document similarity, particularly as a
binary decision problem in which an article is policy related or not. We rely
on one probabilistic method, LDA, which regards documents as distribution over
topics and topics as distribution over words. So, we assume that policy-
related articles have similar topic distributions.

The Hellinger distance can be applied to compute the distance between two
distributions. For document $p$ and document $q$, the distributions of topics
are $z_{p}=\left(z_{p,1},\dots,z_{p,k},\dots,z_{p,K}\right)$ and
$z_{q}=\left(z_{q,1},\dots,z_{q,k},\dots,z_{q,K}\right)$ respectively.
Hellinger Distance $d_{H}$ between those two news with $K$ topics is given as
follows:

$d_{H}\left(z_{p},z_{q},X\right)=\frac{1}{\sqrt{2}}\sqrt{\sum_{k=1}^{K}(\sqrt{z_{p,k}}-\sqrt{z_{q,k}})^{2}}$
(7)

There are reasons that we choose Hellinger distance rather than other
distances to calculate news similarity. First, if we denote $\hat{f}(x)$ as a
kernel density estimator, the asymptotic distribution of
$\sqrt{nh}(\hat{f}(x)-f(x))$ depends on $f(x)$, the true distribution,
however, after taking square root, the asymptotic distribution of
$\sqrt{nh}(\sqrt{\hat{f}(x)}-\sqrt{f(x)})$ eliminates its dependency with
$f(x)$ (see the proof in Appendix). Second, when applying Hellinger distance
in a binary decision criterion, it is not sensitive to the class skew (Cieslak
and Chawla, 2008), implying that it performs well with imbalance data.
Besides, the results from Hellinger distance is bounded by $[0,1]$ for all
values of $z_{p,k}$ and $z_{q,k}$. It is easy to read and compare. The highest
value, 1, indicates the maximized distance and therefore means that the
compared two distributions differ from each other significantly, whereas the
value 0 implies the highest similarity and shortest distances. Meanwhile,
$d_{H}$ is symmetric, meaning $d_{H}\left(z_{p},z_{q}\right)$ =
$d_{H}\left(z_{1},z_{p}\right)$.

As the next step, we calculate the average distance between the article $i$
and all policy-related news ${d}_{i}$:

$\bar{d}_{l,i}=\frac{1}{N_{r}}\sum_{j=1}^{N_{r}}d_{H}(z_{l,i},z_{r,j})$ (8)

where $N_{r}$ is the number of regulatory news, $z_{u,i}$ denotes the topic
distribution of article $i$ and $l=\\{r,\text{ }non\text{ }or\text{ }u\\}$
meaning that the article $i$ is regulatory news “$r$”, non-regulatory news
“$non$” or unclassified news “$u$”. $z_{r,j}$ represents topic distribution of
regulatory new $j$ ($j=1,...,N_{r}$).

Since we have the assumption that regulatory news have smaller distances
between each other than the other news, the average distance
$\bar{d}_{r}=\\{\bar{d}_{r,1},...,\bar{d}_{r,N_{r}}\\}$ for all policy-related
news should be relatively smaller than
$\bar{d}_{non}=\\{\bar{d}_{n,1},...,\bar{d}_{n,N_{non}}\\}$ for all non-
policy-related news. Then if $\bar{d}_{u,i}$ of the unclassified article $i$
is small and close to $\bar{d}_{r}$, we mark that news as policy-related. In
this paper, we set a threshold $\underline{d}$ equals to $\tau^{th}$ quantile
of $\bar{d}_{r}$ and $\tau=0.95$ in this paper.

###  4.4 Construction of CRRIX

As mentioned before, the construction of CRRIX is simply the coverage
frequency of policy-related news, as followed:

$CRRIX^{s}_{t}=\frac{N^{s}_{t,reg}}{N^{s}_{t,all}}$ (9)

where $s$ is the periodicity and $s=\\{daily,\text{ }weekly\text{ }or\text{
}monthly\\}$. $N_{t,reg}$ and $N_{t,all}$ are the number of regulatory news
and all news at time $t$.

##  5 Empirical Results

First we did some pre-processing of the data (words) : stopwords are
eliminated (words that do not informatively or semantically contribute to an
article, e.g. “at”, “or”, “and”); all words have been converted to lower case.
We calculate the coherence value of LDA models with different topic numbers
from $2$ to $25$ given an automatic generated hyperparameter $\alpha$
($\alpha$=0.01) and $\beta=0.1$. The Figure 5 indicates that the best
performanced model with optimal number of topics for the corpus in this paper
is the model with $K=14$. When $K=14$, the model has the highest coherence
value and when the number of topics increases after the optimal choice, the
coherence value turns to relatively stable. We also test the robustness with
different $\alpha$ and $\beta$ from 0.01 to 0.3. The above mentioned
combination performances best but the coherence value doesn’t change much for
given topic number.

Figure 5:  Coherence value for different numbers of topics $K$

CRRIX

Figure 6: The distances between topics using Jaccard distance (left) and
Hellinger distance (right)

Another criterion for determining the quality of the LDA model is to concern
the diversification of topics. Ideally, we would like to see topics as
different as possible. The heating maps in the Figure 6 exhibit the distances
between topics ($K=14$). The red cell represents strongly uncorrelated topics,
while the blue cell indicates high correlation. The left diagram was generated
using Jaccard distance and for the right one, we apply Hellinger distance to
calculate the differences between topics. All elements except those in the
diagonal in both diagrams are red or reddish, which means the 14 topics are
relatively different from each other and our LDA model performance well in
this aspect. However, compared with Hellinger distance, even though Jaccard
distance is robust and wildly used in ML methodologies, it is less sensitive
than Hellinger distance. Therefore, in the later discussion, we only apply
Hellinger’s method in the distance calculation.

Table 1: Categories (Coindesk.com) matched by LDA topics

Coindesk Subcategory LDA Topic Top Keywords Opinions Opinions Bitcoin, say,
people, make, go get, take, would, could, way Tech Technology System,
blockchain, use, transaction, chain Technology, security, work, datum, network
Business Business Company, say, business, new, service base, startup, firm,
founder, CEO Policy & Regulation Regulation Currency, business, virtual, law,
state Regulation, money, digital, exchange, tax Market Investment Bitcoin,
market, currency, price, exchange value, investor, Litecoin, trade, investment
Trading and Exchange Exchange, BTC, account, customer, trading User, deposit,
page, trade, fund Feature Mining mine, power, asic, block, hash chip, network,
unit, hardware, pool Coins Coin, project, Dogecoin, game, Altcoin developer,
community, donate, crowdfunder, token

In order to further show the performance of the trained LDA model, we try to
compare the topics with other sources. In the leading Cryptocurrency news
platform, Coindesk, the news are labelled as those categories: “Opinions”,
“Tech”, “Business”, “Policy & Regulations”, “Market” and “Feature”. These
categories appear in Table 1 (column 1) together with their equivalent topic
(column 2) which is generated by our LDA model and the list of representative
words for each topic (column 3). From the table we can see that for the major
categories in the popular platforms, we could find the corresponding topics in
our model. In the case of “Market”, topics go beyond the categories proposed
by the platform. We must admit that parts of the category “Business” overlaps
with the category “Market”. Even though we select the topic “Trading and
Exchange” to match the category “Market”, it could also be put under a bigger
concept of “Business”. In this sense, the machine learning LDA technique
performs better and clearly identifies topics which keep distances with each
other.

We use the trained LDA model to calculate the Hellinger distances. We find
that the distributions of average distances between each regulatory news and
all other regulatory news $\bar{d}_{r}$ and of average distances between each
non-regulatory news and all regulatory news $\bar{d}_{non}$ are significantly
different. Policy-related news are similar with smaller distances, whereas the
most non-policy-related news are further away. Then we calculate the average
Hellinger distance $\bar{d}_{u,i}$ for each unclassified article $i$. Those,
which are smaller than 0.392, the 0.95 quantile of $\bar{d}_{r}$, will be
classified to the policy-related group.

The classification results of our method based on LDA was compared with those
generated by Naive Bayes and SVM, two broadly used supervised ML
classification methods. The confusion matrices of classification results and
the manually classified data can be found in Table 2. “True” value is given by
human involved annotation, and “Pred” value is predicted by the here applied
ML technique. Number $1$ means that the given news is labelled as policy-
related and number those marked by $0$ is non-policy-related.

Table 2: Confusion matrix of classification for three methods (Naive Bayes, Class-weighted SVM and LDA) True\Pred | NB |  SVMcw | LDA | Total
---|---|---|---|---
1 | 0 | 1 | 0 | 1 | 0
1 | 0 | 582 | 0 | 582 | 361 | 221 | 582
0 | 0 | 4004 | 0 | 4004 | 188 | 3816 | 4004
Total | 0 | 4586 | 0 | 4586 | 549 | 4037 | 4586
Accuracy | 0.873 | 0.873 | 0.907 |

The accuracies of all three methods are relatively high, over 0.87 (seen in
Table 2). However, for supervised ML methods Naive Bayes and Class-weighted
SVM, they simply label all articles non-policy-related. Even with the high
accuracy, those methods can not help with our research question. The purpose
of classification in this paper is to find the ratio of policy-related news
over all. With the increase of news taken in to the calculation, but 0 policy-
related news was identified, the index will go towards destruction.

Meanwhile, the accuracy of our methods is higher, 0.91. Although from the
Table 2 we can read that the type I error for LDA classification are also
high, almost $40$ percent, it still could contribute to build the index and in
this sense performances much better than NB and SVM.

Figure 7: CRRIX (Monthly) with news and price highlights

Multiple reasons could contribute to the misclassification. One could be that
the LDA based criteria are relatively strict. News, which have the distance to
all policy-related news as close as that of 95% the pre-identified regulatory
news, would be counted for regulatory news. Those with a bit larger Hellinger
distance are all excluded. Another reason could come from the data itself.
Cryptocurrency market is young and the core policies, which were discussed by
the public and announced by the governments, were time-varying. Using all time
slot from the year 2013 to 2019 might be biased. A dynamic LDA with rolling
window will solve this problem but it requires sufficient data points. In this
paper, we didn’t consider this method, since the data is limited.

Figure 8: VCRIX (in red) and Regulatory Risk Index of Cryptocurrency Market
(in blue)

CRRIX

After each article was labeled according to its distance with other pre-
identified policy-related news, we simply count the number of articles under
the class “regulation” for a give time slot and divide it by the number of all
articles for the same time. Since the daily time series is too noisy,
especially at the early stage of the market, we only consider weekly time
steps. Figure 7 indicates that the peaks and jumps of the index are mainly led
by big policy changes. The red arrow means positive policy which brought an
increase to the price of Bitcoin, whereas the green arrow is vice versa. The
number next to arrows is the weekly return rate (positive with red arrow and
negative with green arrow).

Figure 7 reveals that the changes of policies are accompanied with drastic
price fluctuations which bring high risk to the market. Out index successfully
captures those big changing moments.

Figure 8 shows that our regulatory risk index is closely related to VCRIX, a
volatility index for CCs market. Especially for the period from Sep 2017 to
March 2018, the extremely high volatility is driven by the policy uncertainty.
The movements for both VCRIX and the regulatory risk index are synchronous.
The correlation between these two indices is 0.44712. Our regulatory risk
index could contribute to forecast the market movement.

We further test the causality between CRRIX and VCRIX. First we do Dicky
Fuller test to confirm stationary of both time series. The results reject the
non-stationary hypothesis ($p-value$ equals to $0.000895$ for VCRIX and
$0.004617$ for CRRIX). Here we only show the Granger causality test results
for lag 1 in the Table 3. The null hypothesis for Granger causality test is
that the time series CRRIX, does NOT Granger cause the time series VCRIX.
Here, for lag 1, we reject the null hypothesis. This means that the past (lag
1) values of CRRIX (lag 1) have a statistically significant effect on the
current value of VCRIX. The results hold for lag 1 to 7.

Table 3: Granger causality test results for lag $1$ number of lags (no zero) 1 |  |  |  |
---|---|---|---|---
ssr based F test: | F=23.1736 | p=0.0000 | df_denom=825 | df_num=1
ssr based chi2 test: | chi2=23.2579 | p=0.0000 | df=1 |
likelihood ratio test: | chi2=22.9372 | p=0.0000 | df=1 |
parameter F test: | F=23.1736 | p=0.0000 | df_denom=825 | df_num=1

##  6 Conclusion

In this paper, via the machine learning tool LDA, we quantify the risks
originating from introducing regulations on the cryptocurrency market and
identify their impact on the cryptocurrency investments. Indices have been
constructed to track the Cryptocurrency markets, however, none of these
indices directly address regulatory risks. The indices introduced in Baker et
al. (2016) focus on economic policy uncertainty in general. Similar to that,
we construct a regulatory risk index for cryptocurrencies that is based on the
policy-related news coverage frequency. Unlike the classical annotation
approach, which involves a meticulous manual process, we employ the ML-LDA
rather costless and efficient method to classify policy-related news.

We first generally reviewed the development of cryptocurrencies and the trend
of regulatory dynamics. Based on that, we discussed the research questions:
What exactly is the regulatory risk for Cryptocurrencies? How to construct an
index of regulatory risk for Cryptocurrency market based on news data? What is
the impact of regulatory risk to the market?

In order to address the answer to those questions, we first collected news
data from the top online cryptocurrency news platform (Guides, 2018), Coindesk
and Bitcoin Magazine, via a dynamic web scraper. In addition, the CRIX and
VCRIX is chosen to represent the value and the volatility of the entire
cryptocurrency market for the later analysis.

To calculate the coverage frequency, we tried to solve the problem of semantic
similarity as a binary decision problem in which an article is policy related
or not, using Latent Dirichlet Allocation (LDA), which models the underlining
topics for a corpus of documents, where each topic is a mixture over words and
each document is a mixture over topics.

The topics given by LDA were comparable with the leading Cryptocurrency news
platform. For the major categories in the popular platforms, we could find the
corresponding topics in our model and the clearly identified topics kept
distances with each other. According to our model, the top words for
regulation topic are: currency, business, virtual, law, state, regulation,
money, digital, exchange and tax.

We use the trained LDA model to calculate the Hellinger distances. Those with
small average distance will be classified to the group of policy-related news.
The results were compared with that of Naive Bayes and Class-weighted SVM
methods. Since our data is very imbalanced, the performance of those to
supervised ML methods were not helpful. But our LDA based distance
classification turned to a high accuracy 0.91 and could contribute to
construct the index.

The final results of the regulatory risk index are shown in Figure 7 and in
Figure 8. Our index successfully captures those big policy changing moments.
The movements for both VCRIX and the regulatory risk index are synchronous,
and the Granger test proved the causality of CRRIX to the market volatility.

## References

  * (1)
  * Auer and Claessens (2020) Auer, Raphael and Stijn Claessens, “Cryptocurrency market reactions to regulatory news,” Discussion Paper DP14602, CEPR April 2020.
  * Azqueta-Gavaldón (2017) Azqueta-Gavaldón, Andrés, “Developing news-based economic policy uncertainty index with unsupervised machine learning,” Economics Letters, 2017, 158, 47–50.
  * Bai and Wang (2015) Bai, Yiqi and Jie Wang, “News classifications with labeled LDA,” in “2015 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K),” Vol. 1 IEEE 2015, pp. 75–83.
  * Baker et al. (2012) Baker, Scott R, Nicholas Bloom, and Steven J Davis, “Has economic policy uncertainty hampered the recovery?,” Government policies and the delayed economic recovery, 2012, 70.
  * Baker et al. (2016) ,  , and  , “Measuring economic policy uncertainty,” The quarterly journal of economics, 2016, 131 (4), 1593–1636.
  * Barrdear and Kumhof (2016) Barrdear, John and Michael Kumhof, “The macroeconomics of central bank issued digital currencies,” Staff Working Paper 605, Bank of England July 2016.
  * Baur et al. (2018) Baur, Dirk G, Kihoon Hong, and Adrian D Lee, “Bitcoin: Medium of exchange or speculative assets?,” Journal of International Financial Markets, Institutions and Money, 2018, 54, 177–189.
  * Binder (1985) Binder, John J, “Measuring the effects of regulation with stock price data,” The RAND Journal of Economics, 1985, pp. 167–183.
  * Blandin et al. (2019) Blandin, Apolline, Ann Sofie Cloots, Hatim Hussain, Michel Rauchs, Rasheed Saleuddin, Jason G Allen, Katherine Cloud, and Bryan Zheng Zhang, “Global cryptoasset regulatory landscape study,” Available at SSRN, 2019.
  * Blei (2012) Blei, David M, “Probabilistic topic models,” Communications of the ACM, 2012, 55 (4), 77–84.
  * Blei et al. (2003) , Andrew Y Ng, and Michael I Jordan, “Latent dirichlet allocation,” Journal of machine Learning research, 2003, 3 (Jan), 993–1022.
  * Borke and Härdle (2018) Borke, Lukas and Wolfgang K Härdle, “Q3-D3-LSA: D3. js and Generalized Vector Space Models for Statistical Computing,” in “Handbook of Big Data Analytics,” Springer, 2018, pp. 377–424.
  * Buckland and Fraser (2001) Buckland, Roger and Patricia Fraser, “Political and regulatory risk in water utilities: Beta sensitivity in the United Kingdom,” Journal of Business Finance & Accounting, 2001, 28 (7-8), 877–904.
  * Caytas (2018) Caytas, Joanna Diane, “Regulation of Cryptocurrencies and Initial Coin Offerings in Switzerland: Declared Vision of a ‘Crypto Nation’,” Includes Chapter News, 2018, 31 (1), 53.
  * Chen et al. (2009) Chen, Jingnian, Houkuan Huang, Shengfeng Tian, and Youli Qu, “Feature selection for text classification with Naïve Bayes,” Expert Systems with Applications, 2009, 36 (3), 5432–5435.
  * Chen et al. (2015) Chen, Xingyuan, Yunqing Xia, Peng Jin, and John Carroll, “Dataless text classification with descriptive LDA,” in “Twenty-Ninth AAAI Conference on Artificial Intelligence” 2015.
  * Cieslak and Chawla (2008) Cieslak, David A and Nitesh V Chawla, “Learning decision trees for unbalanced data,” in “Joint European Conference on Machine Learning and Knowledge Discovery in Databases” Springer 2008, pp. 241–256.
  * da Gama Silva et al. (2019) da Gama Silva, Paulo Vitor Jordão, Marcelo Cabus Klotzle, Antonio Carlos Figueiredo Pinto, and Leonardo Lima Gomes, “Herding behavior and contagion in the cryptocurrency market,” Journal of Behavioral and Experimental Finance, 2019, 22, 41–50.
  * Dnes and Seaton (1999) Dnes, Antony W and Jonathan S Seaton, “The regulation of British Telecom: An event study,” Journal of Institutional and Theoretical Economics (JITE)/Zeitschrift für die gesamte Staatswissenschaft, 1999, pp. 610–616.
  * Dolan (2020) Dolan, Shelagh, “How the laws & regulations affecting blockchain technology and cryptocurrencies, like Bitcoin, can impact its adoption,” https://www.businessinsider.com/blockchain-cryptocurrency-regulations-us-global?r=DE&IR=T 2020\.  Accessed March 8 2020.
  * Fiorillo (2018) Fiorillo, Steve, “Bitcoin History: Timeline, Origins and Founder,” The Street, 2018.
  * Flamary and Courty (2017) Flamary, R’emi and Nicolas Courty, “POT Python Optimal Transport library,” 2017.
  * George et al. (2020) George, Ammu, Taojun Xie, and Joseph Alba, “Central Bank Digital Currency with Adjustable Interest Rate in Small Open Economies,” Technical Working Paper, Asia Competitiveness Institute June 2020.
  * Geva and Zahavi (2014) Geva, Tomer and Jacob Zahavi, “Empirical evaluation of an automated intraday stock recommendation system incorporating both market data and textual news,” Decision support systems, 2014, 57, 212–223.
  * Ghiassi et al. (2013) Ghiassi, Manoochehr, James Skinner, and David Zimbra, “Twitter brand sentiment analysis: A hybrid system using n-gram analysis and dynamic artificial neural network,” Expert Systems with applications, 2013, 40 (16), 6266–6282.
  * Glaser et al. (2014) Glaser, Florian, Kai Zimmermann, Martin Haferkorn, Moritz Christian Weber, and Michael Siering, “Bitcoin-asset or currency? revealing users’ hidden intentions,” Revealing Users’ Hidden Intentions (April 15, 2014). ECIS, 2014.
  * Gold and McBride (2019) Gold, Zack and Megan McBride, “Cryptocurrency: A Primer for Policy-Makers,” 2019.
  * Griffiths and Steyvers (2004) Griffiths, Thomas L and Mark Steyvers, “Finding scientific topics,” Proceedings of the National academy of Sciences, 2004, 101 (suppl 1), 5228–5235.
  * Guides (2018) Guides, Trading Strategy, “Top 10 Cryptocurrency Blogs (You Should Follow in 2019),” 2018.
  * Gulen and Ion (2016) Gulen, Huseyin and Mihai Ion, “Policy uncertainty and corporate investment,” The Review of Financial Studies, 2016, 29 (3), 523–564.
  * Hafner (2018) Hafner, Christian, “Testing for bubbles in cryptocurrencies with time-varying volatility,” Available at SSRN 3105251, 2018.
  * Hafner (2020) Hafner, Christian M, “Testing for bubbles in cryptocurrencies with time-varying volatility,” Journal of Financial Econometrics, 2020, 18 (2), 233–249.
  * Härdle et al. (2017) Härdle, Wolfgang Karl, Cathy Yi-Hsuan Chen, and Ludger Overbeck, Applied quantitative finance, Springer, 2017.
  * Houben and Snyers (2018) Houben, Robby and Alexander Snyers, Cryptocurrencies and blockchain: Legal context and implications for financial crime, money laundering and tax evasion 2018\.
  * Kang and Ratti (2013) Kang, Wensheng and Ronald A Ratti, “Oil shocks, policy uncertainty and stock market return,” Journal of International Financial Markets, Institutions and Money, 2013, 26, 305–318.
  * Kim et al. (2019) Kim, Alisa, Simon Trimborn, and Wolfgang K Härdle, “VCRIX-A Volatility Index for Crypto-Currencies,” Available at SSRN 3480348, 2019\.
  * Klein et al. (2018) Klein, Tony, Hien Pham Thu, and Thomas Walther, “Bitcoin is not the New Gold–A comparison of volatility, correlation, and portfolio performance,” International Review of Financial Analysis, 2018, 59, 105–116.
  * Lee and Deng (2018) Lee, David and Robert H Deng, “Handbook of blockchain, digital finance, and inclusion: Cryptocurrency, FinTech, InsurTech, and regulation,” 2018.
  * Lee et al. (2015) Lee, Seonggyu, Jinho Kim, and Sung-Hyon Myaeng, “An extension of topic models for text classification: A term weighting approach,” in “2015 International Conference on Big Data and Smart Computing (BIGCOMP)” IEEE 2015, pp. 217–224.
  * Lin et al. (2013) Lin, Yung-Shen, Jung-Yi Jiang, and Shie-Jue Lee, “A similarity measure for text classification and clustering,” IEEE transactions on knowledge and data engineering, 2013, 26 (7), 1575–1590.
  * Linton et al. (2017) Linton, Marco, Ernie Gin Swee Teo, Elisabeth Bommes, CY Chen, and Wolfgang Karl Härdle, “Dynamic topic modelling for cryptocurrency community forums,” in “Applied Quantitative Finance,” Springer, 2017, pp. 355–372.
  * Mimno et al. (2011) Mimno, David, Hanna M Wallach, Edmund Talley, Miriam Leenders, and Andrew McCallum, “Optimizing semantic coherence in topic models,” in “Proceedings of the conference on empirical methods in natural language processing” Association for Computational Linguistics 2011, pp. 262–272.
  * Nakamoto (2019) Nakamoto, Satoshi, “Bitcoin: A peer-to-peer electronic cash system,” Technical Report, Manubot 2019.
  * Nakamoto et al. (2008) et al., “Bitcoin: A peer-to-peer electronic cash system.(2008),” 2008.
  * Nassirtoussi et al. (2014) Nassirtoussi, Arman Khadjeh, Saeed Aghabozorgi, Teh Ying Wah, and David Chek Ling Ngo, “Text mining for market prediction: A systematic review,” Expert Systems with Applications, 2014, 41 (16), 7653–7670.
  * Newman et al. (2010) Newman, David, Jey Han Lau, Karl Grieser, and Timothy Baldwin, “Automatic evaluation of topic coherence,” in “Human language technologies: The 2010 annual conference of the North American chapter of the association for computational linguistics” Association for Computational Linguistics 2010, pp. 100–108.
  * Niraula et al. (2013) Niraula, Nobal, Rajendra Banjade, Dan Ştefănescu, and Vasile Rus, “Experiments with semantic similarity measures based on lda and lsa,” in “International conference on statistical language and speech processing” Springer 2013, pp. 188–199.
  * Prager (1992) Prager, Robin A, “The effects of deregulating cable television: evidence from the financial markets,” Journal of Regulatory Economics, 1992, 4 (4), 347–363.
  * Redondo (n.d.) Redondo, Connor Lamon Eric Nielsen Eric, “Cryptocurrency Price Change Prediction Using News.”
  * Röder et al. (2015) Röder, Michael, Andreas Both, and Alexander Hinneburg, “Exploring the space of topic coherence measures,” in “Proceedings of the eighth ACM international conference on Web search and data mining” 2015, pp. 399–408.
  * Schwert (1981) Schwert, G William, “Measuring the effects of regulation: evidence from the capital markets,” Journal of Law and Economics, 1981, 24 (1), 121–58.
  * Sebastiani (2002) Sebastiani, Fabrizio, “Machine learning in automated text categorization,” ACM computing surveys (CSUR), 2002, 34 (1), 1–47.
  * Teh et al. (2004) Teh, Yee Whye, Michael I Jordan, Matthew J Beal, and David M Blei, “Sharing clusters among related groups: Hierarchical dirichlet processes,” in “Proceedings of the 17th International Conference on Neural Information Processing Systems” MIT Press 2004, pp. 1385–1392.
  * Tong and Koller (2001) Tong, Simon and Daphne Koller, “Support vector machine active learning with applications to text classification,” Journal of machine learning research, 2001, 2 (Nov), 45–66.
  * Trimborn and Härdle (2018) Trimborn, Simon and Wolfgang Karl Härdle, “CRIX an Index for cryptocurrencies,” Journal of Empirical Finance, 2018, 49, 107–122.
  * Trimborn et al. (2018) , Mingyang Li, and Wolfgang K Härdle, “Investing with cryptocurrencies-A liquidity constrained investment approach,” Published version: Trimborn, S., Li, M. and WK Härdle (2019)” Investing with Cryptocurrencies—a Liquidity Constrained Investment Approach” Journal of Financial Econometrics, doi. org/10.1093/jjfinec/nbz016, 2018.
  * Tu and Härdle (2018) Tu, Jun and Wolfgang Karl Härdle, “Information Arrival, News Sentiment, Volatilities and Jumps of Intraday Returns Ya Qian,” 2018.
  * Wallach et al. (2009) Wallach, Hanna M, David M Mimno, and Andrew McCallum, “Rethinking LDA: Why priors matter,” in “Advances in neural information processing systems” 2009, pp. 1973–1981.
  * Zhong (2019) Zhong, Raymond, “China’s Cryptocurrency Plan Has a Powerful Partner: Big Brother,” https://www.nytimes.com/2019/10/18/technology/china-cryptocurrency-facebook-libra.html?_ga=2.218041619.426277731.1583491045-877433028.1583491045 2019\.  Accessed March 6, 2020.

Appendix

##  Appendix A Proofs for Hellinger Distance

###  A.1 Asymptotic distribution of $\sqrt{nh}[\hat{f}(x)-f(x)]$ depends on
$f(x)$.

  * Proof

Denote $\hat{f}(x)$ is a kernel density estimator,

$\sqrt{nh}[\hat{f}(x)-f(x)]=\sqrt{nh}\\{\hat{f}(x)-\operatorname{E}[\hat{f}(x)]\\}+\sqrt{nh}\\{\operatorname{E}[\hat{f}(x)]-f(x)\\}$
(10)

For the second part,

$\displaystyle\operatorname{Bias}\left\\{\hat{f}_{h}(x)\right\\}$
$\displaystyle=\operatorname{E}\left\\{\hat{f}_{h}(x)\right\\}-f(x)$ (11)
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\operatorname{E}\left\\{K_{h}\left(x-X_{i}\right)\right\\}-f(x)$
$\displaystyle=\operatorname{E}\left\\{K_{h}(x-X)\right\\}-f(x)$
$\displaystyle=\int\frac{1}{h}K\left(\frac{x-u}{h}\right)f(u)du-f(x)$

The transformation $s=\frac{u-x}{h}$, i.e. $u=hs+x$,
$\left|\frac{ds}{du}\right|=\frac{1}{h}$. A second-order Taylor expansion of
$f(u)$ around $x$ is given by

$f(x+hs)=f(x)+f(x)^{\prime}hs+\frac{1}{2}f^{\prime\prime}(x)h^{2}s^{2}+o\left(h^{2}\right)$
(12)

Then,

$\displaystyle\operatorname{Bias}\left\\{\hat{f}_{h}(x)\right\\}$
$\displaystyle=\int\frac{1}{h}K(-s)f(x+hs)hds-f(x)$ (13) $\displaystyle=\int
K(s)[f(x)+f(x)^{\prime}hs+\frac{1}{2}f^{\prime\prime}(x)h^{2}s^{2}+o\left(h^{2}\right)]ds-f(x)$
$\displaystyle=f(x)\int K(s)ds+f(x)^{\prime}h\int
sK(s)ds+\frac{1}{2}f^{\prime\prime}(x)h^{2}\int
s^{2}K(s)ds-f(x)+o\left(h^{2}\right)$
$\displaystyle=\frac{h^{2}}{2}f^{\prime\prime}(x)\mu_{2}(K)+o\left(h^{2}\right),\quad\text{
as }h\rightarrow 0$

where $\int K(s)ds=1$, $\int sK(s)ds=0$ and $\int s^{2}K(s)ds=\mu_{2}(K)$.

For the first part,

$\displaystyle\operatorname{Var}\left\\{\hat{f}_{h}(x)\right\\}$
$\displaystyle=\operatorname{Var}\left\\{\frac{1}{n}\sum_{i=1}^{n}K_{h}\left(x-X_{i}\right)\right\\}$
(14)
$\displaystyle=\frac{1}{n^{2}}\sum_{i=1}^{n}\operatorname{Var}\left\\{K_{h}\left(x-X_{i}\right)\right\\}$
$\displaystyle=\frac{1}{n}\operatorname{Var}\left\\{K_{h}(x-X)\right\\}$
$\displaystyle=\frac{1}{n}\left\\{\operatorname{E}\left[K_{h}^{2}(x-X)\right]-\left\\{\operatorname{E}\left[K_{h}(x-X)\right]\right\\}^{2}\right\\}$
$\displaystyle=\frac{1}{n}\int\frac{1}{h^{2}}K\left(\frac{x-t}{h}\right)^{2}f(t)dt-\frac{1}{n}\left(\frac{1}{h}\int
K\left(\frac{x-t}{h}\right)f(t)dt\right)^{2}$
$\displaystyle=\frac{1}{n}\int\frac{1}{h^{2}}K\left(\frac{x-t}{h}\right)^{2}f(t)dt-\frac{1}{n}(f(x)+\operatorname{Bias}(\hat{f}(x)))^{2}$

Substituting $s=\frac{u-x}{h}$,

$\operatorname{Var}\left\\{\hat{f}_{h}(x)\right\\}=\frac{1}{nh}\int
K(s)^{2}f(x+hs)ds-\frac{1}{n}\left(f(x)+o\left(h\right)\right)^{2}$ (15)

Applying a Taylor approximation yields

$\displaystyle\operatorname{Var}\left\\{\hat{f}_{h}(x)\right\\}$
$\displaystyle=\frac{1}{nh}\int
K(z)^{2}\left(f(x)+hsf^{\prime}(x)+\frac{1}{2}f^{\prime\prime}(x)h^{2}s^{2}+o(h^{2})\right)ds-\frac{1}{n}\left(f(x)+o\left(h\right)\right)^{2}$
(16)
$\displaystyle=\frac{1}{nh}\|K\|_{2}^{2}f(x)+o\left(\frac{1}{nh}\right),\quad\text{
as }nh\rightarrow\infty$

where $\int K^{2}(s)ds=\|K\|_{2}^{2}$. With
$\operatorname{Var}(\hat{f}(x))\rightarrow 0\text{ as }nh\rightarrow\infty$

$\frac{\hat{f}(x)-\operatorname{E}[\hat{f}(x)]}{\sqrt{\operatorname{Var}(\hat{f}(x))}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,1)$ (17)

Substituting the expression for $\operatorname{Var}(\hat{f}(x))$

$\sqrt{nh}\\{\hat{f}(x)-\operatorname{E}[\hat{f}(x)]\\}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,f(x)\|K\|_{2}^{2}\right)$ (18)

If the bandwidth tends to zero faster than the optimal rate, then

$\sqrt{nh}\\{\operatorname{E}[\hat{f}(x)]-f(x)\\}\rightarrow 0$ (19)

and the bias term vanishes from the asymptotic distribution,

$\sqrt{nh}[\hat{f}(x)-f(x)]\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,f(x)\|K\|_{2}^{2}\right)$ (20)

###  A.2 Asymptotic distribution of $\sqrt{nh}[\sqrt{\hat{f}(x)}-\sqrt{f(x)}]$
does not depend on $f(x)$

  * Proof

From transform theorems, we know that if $\text{ If
}\sqrt{n}(t-\mu)\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}N_{p}(0,\Sigma)$,

$\sqrt{n}[f(t)-f(\mu)]\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}N_{q}\left(0,\mathcal{D}^{\top}\Sigma\mathcal{D}\right)\quad\text{
for }n\longrightarrow\infty$ (21)

Denote $g(x)=x^{1/2}$, then $\frac{dg}{dx}=\frac{1}{2}x^{-1/2}$. With
$\sqrt{nh}(\hat{f}(x)-f(x))\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,f(x)\|K\|_{2}^{2}\right)$, then

$\sqrt{nh}[\sqrt{\hat{f}(x)}-\sqrt{f(x)}]\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,\frac{1}{4}\|K\|_{2}^{2}\right)$ (22)

The asymptotic distribution of $\sqrt{nh}[\sqrt{\hat{f}(x)}-\sqrt{f(x)}]$ does
not depend on $f(x)$