# Exploring Radial Oscillations in Slow Stable and Hybrid Neutron Stars

Sayantan Ghosh1 ID Sailesh Ranjan Mohanty1 ID Tianqi Zhao2 ID Bharat Kumar1 ID
<EMAIL_ADDRESS>1 Department of Physics and Astronomy, National Institute
of Technology, Rourkela 769008, India 2Department of Physics and Astronomy,
Ohio University, Athens, Ohio 45701, USA

###### Abstract

In the era of gravitational wave astronomy, radial oscillations hold
significant potential for not only uncovering the microphysics behind the
internal structure but also investigating the stability of neutron stars
(NSs). We start by constructing families of static NSs following nucleonic,
quarkyonic, and hybrid equations of state and then subject them to radial
perturbations in order to explore the stability of these stars. Unlike other
literature where the fluid elements are assumed to be in chemical equilibrium,
we consider the out-of-equilibrium effects on the chemical composition of
fluid elements for the calculation of radial modes. Taking these
considerations into account, we observe that the sound speed ($c^{2}_{s}$) and
adiabatic index ($\gamma$) avoid singularities and discontinuities over the
equilibrium case. We elucidate the response of the fundamental radial modes by
examining the out-of-equilibrium matter distribution scenario, offering
insights into its dynamic variations. We also demonstrate that this approach
extends the stable branches of stellar models, enabling stars to sustain
stable higher-order mass doublets, shedding some light on observation and
existence of PSR J0740+6620.

##  I Introduction

Neutron stars (NSs), remnants of massive star cores, provide a unique
laboratory for nuclear physics due to their dense composition. Pulsations
called quasi-normal modes (QNMs)Kokkotas and Schmidt (1999); Blázquez-Salcedo
_et al._ (2021); Nollert (1999); Berti  _et al._ (2009); Konoplya and Zhidenko
(2011) provide insights into the star’s internal composition, with pressure as
the primary restoring force. Studying QNMs helps probe the equation of state
(EoS) of dense nuclear matter, crucial for understanding NS behaviour. While
radial oscillations can not directly emit gravitational waves (GWs), their
interaction with non-radial oscillations can amplify GW emissions, and create
a stronger GW that might be detected Passamonti  _et al._ (2006, 2007). The
emission of short gamma-ray bursts (SGRBs)D’Avanzo (2015); Berger (2014);
Jordana-Mitjans  _et al._ (2022); Dichiara  _et al._ (2020) in the process of
forming hyper-massive NSs through binary NS mergers, can be influenced by the
modulation of radial oscillations Chirenti  _et al._ (2019). Consequently, it
holds significance not only for comprehending the physics of dense nuclear
matter within a NS but also holds significance in the realm of gravitational
wave (GW)Coulter  _et al._ (2017); Abbott  _et al._ (2017a); Abbott (2019,
2020) physics.

The Sturm–Liouville problem concerning the linear radial oscillations of
variable stars under Newtonian gravity was thoroughly investigated by Ledoux
and Walraven in 1958 Wrubel  _et al._ (1958). When the square of the
eigenfrequency in the radial oscillation equations becomes negative, the fluid
displacement shows exponential growth, resulting in stellar instability.
Chandrasekhar Chandrasekhar (1964) extended these principles to derive and
solve similar radial oscillation equations in the context of general
relativity. This instability is anticipated to occur prior to the
Schwarzschild limit, particularly relevant for compact objects like white
dwarfs (WDs) and NSs.

Conversely, Wheeler et al. proposed another stability criteria based on the
mass-radius diagram, determined by solving the TOV equationsHarrison  _et al._
(1965). They argued that when the total mass decreases with increasing the
central density, e.g. between stable branches of WDs and NSs, the star is no
longer stable under self-gravity, leading to a collapseHarrison  _et al._
(1965). Remarkably, for barotropic monotonic EoSs, these two methods of
determining stellar instability criteria were found to be equivalent Bardeen
_et al._ (1966).

Zero-temperature catalyzed NSs can be accurately described using a barotropic
EoS, where thermal entropy is disregarded, and the chemical composition is
determined by beta equilibrium. This assumption holds true when the relaxation
time for beta equilibrium is significantly shorter than the oscillation
period, which ensures that NS matter remains catalyzed throughout compression
and expansion oscillations. At lower temperatures, when the temperature is
less than 1 MeV, the neutrino relaxation time becomes longer compared to the
oscillation time, preventing the beta-equilibrium condition from being met
during oscillations Alford and Harris (2019). Consequently, matter composition
becomes frozen under compression, leading to a higher adiabatic index. This
effect substantially extends the range of maximally stable configurations for
WDs, from $\rho_{c}=1.2\times 10^{9}$ g.cm-3 to $\rho_{c}=4\times 10^{10}$
g.cm-3, corresponding to the maximum mass configuration of a WD Chanmugam
(1977).

The influence of the slow weak process differs significantly between WDs and
NSs due to variations in the beta-equilibrium composition, which ranges from a
large proton fraction of approximately 0.5 on the stellar surface to
approximately 0.05 around the crust-core transition density. The assumption of
a small proton fraction at the center of a NS aligns the adiabatic index of
the beta-equilibrium matter with that of matter involved in a slow weak
process. However, this assumption may not hold in EoS models where the proton
fraction rises in the core of a NS, reaching values greater than 0.2 Zhao and
Lattimer (2020).

Furthermore, when a phase transition occurs, the adiabatic index differs
significantly between catalyzed matter and matter with a fixed chemical
composition Jaikumar  _et al._ (2021). Specifically, when there is a density
discontinuity due to a first-order phase transition between hadronic matter
and quark matter, this effect can be manifested through slow and rapid
junction conditions Pereira  _et al._ (2018), as well as intermediate junction
conditions Pereira  _et al._ (2018); Rau and Sedrakian (2023); Rau and Salaben
(2023). Although the barotropic EoS is employed for both hadronic matter and
quark matter, it results in two distinct compositions with an infinite
composition gradient Constantinou  _et al._ (2023). This significant
composition gradient supports non-radial g-mode oscillations, which may be
observable in the third generation of gravitational observationsZhao  _et al._
(2022a). In the case of radial oscillations, a large density discontinuity
with slow junction conditions leads to a long branch of a slow-stable hybrid
star Lugones  _et al._ (2023); Ranea-Sandoval  _et al._ (2023), while
sequential density discontinuities result in multiple slow-stable branches
Gonçalves and Lazzari (2022); Rau and Sedrakian (2023); Rau and Salaben
(2023).

This work is the first study of slow, stable NSs (hadronic and hybrid) with no
density discontinuity, where radial oscillations are solved under the
assumption of a fixed chemical composition in the slow transition limit.
However, previous works on non-equilibrium (adiabatic) radial oscillations
focus on hadronic NSs Chanmugam (1977); Gourgoulhon  _et al._ (1995) or hybrid
NSs with a first-order phase transition Lugones  _et al._ (2023); Ranea-
Sandoval  _et al._ (2023); Rau and Sedrakian (2023); Rau and Salaben (2023).

This paper is organized as; in Section (II), we describe theoretical
formalism, where in Sub-sections II.1, II.2, and II.3, we have discussed sound
speed, hydrostatic equilibrium, and radial oscillation, respectively. We
present our results and discussion part in Section (III), and finally in
Section (IV) we conclude our work.

Throughout this paper, we adopt mostly positive signatures (-, +, +, +) and
utilize a geometrized unit system (G=c=1).

##  II Theoretical Framework

We utilize the Zhao-Lattimer (ZL) model Zhao and Lattimer (2020) to
characterize nucleonic matter. The nuclear saturation properties is set as:
saturation density ($n_{sat}$) at 0.16 fm-3, the binding energy ($E_{sat}$) at
-16 MeV, the compression modulus ($K_{sat}$) at 230 MeV, and the symmetry
energy ($S_{v}$) at 31 MeV and the symmetry energy slope ($L$) at 70 MeV,
while keeping the power-law index ($\gamma_{1}$) constant at a value of 2.

For the quark EoS, we use the vMIT bag model Gomes  _et al._ (2019); Klähn and
Fischer (2015) with Lagrangian density

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{q=u,d,s}\left[\bar{\psi}_{q}\left(i\gamma^{\mu}\partial_{\mu}-m_{q}-B\right)\psi_{q}\right]+\mathcal{L}_{\mathrm{vec}},$
(1) $\displaystyle\mathcal{L}_{\text{vec }}$ $\displaystyle=$ $\displaystyle-
G_{v}\sum_{q}\bar{\psi}\gamma_{\mu}V^{\mu}\psi+\left(m_{V}^{2}/2\right)V_{\mu}V^{\mu}.$
(2)

where $V_{\mu}$ represent vector meson field with repulsive interactions
$(G_{v}/m_{V})^{2}=0.2$ fm-2. $B^{1/4}=180$ MeV is a constant reflecting the
cost of confining the quarks inside the bag, and the $m_{q}$ are the current
quark masses (here, $m_{u}=5$ MeV, $m_{d}=7$ MeV, and $m_{s}=150$ MeV).

Radial oscillation and stability of slow stable hybrid stars have been studied
under first-order phase transition with Maxwell construction. Here we study
hybrid NS with crossover phase transitionKapusta and Welle (2021) and first
order phase transition with Gibbs constructionConstantinou  _et al._ (2023).
Details for the construction of hybrid EoS are described in Zhao  _et al._
(2022b).

###  II.1 Sound speed

Phase transitions or crossovers occurring in dense matter within NSs have a
direct impact on the speed of sound, $c_{s}$($n_{B}$), in relation to baryon
density ($n_{B}$). The alterations in internal composition or structure,
prompted by extreme conditions, result in unique signatures discernible in the
propagation of sound Brandes  _et al._ (2023); Ecker and Rezzolla (2022);
Chatterjee  _et al._ (2023); Reed and Horowitz (2020); Altiparmak  _et al._
(2022); Bedaque and Steiner (2015); Ferrer and Hackebill (2023); Yao  _et al._
(2023). The examination of variations in the speed of sound serves as a
crucial tool for probing EoSs and gaining a comprehensive understanding of the
fundamental properties of NSs, including the exotic states of matter they
contain.

The adiabatic sound speed is obtained by Zhao  _et al._ (2022b)

$c^{2}_{ad}(n_{B},y_{i})=\left(\frac{\partial{P}}{\partial{n_{B}}}\right)_{y_{i}}\left(\frac{\partial{\mathcal{E}}}{\partial{n_{B}}}\right)_{y_{i}}^{-1}$
(3)

where $P$ and $\mathcal{E}$ are the total pressure and energy density
respectively and $y_{i}$ is the particle fraction in favour of the total
baryon number density($n_{B}$) [$y_{i}\rightarrow y_{i,\beta}(n_{B})$,
$i=n,p,u,d,s,e,\mu$]. Then evaluating it in $\beta$-equilibrium

$c^{2}_{ad,\beta}(n_{B})=c^{2}_{ad}[n_{B},y_{i,\beta}(n_{B})].$ (4)

The equilibrium sound speed is expressed as the total derivative of the
pressure and energy density concerning the baryon density, following the
implementation of $\beta$-equilibrium,

$c^{2}_{eq}=\left(\frac{dP_{\beta}}{dn_{B}}\right)\left(\frac{d\mathcal{E}_{\beta}}{dn_{B}}\right)^{-1}$
(5)

###  II.2 Hydrostatic equilibrium

A non-rotating NS’s intense gravitational forces lead to a nearly perfect
spherical equilibrium. This symmetry allows for a reasonable assumption of
spherical symmetry. The Schwarzschild metric accurately describes the
gravitational field of such a bodySchwarzschild (1916):

$ds^{2}=-e^{2\nu}dt^{2}+e^{2\lambda}dr^{2}+r^{2}(d\theta^{2}+\text{sin}^{2}\theta
d\phi^{2}),$ (6)

where $\lambda\equiv\lambda(r)$ and $\nu\equiv\nu(r)$ are metric functions,
each following their respective set of equations. Here, the energy-momentum
tensor $T_{\mu\nu}$ takes the form of a perfect fluid:

$T_{\mu\nu}=(P+\mathcal{E})u_{\mu}u_{\nu}+Pg_{\mu\nu},$ (7)

where $P$ is pressure, $\mathcal{E}$ is energy density, and $u_{\mu}$
represents covariant velocity. In the context of spherical symmetry, the
components $u_{0}$ and $u_{1}$ exhibit non-zero values along the radial
direction.

Applying Einstein’s field equations to the Schwarzschild metric in eq. (6)
under equilibrium conditions and utilizing the boundary condition
$\lambda(r=0)=0$, we obtain:

$e^{-2\lambda(r)}=\left(1-\frac{2m}{r}\right),$ (8)

The mass $m$ can be determined through integration using:

$\frac{dm}{dr}=4\pi r^{2}\mathcal{E}.$ (9)

Likewise, employing the law of conservation of momentum, we obtain Landau and
Lifshitz (1967):

$\frac{d\nu}{dr}=-\frac{1}{P+\mathcal{E}}\frac{dP}{dr}.$ (10)

Ultimately, employing equation (10) and Einstein’s field equations leads to
Oppenheimer and Volkoff (1939); Tolman (1939):

$\frac{dP}{dr}=-\frac{m}{r^{2}}\frac{\left(P+\mathcal{E}\right)\left(1+\frac{4\pi
r^{3}P}{m}\right)}{\left(1-\frac{2m}{r}\right)}.$ (11)

Eqs. (8) and (10) describe the behaviour of the metric functions within the NS
where $r<R$. At the surface, i.e. at $r=R$, they satisfy the boundary
condition,

$e^{2\nu(R)}=e^{-2\lambda(R)}=\left(1-\frac{2M}{R}\right).$ (12)

Eq. (12) remains valid even beyond the star, where $R$ is replaced by $r$ for
$r>R$, taking on the familiar form of the Schwarzschild solution.

###  II.3 Radial oscillation equations

Maintaining the spherical symmetry of the background equilibrium
configuration, we perturb both fluid and spacetime variables. We assume a
harmonic time dependence for the radial displacement of the fluid element at
position $r$ in the unperturbed model,

$\delta r(r,t)=X(r)e^{i\omega t},$ (13)

The equations for linearized radial perturbations can be expressed as
Chandrasekhar (1964); Kokkotas and Ruoff (2001)

$\displaystyle
c_{s}^{2}X^{\prime\prime}+\left(\left(c_{s}^{2}\right)^{\prime}-Z+{4\pi}r\gamma
Pe^{2\lambda}-\nu^{\prime}\right)X^{\prime}$ (14)
$\displaystyle+{\left[2\left(\nu^{\prime}\right)^{2}+\frac{2m}{r^{3}}e^{2\lambda}-Z^{\prime}-4\pi(P+\mathcal{E})Zre^{2\lambda}+\omega^{2}e^{2\lambda-2\nu}\right]X}$
$\displaystyle=0,$

Here, primes represent differentiation with respect to the radial coordinate
$r$, and $c_{s}^{2}$ denotes the square of the speed of sound. The adiabatic
index, $\gamma$, is related to the speed of sound as follows:

$\gamma=\left(\frac{P+\mathcal{E}}{P}\right)c_{s}^{2}.$ (15)

Now we can define the $\gamma$ by considering the two different kinds of sound
speeds, as

$\gamma_{eq}=\left(\frac{P+\mathcal{E}}{P}\right)c_{eq}^{2},\\\
\gamma_{ad}=\left(\frac{P+\mathcal{E}}{P}\right)c_{ad}^{2},$ (16)

where the suffix ”eq” and ”ad” correspond to the equilibrium and adiabatic
case respectively.

$Z(r)=c_{s}^{2}\left(\nu^{\prime}-\frac{2}{r}\right).$ (17)

Now, we re-define the displacement function as

$\zeta=r^{2}e^{-\nu}X.$ (18)

Eq. (14) can be rewritten for $\zeta$ as

$\frac{d}{dr}\left(H\frac{d\zeta}{dr}\right)+\left(\omega^{2}W+Q\right)\zeta=0,$
(19)

with

$\displaystyle H$
$\displaystyle=r^{-2}(P+\mathcal{E})e^{\lambda+3\nu}c_{s}^{2}$ (20a)
$\displaystyle W$ $\displaystyle=r^{-2}(P+\mathcal{E})e^{3\lambda+\nu}$ (20b)
$\displaystyle Q$
$\displaystyle=r^{-2}(P+\mathcal{E})e^{\lambda+3\nu}\left(\left(\nu^{\prime}\right)^{2}+\frac{4}{r}\nu^{\prime}-{8\pi}e^{2\lambda}P\right).$
(20c)

$H$, $W$, and $Q$ are dependent on the radial coordinate $r$ and can be
computed using the unperturbed background configuration. Notably, eq. (19)
explicitly reveals its self-adjoint nature. The Lagrangian variation of the
pressure now is

$\Delta P=-r^{-2}e^{\nu}(P+\mathcal{E})c_{s}^{2}\zeta^{\prime}.$ (21)

As radial oscillations cannot displace the fluid element at the center, the
boundary condition at the center is

$X(r=0)=0.$ (22)

At the stellar surface, the Lagrangian variation of pressure should vanish,
expressed as

$\Delta P(R)=0.$ (23)

The differential equation (19), subject to the boundary conditions in eqs.
(22) and (23), constitutes a Sturm-Liouville eigenvalue problem Sen  _et al._
(2023); Hladík  _et al._ (2020); Routaray  _et al._ (2023a, b).

Figure 1:  On the left panel, the squared sound speeds ($c^{2}_{s}$) are
presented as functions of baryon number density ($n_{B}$), delineating the
relationship between the local speed of sound and the density of baryonic
matter. The right panel exhibits the adiabatic index ($\gamma$) as a function
of baryon number density ($n_{B}$) for the specified models detailed in the
inset. Solid lines represent the equilibrium case, while dashed-dot lines
depict the adiabatic scenario.

Figure 2: On the left panel EoSs are represented as pressure $P$ versus energy
density $\mathcal{E}$. The composition of the three employed models is as
follows: for ZL, nucleons and leptons; for Gibbs, nucleons, quarks, and
leptons in a soft first-order phase transition description; and for KW, the
same as for Gibbs, but in a cross-over description. The right panel displays
the Mass-Radius relationship obtained for our choice of EoSs. The
astrophysical observable constraints on mass and radius from PSR J0740+6620
Miller  _et al._ (2021); et. al. (2021), and NICER data for PSR J0030+0451
Miller  _et al._ (2019); Riley  _et al._ (2019) are represented by coloured
regions. The outer and inner regions of the blue and orange butterfly
structured plot indicate the 90% (solid) and 50% (dashed) confidence intervals
based on the LIGO-Virgo analysis for Binary Neutron Star (BNS) components of
the GW170817 eventAbbott  _et al._ (2017b); Fattoyev  _et al._ (2018); Patra
_et al._ (2023). Figure 3:  On the left panel, the $f$-mode frequency is
depicted as a function of central energy density ($\mathcal{E}_{c}$), offering
insights into the dynamic behaviour in response to varying internal energy
conditions. The right panel illustrates the $f$-mode frequency against the
mass ($M$).

##  III Results and Discussions

In this section, we discuss numerical results obtained by considering three
different EoSs such as Gibbs40, KW55, and ZL70. Firstly, we have studied the
variation of the sound speed and adiabatic index under different conditions.
Additionally, our investigation focuses on the mass-radius relation of NSs.
Furthermore, our attention extends to the $f$-mode oscillation, to understand
the dynamics of NS.

In the left panel of Fig. 1, we depict the variation of sound speed with
baryon density. Notably, the curves corresponding to the purely nucleonic
model (ZL) consistently exhibit a steep rise as baryon density increases.
Conversely, when quarks are introduced, as illustrated by the KW and Gibbs
models, a distinct behaviour emerges, showing up less dramatically, but still
visibly in terms of a more or less pronounced change of slope in
$c_{s}$($n_{B}$). The presence or enhancement of quarks introduces a notable
deviation from the monotonic trend observed in the purely nucleonic scenario.

In the case of the KW model, the growth in sound speed is either slowed down
or reversed, creating a smoother transition compared to the Gibbs model. For
the Gibbs model, the effect is more abrupt and discontinuous. The enhancement
of quarks in the Gibbs model leads to a sharp alteration in the growth pattern
of sound speed, introducing a distinctive discontinuity in the curves.

When calculating the equilibrium sound speed, we are doing the total pressure
derivative concerning the energy density. However in the adiabatic case, it is
different, here we are calculating the corresponding sound speed for each
composition. So one noteworthy observation is, that the curves become slightly
smoother when considering the adiabatic case Constantinou  _et al._ (2021), in
comparison to the equilibrium case. Now, examining the right panel of Fig. 1,
we observe a similar effect on the adiabatic index ($\gamma$) as seen in the
left panel, reflecting the influence of quark content on both sound speed and
$\gamma$.

The Mass-Radius relation, depicted in the right panel of Fig. 2, illustrates
the diverse behaviour of NSs under different EoSs. At lower masses the NS
radius is larger, reflecting the pressure support provided by the nuclear and
sub-nuclear constituents resisting gravitational collapse. As the mass
increases, the star’s gravity becomes stronger, leading to a more compact
configuration with a smaller radius. However, there is a limit to how much
mass a NS can support against gravity. This upper limit is the maximum mass,
shown in the M-R curve with dot markers. If the mass exceeds this limit, the
NS is unstable in the equilibrium case but is still stable in the adiabatic
case. Star markers denote the NS becomes unstable even in adiabatic cases. In
the regions between the star and the dot markers are known as the slow stable
NS. In regions beyond the star markers (the dotted portion), the NS go through
gravitational collapse into a Black hole (BH). In this representation,
distinct markers convey specific characteristics:

  * •

The star marker signifies the point where the $f$-mode frequency (adiabatic)
tends to zero, occurring beyond the maximum mass configuration.

  * •

The dot marker corresponds to the $f$-mode frequency (equilibrium) reaching
zero, typically aligning with the configuration of maximum mass.

So to study the instability in NS we have to consider the adiabatic case since
it is in the unstable region. Upon closer examination of the figure, a notable
observation is that the maximum mass associated with the ZL EoS surpasses that
of the other two EoSs. The Gibbs EoS exhibits a marginally higher maximum mass
compared to the KW EoS. These discernible differences in maximum mass values
are quantified in Table 1. The differences between the curves are due to the
absence (ZL is purely hadronic) or presence (Gibbs, KW) of quarks, and quark
matter can be more compressible than nuclear matter, leading to a softer EoS.

Table 1: NS parameters of the maximally stable NS corresponding to both
equilibrium and adiabatic cases for chosen EoSs. Central energy density
$(\mathcal{E}_{c})$ is in $(\mathrm{MeV\cdot fm^{-3}})$, Mass in $(M_{\odot})$
and Radius in $(\mathrm{km})$.

| $f_{\mathrm{eq}}=0$ |  | $f_{\mathrm{ad}}=0$
---|---|---|---
EoS | $\mathcal{E}_{c}$ | $M$ | $R$ |  | $\mathcal{E}_{c}$ | $M$ | $R$
Gibbs40 | 1419 | 2.035 | 10.492 |  | 1944 | 2.009 | 10.035
KW55 | 1247 | 2.016 | 11.228 |  | 1583 | 1.995 | 10.678
ZL70 | 1308 | 2.103 | 11.180 |  | 1727 | 2.081 | 10.545

In Fig. 3, the variation of the $f$-mode frequency with central energy density
($\mathcal{E}_{c}$) is presented in the Left panel, while the Right panel
displays the variation with Mass ($M$), aiming to explore radial oscillation
and stability. The curves for all EoSs exhibit a consistent trend. Notably, a
closer examination of the Left panel reveals that at a specific central
density (approximately 300 $\mathrm{MeV\cdot fm^{-3}}$) for Gibbs and KW, the
$f$-mode frequency surpasses that of ZL, indicating the point at which quarks
start to leak out from the nucleons. Now, let’s examine the distinctive
behaviour of the $f$-mode frequency curves corresponding to the equilibrium
and adiabatic sound speed case, arising from the contrasting timescales for
reaching equilibrium. considering the equilibrium sound speed case,
represented by the ”solid” line, the NS matter quickly approaches the ground
state, rapidly converging to a new equilibrium state and causing the $f$-mode
frequency to vanish at a lower central density. This immediate response
signifies the prompt stabilization of the NS in the equilibrium scenario.
Conversely, for the adiabatic sound speed case, depicted by the ”dashed-dot”
line, the process unfolds more gradually, necessitating a more extended
duration for the NS to attain equilibrium. Consequently, the $f$-mode
frequency diminishes at a higher central density, highlighting the prolonged
timescale associated with the adiabatic case. Moving to the Right panel of
this figure, in proximity to 1.4 $M_{\odot}$, the $f$-mode frequency of Gibbs
surpasses that of KW, and for KW, it exceeds that of ZL [refer to Table 1].
Remarkably, the $f$-mode frequency for ZL diminishes at a later stage compared
to Gibbs and KW. This suggests that the hadronic EoS maintains constancy until
it attains its maximum mass.

##  IV Summary & Conclusions

In this study, we thoroughly examined three distinct EoSs, each distinguished
by a unique symmetry energy slope (L): ZL70 representing a nucleonic EoS, and
Gibbs40 & KW55 representing hybrid EoSs, with KW55 incorporating crossover
matter. The differences between primary focus centred around an in-depth
analysis of the adiabatic sound speed in comparison to the equilibrium
scenario. A prominent observation emerged, showing smoother trends in the
sound speed dynamics when considering the adiabatic case over the equilibrium
one. Similarly, the adiabatic index ($\gamma$) manifests smoother trends in
the adiabatic case than in equilibrium. Then we studied the mass-radius
relation of NS with those aforementioned EoSs and found some different
behaviours like, ZL70 showcasing heightened stiffness compared to the other
two. Additionally, Gibbs40 exhibited more stiffness than KW55. Within the
Mass-Radius curve, we meticulously identified stable and unstable branches of
different EoSs. Shifting our focus to radial oscillations, we probed the
behaviour of the fundamental mode frequency ($f$-mode). Considering the
different EoSs, Gibbs40 has a larger frequency than the other two at a
particular mass (say 1.4 $M_{\odot}$) of NS. We have seen that for the
equilibrium case, $f$\- mode frequencies vanish at lower central energy
densities, however, for the adiabatic case, it vanishes at much larger central
energy densities. As a result, the stable branch can be extended beyond the
maximum mass configuration in an adiabatic case. Although this slow stable
branch has a similar mass as the maximum mass, the radius can be 0.6 km
smaller. Since maximum mass configuration usually corresponds to the lower
bound of radius, the slow stable branch extends the lower bound of NS radius,
which is particularly relevant to radius measurement of high mass NSs such as
PSR J0740+6620 Miller  _et al._ (2021); et. al. (2021). A radius measurement
of $R<11$km for two solar mass neutrons might indicate a NS on the slow stable
branch, while PSR J0740+6620 is more likely to be on the regular stable branch
based on the NICER observation. Our investigation sought to unravel how the
$f$-mode frequency responds by considering the adiabatic sound speed case,
providing insights into its dynamic changes. Our stability analysis, focused
on linearized oscillation where the perturbation is small, acknowledges the
potential influence of higher-order terms that could render slow stars
unstable at large oscillation displacement, as discussed by Gourgoulhon et al.
Gourgoulhon  _et al._ (1995).

##  V Acknowledgments

B.K. acknowledges partial support from the Department of Science and
Technology, Government of India, with grant no. CRG/2021/000101. TZ is
supported by the Network for Neutrinos, Nuclear Astrophysics and Symmetries
(N3AS), through the National Science Foundation Physics Frontier Center Grant
No. PHY-2020275.

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*

#  Unified Inference for Dynamic Quantile Predictive Regression

Christis Katsouris
University of Helsinki

This version: Working paper Postdoctoral Researcher, Department of Economics,
University of Helsinki, P.O. Box 17, Arkadiankatu 7 Helsinki, FI-00014
Finland. E-mail<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

This paper develops unified asymptotic distribution theory for dynamic
quantile predictive regressions which is useful when examining quantile
predictability in stock returns under possible presence of nonstationarity.

Keywords: persistence; local-to-unity; dynamic regression; quantile
predictability.

JEL Classification: C12, C22

##  Introduction

A major concern of time series and financial econometricians when it comes to
modelling financial and macroeconomic data is the presence of features such as
structural breaks, nonstationarity in the form of near unit roots, stochastic
trends as well as cointegration dynamics. In addition, more recently as
explained by Katsouris, (2021), there has been a growing interest to develop
robust estimation and inference techniques for the risk measures of
$\mathsf{VaR}$ and $\mathsf{CoVaR}$, using the local-to-unity parametrization,
with the aim to incorporate to the information set the persistence properties
of regressors used to obtain the forecasts of these risk measures.

From the perspective of existing methodologies currently presented in the
literature, a framework for unified inference in dynamic quantile predictive
regression models still remains an open research problem. In particular, limit
results for the predictive regressions with conditional mean (see, Chen et
al., (2013), Kostakis et al., (2015), Kasparis et al., (2015), Li et al.,
(2017), Yang et al., (2023), Harvey et al., (2023)) and conditional quantile
functional forms (see, Lee, (2016), Fan and Lee, (2019) and Liu et al.,
(2023)), have already been extensively examined under various econometric
conditions. Similarly unified inference in dynamic predictive regression with
a conditional mean functional form has been proposed by Yang et al., (2021).
This leaves the door open for us to study relevant asymptotic theory results
when considering conditional quantile functional forms in dynamic quantile
predictive regressions. Our asymptotic theory here, is also useful when
extending our framework to the robust estimation and inference of the risk
measures pair $(\mathsf{VaR},\mathsf{CoVaR})$ in the spirit of Adrian and
Brunnermeier, (2016) and Härdle et al., (2016), but under possible predictors
nonstationarity, as proposed in the Ph.D. thesis of the author.

The first rigorous work to define and construct tests which are asymptotically
optimal was Wald, (1943), who argued that maximum likelihood estimators may be
asymptotically sufficient for detecting local deviations from the null
hypothesis. In other words, Wald tests have an "optimal" asymptotic envelope
power function in the minimax sense over local neighborhoods of the null
hypothesis. Moreover, Wald tests can be used for more general problems if we
can find asymptotically efficient estimates for the parameters of interest -
an estimator $\hat{\vartheta}_{n}$ with
$\sqrt{n}\big{(}\hat{\vartheta}_{n}-\vartheta_{n}\big{)}$ asymptotically
$\mathcal{N}\left(0,B^{*-1}\right)$ under every
$\big{(}\vartheta_{n}(h_{\vartheta}),\eta_{n}(h_{n})\big{)}$. Then, the
characterization of asymptotically uniformly most powerful (AUMP), unbiased
and invariant tests are constructed using the properties of the asymptotic
local power function. Sufficient condition for optimality is that a test be
equivalent to a canonical effective score test - an optimal test requiring
knowledge of nuisance parameters. Furthermore, Stein’s notion of adaptation
implies replacing, in tests which are optimal when certain nuisance parameters
are known, these parameters by estimates without affecting the asymptotic
performance of the test. The particular aspect is the equivalent of large-
sample studentization, a variance parameter can be replaced by an estimate
without large-sample penalty. Lastly, a testing problem is invariant under
locally linear transformation of the parameter of interest, which implies
standardized effective score tests to be rotation invariant (asymptotically).

###  Literature Review

Due the unknown persistence properties of regressors, imposing a local-to-
unity specification111The limit theory for moderate deviations from a unit
root in autoregressive processes have been examined by Giraitis and Phillips,
(2006), Jansson and Moreira, (2006), Phillips and Magdalinos, (2007), Aue and
Horváth, (2007) and Huang et al., (2014) among many others. allows to capture
various nonstationarity regimes; although with the challenge that one has to
ensure asymptotic theory is nuisance parameter-free. Both the empirical
likelihood approach (see, Zhu et al., (2014) and Cai and Wang, (2014)) as well
as the IVX filtration method (see, Phillips and Magdalinos, (2009) and
Kostakis et al., (2015)) ensure that statistical inference is uniform
regardless of whether the predicting variable is stationary or nonstationary
or has an infinite variance. Inference for a multivariate nonstationary
regressor or even within a high-dimensional setting still involves challenges
in both cases but is beyond the scope of this paper.

The predictive regression model has been widely used for investigating the
predictability of asset returns. Applications include estimation and inference
methodologies for predictive regression model based on a parametric (see,
Kostakis et al., (2015)) or nonparametric (see, Kasparis et al., (2015))
conditional mean functional form. The current literature mainly covers linear
conditional mean autoregressive and predictive regression models as well as
dynamic conditional mean predictive regression models. Thus, when one is
interested for testing the presence of quantile predictability, related
studies have considered the implementation of the model based on the
conditional quantile functional form. Related studies include Maynard et al.,
(2023), Lee, (2016), Fan and Lee, (2019), Katsouris, (2022), Cai et al.,
(2022) and Liu et al., (2023). Furthermore, the inclusion of model intercepts
in both the predictive regression model as well as the autoregressive model
either based on a conditional mean or on a conditional quantile functional
forms imposes additional challenges when developing inference techniques for
the $\beta$ parameter based on the usual likelihood ratio test or Wald-type
statistics; especially when the autoregressive coefficient is nearly-
integrated (see, Romano and Wolf, (2001), Liu and Peng, (2019), Wang et al.,
(2022), Yang et al., (2023)).

Various studies in the literature employ the likelihood-based approach to
investigate the properties and asymptotic theory of corresponding estimators
and test statistics for nonstationary autoregressive processes and predictive
regression models. These likelihood approaches include the weighted least
squares, the empirical likelihood methodology as well as the profile
likelihood approach. In particular, Zhu et al., (2014), Liu et al., 2019a ;
Liu et al., 2019b  and Yang et al., (2021) consider an application of the
empirical likelihood approach which induces uniform inference across the
different degrees of persistence and confidence regions for the coefficients
of the linear predictive regression. Moreover, the framework of Chen et al.,
(2013) proposes a uniform inference approach (see, also Chen and Deo, 2009a
222Related studies include Chen and Deo, 2009b , Chen and Deo, (2012) as well
as Zhao, (2000).) which is valid regardless of the persistence of regressors
using the weighted least squares approximate restricted likelihood (WLSRL).
Liu et al., 2019b  propose predictability tests that correspond to testing the
null hypothesis of no statistical significance for the model coefficients of a
predictive regression model with additional covariates the difference of the
predicting variable. These predictability tests are based on the unified
empirical likelihood methodology that rejects the null hypothesis
$H_{0}:\beta_{1}=\beta_{2}=0$ or testing the hypothesis $H_{0}:\beta_{1}=0$
and $H_{0}:\beta_{2}=0$. Under the presence of known nonzero model intercept
in the predictive regression model using weighted score equations and the
sample-splitting methodology it can be proved that nuisance parameter-free
inference holds regardless of the persistence properties of regressors.
However, implementing the sample-split approach into a multivariate
nonstationary vector of regressors remains a computational challenging task as
this would require to split the data into blocks equal to the dimension of the
vector-valued regressor and then use each block to construct score equations
with respect to each variable in the vector. A similar difficulty exists when
one considers the IVX filtration to a multivariate or even high-dimensional
setting (see, Xu and Guo, (2022)).

Both methodologies are fundamentally aiming to develop asymptotic theory which
is valid to the presence of model intercept in the predictive regression model
as it is well-known that nonstandard limit distribution results appear when
$\left\\{X_{t}\right\\}_{t=1}^{n}$ is nonstationary. The empirical likelihood
method takes the first difference of the paired data such that
$\left\\{Y_{t+1}-Y_{t}\right\\}_{t=1}^{n}$ and
$\left\\{X_{t+1}-X_{t}\right\\}_{t=1}^{n}$, but a new issue arises as the
corresponding error sequence $\left\\{U_{t+1}-U_{t}\right\\}_{t=1}^{n}$
violates the independence assumption. Therefore, to fix this issue a split
sample methodology is applied and the first difference transformation is
applied based on a larger lag (i.e., the value of $m=\lfloor n/2\rfloor$)
which ensures $m-$dependence holds (i.e., approximate or asymptotic
independence as $m\to\infty$), and lastly the empirical likelihood estimation
is applied. The asymptotic theory for the empirical likelihood confidence
interval for $\beta_{0}$ is then proved to follow standard distribution
results (i.e., converges to a $\chi^{2}-$distribution). The IVX filtration is
an endogenous instrumentation procedure which is used as an alternative to the
OLS estimator for the unknown model coefficient $\beta_{0}$. In particular,
the IVX instruments are constructed with the main consideration of obtaining
less persistence regressors than the original series $\left\\{X_{t}\right\\}$
without having to consider a first-difference transformation based on a large
value of the lag as well as the application of sample splitting to ensure that
the distribution of the innovations of the predictive regression have
independent realizations. This methodology implies that asymptotic theory for
the unknown model coefficient follows a mixed gaussian distribution and the
corresponding Wald test follow a $\chi^{2}-$distribution.

Uniform inference for unit root moderate deviations (based on the local-to-
unity parametrization) holds in both cases. The empirical likelihood plus
weighted least squares induces uniform inference regardless of persistence
properties or the presence of an intercept in the autoregressive equation.
Likewise, robustness to the abstract degree of persistence holds for the IVX
estimator and uniform inference can be established based on the mixed
gaussianity property (see, Kostakis et al., (2015)). Recently Magdalinos and
Petrova, (2022) show that the IVX-instrumentation is indeed robust to the
presence of an intercept under general autoregressive dependence. Similar
properties hold for conditional quantile functional forms; Lee, (2016)
develops a framework for the implementation of the IVX instrumentation in
quantile predictive regressions, which is found to have standard asymptotic
theory while Liu et al., (2023) proved that their approach is indeed robust to
both the presence of intercept in the autoregressive equation and
nonstationarity of the $\left\\{X_{t}\right\\}$ process. Currently, Katsouris,
(2022) investigates the extension of the framework proposed by Magdalinos and
Petrova, (2022) to quantile predictive regressions.

Notation All random elements are defined with a suitable probability space,
denoted by $\left(\Omega,\mathcal{F},\mathbb{P}\right)$. Throughout the paper,
all limits are taken as $n\to\infty$, where $n$ is the sample size. The symbol
$"\Rightarrow"$ is used to denote the weak convergence of the associated
probability measures as $n\to\infty$. The symbol $\overset{d}{\to}$ denotes
convergence in distribution and $\overset{\text{plim}}{\to}$ denotes
convergence in probability, within the probability space (see, Billingsley,
(1968)).

Structure The rest of the paper is organized as follows. Section 2, presents
the model estimation procedure and main assumptions of our framework. Section
3, examines the finite-sample performance of the proposed unified tests using
a Monte Carlo simulation study and Section 4 illustrates the implementation of
the method using real data. Section 5 summarizes the main conclusions. Proofs
of the main results can be found in the Appendix of the paper.

##  Econometric Model of Dynamic Quantile Predictive Regression

###  Classical Predictive Regression

Let $y_{t}$ be a realization of the random variable
$\left\\{Y_{t}\right\\}_{t=1}^{n}$, which corresponds to a univariate
predictant (such as stock return) that is assumed to be stationary, and
$x_{t}$ a realization of the random variable
$\left\\{X_{t}\right\\}_{t=1}^{n}$, which corresponds to be a predictive
variable which is assumed to be possibly nonstationary. The pair
$\left\\{(Y_{t},X_{t-1})\right\\}_{t=1}^{n}$ is assumed to follow the
predictive regression structure:

$\displaystyle Y_{t}$ $\displaystyle=\alpha_{n}+\beta_{n}X_{t-1}+U_{t},\ \ \
\text{for}\ \ 1\leq t\leq n$ (2.1) $\displaystyle X_{t}$
$\displaystyle=\mu_{n}+\rho_{n}X_{t-1}+V_{t}$ (2.2)

where $\rho_{n}=\left(1+\frac{c}{k_{n}}\right)$ with $k_{n}=n^{\gamma}$, such
that $\gamma=0$, $\gamma=1$ or $\gamma\in(0,1)$. Moreover, assume that the
initial condition $X_{0}$ is constant and $\left(U_{t},V_{t}\right)$ are
independently and identically distributed errors with mean zero and finite
variance. Consider the vector $W_{t}=\left(U_{t},V_{t}\right)^{\prime}$ such
that $W_{t}$ is a strictly stationary martingale difference sequence. Then,
the partial sums process for the sequence $W_{t}$ satisfy the invariance
principle as proposed by Phillips, 1987a ; Phillips, 1987b  (see, Phillips and
Solo, (1992))

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
ns\rfloor}W_{t}\Rightarrow\mathbf{B}_{z}(s):=\big{[}B_{u}(s),B_{v}(s)\big{]}^{\top}\equiv\text{BM}\left(\mathbf{\Sigma}_{ww}\right),\
\ \mathbf{\Sigma}_{ww}:=\begin{bmatrix}\Sigma_{uu}&\Sigma_{uv}\\\
\Sigma_{uv}&\Sigma_{vv}\end{bmatrix}$ (2.3)

where $\mathbf{B}_{w}(s)$ is a vector Brownian motion with a positive-definite
matrix $\mathbf{\Sigma}_{ww}$ such that $\Sigma_{uu}>0$, $\Sigma_{uu}>0$ and
$\Sigma_{vv}>0$.

####  Variable-Augmented Predictive Regression

The econometric specification for the predictive regression model with
variable-augmentation is:

$\displaystyle Y_{t}$
$\displaystyle=\alpha_{n}+\gamma_{n}Y_{t-1}+\beta_{n}X_{t-1}+U_{t},\ \ \
\text{for}\ \ 1\leq t\leq n$ (2.4) $\displaystyle X_{t}$
$\displaystyle=\mu_{n}+\rho_{n}X_{t-1}+V_{t}$ (2.5)

In terms of statistical inference, based on the above conditional mean
functional form, the unknown parameters of interest are
$\theta_{n}=(\beta_{n},\gamma_{n})$. A relevant hypothesis testing formulation
is the joint existence of the lagged predicted variables and predictability,
which implies that under the null hypothesis it holds that
$\mathcal{H}_{0}:\left\\{\beta_{n}=0\right\\}\ \bigcap\
\left\\{\gamma_{n}=0\right\\}$. Unified tests which are robust to the unknown
degree of persistence, imply that uniform inference regardless of regressors
being stationary or nearly intergated as well as in the case that the
autoregressive equation is specified with or without an intercept holds. In
particular, Li et al., (2017) assumes that $\gamma$ is unknown and tests
either $\mathcal{H}_{0}:\gamma=0$ or $\mathcal{H}_{0}:\beta=0$ without
requiring that $\gamma=0$. A simplified approach is the case that one assumes
that $\gamma=0$ and formulates the null hypothesis $\mathcal{H}_{0}:\beta=0$
(see, Demetrescu, (2014)). However, is more economically meaningful to avoid
imposing the assumption that the model parameter $\gamma=0$ a priori. Then,
the main interest is to construct a robust inference method for testing the
null hypothesis that $\mathcal{H}_{0}:\gamma=0$ and $\mathcal{H}_{0}:\beta=0$,
regardless of the following cases:

  * •

$\left\\{X_{t}\right\\}$ is stationary: $|\rho_{0}|<1$

  * •

$\left\\{X_{t}\right\\}$ is nearly-integrated:
$\rho_{0}=\left(1-\frac{c}{n}\right)$ for some $c\neq 0$.

  * •

$\left\\{X_{t}\right\\}$ is unit root: $\rho_{0}=1$.

Under the assumption that $\alpha_{0}=0$ then, the least squares estimator for
$(\gamma,\beta)^{\top}$ can be obtained by solving the following score
equations:

$\displaystyle\frac{\partial S_{t}}{\partial\gamma}$
$\displaystyle=\sum_{t=1}^{n}\big{(}Y_{t}-\alpha_{0}-\gamma Y_{t-1}-\beta
X_{t-1}\big{)}Y_{t-1}\equiv 0$ (2.6) $\displaystyle\frac{\partial
S_{t}}{\partial\beta}$
$\displaystyle=\sum_{t=1}^{n}\big{(}Y_{t}-\alpha_{0}-\gamma Y_{t-1}-\beta
X_{t-1}\big{)}X_{t-1}=0$ (2.7)

Therefore, we can construct an inference procedure about the model parameters
$\beta$ and $\gamma$ by directly applying the empirical likelihood method
based on estimating equations. However, the asymptotic theory of the empirical
likelihood function is nonstandard when $\left\\{X_{t}\right\\}$ is nearly-
integrated.

####  Empirical Likelihood Estimation Approach

The empirical likelihood approach is defined as below:

$\displaystyle
L_{n}(\beta)=\mathsf{sup}\left\\{\prod_{t=1}^{n}(np_{t}):p_{1}\geq
0,...,p_{n}\geq
0,\sum_{t=1}^{n}p+t=1,\sum_{t=1}^{n}p_{t}Z_{t}(\beta)\right\\}$ (2.8)

where $Z_{t}(\beta)=\big{(}Y_{t}-\beta
X_{t-1}\big{)}X_{t-1}\big{/}\sqrt{1+X^{2}_{t-1}}$. Then, based on the Lagrange
multiplier optimization method it follows that

$\displaystyle\ell_{n}(\beta):=-2\mathsf{log}L_{n}(\beta)=2\sum_{t=1}^{n}\mathsf{log}\big{\\{}1+\lambda
Z_{t}(\beta)\big{\\}}$ (2.9)

Then, taking the first-order-condition for some $\lambda=\lambda(\beta)$
satisfies

$\displaystyle\sum_{t=1}^{n}\frac{Z_{t}(\beta)}{1+\lambda Z_{t}(\beta)}\equiv
0.$ (2.10)

###  Dynamic Quantile Predictive Regression Model

Recall that the quantile predictive regression model has the advantage that
the innovation sequences of the predictive regression based on a conditional
quantile functional form, denote with $\psi_{\uptau}(u_{t}(\uptau))$ are only
contemporaneously correlated with the innovations of unit root sequences. Our
main contribution to the literature is to propose a unified inference
procedure for Dynamic Quantile Predictive Regressions (DQPRs), which can be
considered as further extension to the frameworks proposed by Liu et al.,
(2023) and Yang et al., (2021).

Consider the dynamic quantile predictive regression model:

$\displaystyle y_{t}$
$\displaystyle=\alpha_{\tau}+\beta_{\tau}x_{t-1}+\gamma_{\tau}y_{t-1}+\epsilon_{t\tau},\
\ \epsilon_{t\tau}=u_{t}-\sigma_{t}Q_{\eta}(\tau)$ (2.11) $\displaystyle
x_{t}$ $\displaystyle=\mu+\rho x_{t-1}+u_{t}$ (2.12)

Moreover, consider the following quantile moment conditions as expressed below

$\displaystyle\mathbb{E}\big{[}\big{(}\boldsymbol{1}\left\\{y_{t}\leq\beta
x_{t-1}\right\\}-\tau\big{)}\cdot x_{t-1}\big{]}=0$ (2.13)

Following the framework proposed by Whang, (2006) we replace the indicator
function which is not differentiable in the whole of the support of the model
parameter $\beta$ with a smooth function.

Define the following two score functions:

$\displaystyle Z^{\alpha}_{t,\tau}(\beta,\gamma)$
$\displaystyle:=\psi_{\tau}\big{(}Y_{t}-\alpha_{0}-\beta X_{t-1}-\gamma
Y_{t-1}\big{)}\equiv 0$ (2.14) $\displaystyle
Z^{\beta}_{t,\tau}(\beta,\gamma)$
$\displaystyle:=\psi_{\tau}\big{(}Y_{t}-\alpha_{0}-\beta X_{t-1}-\gamma
Y_{t-1}\big{)}w_{t}\equiv 0,\ \ w_{t}=\frac{X_{t-1}}{\sqrt{1+X^{2}_{t-1}}}$
(2.15) $\displaystyle Z^{\gamma}_{t,\tau}(\beta,\gamma)$
$\displaystyle:=\psi_{\tau}\big{(}Y_{t}-\alpha_{0}-\beta X_{t-1}-\gamma
Y_{t-1}\big{)}\widetilde{w}_{t}\equiv 0,\ \ \widetilde{w}_{t}=(Y_{t-1}-\beta
X_{t-1}).$ (2.16)

In particular the random variable $X_{t-1}$ is replaced by a weighted random
variable $\displaystyle\frac{X_{t-1}}{\phi(|X_{t-1}|)}$ where the weight
function $\phi(t)$ satisfies the limit $\frac{t}{\phi(t)}\to 1$ as
$t\to\infty$. Thus a simple choice which satisfies these requirements is
$\phi(t)=\sqrt{1+t^{2}}$ which implies the weight
$w(t)=\frac{t}{\sqrt{1+t^{2}}}$ that is indeed a bounded continuous function.
Notice that self-weighted M-estimators are proved in the literature to have
normal asymptotic theory. Furthermore, the purpose of weight $w_{t}$ is to
downweight the leverage points in $X_{t}$ ensuring this way the finiteness
property of covariance matrices under the presence of heavy-tailed errors.
Furthermore, the weight is added in the second estimating equation to ensure
that $\frac{1}{n}\sum_{t=1}^{n}\frac{X^{2}_{t-1}}{1+X^{2}_{t-1}}$ converges in
probability to a constant regardless the nonstationarity of the regressor
series $\left\\{X_{t}\right\\}$.

Then we define with

$\displaystyle\boldsymbol{Z}_{t}(\beta,\gamma)=\begin{pmatrix}Z^{\alpha}_{t,\tau}(\beta,\gamma)\\\
Z^{\beta}_{t,\tau}(\beta,\gamma)\\\
Z^{\gamma}_{t,\tau}(\beta,\gamma)\end{pmatrix}$ (2.17)

Therefore, the empirical likelihood approach satisfies the following equation:

$\displaystyle\sum_{t=1}^{n}\frac{\boldsymbol{Z}_{t}(\beta,\gamma)}{1+\boldsymbol{\lambda}^{\top}\boldsymbol{Z}_{t}(\beta,\gamma)}\equiv
0.$ (2.18)

Furthermore define the following matrices
$\boldsymbol{V}_{n}:=\mathbb{E}\big{[}\boldsymbol{Z}_{t}(\beta,\gamma)\boldsymbol{Z}^{\prime}_{t}(\beta,\gamma)\big{]}$.
Similarly defining the standardized Lagrangian multiplies
$\tilde{\lambda}=\boldsymbol{V}_{n}^{1/2}\lambda$ and the standardized score
function
$\boldsymbol{W}_{t}(\beta,\gamma)=\boldsymbol{V}_{n}^{1/2}\boldsymbol{Z}_{t}(\beta,\gamma)$
we have the following log-likelihood function.

$\displaystyle\ell_{n}(\beta,\gamma)=2\sum_{t=1}^{n}\mathsf{log}\big{(}1+\tilde{\lambda}\cdot\boldsymbol{W}_{t}(\beta,\gamma)\big{)}.$
(2.19)

Then, developing a Taylor expansion for $\tilde{\lambda}$ for
$\tilde{\lambda}$ and $\ell_{h}(\beta_{0})$ implies that around the FOC:

$\displaystyle 0$
$\displaystyle=\frac{1}{n}\sum_{t=1}^{n}\frac{\boldsymbol{W}_{t}(\beta,\gamma)}{1+\tilde{\lambda}\cdot\boldsymbol{W}_{t}(\beta,\gamma)}$
$\displaystyle=\frac{1}{n}\sum_{t=1}^{n}\boldsymbol{W}_{t}(\beta,\gamma)-\left(\frac{1}{n}\sum_{t=1}^{n}\boldsymbol{W}_{t}(\beta,\gamma)\boldsymbol{W}_{t}(\beta,\gamma)^{\prime}\right)\lambda+\frac{1}{n}\sum_{t=1}^{n}\big{(}\lambda^{\prime}\boldsymbol{W}_{t}(\beta,\gamma)\big{)}^{2}\boldsymbol{W}_{t}(\beta,\gamma)-....$

Therefore, it can be proved that:

$\displaystyle\ell_{h}(\beta_{0})$
$\displaystyle=\left[\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\big{(}\boldsymbol{Z}_{t}(\beta,\gamma)-\mathbb{E}\boldsymbol{Z}_{t}(\beta,\gamma)\big{)}+\mathsf{g}(n)\mathbb{E}\boldsymbol{Z}_{t}(\beta,\gamma)\right]^{\prime}\times\boldsymbol{V}_{n}^{-1}$
$\displaystyle\times\left[\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\big{(}\boldsymbol{Z}_{t}(\beta,\gamma)-\mathbb{E}\boldsymbol{Z}_{t}(\beta,\gamma)\big{)}+\mathsf{g}(n)\mathbb{E}\boldsymbol{Z}_{t}(\beta,\gamma)\right]$

where $\mathsf{g}(n)=n^{1/2}$. In the stationary case when $|\rho|<1$ we have
that as $n\to\infty$

$\displaystyle\boldsymbol{D}_{1,m}\begin{pmatrix}\sqrt{m}\left(\hat{\alpha_{\tau}}-\alpha_{\tau}\right)\\\
\sqrt{m}\left(\hat{\beta_{\tau}}-\beta_{\tau}\right)\end{pmatrix}\overset{d}{\to}\mathcal{N}\big{(}0,\boldsymbol{\Sigma}_{1}\big{)}.$
(2.20)
$\displaystyle\boldsymbol{D}_{1,m}=f_{\epsilon}(0)\begin{pmatrix}1&\displaystyle\frac{1}{m}\sum_{t=1}^{m}x_{t-1}\\\
\displaystyle\frac{1}{m}\sum_{t=m+1}^{2m}w_{t}&\displaystyle\frac{1}{m}\sum_{t=m+1}^{2m}w_{t}x_{t-1}\end{pmatrix}\
\ \ \text{and}\ \ \ \boldsymbol{\Sigma}_{1}=\tau(1-\tau)\begin{pmatrix}1&0\\\
0&\displaystyle\mathbb{E}\left(\frac{x_{1}^{2}}{1+x_{1}^{2}}\right)\end{pmatrix}$
(2.21) $\displaystyle
A_{1,m}\times\sqrt{m}\left(\hat{\beta}_{\tau}-\beta_{\tau}\right)\overset{d}{\to}\mathcal{N}\left(0,\boldsymbol{\gamma}_{1}^{\top}\boldsymbol{\Sigma}_{1}\boldsymbol{\gamma}_{1}\right)$
(2.22)

where

$\displaystyle\boldsymbol{\gamma}_{1}=\begin{pmatrix}\displaystyle-\mathbb{E}\left(\frac{x_{t}^{2}}{\sqrt{1+x_{t}^{2}}}\right)\\\
1\end{pmatrix},\
A_{1,m}=\left\\{\frac{1}{m}\sum_{t=m+1}^{2m}x_{t-1}w_{t}-\left(\frac{1}{m}\sum_{t=m+1}^{2m}w_{t}\right)\left(\frac{1}{m}\sum_{t=m+1}^{2m}x_{t-1}\right)\right\\}f_{\epsilon}(0).$

Denote with $\boldsymbol{\Lambda}_{t}=(1,x_{t})^{\top}$ such that

$\displaystyle\boldsymbol{Z}_{m}(\boldsymbol{v})=\begin{pmatrix}\displaystyle\frac{1}{\sqrt{m}}\sum_{t=1}^{m}\psi_{\tau}\big{(}\varepsilon_{t}-\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{)}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\psi_{\tau}\big{(}\varepsilon_{t}-\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{)}w_{t}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\psi_{\tau}\big{(}\varepsilon_{t}-\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{)}\widetilde{w}_{t}\end{pmatrix}$
(2.23)

Then, for any given $\boldsymbol{v}\in\mathbb{R}^{2}$ we have that

$\displaystyle\boldsymbol{Z}_{m}(\boldsymbol{v})=\begin{pmatrix}\displaystyle\frac{1}{\sqrt{m}}\sum_{t=1}^{m}\psi_{\tau}(\varepsilon_{t})\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\psi_{\tau}(\varepsilon_{t})w_{t}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\psi_{\tau}(\varepsilon_{t})\widetilde{w}_{t}\end{pmatrix}-\boldsymbol{D}_{m}\boldsymbol{v}+o_{p}(1),\
\text{as}\ n\to\infty,$ (2.24)

###### Proof.

Since it holds that $\psi_{\tau}=\tau-\boldsymbol{1}\left\\{u\leq 0\right\\}$,
we have that

$\displaystyle\boldsymbol{Z}_{m}(\boldsymbol{v})-\begin{pmatrix}\displaystyle\frac{1}{\sqrt{m}}\sum_{t=1}^{m}\psi_{\tau}(\varepsilon_{t})\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\psi_{\tau}(\varepsilon_{t})w_{t}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\psi_{\tau}(\varepsilon_{t})\widetilde{w}_{t}\end{pmatrix}$
$\displaystyle=-\begin{pmatrix}\displaystyle\frac{1}{\sqrt{m}}\sum_{t=1}^{m}\left[\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{\\}}-\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq
0\big{\\}}\right]\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\left[\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{\\}}-\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq
0\big{\\}}\right]\cdot w_{t}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\left[\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{\\}}-\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq
0\big{\\}}\right]\cdot\widetilde{w}_{t}\end{pmatrix}$
$\displaystyle=-\begin{pmatrix}\displaystyle\frac{1}{\sqrt{m}}\sum_{t=1}^{m}\mathbb{E}\big{[}\xi_{1,t}|\mathcal{F}_{t-1}\big{]}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\mathbb{E}\big{[}\xi_{2,t}|\mathcal{F}_{t-1}\big{]}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\mathbb{E}\big{[}\xi_{3,t}|\mathcal{F}_{t-1}\big{]}\end{pmatrix}-\begin{pmatrix}\displaystyle\frac{1}{\sqrt{m}}\sum_{t=1}^{m}\xi_{1,t}-\mathbb{E}\big{[}\xi_{1,t}|\mathcal{F}_{t-1}\big{]}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\xi_{2,t}-\mathbb{E}\big{[}\xi_{2,t}|\mathcal{F}_{t-1}\big{]}\\\
\displaystyle\frac{1}{\sqrt{m}}\sum_{t=m+1}^{2m}\xi_{3,t}-\mathbb{E}\big{[}\xi_{3,t}|\mathcal{F}_{t-1}\big{]}\end{pmatrix}$

where we denote with

$\displaystyle\xi_{1,t}$
$\displaystyle=\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{\\}}-\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq
0\big{\\}}$ (2.25) $\displaystyle\xi_{2,t}$
$\displaystyle=\left[\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{\\}}-\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq
0\big{\\}}\right]\cdot w_{t}$ (2.26) $\displaystyle\xi_{3,t}$
$\displaystyle=\left[\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq\boldsymbol{v}^{\top}\mathbb{M}_{m}^{-1}\boldsymbol{\Lambda}_{t-1}\big{\\}}-\boldsymbol{1}\big{\\{}\varepsilon_{t}\leq
0\big{\\}}\right]\cdot\widetilde{w}_{t}$ (2.27)

∎

Notice that the following invariance principles hold:

$\displaystyle\frac{1}{\sqrt{m}}x_{\lfloor
2mr\rfloor}\overset{d}{\to}J_{c}(r)=\int_{0}^{r}e^{(r-s)}dW_{u}(s)$ (2.28)
$\displaystyle\frac{1}{m}\sum_{t=1}^{m}\frac{x_{t-1}}{\sqrt{m}}\overset{d}{\to}\int_{0}^{1}J_{c}\left(\frac{r}{2}\right)dr$
(2.29)

##  Monte Carlo Simulation Study

The numerical studies of our paper include both a Monte Carlo simulation
experiments as well as an empirical application to validate the performance of
the proposed statistical methodologies on a real data set. In this Section, we
examine the finite sample performance of the proposed estimators and
predictability tests via an extensive Monte Carlo simulation study. We
consider the finite sample size and power performance of the proposed QR
method.

###  Empirical size and power analysis of predictability tests

Before illustrating insights from Monte Carlo simulation study it is useful to
discuss main findings presented in the literature. In particular, from Table 1
in Yang et al., (2021) we summarize the following observations. Under the
presence of a non-zero augmented variable for large samples the IVX estimator
for testing the null hypothesis $H_{0}:\beta=0$ has an excellent empirical
size comparable with the empirical likelihood approach given a zero model
intercept in the autoregressive equation. On the other hand, under the
presence of a non-zero model intercept the size of the IVX-based test
statistic detiorates while the corresponding test statistic based on the
empirical likelihood approach seems to preserve a good control of size in both
finite and large samples. Furthermore, Hosseinkouchack and Demetrescu, (2021)
propose finite-sample size control for the IVX instrumentation by considering
recursive demeaning as a possible correction. Specifically, by using a
recursively demeaned regressor and forward demeaning for the regressand
implies that the recursively demeaned regressor and the forward demeaned
distrurbance are orthogonal irrespective of the correlation between the
innovation sequences $u_{t}$ and $v_{t}$, thereby ensuring robustness to the
degree of endogeneity regardless of the presence of model intercepts in the
two functional forms.

####  Experimental Design

The data generating mechanism is given by the following expression:

$\displaystyle y_{t}$
$\displaystyle=\alpha(\tau)+\beta(\tau)x_{t-1}+u_{t}(\tau)$ (3.1)
$\displaystyle x_{t}$ $\displaystyle=\mu+\rho x_{t-1}+v_{t},\ \
\rho=\left(1-\frac{c}{n}\right)$ (3.2)

####  Weighted Bootstrap

Since the random weighted bootstrap method neither infers the heteroscedastic
errors nor generates samples from the fitted model via resampling residuals
like residual-based bootstrap methods, the developed test is robust against
conditional heteroscedasticity and computational convenient.

##  Empirical Application

The stock return predictability is a major puzzle in financial economics. The
literature goes back several decades; we briefly highlight important studies.
The seminal paper of Perron, (1989) discuss the unit root hypothesis around
periods of financial turbulence. DeJong and Whiteman, (1991) study the debate
whether divided are trend-stationary or integrated processes while Stein and
Stein, (1991) propose a framework for stock price distributions and stochastic
volatility. We focus on examining the presence of predictability for monthly
US stock market excess returns over the period 1990-2019 using the set of
variables considered in Welch and Goyal, (2008)333The dataset can be retrieved
from Amit Goyal’s website at http://www.hec.unil.ch/agoyal/. Detailed
descriptions of variables can be found in the Online Appendix of Welch and
Goyal, (2008).  which capture economic and financial conditions for the US
economy. Following the common practise in the predictability literature, we
use value weighted returns for the with monthly sampling frequency from
Jan-1990 to Dec-2020. Therefore, we employ the predictive regression model
which is an econometric environment that allows for the two innovations of the
system to be correlated a more realistic assumption when modelling the
presence of cointegrating dynamics between stock returns and macroeconomic
and/or financial variables in the economy.

##  Conclusion

The use of optimization techniques such as the empirical likelihood estimation
procedure or the profile likelihood approach can be used for estimating
predicting regression models and are found to be robust to unknown degree of
persistence is a similar spirit as when the IVX instrumentation is applied.
Although these two estimation methodologies work differently there is a common
denominator: regardless of the persistence properties of regressors uniform
inference can be applied which provides a solution to the challenges that
other methodologies in the literature have (such as Bonferonni corrections and
the alike). Both procedures are practical in empirical applications and it is
up to the econometrician to decide which methodology is preferable especially
when developing asymptotic theory results for constructing test statistics.

In this article we develop a framework for unified statistical inference for
dynamic quantile predictive regression models regardless of zero on nonzero
intercept, without any prior information on the unknown degree of persistence
in modeling possibly nonstationary predictors. Our test statistics are
constructed using weighted quantile regression estimation which preserves the
asymptotic mixed-Gaussianity (AMG) property of the estimators. Consequently,
we can obtain a unified predictability test whose statistic follows a Chi-
square distribution. Although the empirical likelihood approach is not always
a preferable approach by econometricians it does provide valid uniform
inference methodology. On the other hand the IVX filtration is also a robust
uniform methodology while allows to develop and extend relevant econometric
theory results in various directions.

##### Conflicts of interest

The author declares that there are no known conflicts of interest.

##### Acknowledgments

The author would like to thank Professor Markku Lanne and Professor Mika Meitz
at the Department of Economics, Faculty of Social Sciences, University of
Helsinki for helpful suggestions that remarkably improve the article. In
addition, the author sincerely thank Dr. Ji Hyung Kee for generously sharing
the replication code for the IVX-QR procedure which has been adapted in this
article.

##### ORCID

Dr. Christis Katsouris  ID http://www.orcid.org/0000-0002-8869-6641

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