# Time-Varying Poisson Autoregression

Giovanni Angelini  Giuseppe Cavaliere  Enzo D’Innocenzo  Luca De Angelis
Department of Economics, University of Bologna  School of Business and
Economics, Vrije Universiteit Amsterdam  Essex Business School, University of
Essex

###### Abstract

In this paper we propose a new time-varying econometric model, called Time-
Varying Poisson AutoRegressive with eXogenous covariates (TV-PARX), suited to
model and forecast time series of counts. We show that the score-driven
framework is particularly suitable to recover the evolution of time-varying
parameters and provides the required flexibility to model and forecast time
series of counts characterized by convoluted nonlinear dynamics and structural
breaks. We study the asymptotic properties of the TV-PARX model and prove
that, under mild conditions, maximum likelihood estimation (MLE) yields
strongly consistent and asymptotically normal parameter estimates. Finite-
sample performance and forecasting accuracy are evaluated through Monte Carlo
simulations. The empirical usefulness of the time-varying specification of the
proposed TV-PARX model is shown by analyzing the number of new daily COVID-19
infections in Italy and the number of corporate defaults in the US.

##  1 Introduction

In the last years time series of counts has been experiencing an increasing
interest among researchers and practitioners. Applications on count data span
over several different areas, including finance (Fokianos and Tjøstheim,,
2011), corporate defaults (Agosto et al.,, 2016), environmental economics
(Wang et al.,, 2014), epidemiology (Davis et al.,, 2003), COVID-19 infections
and deaths (Khismatullina and Vogt,, 2020; Li and Linton,, 2021), sports
(Angelini and De Angelis,, 2017; Koopman and Lit,, 2019). Models for time
series of counts are generally based on the assumption that the observations
follow a Poisson distribution conditional on past observations and explanatory
variables, see Davis et al., (2016) for an overview of recent developments on
the econometric models for discrete values. In particular, Davis et al.,
(2003) propose an observation-driven Poisson model for which they derive the
conditions for stationarity and ergodicity and the large-sample properties of
the MLE. They apply their method to daily counts of asthma presentations at a
Sydney hospital. Fokianos et al., (2009) develop and establish the asymptotic
theory of a Poisson autoregression (PAR) model where the conditional
distribution of the observations $\\{y_{t}\\}_{t\in\mathbb{Z}}$ given the
intensity process $\\{\lambda_{t}\\}_{t\in\mathbb{Z}}$ is Poisson with
parameter $\lambda_{t}$ and they apply this new model to describe the number
of transactions per minute of the stock Ericsson B. Agosto et al., (2016)
generalise the PAR model by including a set of exogenous covariates in the
dynamic specification of the intensity process (PARX) to improve the model
performance for forecasting the number of aggregate corporate default counts
in US. Wang et al., (2014) propose a self-excited threshold structure in the
PAR model specification by considering a two regime-switching approach
according to the magnitude of the lagged observations, in order to model the
number of major earthquakes in the world.

In this paper we contribute to this literature by introducing a new class of
Poisson AutoRegressive models allowing for time-variation in the intensity and
in the parameters which is particularly suitable to capture several (non-
standard) features, including structural breaks, regime-switching and non-
linearities, overdispersion and negative correlation, thus providing better
fitting and forecasting performance than the original PARX model. The
extension to the time-varying parameter specification of the PARX model,
labeled TV-PARX model in what follows, is achieved using a score-driven
approach. Originally proposed by Creal et al., (2013) and Harvey, (2013) to
estimate the evolution of time-varying parameters, the Generalized
Autoregressive Score (GAS) framework is now a popular methodology to develop
time-varying econometrics models. Blasques et al., (2019) propose a new class
of GAS models for continuous time series allowing for time-varying parameters
and provide the theoretical foundations for their acceleration method by
showing optimality in terms of Kullback-Leibler divergence.

The literature on GAS models for time series of counts has been growing
remarkably in the very recent years. Gorgi, (2020) uses the beta–negative
binomial distribution for the narcotics trafficking reports in Sydney and the
euro–pound sterling exchange rate, Koopman and Lit, (2019) use the bivariate
Poisson distribution for the number of goals in football matches and the
Skellam distribution for the score difference, and Gorgi, (2018) uses the
Poisson distribution as well as the negative binomial distribution for the
number of offensive conduct reports in the city of Blacktown, Australia.

The TV-PARX model we propose generalizes the accelerated score-driven model by
Blasques et al., (2019) in the framework of time series of counts and allowing
for the inclusion of exogenous covariates and deterministic components in the
model specification. We provide conditions for stationarity and ergodicity, as
well as asymptotic results for the MLE. Moreover, we show the relevance of our
modeling approach by focusing on a Monte Carlo simulation study and two main
applications. First, we analyze the dynamics of new COVID-19 confirmed cases
in Italy and, as in Li and Linton, (2021), we found that the time-varying
model specification is required. Indeed, the model parameters substantially
change over time due to the different ‘waves’ of the coronavirus infections
and their association with the introduction of government’s measures to
control the spread of the virus, e.g. lockdown, curfew, and mass vaccination.
Second, we study the dynamics of US corporate default counts. Using the data
set analyzed by Agosto et al., (2016), we show that the time-varying
specification of the PARX model allows to better capture and forecast the
number of corporate defaults and to disentangle the main drivers of the US
recessions.

The paper is organized as follows. Section 2 describes the new TV-PARX model
and discusses the properties of TV-PARX. Section 3 presents the MLE for the
TV-PARX. Section 4 shows the finite sample properties and Section 5 reports
the empirical illustrations on COVID-19 data and on corporate defaults.
Section 6 concludes the paper. Details on notation and all the proofs are
provided in A.

##  2 Time-Varying Poisson Autoregression

Consider a time series of counts $\\{y_{t}\\}_{t\in\mathbb{Z}}$ with a Poisson
conditional distribution,

$\displaystyle
y_{t}|\mathcal{F}_{t-1}\sim\mathcal{P}\textit{ois}(\lambda_{t})\hskip
28.45274ptt\in\mathbb{Z},$ (1)

where $\mathcal{F}_{t-1}=\sigma\\{y_{t-1},y_{t-2},y_{t-3},\dots\\}$ denotes
the filtration generated by the count process $\\{y_{t-i}\\}_{i>0}$ up to time
$t-1$, whereas $\lambda_{t}:=\mathbb{E}[y_{t}|\mathcal{F}_{t-1}]$ is the
conditional mean of the count process, which is allowed to vary over time.
Examples are the Poisson autoregressive models by Fokianos et al., (2009) and
Agosto et al., (2016). While these papers consider a time-invariant parameter
approach, we assume that the evolution of the conditional mean is described by
a time-varying coefficient model representation using the score-driven
framework $-$ see e.g. Creal et al., (2013) and Harvey, (2013) for a detailed
review $-$ thus allowing for time variation in the parameters of the dynamic
equation for $\lambda_{t}$. Koopman et al., (2008) show that a score-driven
model for Poisson time series of counts encompasses most of the observation-
driven models considered by Davis et al., (2003). Then, our contribution is to
generalize the score-driven model for the intensity process and formally
derive the stochastic properties of our Poisson score-driven model and the
asymptotic theory of the maximum likelihood estimator. In particular, we
consider a score-driven model with the observation equation in (1), which
implies that the conditional $\log$-density of $y_{t}$ is given up to a
constant by

$\displaystyle\log
p(y_{t}|\mathcal{F}_{t-1})={y_{t}}\log\lambda_{t}-\lambda_{t}.$ (2)

We propose the following nonlinear Poisson autoregressive model with exogenous
covariates and deterministic components defined as

$\displaystyle\log\lambda_{t+1}$ $\displaystyle=$
$\displaystyle\omega+\beta\log\lambda_{t}+\alpha_{t+1}(y_{t}-\lambda_{t})\lambda_{t}^{-1}+\gamma_{t+1}^{\prime}x_{t}+\psi^{\prime}d_{t}$
(3)

where $x_{t}\in\mathbb{R}^{m}$ denotes $m$ exogenous covariates, whereas in
$d_{t}\in\mathbb{R}^{d}$ are stacked the deterministic components, e.g.
seasonal dummies, impulse and step dummies, trend functions, etc. In the
following, the model in (3) is called Time-Varying Poisson AutoRegressive with
eXogenous covariates, TV-PARX. Indeed, while $(\omega,\beta)\in\mathbb{R}^{2}$
and $\psi\in\mathbb{R}^{d}$ are unknown static parameters, the key feature of
model (3) is that $\alpha_{t}$ and $\gamma_{t}$ are time-varying.

Similarly to Blasques et al., (2019), we characterize $\alpha_{t}$ by the
following updating equation

$\displaystyle\alpha_{t+1}$
$\displaystyle=\delta_{\alpha}+\phi_{\alpha}\alpha_{t}+\kappa_{\alpha}(y_{t}-\lambda_{t})\lambda_{t}^{-1}(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1},$
(4)

where $\delta_{\alpha}$, $\phi_{\alpha}$ and $\kappa_{\alpha}$ are treated as
fixed and unknown parameters. We note that, analogously to the models for
continuous time series considered in Blasques et al., (2019), the law of
motion of $\alpha_{t}$ is driven by the product of current and past
innovations. Moreover, by scaling the score innovations with the inverse of
information quantity $\mathcal{I}_{t}=\lambda_{t}$, Harvey, (2013) shows that
the law motion for $\log\lambda_{t}$ is given by the first-order
autoregressive process in (3) in the case of no exogenous covariates
($x_{t}=0$) and no deterministic components ($d_{t}=0$). With the same
rationale of $\alpha_{t}$, we define the score-driven-based updating of the
parameters for the exogenous covariates by

$\displaystyle\gamma_{t+1}$ $\displaystyle=$
$\displaystyle\delta_{\gamma}+\phi_{\gamma}\gamma_{t}+\kappa_{\gamma}(y_{t}-\lambda_{t})\lambda_{t}^{-1}x_{t}$
(5)

where $\delta_{\gamma}$, $\phi_{\gamma}$ and $\kappa_{\gamma}$ are fixed and
unknown parameters. Allowing for time-varying coefficients of the exogenous
covariates is particularly useful to capture dynamic effects of the covariates
on the conditional mean of the count process. In particular, this feature of
the model enables a much better fit of the structural breaks and time-varying
causality due to the occurrence of exogenous shocks, which is particularly
recurrent in economic and financial phenomena.

Note also that in (3) we adopt an exponential link function, which allows the
conditional mean $\lambda_{t}$ to be always positive, i.e. there is no need to
impose further restrictions to have $\lambda_{t}>0$, for any $t$. Moreover,
there is no need to transform the $m$ exogenous covariates in $x_{t}$ to make
them positive as done, e.g., in Agosto et al., (2016) (see Section 5.2 for an
empirical illustration).

Remark 1. Several models discussed in the literature are special cases of the
model proposed here. For instance, the model proposed by Harvey, (2013) is a
special case of the model in (3) with no covariates and deterministics, i.e.
$x_{t}=0$ and $d_{t}=0$, and where the parameter $\alpha$ is time-invariant.
Also, the resulting model is equivalent to the Poisson model of Davis et al.,
(2003) for the case of time-invariant coefficients of the exogenous
covariates. $\square$

Despite the time-varying specification of the model in (3)-(5) is in the same
vein of Blasques et al., (2019), i.e. the coefficient $\alpha$ is allowed to
change at each time period by means of the score-driven approach, with respect
to Blasques et al., (2019) our proposal is developed to model time series of
counts instead of continuous responses and we can include $m$ exogenous
covariates in the model specification. Moreover, our proposal extends the
Poisson autoregressive model by Fokianos et al., (2009) and static PARX model
by Agosto et al., (2016) by allowing for time-varying parameters, including
the coefficients for the exogenous covariates.

We complete this section by obtaining sufficient conditions for geometric
ergodicity of the Poisson autoregressive model described above. As usual in
the literature, let $\\{N_{t}(\,\cdot\,)\\}_{t\in\mathbb{Z}}$ denotes a
sequence of independent Poisson processes with unity intensity, so that the
process $\\{y_{t}\\}_{t\in\mathbb{Z}}$ in (2) can be expressed as

$\displaystyle y_{t}=N_{t}(\lambda_{t}),$

where $\lambda_{t}$ follows the dynamics given in (3) and (5), see e.g.
Fokianos et al., (2009). We begin by introducing the first result.

######  Proposition 1.

Consider the model in (3) and (5) under the condition $|\phi_{\alpha}|<1$.
Then, $\\{\alpha_{t}\\}_{t\in\mathbb{Z}}$ admits a strictly stationary and
ergodic solution, with
$\bar{\alpha}:=\mathbb{E}[\alpha_{t}]=\delta_{\alpha}/(1-\phi_{\alpha})<\infty$.
Moreover, under the additional conditions that $0<\beta<1$ and
$\beta|\beta+\bar{\alpha}|<1$, then, $\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$
is geometrically ergodic and, moreover, $\\{\lambda_{t}\\}_{t\in\mathbb{Z}}$
is geometrically ergodic with $\mathbb{E}[\lambda_{t}]<\infty$.

Remark 2. In line with the $\log$-linear Poisson autoregressive model of
Fokianos et al., (2009), by considering the full model given by equations (3)
and (5) instead of the restricted on with no covariates and dummy variables
$x_{t}$ and $\delta_{t}$, respectively, does not crucially affect the
conditions for stationarity and ergodicity stated in Proposition 1. In fact,
since $\delta_{t}$ are deterministic, as soon as the exogenous covariate
processes $\\{x_{t}\\}_{t\in\mathbb{Z}}$ are Markov chains, we only need to
retrive a set of separate conditions for their transition mechanism, together
with need the additional assumption that $|\phi_{\gamma}|<1$.

It is well-known that when dealing with nonlinear time series models it is
usually not easy to establish clear and/or simple stationarity conditions.
However, we note that the sufficient conditions given in Proposition 1 are
remarkably easy to check. Moreover, a direct consequence of Proposition 1 is
that if the process $\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$ is initialized at
its stationary distribution, then the process $\\{y_{t}\\}_{t\in\mathbb{Z}}$
will also be stationary and ergodic with a finite first moment
$\mathbb{E}[y_{t}]=\mathbb{E}[\lambda_{t}]<\infty$; see Davis et al., (2003).

We conclude this section with a new proposition which gives sufficient
conditions for $\mathbb{E}[y_{t}^{k}]<\infty$, with $k$ is a strictly positive
number.

######  Proposition 2.

Under the assumptions of proposition 1
$\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$ and $\\{y_{t}\\}$ have finite moments
of any order, i.e. for any $k>0$, $\mathbb{E}[|\log\lambda_{t}|^{k}]<\infty$
and $\mathbb{E}[y_{t}^{k}]<\infty$.

It is worth to note that Proposition 2 we do not need stricter contraction
conditions in order to obtain the finiteness of the higher-order moments. This
feature of our model is particularly useful since it allow us to derive bounds
for proving the asymptotic properties of the MLE, which is the main argument
of the next Section, without imposing further restrictions on the data
generating process.

##  3 Maximum likelihood estimation

The TV-PARX model can be easily estimated by standard ML, since the predictive
log-likelihood is available in closed form. In the following, for the sake of
simplicity, we consider the case of no exogenous covariates and no
deterministic components, i.e.

$\displaystyle\log\lambda_{t+1}$ $\displaystyle=$
$\displaystyle\omega+\beta\log\lambda_{t}+\alpha_{t+1}(y_{t}-\lambda_{t})\lambda_{t}^{-1}$
(6)

which corresponds to (3) where $x_{t}=0$ and $d_{t}=0$, and is labeled the TV-
PAR model. Note that in the case where $x_{t}\neq 0$ and $d_{t}\neq 0$, the
asymptotic theory developed for our MLE below, could be adapted
straightforwardly using partial likelihood theory, see Wong, (1986).

Denote the parameter vector
$\boldsymbol{\theta}=(\boldsymbol{\xi}^{\prime},\boldsymbol{\psi}^{\prime})^{\prime}\in\boldsymbol{\Theta}\subset\mathbb{R}^{5}$,
where $\boldsymbol{\xi}=(\omega,\beta)^{\prime}$,
$\boldsymbol{\psi}=(\delta_{\alpha},\phi_{\alpha},\kappa_{\alpha})^{\prime}$,
and $\boldsymbol{\Theta}$ is a compact parameter space. The true value of the
combined parameter vector is denoted by $\boldsymbol{\theta}_{0}$, and we
assume that $\\{y_{t}\\}_{t=1}^{T}$ is generated according to the TV-PAR
process described by equations (1), (6) and (4), evaluated at
$\boldsymbol{\theta}_{0}$. The $\log$-likelihood function is given by

$\displaystyle\hat{L}_{T}(\boldsymbol{\theta})=\frac{1}{T}\sum_{t=1}^{T}\hat{l}_{t}(\boldsymbol{\theta}),$
(7)

where

$\displaystyle\hat{l}_{t}(\boldsymbol{\theta})=y_{t}\log\hat{\lambda}_{t}(\boldsymbol{\theta})-\hat{\lambda}_{t}(\boldsymbol{\theta})$
(8)

and

$\displaystyle\log\hat{\lambda}_{t+1}(\boldsymbol{\theta})$ $\displaystyle=$
$\displaystyle\omega+\beta\log\hat{\lambda}(\boldsymbol{\theta})+\hat{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}-\hat{\lambda}_{t}(\boldsymbol{\theta}))\hat{\lambda}^{-1}_{t}(\boldsymbol{\theta})$
(9) $\displaystyle\hat{\alpha}_{t+1}(\boldsymbol{\theta})$ $\displaystyle=$
$\displaystyle\delta_{\alpha}+\phi_{\alpha}\hat{\alpha}_{t}(\boldsymbol{\theta})+\kappa_{\alpha}(y_{t}-\hat{\lambda}_{t}(\boldsymbol{\theta}))\hat{\lambda}_{t}^{-1}(\boldsymbol{\theta})(y_{t-1}-\hat{\lambda}_{t-1}(\boldsymbol{\theta}))\hat{\lambda}_{t-1}^{-1}(\boldsymbol{\theta}).$
(10)

Note that the time-varying parameters in (9) and (10) are obtained recursively
by using some fixed starting values
$\hat{\lambda}_{1}(\boldsymbol{\theta})\in\mathbb{R}_{+}$,
$\hat{\alpha}_{1}(\boldsymbol{\theta})\in\mathbb{R}$ and the observations
$\\{y_{t}\\}_{t=1}^{T}$.

The MLE $\hat{\boldsymbol{\theta}}_{T}$ of $\boldsymbol{\theta}$ is defined as

$\displaystyle\hat{\boldsymbol{\theta}}_{T}=\operatorname*{argmax}_{\theta\in\boldsymbol{\Theta}}\hat{L}_{T}(\boldsymbol{\theta}).$
(11)

The conditions stated in Proposition 1 implies that $\alpha_{t}$ and
$\log\lambda_{t}$ have stationary representations. Now, for the likelihood
analysis and the asymptotic properties of the MLE we need to derive the
stochastic limit properties of the filtered parameters
$\\{\hat{\lambda}_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{N}}$ and
$\\{\hat{\alpha}_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{N}}$, which can be
seen as two stochastic functions. In particular, because of the
initializations $\hat{\lambda}_{1}(\boldsymbol{\theta})$ and
$\hat{\alpha}_{1}(\boldsymbol{\theta})$, the filtered parameters in (9) and
(10), and therefore the approximate $\log$-likelihood function in (7), are in
general non-stationary. Therefore, in the next proposition, we derive
sufficient conditions, which ensure that the effect that
$\hat{\lambda}_{1}(\boldsymbol{\theta})$ and
$\hat{\alpha}_{1}(\boldsymbol{\theta})$ have on the approximate
$\log$-likelihood function
$\\{\hat{L}_{T}(\boldsymbol{\theta})\\}_{t\in\mathbb{N}}$ vanish almost surely
and exponentially fast (e.a.s.) uniformly over $\boldsymbol{\Theta}$. This
phenomenon is well-known in the literature of nonlinear time series models,
which is usually referred to the notion of invertibility, see Straumann and
Mikosch, (2006). Stationarity and ergodicity of $\\{\log\lambda_{t}\\}$ (as
established in Proposition 1), and Proposition 3 below, are the key
ingredients for establish asymptotic properties of the MLE.

######  Proposition 3.

Consider the model in (6) and (10) under the conditions of Proposition 1. Let
$\boldsymbol{\Theta}$ be compact with $\kappa_{\alpha}>0$ and assume that

$\displaystyle\mathbb{E}\Big{[}\log\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{|}\beta\exp\Big{\\{}\omega+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)-\ell(1-\beta)\Big{\\}}\Big{|}\Big{]}<0,$
(12)

where

$\displaystyle\bar{\alpha}_{t+1}(\boldsymbol{\theta})$
$\displaystyle:=\delta_{\alpha}+\phi_{\alpha}\bar{\alpha}_{t}(\boldsymbol{\theta})+\kappa_{\alpha}(y_{t}e^{-\ell}-1)(y_{t-1}e^{-\ell}-1),$
$\displaystyle\ell$
$\displaystyle:=\frac{\omega-\frac{\delta_{\alpha}+\kappa_{\alpha}}{1-\phi_{\alpha}}}{1-\beta}$

Then,

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{\alpha}_{t}(\boldsymbol{\theta})-{\alpha}_{t}(\boldsymbol{\theta})|\xrightarrow[]{e.a.s.}0\hskip
28.45274pt\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{\lambda}_{t}(\boldsymbol{\theta})-{\lambda}_{t}(\boldsymbol{\theta})|\xrightarrow[]{e.a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,t\rightarrow\infty,$
(13)

where $\\{{\alpha}_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{Z}}$ and
$\\{{\lambda}_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{Z}}$ are stationary and
ergodic.

Moreover, if

$\displaystyle\sup_{(\boldsymbol{\theta}\times
y)\in(\boldsymbol{\Theta}\times\mathbb{N}_{0})}\Big{|}\beta\exp\Big{\\{}\omega+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(ye^{-\ell}-1)-\ell(1-\beta)\Big{\\}}\Big{|}<1,$
(14)

then $\exists m>0$ such that
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}(\boldsymbol{\theta})|^{m}]<\infty$.

Condition (12) is a sufficient condition which ensures the invertibility of
the TV-PARX model by an application of Proposition 5.2.12 in Straumann,
(2005), or equivalently Theorem 2.8 in Straumann and Mikosch, (2006). It is
different from the conditions for stationarity and ergodicity given in Theorem
1, and moreover, it is only concerned with the filtering recursion relative to
$\lambda_{t}(\boldsymbol{\theta})$ because the stationarity condition imposed
in Proposition 1 for $\alpha_{t}(\boldsymbol{\theta})$ is also sufficient for
filter invertibility. The complex form of the contraction condition in (12) is
due to the exponential function. A similar problem with related discussion can
be found in Wintenberger, (2013) for the EGARCH(1,1) model. Nevertheless,
despite its sufficiency, this condition it is still important as it underline
that the invertibility region is not degenerate.

Finally, by slightly reinforcing the contraction condition in (12) as in (14),
Proposition 3 also ensures the existence of an arbitrary large number of
unconditional moments for the stationary and ergodic intensity process
$\\{\lambda_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{Z}}$. This is a crucial
property which will be useful for proving the consistency and asymptotic
normality of the MLE.

To derive the asymptotic properties of the MLE we follow the classic theory in
White, (1994). First, for consistency we consider the almost sure uniform
convergence of the approximate $\log$-likelihood function in (7). In
particular, Lemma 3.1 below ensures that the average $\log$-likelihood
$\hat{L}_{T}(\boldsymbol{\theta})$ converges uniformly to a function
$L(\boldsymbol{\theta})$.

######  Lemma 3.1.

Let the conditions of Propositions 1-3 hold true. Then,

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}(\boldsymbol{\theta})|\xrightarrow[]{a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,T\rightarrow\infty,$

where
${L}(\boldsymbol{\theta}):=\mathbb{E}[l_{1}(\boldsymbol{\theta})]=\mathbb{E}[y_{1}\log\lambda_{1}(\boldsymbol{\theta})-\lambda_{1}(\boldsymbol{\theta})]$.

Second, we verify the identifiability of the model parameterization, which is
the argument of the next Lemma.

######  Lemma 3.2.

The true parameter vector $\boldsymbol{\theta}_{0}$ is the unique maximizer of
${L}(\boldsymbol{\theta})$ in $\boldsymbol{\Theta}$; that is, for any
$\boldsymbol{\theta}\neq\boldsymbol{\theta}_{0}$, then
${L}(\boldsymbol{\theta})<{L}(\boldsymbol{\theta}_{0})$.

As a consequence, given the results obtained in Lemma 3.1 and Lemma 3.2, an
application of Theorem 2.11 of White, (1994) implies the strong consistency of
the MLE.

######  Theorem 3.1.

Let the conditions of Propositions 1-3 hold true. Then,
$\hat{\boldsymbol{\theta}}_{T}\xrightarrow[]{a.s.}\boldsymbol{\theta}_{0}$ as
$T\rightarrow\infty.$

Next, we discuss the asymptotic distribution of the MLE. Consider the
normalized score evaluated at $\boldsymbol{\theta}_{0}$

$\displaystyle\boldsymbol{\bar{\eta}}_{T}=\sqrt{T}\frac{\partial
L_{T}(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}}=\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\boldsymbol{\eta}_{t}\
\ ,\ \
\boldsymbol{\eta}_{t}=e_{t}\lambda_{t}\log\lambda^{\boldsymbol{\theta}_{0}}_{t},$
(15)

where
$e_{t}:=(y_{t}-\lambda_{t}(\boldsymbol{\theta}_{0}))\lambda^{-1}_{t}(\boldsymbol{\theta}_{0})$,
$\lambda_{t}:=\lambda_{t}(\boldsymbol{\theta}_{0})$ and
$\log\lambda^{\boldsymbol{\theta}_{0}}_{t}:=\frac{\partial\log\lambda_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}=\boldsymbol{\theta}_{0}}$.
Since
$\boldsymbol{\theta}=(\boldsymbol{\xi}^{\prime},\boldsymbol{\psi}^{\prime})^{\prime}$,
we have that
$\log\lambda^{\boldsymbol{\theta}_{0}}_{t}=((\log\lambda^{\boldsymbol{\xi}_{0}}_{t})^{\prime},(\log\lambda^{\boldsymbol{\psi}_{0}}_{t})^{\prime})^{\prime}$,
where
$\log\lambda^{\boldsymbol{\xi}_{0}}_{t}=\left(\frac{\partial\log\lambda_{t}(\boldsymbol{\theta}_{0})}{\partial\omega}^{\prime},\frac{\partial\log\lambda_{t}(\boldsymbol{\theta}_{0})}{\partial\beta}^{\prime}\right)^{\prime}$
and $\log\lambda^{\boldsymbol{\psi}}_{t}=\alpha^{\boldsymbol{\psi}}_{t}e_{t}$,
with
$\alpha^{\boldsymbol{\psi}_{0}}_{t}=\left(\frac{\partial\alpha_{t}(\boldsymbol{\theta}_{0})}{\partial\delta_{\alpha}}^{\prime},\frac{\partial\alpha_{t}(\boldsymbol{\theta}_{0})}{\partial\phi_{\alpha}}^{\prime},\frac{\partial\alpha_{t}(\boldsymbol{\theta}_{0})}{\partial\kappa_{\alpha}}^{\prime}\right)$,
as defined in Lemma A.1 in the Appendix.

Asymptotic normality then follows by a standard central limit theorem for
martingale difference sequences, see e.g. Corollary 3.1 of Hall and Heyde,
(1980), by showing that

$\displaystyle\sum_{t=1}^{T}\mathbb{E}[\boldsymbol{\eta}_{t}\boldsymbol{\eta}_{t}^{\prime}|\mathcal{F}_{t-1}]\xrightarrow[]{P}\boldsymbol{V},$
(16)

where

$\displaystyle\boldsymbol{V}=\lim_{T\rightarrow\infty}\frac{1}{T}\sum_{t=1}^{T}\lambda_{t}(\log\lambda^{\boldsymbol{\theta}_{0}}_{t}\log\lambda^{\boldsymbol{\theta}_{0}^{\prime}}_{t}),$
(17)

and, the Lindberg condition holds, i.e., $\forall\epsilon>0$,

$\displaystyle\mathbb{E}[\boldsymbol{\eta}_{t}\boldsymbol{\eta}_{t}^{\prime}\mathbbm{1}(\|\boldsymbol{\eta}_{t}\|>\epsilon)|\mathcal{F}_{t-1}]\xrightarrow[]{P}\boldsymbol{0}.$
(18)

These conditions are verified in the following proposition.

######  Proposition 4.

Let the conditions of Propositions 1-3 hold true. Moreover, assume that
$\mathbb{E}[|A_{t}|^{2}]<1$ and $0<|\phi_{\alpha}|<1$, then (15), (16) and
(17) hold and the score function in (15) satisfies

$\displaystyle\boldsymbol{\bar{\eta}}_{T}\overset{d}{\to}\mathcal{N}(\boldsymbol{0},\boldsymbol{V}).$

Denote the information matrix at the true value by $\boldsymbol{J}$, i.e.

$\displaystyle\boldsymbol{J}=\mathbb{E}\bigg{[}\frac{1}{\lambda_{t}}\frac{\partial\lambda_{t}(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}}\frac{\partial\lambda_{t}(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}^{\prime}}\bigg{]}.$
(19)

Then, we finally state the following theorem.

######  Theorem 3.2.

Let the conditions of Propositions 1-4 hold true. Moreover, assume that the
true parameter vector $\boldsymbol{\theta}_{0}$ is in the interior of
$\boldsymbol{\Theta}$. Then, the MLE $\hat{\boldsymbol{\theta}}_{T}$ is
asymptotically normal,

$\displaystyle\sqrt{T}(\hat{\boldsymbol{\theta}}_{T}-{\boldsymbol{\theta}}_{0})\Rightarrow\mathcal{N}(\boldsymbol{0},\boldsymbol{J}^{-1}).$

Remark 3. It is worth noting that the results obtained for the MLE
$\hat{\boldsymbol{\theta}}_{T}$ in this section hold true even if the
conditional distribution is misspecified. This is a consequence of the fact
that the Poisson distribution belongs to the linear exponential family. In
particular, as proved by White, (1982) and Gourieroux et al., (1984), under
high-level assumptions the MLE is a QMLE and maintain the same properties, the
only change will occur on the asymptotic variance-covariance matrix of
$\sqrt{T}(\hat{\boldsymbol{\theta}}_{T}-{\boldsymbol{\theta}}_{0})$, which
will assume the classic sandwitch form, that is

$\displaystyle\sqrt{T}(\hat{\boldsymbol{\theta}}_{T}-{\boldsymbol{\theta}}_{0})\Rightarrow\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}),$

where
$\boldsymbol{\Sigma}=\boldsymbol{J}^{-1}\boldsymbol{I}\boldsymbol{J}^{-1}$,
with $\boldsymbol{J}$ as in equation (19) and
$\boldsymbol{I}=\mathbb{E}\Big{[}\frac{\mathbb{V}[y_{t}|\mathcal{F}_{t-1}]}{\lambda^{2}_{t}(\boldsymbol{\theta}_{0})}\frac{\partial\lambda_{t}(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}}\frac{\partial\lambda_{t}(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}^{\prime}}\Big{]}$.
See also Ahmad and Francq, (2016) or Aknouche and Francq, (2021).

##  4 Finite sample performance

In this section we investigate the empirical performance in finite samples of
the TV-PAR model compared to the standard PAR model. In particular, the aim of
this Monte Carlo exercise is to investigate how quickly the $\lambda_{t}$ can
adapt to (structural) changes. The time series are generated from the
following DGP

$y_{t}=N_{t}(\lambda_{t}^{0}),$ (20)

where $N_{t}(\cdot)$ denotes a sequence of independent Poisson processes with
unity intensity. Similarly to Blasques et al., (2019), for the time-varying
$\lambda_{t}^{0}$, we consider a deterministic step-function

$\displaystyle\lambda_{t}^{0}=\begin{cases}2&\hskip 28.45274pt\text{if}\hskip
28.45274pt\sin(\gamma^{-1}10^{-2}(\pi t-1)\geq 0\\\ 2+\delta&\hskip
28.45274pt\text{if}\hskip 28.45274pt\sin(\gamma^{-1}10^{-2}(\pi
t-1)<0\end{cases}$ (21)

where $\delta\in\\{0,2,4\\}$ and $\gamma\in\\{1,1.5,2\\}$. In this Monte Carlo
exercise we generate $m=1000$ time series of sample sizes
$T\in\\{250,500,1000\\}$. Figure 1 reports the time varying intensity
$\lambda_{t}^{0}$ for all the combinations of $\delta$ and $\gamma$
considered. For each series we estimate the time-varying PAR in (6) and the
static PAR, i.e. where $\alpha_{t}=\alpha$ in (6). Table 1 reports the Root
Mean Squared Errors (RMSE) for all the cases considered. The results in Table
1 show that the RMSE of the TV-PAR is smaller for all DGPs considered, except
for, unsurprinsigly, the case of no steps. However, in the case of no steps,
i.e. when $\delta=0$, the difference in RMSE between the time-varying and
static models is relatively small in all the cases considered.

Conversely, when the $\delta>0$, the RMSE of the TV-PAR model is smaller than
the one of the standard PAR model for the all cases considered. In particular,
the higher the $\delta$, the better the performance of TV-PAR model compared
to the standard PAR model, regardless of the sample size considered.
Therefore, with more convoluted DGP dynamics, the time-varying model
specification works better than to its static counterpart.

To illustrate, we report in Figure 2 the cases of the DGPs with $\delta=4$ and
$\gamma=1$ for $T=250,500,$ and 1000. This figure shows the mean
$\hat{\lambda}_{t}$ and the corresponding 95% confidence bands for the TV-PAR
(in green) and the static PAR (in red) models. From Figure 2, it clearly
emerges that the time-varying model specification provides a much faster
reaction and better fit to the jumps created by the step-function in (21).

##  5 Empirical illustrations

In this section, we show the usefulness of the time-varying specification of
the TV-PARX model by considering two empirical illustrations. Section 5.1
deals with the daily counts of infections of SARS-COV-2 virus in Italy until
the end of May 2021. Section 5.2 analyzes the monthly corporate default counts
in the US as previously analyzed by Agosto et al., (2016).

###  5.1 COVID-19

The COVID-19 pandemic, started in 2019 in the city of Wuhan in China’s region
of Hubei and spread all around the world in 2020, has caused a tremendous
public health emergency and a huge negative shock on the business cycle. Since
then, researchers of various fields have started to work on that topic in
order to help policy makers in their decisions. Are the number of cases
(deaths) predictable? Are the evolution of the number of cases (deaths)
similar across countries? These questions have increased the interest among
researchers and practitioners for new models suited for time series of counts.
Epidemiologists, for instance, use reduced-form models to produce forecasts of
the number of infections (or deaths), see for instance Batista, (2020) for a
logistic growth model and Giordano et al., (2020) and Hethcote, (2000) for
reviews on several variants of the Susceptible, Infectious and Removed (SIR)
model.

Due to the convoluted time evolution of the COVID-19 pandemic (see the number
of new cases depicted in blue in Figure 3), the time-varying specification of
the model is expected to capture the different waves of the pandemic and its
non-standard dynamics are likely to outperforms much better time-invariant
alternatives. Among others, Li and Linton, (2021) found that the model
parameters change both over time and across countries. Country-specific
behaviour and local, transient outbreaks of COVID-19 is also found by
Khismatullina and Vogt, (2020) and Dong et al., (2020) using non-parametric
approaches and by Palmer et al., (2021) using Bayesian hierarchical models.

Therefore, we fit the TV-PARX model in (3)-(5) with $\gamma_{t}=0$ to the
daily counts of confirmed new cases in Italy from the beginning of the
COVID-19 pandemic outbreak (March 2020) to the end of May 2021, which involves
three or four main waves of cases (see Figure 3). The counts dynamics in
Figure 3 shows a seasonal component mainly due to reporting. Hence, we
consider the following TV-PARX model

$\displaystyle\log\lambda_{t+1}$ $\displaystyle=$
$\displaystyle\omega+\beta\log\lambda_{t}+\alpha_{t+1}(y_{t}-\lambda_{t})\lambda_{t}^{-1}+\psi^{\prime}d_{t}$
$\displaystyle\alpha_{t+1}$ $\displaystyle=$
$\displaystyle\delta_{\alpha}+\phi_{\alpha}\alpha_{t}+\kappa_{\alpha}(y_{t}-\lambda_{t})\lambda_{t}^{-1}(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1}$
(22)

where $d_{t}$ denotes seasonal (daily) dummies.

For comparison purposes, we also fit a time-invariant PARX model where
$\phi_{\alpha}=\kappa_{\alpha}=0$ in (22). Table 2 shows the estimates and
Table 3 reports the differences in the log-likelihoods, information criteria
and Root Mean Squared Errors (RMSE) of the estimated models. In terms of in-
sample performance, these results show that the time-varying specification of
the TV-PARX model is preferred over its time-invariant counterpart. Moreover,
albeit both the time-varying model (TV-PARX) and its time-invariant
counterpart (PARX) provide a good fit to the observed data, from Figure 3 it
emerges that the TV-PARX model approximates better the different waves of the
COVID-19 pandemic in Italy. In particular, the estimated time-varying
parameter $\hat{\alpha}_{t}$ ranges from around -0.2 to 0.8, and the minimum
value virtually corresponds to the end of the first wave (end of June 2020),
while the maximum values are observed at specific time points which are
closely related to the beginning of each wave, e.g. March 2020, September-
November 2020 and January 2021, as well as abrupt changes in
$\hat{\alpha}_{t}$ for the last wave begun in March 2021.

###  5.2 Corporate defaults

In this section, we stress the usefulness of the time-varying specification of
the TV-PARX model compared to the standard approach based on the constant
parameter version of the PARX model by analyzing the count time series of
corporate defaults in US. Agosto et al., (2016) contributed to the literature
on modeling and forecasting time series of corporate defaults, by exploring
the stylized fact that defaults tend to cluster over time and investigating
the causes of this phenomenon. In particular, using a (constant parameter)
PARX model they discriminate between comovements in corporate solvency caused
by common underlying macroeconomic and financial risk factors, i.e.
“systematic risk”, and feedback effects, i.e. “contagion”, where,
conditionally on the common risk factors, the current number of defaults
affects the probability of other firms’ future insolvency. As pointed out in
Agosto et al., (2016), one limitation of their approach is that the measure of
contagion ignores feedback effects from defaults to covariates; that is, the
case when $x_{t}$ is affected by past defaults, which in turn will affect
future defaults. This issue is softened by the time-varying model
specification of the TV-PARX as the time-dependent parameters may allow to
capture the feedback effects from past defaults to the covariates. The data
set on default counts analyzed by Agosto et al., (2016) consists of the number
of bankruptices among Moody’s rated industrial firms in the US collected at
monthly frequency between the 1982 and 2011, for a total of $T=360$
observations, from the Moody’s Credit Risk Calculator (CRC). The blue line in
Figure 5 shows the default count time series. As discussed in Agosto et al.,
(2016), Figure 5 discloses that default counts is a process characterised by
(i) high persistence, (ii) clusters over time, and (iii) overdispersion. These
three stylized facts are satisfactorily captured by the PARX model. However,
the peak of defaults observed in 2008 (Figure 5) as well as the high increase
in bankruptices around the year 2000, possibly interpretable as structural
breaks, may be better captured by the time-varying specification of the TV-
PARX model, which can locally allow for explosive roots rather than standard
PARX model where the stationary condition must hold. Therefore, we consider
the TV-PARX model, described in (3)-(5), with no deterministic componenents,
i.e. $d_{t}=0$. For the sake of comparison, in the TV-PARX model specification
we include the same covariates considered in the PARX model in Agosto et al.,
(2016).111Note that, with the specification in logs of the TV-PARX model, we
are able to consider the whole time series of LI, and not only the negative
values (in modulus to ensure non-negativity) as in Agosto et al., (2016).  In
particular, we consider $x_{t}=(RV_{t},LI_{t})^{\prime}$, where $RV_{t}$ is
the realized volatility computed as a proxy of the S&P 500 monthly realized
volatility using daily squared returns, and $LI_{t}$ is the Leading Index
released by the Federal Reserve. More formally, we consider the following TV-
PARX model

$\displaystyle\log\lambda_{t+1}$ $\displaystyle=$
$\displaystyle\omega+\beta\log\lambda_{t}+\alpha_{t+1}(y_{t}-\lambda_{t})\lambda_{t}^{-1}+\gamma_{1,t+1}RV_{t}+\gamma_{2,t+1}LI_{t}$
$\displaystyle\alpha_{t+1}$ $\displaystyle=$
$\displaystyle\delta_{\alpha}+\phi_{\alpha}\alpha_{t}+\kappa_{\alpha}(y_{t}-\lambda_{t})\lambda_{t}^{-1}(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1}$
$\displaystyle\gamma_{1,t+1}$ $\displaystyle=$
$\displaystyle\delta_{\gamma_{1}}+\phi_{\gamma_{1}}\gamma_{1,t}+\kappa_{\gamma_{1}}(y_{t}-\lambda_{t})\lambda_{t}^{-1}RV_{t}$
$\displaystyle\gamma_{2,t+1}$ $\displaystyle=$
$\displaystyle\delta_{\gamma_{2}}+\phi_{\gamma_{2}}\gamma_{2,t}+\kappa_{\gamma_{2}}(y_{t}-\lambda_{t})\lambda_{t}^{-1}LI_{t}.$
(23)

We provide an analysis for the full sample 1982-2011 and compare the results
for the time-invariant PARX model specification. For comparison purposes, we
also fit a time-invariant PARX model where
$\phi_{\alpha}=\kappa_{\alpha}=\phi_{\gamma_{1}}=\kappa_{\gamma_{1}}=\phi_{\gamma_{2}}=\kappa_{\gamma_{2}}=0$
in (23). The maximum likelihood estimates are reported in Table 2. According
to the information criteria and the RMSE reported in Table 3, the TV-PARX
model provides a better in-sample fit than the PARX model, except for the
BIC.222It is interesting to note that our TV-PARX model outperforms the best
model specification in Agosto et al., (2016), namely (non-logs) PARX(2,1). As
can be noted from Figure 5, both models capture the actual default counts
dynamics well, but the goodness of fit of the TV-PARX model is superior during
the periods where the number of defaults is larger, especially around the year
2009. Figure 6 shows the filtered dynamic parameters, $\alpha_{t}$ and
$\gamma_{t}$. From this figure, it can be noted that the estimated time-
varying parameters are able to adjust according to the observed default count
time series. Interestingly, the time-varying specification allows us to
determine the main driver of a specific crisis. In particular, the dynamics of
the estimated parameters $\gamma_{1,t}$ and $\gamma_{2,t}$ in Figure 6
suggests that financial volatility is the main driver of the spike of the
defaults in the US during the 2008-09 crisis, while during the early 90s
recession the main driver is mainly macroeconomic. Therefore, the time-varying
version of the PARX model proposed in (3)-(5) provides a more flexible
alternative to the time-invariant counterpart.

##  6 Conclusion

In this paper, we have developed a novel class of score-driven models for time
series of counts, allowing for time-varying parameters and the inclusion of
exogenous variables. We have provided conditions for stationarity and
ergodicity, as well as asymptotic results for the MLE. Moreover, we have shown
the relevance of our modeling approach by focusing on a Monte Carlo simulation
study. We have considered two empirical illustrations where we apply our
nonlinear time-varying Poisson autoregressive model: the first applies the TV-
PARX model with deterministic components to the number of COVID-19 infections
in Italy; the second focuses on count time series of corporate defaults in the
US by including (time-varying) exogenous covariates in the TV-PARX model
specification. For these relevant illustrations, we find that the proposed
time-varying framework is capable of improving the in-sample fit. Further
issues are left to future research. In particular, in this paper we have
specified an univariate TV-PARX model. A multivariate extension could be
extremely useful to capture the relationships between different time series of
counts (e.g. COVID-19 cases or number of defaults in different countries) and
is currently under investigation by the authors. Moreover, the time-varying
specification of $\alpha_{t}$ could be used to construct a new test for the
presence of structural breaks for time series of counts.

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| $\gamma=1$ | $\gamma=1.5$ | $\gamma=2$
---|---|---|---
| PARX | TV-PARX | PARX | TV-PARX | PARX | TV-PARX
$T=250$
$\delta=0$ | 0.0077 | 0.0086 | 0.0077 | 0.0086 | 0.0077 | 0.0086
$\delta=2$ | 0.4052 | 0.3686 | 0.3317 | 0.2630 | 0.3358 | 0.2776
$\delta=4$ | 0.6331 | 0.5928 | 0.5034 | 0.3749 | 0.5002 | 0.3842
$T=500$
$\delta=0$ | 0.0033 | 0.0049 | 0.0033 | 0.0049 | 0.0033 | 0.0049
$\delta=2$ | 0.4097 | 0.3914 | 0.3729 | 0.3477 | 0.3371 | 0.3115
$\delta=4$ | 0.6324 | 0.6092 | 0.5726 | 0.5359 | 0.5198 | 0.4810
$T=1000$
$\delta=0$ | 0.0026 | 0.0031 | 0.0026 | 0.0031 | 0.0026 | 0.0031
$\delta=2$ | 0.4245 | 0.4121 | 0.3784 | 0.3644 | 0.3373 | 0.3212
$\delta=4$ | 0.6529 | 0.6363 | 0.5863 | 0.5626 | 0.5224 | 0.4931
Table 1: Root Mean Squared Error (RMSE) where the error is between the true
$\lambda_{t}^{0}$ and the filtered parameter $\lambda_{t}$ from TV-PARX and
PARX models for different true values of $\delta$ and $\gamma$.

| SARS-COV-2 | Defaults
---|---|---
| TV-PARX | PARX | TV-PARX | PARX
$\omega$ | $\underset{(0.0002)}{0.0045}$ | $\underset{(0.0002)}{0.0039}$ | $\underset{(0.0323)}{-0.0227}$ | $\underset{(0.0056)}{0.0219}$
$\beta$ | $\underset{(0.0001}{0.9990}$ | $\underset{(0.0001)}{0.9990}$ | $\underset{(0.0386)}{0.9423}$ | $\underset{(0.0038)}{0.9441}$
$\psi_{1}$ | $\underset{(0.0197)}{-0.0514}$ | $\underset{(0.0002)}{-0.0416}$ | - | -
$\psi_{2}$ | $\underset{(0.0071)}{-0.1310}$ | $\underset{(0.0001)}{-0.1246}$ | - | -
$\psi_{3}$ | $\underset{(0.0004)}{-0.2767}$ | $\underset{(0.0001)}{-0.2724}$ | - | -
$\psi_{4}$ | $\underset{(0.0106)}{0.1739}$ | $\underset{(0.0005)}{0.1794}$ | - | -
$\psi_{5}$ | $\underset{(0.0065)}{0.1578}$ | $\underset{(0.0002)}{0.1602}$ | - | -
$\psi_{6}$ | $\underset{(0.0100)}{0.1043}$ | $\underset{(0.0006)}{0.1083}$ | - | -
$\delta_{\alpha}$ | $\underset{(0.0017)}{-0.0013}$ | $\underset{(0.0021)}{0.7033}$ | $\underset{(0.0061)}{-0.0012}$ | $\underset{(0.0021)}{0.2369}$
$\phi_{\alpha}$ | $\underset{(0.0004)}{0.9984}$ | - | $\underset{(0.0227)}{0.9946}$ | -
$\kappa_{\alpha}$ | $\underset{(0.0250)}{0.5000}$ | - | $\underset{(0.0798)}{0.0157}$ | -
$\delta_{\gamma_{1}}$ | - | - | $\underset{(2.6856)}{1.1954}$ | $\underset{(0.1994)}{11.8267}$
$\phi_{\gamma_{1}}$ | - | - | $\underset{(0.1418)}{0.9498}$ | -
$\kappa_{\gamma_{1}}$ | - | - | $\underset{(273.6449)}{917.6204}$ | -
$\delta_{\gamma_{2}}$ | - | - | $\underset{(0.0062)}{-0.0004}$ | $\underset{(0.0018)}{0.0010}$
$\phi_{\gamma_{2}}$ | - | - | $\underset{(0.3673)}{0.9766}$ | -
$\kappa_{\gamma_{2}}$ | - | - | $\underset{(0.0021)}{0.0070}$ | -
Table 2: Estimated parameters, standard errors in brackets.

| COVID-19 | Defaults
---|---|---
| TV-PARX $-$ PARX | TV-PARX $-$ PARX
log-likelihood | 422.53 | 15.11
AIC | -841.05 | -18.21
HQC | -837.79 | -8.94
BIC | -832.78 | 5.10
RMSE | -14.29 | -0.01
Table 3: Difference between the information criteria and the Root Mean Squared
Error (RMSE) of the estimated TV-PAR and PAR models for the empirical analysis
of the daily number of new COVID-19 cases in Italy and the number of monthly
corporate defaults in the US.

$T=250$
---
|  |
|  |
|  |
$T=500$
|  |
|  |
|  |
$T=1000$
|  |
|  |
|  |
Figure 1: Dynamic of $\lambda_{t}^{0}$ for $\delta\in\\{0,2,4\\}$ and
$\gamma\in\\{1,1.5,2\\}$.

$T=250$
---
$T=500$
$T=1000$
Figure 2: Estimation of $\lambda_{t}$ for $\delta=4$ and $\gamma=1$ with the
95 % confidence bands obtained with the TV-PARX (in green) and the static PARX
(in red).

Figure 3: Fitted vs actual daily number of cases of Covid-19 in Italy.

Figure 4: Estimated time-varying parameter $\alpha_{t}$ in (22) for the daily
number of cases of Covid-19 in Italy. Shaded areas are the 95% confidence
intervals.

Figure 5: Fitted vs actual monthly number of corporate defaults in US.

Figure 6: Estimated time-varying parameter $\alpha_{t}$, $\gamma_{1,t}$ and
$\gamma_{2,t}$ in (23) for the monthly number of corporate defaults in US.
Shaded areas are the 95% confidence intervals.

##  Appendix A Proofs

### Proof of Proposition 1

Following Davis et al., (2003), we define
$e_{t}:=(y_{t}-\lambda_{t})\lambda^{-1}_{t-1}$. Under the initial conditions
that $e_{s}=0$ and $y_{s}=0$ for $s\leq 0$, it is shown in Davis et al.,
(2003) that $\\{e_{t}\\}_{t\in\mathbb{Z}}$ is a martingale difference
sequence, so that $\mathbb{E}[e_{t}|\mathcal{F}_{t-1}]=0$, and
$\mathbb{E}[e_{t}^{2}]=\mathbb{E}[\lambda_{t}^{-1}]$. Therefore, as long as
$|\phi_{\alpha}|<1$, by unfolding the recursion in (4) we obtain that

$\displaystyle\alpha_{t+1}=\frac{\delta_{\alpha}}{1-\phi_{\alpha}}+\sum_{i=0}^{\infty}\kappa_{\alpha}\phi_{\alpha}^{i}e_{t-i}e_{t-i-1}.$
(24)

Thus, it is straightforward to see that
$\bar{\alpha}=\mathbb{E}[\alpha_{t}]=\delta_{\alpha}/(1-\phi_{\alpha})$. Now,
we can focus on the recursion in (6) and to prove stability and geometric
ergodicity we use the Meyn-Tweedie’s criterion with the test function
$V(x)=|x|$, see Meyn and Tweedie, (2009), and we substitute $\alpha_{t}$ with
$\bar{\alpha}$ in (6), in order to distinguish four different cases:

_(i)_ $\bar{\alpha}>0$. We have

$\displaystyle\mathbb{E}[|\log\lambda_{t+1}|\,|\log\lambda_{t}=\log\lambda]=\mathbb{E}[|\omega+\beta\log\lambda+\bar{\alpha}(y_{t}-\lambda)\lambda^{-1}|\,|\log\lambda_{t}=\log\lambda].$

If $\log\lambda_{t}=\log\lambda>0$ and $\log\lambda\rightarrow\infty$, then
$\lambda\rightarrow\infty$ and note that
$\lim_{\lambda\rightarrow\infty}(y_{t}-\lambda)\lambda^{-1}=\lim_{\lambda\rightarrow\infty}(N_{t}(\lambda)/\lambda-1)=0$
almost surely, by Lemma A.1 of Wang et al., (2014). Thus

$\displaystyle\mathbb{E}[|\log\lambda_{t+1}|\,|\log\lambda_{t}=\log\lambda]\approx\beta\log\lambda,$

which implies that the condition $0<\beta<1$ ensures that the process
$\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$ is geometrically ergodic. If
$\log\lambda_{t}=\log\lambda<0$ and $|\log\lambda|\rightarrow\infty$ we take
the process in (6) two steps forward and write

$\displaystyle\log\lambda_{t+1}=\omega+\beta\omega+\beta[\beta\log\lambda_{t-1}+\bar{\alpha}(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1}]+\bar{\alpha}(y_{t}-\lambda_{t})\lambda_{t}^{-1}.$
(25)

Now, the fact that $\log\lambda_{t}=\log\lambda<0$ with
$|\log\lambda|\rightarrow\infty$ implies that $\lambda_{t}=\lambda$ becomes
very small, so that $(y_{t}-\lambda)\lambda^{-1}\rightarrow 0$, because
$y_{t}$ is Poisson distributed with intensity $\lambda$. Moreover, as
$\log\lambda_{t}=\log\lambda<0$

$\displaystyle\log\lambda_{t}=\omega+\beta\log\lambda_{t-1}+\bar{\alpha}(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1},$

we also obtain that $\log\lambda_{t-1}$ must be negative. Accordingly,
$(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1}$ could be made arbitrarily close to
zero by simply choosing $\log\lambda_{t-1}$ negative and small enough. Thus,
recalling that $e_{t-1}=(y_{t-1}-\lambda_{t-1})\lambda_{t-1}^{-1}$, the
following Taylor expansion is valid

$\displaystyle e_{t-1}=\log
1+\frac{1}{e_{t-1}+1}\bigg{|}_{e_{t-1}=0}e_{t-1}\approx\log({e}_{t-1}+1)\approx\log\bigg{(}\frac{\max(y_{t-1},c)}{\lambda_{t-1}}\bigg{)},$

for some $c\in(0,1]$. Hence, we may approximate (25) with

$\displaystyle\log\lambda_{t+1}$
$\displaystyle\approx\omega+\beta\omega+\beta^{2}\log\lambda_{t-1}+\beta\bar{\alpha}\log\bigg{(}\frac{\max(y_{t-1},c)}{\lambda_{t-1}}\bigg{)}$
$\displaystyle=\omega+\beta\omega+\beta^{2}\log\lambda_{t-1}+\beta\bar{\alpha}\log(\max(y_{t-1},c))-\beta\bar{\alpha}\log\lambda_{t-1}.$
(26)

In conclusion, for $\log\lambda_{t-1}$ negative and small enough it holds that

$\displaystyle\mathbb{E}[|\log\lambda_{t+1}|\,|\log\lambda_{t-1}=\log\lambda]\leq
K+\beta(\beta+\bar{\alpha})|\log\lambda|,$

and the condition $\beta(\beta+\bar{\alpha})<1$ ensures the geometric
ergodicity of the process $\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$.

_(ii)_ $\bar{\alpha}<0$. We can easily show that by following similar
arguments of _(i)_ for $\log\lambda_{t}=\log\lambda>0$ large, then

$\displaystyle\mathbb{E}[|\log\lambda_{t+1}|\,|\log\lambda_{t-1}=\log\lambda]\approx\beta\log\lambda.$

On the other hand, if $\log\lambda_{t}=\log\lambda<0$, with $|\log\lambda|$
large, then

$\displaystyle\mathbb{E}[|\log\lambda_{t+1}|\,|\log\lambda_{t-1}=\log\lambda]\approx\beta|\beta+\bar{\alpha}||\log\lambda|.$

It remains only to show that $\\{\lambda_{t}\\}_{t\in\mathbb{Z}}$ is
geometrically ergodic with $\mathbb{E}[\lambda_{t}]<\infty$. Again, we use the
Meyn-Tweedie’s criterion. We define a new test function $V(x)=e^{x}$ and
follow the same arguments as before, and consider the same four different
cases.

For $\log\lambda_{t}=\log\lambda>0$ and $\log\lambda$ large, then

$\displaystyle\mathbb{E}[|\lambda_{t+1}|\,|\log\lambda_{t}=\log\lambda]\leq\exp\\{\beta\log\lambda\\}=\exp\\{(\beta-1)\log\lambda\\}\exp\\{\log\lambda\\},$

which converges since $0<\beta<1$. Analogously, for
$\log\lambda_{t}=\log\lambda<0$ with $|\log\lambda|$ large, then

$\displaystyle\mathbb{E}[|\lambda_{t+1}|\,|\log\lambda_{t}=\log\lambda]\leq\exp\\{(\beta|\beta+\bar{\alpha}|-1)|\log\lambda|\\}\exp\\{|\log\lambda|\\},$

which converges since $0<\beta|\beta+\bar{\alpha}|<1$.

Therefore, the conditions stated above, are also sufficients for the geometric
ergodicity of $\\{\lambda_{t}\\}_{t\in\mathbb{Z}}$ with
$\mathbb{E}[\lambda_{t}]<\infty$.

### Proof of Proposition 2

We first show that the the stationary and ergodic
$\\{\alpha_{t}\\}_{t\in\mathbb{Z}}$ has finite $k$-th moment, $\forall k>0$.
From equation (24) in the proof of Proposition 1 we observe that

$\displaystyle\mathbb{E}[|\alpha_{t+1}|^{k}]\leq$
$\displaystyle\bigg{|}\frac{\delta_{\alpha}}{1-\phi_{\alpha}}\bigg{|}^{k}+\mathbb{E}\bigg{[}\bigg{|}\sum_{i=0}^{\infty}\kappa_{\alpha}\phi_{\alpha}^{i}e_{t-i}e_{t-i-1}\bigg{|}^{k}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\bigg{|}\frac{\delta_{\alpha}}{1-\phi_{\alpha}}\bigg{|}^{k}+\sum_{i=0}^{\infty}|\kappa_{\alpha}|^{k}|\phi_{\alpha}|^{ki}\mathbb{E}[|e_{t-i}e_{t-i-1}|^{k}].$

Since $|\phi_{\alpha}|<1$, it suffices to prove that
$\mathbb{E}[|e_{t}e_{t-1}|^{k}]<\infty$.

We can then use the law of iterated expectations, in order to show that

$\displaystyle\mathbb{E}[|e_{t}e_{t-1}|^{k}]=$
$\displaystyle\mathbb{E}[|e|^{k}_{t}|e_{t-1}|^{k}]=\mathbb{E}[\mathbb{E}[|e_{t}|^{k}|e_{t-1}|^{k}\,|\lambda_{t-1}=\lambda]]=\mathbb{E}[|e_{t}|^{k}\mathbb{E}[|e_{t-1}|^{k}\,|\lambda_{t-1}=\lambda]].$

Now, from Lemma A.1 of Wang et al., (2014), we know that the process
$\\{e^{k}_{t}\\}_{t\in\mathbb{Z}}$ is uniformly integrable for any integer $k$
$\geq 1$. Therefore, $\exists C_{1},C_{2}>0$ such that

$\displaystyle\mathbb{E}[|e_{t}|^{k}\mathbb{E}[|e_{t-1}|^{k}\,|\lambda_{t-1}=\lambda]]\leq
C_{1}\,\mathbb{E}[|e_{t}|^{k}]=C_{1}\,\mathbb{E}[\mathbb{E}[|e_{t}|^{k}\,|\lambda_{t}=\lambda]]\leq
C_{1}C_{2}<\infty.$

We conclude that $\mathbb{E}[|\alpha_{t+1}|^{k}]<\infty$. Now we turn our
attention to the recursion (6). The existence of the $k$-th moment for
$\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$ can be proved by following similar
arguments of Fokianos et al., (2009). In particular, analogously to the proof
of Proposition 2.1 of Fokianos et al., (2009), we define a new test function
$V(x)=1+x^{k}$. Then

$\displaystyle\mathbb{E}[V(\log\lambda_{t+1})\,|\log\lambda_{t}=\log\lambda]=$
$\displaystyle
1+\mathbb{E}[(\omega+\beta\log\lambda+\alpha_{t}(y_{t}-\lambda)\lambda^{-1})^{k}\,|\log\lambda_{t}=\log\lambda]$
$\displaystyle=$ $\displaystyle
1+\sum_{i=0}^{k}\binom{k}{i}(\beta\log\lambda)^{k-i}\mathbb{E}[(\omega+\alpha_{t+1}(y_{t}-\lambda)\lambda^{-1})^{i}\,|\log\lambda_{t}=\log\lambda].$

Note that, $\omega$ do not depend on $\log\lambda$ and hence, it can be
neglected as $\log\lambda\rightarrow\infty$. This means that it suffices to
verify that
$\mathbb{E}[(\alpha_{t+1}(y_{t}-\lambda)\lambda^{-1})^{k}\,|\log\lambda_{t}=\log\lambda]<\infty$.
However, it can be seen that by the Hölder’s inequality

$\displaystyle\mathbb{E}[|\alpha_{t+1}(y_{t}-\lambda)\lambda^{-1}|^{k}]\leq\Big{(}\mathbb{E}[|\alpha_{t+1}|^{2k}]\Big{)}^{\frac{1}{2k}}\Big{(}\mathbb{E}[|(y_{t}-\lambda)\lambda^{-1}|^{2k}]\Big{)}^{\frac{1}{2k}}<\infty,$

which implies that
$\mathbb{E}[(\alpha_{t+1}(y_{t}-\lambda)\lambda^{-1})^{k}\,|\log\lambda_{t}=\log\lambda]\leq
C<\infty$, for some constant $C>0$ and $\forall k>0$. Therefore,

$\displaystyle\mathbb{E}[V(\log\lambda_{t+1})\,|\log\lambda_{t}=\log\lambda]=$
$\displaystyle 1+\beta^{k}\log\lambda+\sum_{j=0}^{k-1}C_{j}(\log\lambda),$

where $C_{j}(\log\lambda)$ depends on the coefficients $\omega$, $\alpha$ and
$\beta$, and, depending on which cases of proof of Proposition 1 we are taking
into account, possibly on $\log\lambda$. In particular, from the arguments
discussed above it can be see that for $\log\lambda_{t}=\log\lambda>0$ with
$\log\lambda\rightarrow\infty$, then
$\mathbb{E}[|\log\lambda_{t+1}|^{k}\,|\log\lambda_{t}=\log\lambda]\approx|\beta|^{k}|\log\lambda|^{k}$,
whereas for $\log\lambda_{t}=\log\lambda<0$ with
$|\log\lambda|\rightarrow\infty$, then
$\mathbb{E}[|\log\lambda_{t+1}|^{k}\,|\log\lambda_{t}=\log\lambda]\approx|\beta|^{k}|\beta+\bar{\alpha}|^{k}|\log\lambda|^{k}$.
Therefore, the conditions stated in Proposition 1 are sufficients for the
existence of the $k$-th order moment of
$\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$.

Finally, by slightly modifying the proof of Proposition 2.2 of Fokianos et
al., (2009) we obtain that, under the condition of Proposition 1, the chain
$\\{y_{t},\log\lambda_{t}\\}_{t\in\mathbb{Z}}$ is geometrically ergodic with
$V(y,\lambda)=1+y^{k}+\log\lambda^{k}$.

### Proof of Proposition 3

To derive the contraction condition stated in Proposition 3 we proceed as
follows. We consider the model defined by the filtering equations:

$\log\lambda_{t+1}(\boldsymbol{\theta})=\omega+\beta\log\lambda_{t}(\boldsymbol{\theta})+\alpha_{t+1}(\boldsymbol{\theta})(y_{t}\lambda^{-1}_{t}(\boldsymbol{\theta})-1),$

$\alpha_{t+1}(\boldsymbol{\theta})=\delta_{\alpha}+\phi_{\alpha}\alpha_{t}(\boldsymbol{\theta})+\kappa_{\alpha}(y_{t}\lambda^{-1}_{t}(\boldsymbol{\theta})-1)(y_{t-1}\lambda^{-1}_{t-1}(\boldsymbol{\theta})-1).$

First note that by iterating the recursion for
$\alpha_{t+1}(\boldsymbol{\theta})$, we obtain

$\displaystyle\alpha_{t+1}(\boldsymbol{\theta})=\frac{\delta_{\alpha}}{1-\phi_{\alpha}}+\phi_{\alpha}^{t-1}\alpha_{1}(\boldsymbol{\theta})+\sum_{k=0}^{t-1}\kappa_{\alpha}\phi_{\alpha}^{k}(y_{t-k}\lambda^{-1}_{t-k}(\boldsymbol{\theta})-1)(y_{t-k-1}\lambda^{-1}_{t-k-1}(\boldsymbol{\theta})-1),$
(27)

and so, by applying the Cauchy rule, it is well-known that there exists a
stationary, ergodic and non anticipative solution to the stochastic recursion
in (27) if $0\leq\phi_{\alpha}<1$, which is implied by assumption. Thus,

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{\alpha}_{t+1}(\boldsymbol{\theta})-{\alpha}_{t+1}(\boldsymbol{\theta})|\leq\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\phi^{t-1}_{\alpha}|\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{\alpha}_{1}(\boldsymbol{\theta})-{\alpha}_{1}(\boldsymbol{\theta})|\xrightarrow[]{e.a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,t\rightarrow\infty.$

Now, note that $\alpha_{t+1}(\boldsymbol{\theta})$ is bounded below by

$\displaystyle\alpha_{t+1}(\boldsymbol{\theta})\geq\frac{\delta_{\alpha}+\kappa_{\alpha}}{1-\phi_{\alpha}}.$

Therefore

$\displaystyle\log\lambda_{t+1}(\boldsymbol{\theta})\geq$
$\displaystyle\omega+\beta\log\lambda_{t}(\boldsymbol{\theta})+\frac{\delta_{\alpha}+\kappa_{\alpha}}{1-\phi_{\alpha}}(y_{t}\lambda^{-1}_{t}(\boldsymbol{\theta})-1)$
$\displaystyle\geq$
$\displaystyle\omega+\beta\log\lambda_{t}(\boldsymbol{\theta})-\frac{\delta_{\alpha}+\kappa_{\alpha}}{1-\phi_{\alpha}}.$

Since $\omega\in\mathbb{R}$ and $0\leq\beta<1$, by recursive substitutions, we
obtain that

$\displaystyle\log\lambda_{t+1}(\boldsymbol{\theta})\geq$
$\displaystyle\frac{\omega-\frac{\delta_{\alpha}+\kappa_{\alpha}}{1-\phi_{\alpha}}}{1-\beta}:=\ell.$
(28)

Clearly, this implies that

$\displaystyle\alpha_{t+1}(\boldsymbol{\theta})\leq\delta_{\alpha}+\phi_{\alpha}\alpha_{t}(\boldsymbol{\theta})+\kappa_{\alpha}(y_{t}e^{-\ell}-1)(y_{t-1}e^{-\ell}-1):=\bar{\alpha}_{t+1}(\boldsymbol{\theta}),$

and hence

$\displaystyle\lambda_{t+1}(\boldsymbol{\theta})\leq$
$\displaystyle\exp\\{\omega+\beta\log\lambda_{t}(\boldsymbol{\theta})+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)\\}.$

The contraction condition is then obtained by noting that

$\displaystyle\bigg{|}\frac{\partial\lambda_{t+1}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\bigg{|}\leq$
$\displaystyle\exp\\{\omega+\beta\log\lambda_{t}(\boldsymbol{\theta})+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)\\}\frac{\beta}{\lambda_{t}(\boldsymbol{\theta})}$
$\displaystyle=$
$\displaystyle\exp\\{\omega+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)\\}\lambda^{\beta}_{t}(\boldsymbol{\theta})\frac{\beta}{\lambda_{t}(\boldsymbol{\theta})}$
$\displaystyle=$
$\displaystyle\exp\\{\omega+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)\\}\frac{\beta}{\lambda^{1-\beta}_{t}(\boldsymbol{\theta})}.$

Since the inequality in (28) holds true, it must be that

$\displaystyle\bigg{|}\frac{\partial\lambda_{t+1}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\bigg{|}\leq$
$\displaystyle\exp\\{\omega+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)\\}\frac{\beta}{e^{\ell(1-\beta)}_{t}}$
$\displaystyle=$
$\displaystyle\beta\exp\\{\omega+\bar{\alpha}_{t+1}(\boldsymbol{\theta})(y_{t}e^{-\ell}-1)-\ell(1-\beta)\\},$

which yields the contraction condition stated in the Proposition.

In fact, by the mean value theorem it holds that

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{\lambda}_{t+1}(\boldsymbol{\theta})-\lambda_{t+1}(\boldsymbol{\theta})|\leq\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{|}\frac{\partial\hat{\lambda}_{t+1}(\boldsymbol{\theta})}{\partial{\lambda}^{\star}}\bigg{|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{\lambda}_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta})|,$

where
${\lambda}^{\star}\in(\hat{\lambda}_{t}(\boldsymbol{\theta}),\lambda_{t}(\boldsymbol{\theta}))$.
The Proposition follows from

$\displaystyle\mathbb{E}\bigg{[}\log\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{|}\frac{\partial\lambda_{t+1}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\bigg{|}\bigg{]}<0,$

by application of Proposition 5.2.12 in Straumann, (2005). For the application
of Proposition 5.2.12 in Straumann, (2005), we have also checked that
$\mathbb{E}[\log^{+}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t+1}(\boldsymbol{\theta})-\lambda|]<\infty$
for a constant $\lambda>0$ and

$\displaystyle\mathbb{E}\bigg{[}\log^{+}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{|}\frac{\partial\lambda_{t+1}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\bigg{|}\bigg{]}<\infty.$

Note that
$\mathbb{E}\big{[}\log^{+}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\big{|}\frac{\partial\lambda_{t+1}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\big{|}\big{]}<\infty$
directly follows from
$\mathbb{E}\big{[}\log\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\big{|}\frac{\partial\lambda_{t+1}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\big{|}\big{]}<0$.
Moreover, we get
$\mathbb{E}[\log^{+}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t+1}(\boldsymbol{\theta})-\lambda|]<\infty$
for a constant $\lambda>0$ by simply noting that $y_{t}$ has a bounded moment
from Proposition 1.

Finally, we address the second part of the Proposition, i.e., the existence of
an $m>0$ such that
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}(\boldsymbol{\theta})|^{m}]<\infty$.
However, under the stricter contraction condition in (14), this result follows
directly by applying Proposition 3.2 of Blasques et al., (2021), since their
condition $(iv)$ follows from our condition (14).

### Proof of Lemma 3.1

By the triangle inequality, it holds that

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}(\boldsymbol{\theta})|\leq\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}_{T}(\boldsymbol{\theta})|+\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|{L}_{T}(\boldsymbol{\theta})-{L}(\boldsymbol{\theta})|,$
(29)

where ${L}_{T}(\boldsymbol{\theta})=\frac{1}{T}\sum_{t=1}^{T}l_{t}$ is
stationary and ergodic since it is a continuous function of the stationary and
ergodic processes $\\{\lambda_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{Z}}$
and $\\{\alpha_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{Z}}$. We begin by
showing that the first term in the RHS of (29) converges to zero almost
surely. By the mean value theorem we have

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}_{T}(\boldsymbol{\theta})|\leq\frac{1}{T}\sum_{t=1}^{T}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\begin{Vmatrix}\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial\lambda^{\star}_{t}(\boldsymbol{\theta})}\\\
\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial\alpha^{\star}_{t}(\boldsymbol{\theta})}\end{Vmatrix}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\begin{Vmatrix}\hat{\lambda}_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta})\\\
\hat{\alpha}_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta})\end{Vmatrix},$

where
$\lambda^{\star}_{t}(\boldsymbol{\theta})\in(\hat{\lambda}_{t}(\boldsymbol{\theta}),\lambda_{t}(\boldsymbol{\theta}))$,
$\alpha^{\star}_{t}(\boldsymbol{\theta})\in(\hat{\alpha}_{t}(\boldsymbol{\theta}),\alpha_{t}(\boldsymbol{\theta}))$
and the parial derivatives are $\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial\lambda^{\star}_{t}(\boldsymbol{\theta})}=\frac{y_{t}}{\lambda^{\star}_{t}(\boldsymbol{\theta})}-1$
and $\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial\alpha_{t}(\boldsymbol{\theta})}=(\frac{y_{t}}{\lambda_{t}(\boldsymbol{\theta})}-1)(\frac{y_{t-1}}{\lambda_{t-1}(\boldsymbol{\theta})}-1)$.

Note that, from the inequality (28) derived in the proof of Proposition 3 and
the compactness of $\boldsymbol{\Theta}$,
$\log\lambda^{\star}_{t}(\boldsymbol{\theta})$ is bounded below by $\ell$.
Thus $\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\big{|}\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial\lambda^{\star}_{t}(\boldsymbol{\theta})}\big{|}\leq|{y_{t}}e^{-\ell}-1|,$
which implies that

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}_{T}(\boldsymbol{\theta})|\leq\frac{1}{T}\sum_{t=1}^{T}\begin{Vmatrix}({y_{t}}e^{-\ell}-1)\\\
({y_{t}}e^{-\ell}-1)({y_{t-1}}e^{-\ell}-1)\end{Vmatrix}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\begin{Vmatrix}\hat{\lambda}_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta})\\\
\hat{\alpha}_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta})\end{Vmatrix}.$
(30)

From Proposition 3 it is clear that the second term in the RHS of (30)
converges to zero almost surely and exponentially fast. Therefore, if we can
prove that $({y_{t}}e^{-\ell}-1)({y_{t-1}}e^{-\ell}-1)$ has a bounded
$\log$-moment, the desired convergence of the first term in the RHS of (29)
follows by an application of Lemma 2.5.1 of Straumann, (2005). However, the
$\log$-moment condition is trivially verified, since by Proposition 1, Lemma
2.5.3 of Straumann, (2005) and the Jensen’s inequality imply that

$\displaystyle\mathbb{E}[\log^{+}|({y_{t}}e^{-\ell}-1)({y_{t-1}}e^{-\ell}-1)|]\leq\mathbb{E}[\log^{+}|{y_{t}}e^{-\ell}-1|]+\mathbb{E}[\log^{+}|{y_{t-1}}e^{-\ell}-1|]<c_{1}+c_{2}<\infty,$

for any $c_{1},c_{2}>0$ because $y_{t}$ has a bounded moment. Therefore,
$\exists K>0$ such that

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}_{T}(\boldsymbol{\theta})|\leq\frac{1}{T}\sum_{t=1}^{T}K\times\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\begin{Vmatrix}\hat{\lambda}_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta})\\\
\hat{\alpha}_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta})\end{Vmatrix}\xrightarrow[]{a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,T\rightarrow\infty.$

Next, we study the almost sure convergence of the second term in the RHS of
(29). This result, could be obtained by applying the uniform law of large
number for ergodic sequences of Rao, (1962). In particular, we need to prove
that the stationary and ergodic sequence
$\\{l_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{Z}}$ has a uniformly bounded
moment over the compact parameter space, i.e.,
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|l_{1}(\boldsymbol{\theta})|]<\infty$.

Since $\log(x)\leq x-1$ $\forall x>0$, we have

$\displaystyle|\ell_{t}(\boldsymbol{\theta})|=|y_{t}\log\lambda_{t}(\boldsymbol{\theta})-\lambda_{t}((\boldsymbol{\theta}))|\leq|y_{t}(\lambda_{t}(\boldsymbol{\theta})-1)-\lambda_{t}((\boldsymbol{\theta}))|,$

Thus, from the Cauchy-Schwarz inequality we obtain

$\displaystyle\mathbb{E}\bigg{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|l_{t}(\boldsymbol{\theta})|\bigg{]}\leq\sqrt{\mathbb{E}[y_{t}^{2}]\mathbb{E}\bigg{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}^{2}(\boldsymbol{\theta})|\bigg{]}}+\mathbb{E}[y_{t}]+\mathbb{E}\bigg{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}(\boldsymbol{\theta})|\bigg{]}<\infty,$

since $\mathbb{E}[y_{t}]<\infty$ and $\mathbb{E}[y^{2}_{t}]<\infty$ are
ensured by Proposition 2, and
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}^{2}(\boldsymbol{\theta})|]<\infty$
is ensured by Proposition 3. Therefore, by the uniform law of large number of
Rao, (1962)

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-L(\boldsymbol{\theta})|\xrightarrow[]{\text{a.s.}}0\hskip
28.45274pt\text{as}\hskip 28.45274ptT\rightarrow\infty.$

By collecting the results, we obtain from (29)

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\hat{L}_{T}(\boldsymbol{\theta})-{L}(\boldsymbol{\theta})|\leq
o_{p}(1)+o_{p}(1),$

which concludes the proof.

### Proof of Lemma 3.2

Similarly to Wang et al., (2014), we can show that by the law of iterated
expectations and the elementary inequality $\log(x)\leq x-1$ $\forall x>0$, it
holds that

$\displaystyle\mathbb{E}[l_{t}(\boldsymbol{\theta})-l_{t}(\boldsymbol{\theta}_{0})]=$
$\displaystyle\mathbb{E}\bigg{[}{y_{t}}\log\frac{\lambda_{t}(\boldsymbol{\theta})}{\lambda_{t}(\boldsymbol{\theta}_{0})}+\lambda_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta}_{0})\bigg{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\bigg{[}\lambda_{t}(\boldsymbol{\theta}_{0})\bigg{(}\frac{\lambda_{t}(\boldsymbol{\theta})}{\lambda_{t}(\boldsymbol{\theta}_{0})}-1\bigg{)}+\lambda_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta}_{0})\bigg{]}=0,$

if and only if
$\lambda_{t}(\boldsymbol{\theta})=\lambda_{t}(\boldsymbol{\theta}_{0})$ almost
surely. However, the latter equality holds if and only if
${\log\lambda}_{t}(\boldsymbol{\theta})={\log\lambda}_{t}(\boldsymbol{\theta}_{0})$
almost surely.

To prove that this cannot be true, assume that
$\exists\boldsymbol{\theta}\in\boldsymbol{\Theta}$ such that
$\log\lambda_{t}(\boldsymbol{\theta})=\log\lambda_{t}(\boldsymbol{\theta}_{0})$,
then

$\displaystyle\log\lambda_{t}(\boldsymbol{\theta})-\log\lambda_{t}(\boldsymbol{\theta}_{0})=\omega-\omega_{0}+(\beta-\beta_{0})\log\lambda_{t-1}(\boldsymbol{\theta}_{0})+(\alpha_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta}_{0}))e_{t-1},$
(31)

almost surely. From equation (31) above it can be seen that if
$\log\lambda_{t}(\boldsymbol{\theta})=\log\lambda_{t}(\boldsymbol{\theta}_{0})$,
then it must holds that
$\alpha_{t}(\boldsymbol{\theta})=\alpha_{t}(\boldsymbol{\theta}_{0})$ almost
surely. However, note that

$\displaystyle\alpha_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta}_{0})=\delta_{\alpha}-\delta_{\alpha,0}+(\phi_{\alpha}-\phi_{\alpha,0})\alpha_{t}(\boldsymbol{\theta}_{0})+(\kappa_{\alpha}-\kappa_{\alpha,0})e_{t-1}e_{t-2},$
(32)

but the process $\\{e_{t}\\}_{t\in\mathbb{Z}}$ forms a martingale difference
sequence, and thus, since $\kappa_{\alpha}\neq 0$ by assumption, it can be
seen that if
$\alpha_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta}_{0})=0$ almost
surely, we must have
$(\delta_{\alpha},\phi_{\alpha},\kappa_{\alpha})=(\delta_{\alpha,0},\phi_{\alpha,0},\kappa_{\alpha,0})$.
It is evident now from (31) that this result further implies that if
$\alpha_{t}(\boldsymbol{\theta})-\alpha_{t}(\boldsymbol{\theta}_{0})=0$ and
$\log\lambda_{t}(\boldsymbol{\theta})-\log\lambda_{t}(\boldsymbol{\theta}_{0})=0$
almost surely, it must also holds that
$(\omega,\beta)=(\omega_{0},\beta_{0})$, and hence
$\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$.

### Proof of Theorem 3.1

With Lemma 3.1, which yields the uniform convergence of the $\log$-likelihood
function, and Lemma 3.2, which shows the identifiability and uniqueness of the
maximizers $\boldsymbol{\theta}_{0}\in\boldsymbol{\Theta}$, at hand, we
directly obtain strong consistency of the MLE $\hat{\boldsymbol{\theta}}_{T}$
by applying Theorem 3.4 of White, (1994).

### Proof of Proposition 4

To establish the asymptotic normality of the normalized score function (15) we
verify (16) and (18).

First, we note that $\log\lambda_{t}^{\boldsymbol{\theta}_{0}}$ is a
continuous function of $\\{\lambda_{t}\\}_{t\in\mathbb{Z}}$ and
$\\{\alpha_{t}\\}_{t\in\mathbb{Z}}$, that are strictly stationary and ergodic
by Proposition 1. Hence,
$\\{\log\lambda^{\boldsymbol{\theta}}_{t}\\}_{t\in\mathbb{Z}}$ is strictly
stationary and ergodic.

Then, to verify conditions (16), we note that

$\displaystyle\sum_{t=1}^{T}\mathbb{E}[\boldsymbol{\eta}_{t}\boldsymbol{\eta}^{\prime}_{t}|\mathcal{F}_{t-1}]=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}[e_{t}^{2}|\mathcal{F}_{t-1}]\lambda_{t}^{2}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})$
$\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}[\mathbb{E}[e_{t}^{2}|\lambda_{t}]|\mathcal{F}_{t-1}]\lambda_{t}^{2}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})$
$\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}[\lambda^{-1}_{t}|\mathcal{F}_{t-1}]\lambda_{t}^{2}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})$
$\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}}).$

Therefore, if
$\mathbb{E}[\|\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})\|]<\infty$,
we can apply the ergodic theorem in order to obtain

$\displaystyle\lim_{t\rightarrow\infty}\frac{1}{T}\sum_{t=1}^{T}\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})=\mathbb{E}[\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})],$

almost surely. Clearly, the bounded moment condition
$\mathbb{E}[\|\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})\|]<\infty$
is automatically satisfied if
$\mathbb{E}[\|\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\xi}_{0}}\log\lambda_{t}^{\boldsymbol{\xi}_{0}^{\prime}})\|]<\infty$
and
$\mathbb{E}[\|\lambda_{t}(\alpha_{t}^{\boldsymbol{\psi}_{0}}\alpha_{t}^{\boldsymbol{\psi}_{0}^{\prime}})\|]<\infty$,
which in turn is implied by verifying that
$\mathbb{E}[|\lambda_{t}(\log\lambda_{t}^{\omega})^{2}|]<\infty$ and
$\mathbb{E}[|\lambda_{t}(\log\lambda_{t}^{\beta})^{2}|]<\infty$, and moreover
$\mathbb{E}[|\lambda_{t}(\alpha_{t}^{\delta_{\alpha}})^{2}|]<\infty$,
$\mathbb{E}[|\lambda_{t}(\alpha_{t}^{\phi_{\alpha}})^{2}|]<\infty$ and
$\mathbb{E}[|\lambda_{t}(\alpha_{t}^{\kappa_{\alpha}})^{2}|]<\infty$.

Let us denote with $\|\,\cdot\,\|_{2}$ the $L_{2}$-norm. By the Minkowsky’s
inequality, we have

$\displaystyle\|\lambda_{t}^{1/2}\log\lambda_{t}^{\omega}\|_{2}\leq$
$\displaystyle\|\lambda_{t}^{1/2}\|_{2}+\sum_{i=1}^{\infty}\bigg{\|}\lambda_{t}^{1/2}\prod_{j=0}^{i}A_{t-i-1}\bigg{\|}_{2}.$

However, note that $\lambda_{t}$ has a bounded moment,
$\\{A_{t}\\}_{t\in\mathbb{Z}}$ is strictly stationary and ergodic, and
$\mathbb{E}[|A_{t}|^{2}]<1$ by assumption. Therefore,

$\displaystyle\|\lambda_{t}^{1/2}\log\lambda_{t}^{\omega}\|_{2}\leq
c_{1}+c_{1}\sum_{i=1}^{\infty}\|A_{t-i-1}\|^{i}_{2}<\infty.$

By using similar arguments as before, we obtain

$\displaystyle\|\lambda_{t}^{1/2}\log\lambda_{t}^{\beta}\|_{2}\leq
c_{1}c_{2}+c_{1}c_{2}\sum_{i=1}^{\infty}\|A_{t-i-1}\|^{i}_{2}<\infty.$

Now, since $0\leq|\phi_{\alpha}^{2}|<1$ by assumption and, as shown in
Proposition 2, $\\{e_{t}e_{t-1}\\}_{t\in\mathbb{Z}}$ has $k$ bounded moment
for every $k>0$, we can easily show that

$\displaystyle\|\lambda_{t}^{1/2}\alpha_{t}^{\boldsymbol{\psi}}\|_{2}<\infty.$

Therefore, the desired moment bound
$\mathbb{E}[\|\lambda_{t}(\log\lambda_{t}^{\boldsymbol{\theta}_{0}}\log\lambda_{t}^{\boldsymbol{\theta}_{0}^{\prime}})\|]<\infty$
hold true.

The second required condition in (18) can then be easily using the established
moment bound, together with the stationarity and ergodicity of
$\\{\log\lambda_{t}\\}_{t\in\mathbb{Z}}$, see Davis et al., (2005).

### Proof of Theorem 3.2

As done for the $\log$-likehood function in (7), we first need to study the
impact that the initialization of the filters
$\hat{\lambda}_{1}(\boldsymbol{\theta})$ and
$\hat{\alpha}_{1}(\boldsymbol{\theta})$ have on the score function. In
particular, we study the convergence of the non-stationary
$\hat{\bar{\boldsymbol{{\eta}}}}_{T}(\boldsymbol{\theta})=\sqrt{T}\frac{\partial\hat{L}_{T}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$,
which is a continuous function of the filters
$\\{\hat{\lambda}_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{N}}$ and
$\\{\hat{\alpha}_{t}(\boldsymbol{\theta})\\}_{t\in\mathbb{N}}$, to its
stationary and ergodic counterpart
${\bar{\boldsymbol{{\eta}}}}_{T}(\boldsymbol{\theta})=\sqrt{T}\frac{\partial{L}_{T}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$.

By the triangle inequality, we have

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{\bar{\boldsymbol{{\eta}}}}_{T}(\boldsymbol{\theta})-{\bar{\boldsymbol{{\eta}}}}_{T}(\boldsymbol{\theta})\|\leq$
$\displaystyle\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial{\lambda}_{t}(\boldsymbol{\theta})}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}$
$\displaystyle+\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\alpha}_{t}(\boldsymbol{\theta})}\frac{\partial\hat{\alpha}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial{\alpha}_{t}(\boldsymbol{\theta})}\frac{\partial{\alpha}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|},$
(33)

almost surely.

Let us consider the first term in the RHS of inequality (A). By the mean value
theorem, we have

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial{\lambda}_{t}(\boldsymbol{\theta})}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}\leq$
$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}$
$\displaystyle+\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}-\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial{\lambda}_{t}(\boldsymbol{\theta})}\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}$
$\displaystyle\leq$
$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}$
$\displaystyle+\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial^{2}\hat{l}_{t}(\boldsymbol{\theta})}{\partial{\lambda}^{\star}}\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{\lambda}_{t}(\boldsymbol{\theta})-{\lambda}_{t}(\boldsymbol{\theta})\|,$

where
${\lambda}^{\star}\in(\hat{\lambda}_{t}(\boldsymbol{\theta}),{\lambda}_{t}(\boldsymbol{\theta}))$.
As noted in the proof of Lemma 3.1, it is clearly seen that
$\mathbb{E}\Big{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}\Big{\|}\Big{]}<\infty$
and
$\mathbb{E}\Big{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{\|}\frac{\partial^{2}\hat{l}_{t}(\boldsymbol{\theta})}{\partial^{2}\hat{\lambda}_{t}(\boldsymbol{\theta})}\Big{\|}\Big{]}<\infty$.
Thus, the results implied by Proposition 3 and Lemma A.1, together with Lemma
2.5.2 of Straumann, (2005), we obtain that

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial{\lambda}_{t}(\boldsymbol{\theta})}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}\xrightarrow[]{a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,t\rightarrow\infty.$

By using similar arguments, it is also straightforward to see that the same
result hold true for the second term in the RHS of inequality (A), hence

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{l}_{t}(\boldsymbol{\theta})}{\partial\hat{\alpha}_{t}(\boldsymbol{\theta})}\frac{\partial\hat{\alpha}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial
l_{t}(\boldsymbol{\theta})}{\partial{\alpha}_{t}(\boldsymbol{\theta})}\frac{\partial{\alpha}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}\xrightarrow[]{a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,t\rightarrow\infty.$

Putting all together, we thus conclude that the normalized score function
satisfies

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{\bar{\boldsymbol{{\eta}}}}_{T}(\boldsymbol{\theta})-{\bar{\boldsymbol{{\eta}}}}_{T}(\boldsymbol{\theta})\|\leq
o_{p}(1).$

Now, having proved the convergence of the normalized score function, for $T$
large enough, we have

$\displaystyle\boldsymbol{0}=\hat{\bar{\boldsymbol{{\eta}}}}_{T}(\hat{\boldsymbol{\theta}}_{T})=\bar{\boldsymbol{{\eta}}}_{T}(\hat{\boldsymbol{\theta}}_{T})+o_{p}(1)=\bar{\boldsymbol{{\eta}}}_{T}({\boldsymbol{\theta}}_{0})-\boldsymbol{J}_{T}^{\star}\sqrt{T}(\hat{\boldsymbol{\theta}}_{T}-{\boldsymbol{\theta}}_{0}),$
(34)

where

$\displaystyle\boldsymbol{J}_{T}^{\star}=-\frac{\partial^{2}L_{T}(\boldsymbol{\theta}^{\star})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}=-\frac{1}{T}\sum_{t=1}^{T}\frac{\partial^{2}l_{t}(\boldsymbol{\theta}^{\star})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}},$

for some
$\boldsymbol{\theta}^{\star}\in(\hat{\boldsymbol{\theta}}_{T},\boldsymbol{\theta}_{0})$.
Note that, from Proposition 4,
$\bar{\boldsymbol{{\eta}}}_{T}({\boldsymbol{\theta}}_{0})\Rightarrow\mathcal{N}(\boldsymbol{0},\boldsymbol{V})$
as $T\rightarrow\infty$. Therefore, it only remains to show that
$\boldsymbol{J}_{T}^{\star}\xrightarrow[]{a.s.}\boldsymbol{J}$ as
$T\rightarrow\infty$.

However, by combining the strong consistency of the MLE together with Lemma
A.3, and the uniform law of large numbers for stationary and ergodic processes
of Rao, (1962), we note that since
$\hat{\boldsymbol{\theta}}_{T}\xrightarrow[]{a.s.}\boldsymbol{\theta}_{0}$ as
$T\rightarrow\infty$, it must holds that $\hat{\boldsymbol{\theta}}_{T}\in
V({\boldsymbol{\theta}}_{0})$ almost surely, where
$V({\boldsymbol{\theta}}_{0})$ denotes a neighborhood of
${\boldsymbol{\theta}}_{0}$. Then, we obtain that
$\hat{\boldsymbol{J}}_{T}=-\frac{\partial^{2}L_{T}(\hat{\boldsymbol{\theta}}_{T})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\xrightarrow[]{a.s.}\boldsymbol{J}$.
Moreover, note also that when the model is correct it holds that
$\boldsymbol{V}=\boldsymbol{J}$, which follows by standard arguments since
$l(\boldsymbol{\theta}_{0})$ is the true conditional probability mass function
of the Poisson distribution.

Finally, we get that
$\boldsymbol{J}^{\star}_{T}\xrightarrow[]{a.s.}\boldsymbol{J}$ and the
conclusion now simply follows by combining the expansion in equation (34)
together with the results obtained in Proposition 4.

######  Lemma A.1.

Let the assumptions of Proposition 4 hold, then

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}\xrightarrow[]{e.a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,t\rightarrow\infty,$

where
$\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$
is the stationary and ergodic derivative process of
${\lambda}_{t}(\boldsymbol{\theta})$, which satisfies
$\mathbb{E}\Big{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{\|}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\Big{\|}^{2}\Big{]}<\infty$.

###### Proof.

Since
$\boldsymbol{\theta}=(\boldsymbol{\xi}^{\prime},\boldsymbol{\psi}^{\prime})^{\prime}$,
we define the derivative processes evaluated at some
$\boldsymbol{\theta}\in\boldsymbol{\Theta}$ as
$\log\lambda^{\boldsymbol{\theta}}_{t}(\boldsymbol{\theta})=(\log\lambda^{\boldsymbol{\xi}^{\prime}}_{t}(\boldsymbol{\theta}),\log\lambda^{\boldsymbol{\psi}^{\prime}}_{t}(\boldsymbol{\theta}))^{\prime}$,
where

$\displaystyle\log\lambda^{\boldsymbol{\xi}}_{t}(\boldsymbol{\theta})=\begin{bmatrix}\frac{\partial\log\lambda_{t}(\boldsymbol{\theta})}{\partial\omega}\\\
\frac{\partial\log\lambda_{t}(\boldsymbol{\theta})}{\partial\beta}\end{bmatrix}:=\begin{bmatrix}\log\lambda^{\omega}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\beta}_{t}(\boldsymbol{\theta})\end{bmatrix}=$
$\displaystyle\begin{bmatrix}1+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega}_{t-1}(\boldsymbol{\theta})\\\
\log\lambda_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\beta}_{t-1}(\boldsymbol{\theta})\end{bmatrix},$

with

$\displaystyle
A_{t}(\boldsymbol{\theta})=\bigg{[}\beta-\alpha_{t}(\boldsymbol{\theta})y_{t}\lambda^{-1}_{t}(\boldsymbol{\theta})+(y_{t}-\lambda_{t}(\boldsymbol{\theta}))\frac{\partial\alpha_{t}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\bigg{]}\lambda_{t}(\boldsymbol{\theta}).$

Moreover,
$\log\lambda^{\boldsymbol{\psi}}_{t}(\boldsymbol{\theta})=\alpha^{\boldsymbol{\psi}}_{t}(\boldsymbol{\theta})(y_{t}-\lambda_{t}(\boldsymbol{\theta}))\lambda_{t}^{-1}(\boldsymbol{\theta})$,
where

$\displaystyle\alpha^{\boldsymbol{\psi}}_{t}(\boldsymbol{\theta})$
$\displaystyle=\begin{bmatrix}\frac{\partial\alpha_{t}(\boldsymbol{\theta})}{\partial\delta_{\alpha}}\\\
\frac{\partial\alpha_{t}(\boldsymbol{\theta})}{\partial\phi_{\alpha}}\\\
\frac{\partial\alpha_{t}(\boldsymbol{\theta})}{\partial\kappa_{\alpha}}\end{bmatrix}$
$\displaystyle:=$
$\displaystyle\begin{bmatrix}\alpha_{t}^{\delta_{\alpha}}(\boldsymbol{\theta})\\\
\alpha_{t}^{\phi_{\alpha}}(\boldsymbol{\theta})\\\
\alpha_{t}^{\kappa_{\alpha}}(\boldsymbol{\theta})\end{bmatrix}=\begin{bmatrix}1+\phi_{\alpha}\alpha_{t-1}^{\delta_{\alpha}}(\boldsymbol{\theta})\\\
\alpha_{t-1}(\boldsymbol{\theta})+\phi_{\alpha}\alpha_{t-1}^{\phi_{\alpha}}(\boldsymbol{\theta})\\\
(y_{t-1}-\lambda_{t-1}(\boldsymbol{\theta}))\lambda_{t-1}^{-1}(\boldsymbol{\theta})(y_{t-2}-\lambda_{t-2}(\boldsymbol{\theta}))\lambda_{t-2}^{-1}(\boldsymbol{\theta})+\phi_{\alpha}\alpha_{t-1}^{\kappa_{\alpha}}(\boldsymbol{\theta})\end{bmatrix}.$

To verify the uniform convergence
$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}-\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}\xrightarrow[]{e.a.s.}0$
we first check the condition in Theorem 2.6.4. Straumann, (2005), which is
equivalent in only proving that
$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{A}_{t}(\boldsymbol{\theta})-A_{t}(\boldsymbol{\theta})\|\xrightarrow[]{e.a.s.}0$,
where $\hat{A}_{t}(\boldsymbol{\theta})$ and ${A}_{t}(\boldsymbol{\theta})$
are continuous function of $\hat{\lambda}_{t}(\boldsymbol{\theta})$ and
${\lambda}_{t}(\boldsymbol{\theta})$ (and so of
$\hat{\alpha}_{t}(\boldsymbol{\theta})$ and
${\alpha}_{t}(\boldsymbol{\theta})$), respectively, since the uniform
convergence of $\log\hat{\lambda}_{t}(\boldsymbol{\theta})$,
$\hat{\lambda}_{t}(\boldsymbol{\theta})$,
$\hat{\lambda}^{-1}_{t}(\boldsymbol{\theta})$ and
$\hat{\alpha}_{t}(\boldsymbol{\theta})$ are ensured by Proposition 3.

However, from the results of Propositions 1 and 2, and by an application of
the mean value theorem, it is immediate to note that $\exists K>0$ such that

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{A}_{t}(\boldsymbol{\theta})-A_{t}(\boldsymbol{\theta})\|\leq
K\times\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{\lambda}_{t}(\boldsymbol{\theta})-\lambda_{t}(\boldsymbol{\theta})\|\xrightarrow[]{e.a.s.}0.$

The desired uniform convergence of the derivative process
$\frac{\partial\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$
is then a clear consequence of Lemma 2.5.4 of Straumann, (2005).

Finally, the second moment bound
$\mathbb{E}\Big{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{\|}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\Big{\|}^{2}\Big{]}<1$
follows directly since
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|A_{t}(\boldsymbol{\theta})|^{2}]<\infty$
holds by assumption together with
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}(\boldsymbol{\theta})|^{m}]<\infty$
by Proposition 3 and $\mathbb{E}[y_{t}^{2}]<\infty$ by Proposition 2. ∎

######  Lemma A.2.

Let the assumptions of Proposition 4 hold, then

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial^{2}\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}-\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\bigg{\|}\xrightarrow[]{e.a.s.}0\,\,\,\,\,\text{as}\,\,\,\,\,t\rightarrow\infty,$

where
$\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}$
is the stationary and ergodic second derivative process of
${\lambda}_{t}(\boldsymbol{\theta})$, which satisfies
$\mathbb{E}\Big{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{\|}\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\Big{\|}\Big{]}<\infty$.

###### Proof.

Using the same notation of Lemma A.1, the second derivative processes
evaluated at some $\boldsymbol{\theta}\in\boldsymbol{\Theta}$ as
$\text{vech}\log\lambda_{t}^{\boldsymbol{\theta}\boldsymbol{\theta}}(\boldsymbol{\theta})=(\text{vech}\lambda_{t}^{\boldsymbol{\xi}\boldsymbol{\xi}}(\boldsymbol{\theta}),\text{vec}\lambda_{t}^{\boldsymbol{\xi}\boldsymbol{\psi}}(\boldsymbol{\theta}),\text{vech}\lambda_{t}^{\boldsymbol{\psi}\boldsymbol{\psi}}(\boldsymbol{\theta}))^{\prime}$,
where

$\displaystyle\text{vech}\log\lambda^{\boldsymbol{\xi}\boldsymbol{\xi}}_{t}(\boldsymbol{\theta})$
$\displaystyle=\text{vech}\begin{bmatrix}\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\omega^{2}}&\frac{\partial\log\lambda_{t}(\boldsymbol{\theta})}{\partial\omega\partial\beta}\\\
\cdot&\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\beta^{2}}\end{bmatrix}$
$\displaystyle:=$
$\displaystyle\begin{bmatrix}\log\lambda^{\omega\omega}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\omega\beta}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\beta\beta}_{t}(\boldsymbol{\theta})\end{bmatrix}=\begin{bmatrix}B_{t-1}(\boldsymbol{\theta})(\log\lambda_{t-1}^{\omega}(\boldsymbol{\theta}))^{2}+A_{t-1}(\boldsymbol{\theta})\log\lambda_{t-1}^{\omega\omega}(\boldsymbol{\theta})\\\
B_{t-1}(\boldsymbol{\theta})\log\lambda_{t-1}^{\omega}(\boldsymbol{\theta})\log\lambda_{t-1}^{\beta}(\boldsymbol{\theta})+\log\lambda_{t-1}^{\beta}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda_{t-1}^{\omega\beta}(\boldsymbol{\theta})\\\
B_{t-1}(\boldsymbol{\theta})(\log\lambda_{t-1}^{\beta}(\boldsymbol{\theta}))^{2}+2\log\lambda_{t-1}^{\beta}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda_{t-1}^{\beta\beta}(\boldsymbol{\theta})\end{bmatrix},$

with $A_{t}(\boldsymbol{\theta})$ as defined in Lemma A.1 and

$\displaystyle B_{t}(\boldsymbol{\theta})=\bigg{\\{}$
$\displaystyle\bigg{[}\alpha_{t}(\boldsymbol{\theta})y_{t}\lambda^{-2}_{t}(\boldsymbol{\theta})-\frac{\partial\alpha_{t}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}+(y_{t}-\lambda_{t}(\boldsymbol{\theta}))\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\lambda^{2}_{t}(\boldsymbol{\theta})}\bigg{]}\lambda_{t}(\boldsymbol{\theta})$
$\displaystyle+\bigg{[}\beta-\alpha_{t}(\boldsymbol{\theta})y_{t}\lambda^{-1}_{t}(\boldsymbol{\theta})+(y_{t}-\lambda_{t}(\boldsymbol{\theta}))\frac{\partial\alpha_{t}(\boldsymbol{\theta})}{\partial\lambda_{t}(\boldsymbol{\theta})}\bigg{]}\bigg{\\}}\lambda_{t}(\boldsymbol{\theta}).$

Now,

vec
$\displaystyle\log\lambda^{\boldsymbol{\xi}\boldsymbol{\psi}}_{t}(\boldsymbol{\theta})=\text{vec}\begin{bmatrix}\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\omega\partial{\delta_{\alpha}}}&\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\omega\partial{\phi_{\alpha}}}&\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\omega\partial{\kappa_{\alpha}}}\\\
\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\beta\partial{\delta_{\alpha}}}&\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\beta\partial{\phi_{\alpha}}}&\frac{\partial^{2}\log\lambda_{t}(\boldsymbol{\theta})}{\partial\beta\partial{\kappa_{\alpha}}}\end{bmatrix}:=\begin{bmatrix}\log\lambda^{\omega\delta_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\omega\phi_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\omega\kappa_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\beta\delta_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\beta\phi_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\log\lambda^{\beta\kappa_{\alpha}}_{t}(\boldsymbol{\theta})\end{bmatrix}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}C_{t-1}(\boldsymbol{\theta})\alpha^{\delta_{\alpha}}_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega}_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega\delta_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
C_{t-1}(\boldsymbol{\theta})\alpha^{\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega}_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
C_{t-1}(\boldsymbol{\theta})\alpha^{\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega}_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\alpha_{t-1}^{\delta_{\alpha}}(\boldsymbol{\theta})(y_{t-1}-\lambda_{t-1}(\boldsymbol{\theta}))\lambda^{-1}_{t-1}(\boldsymbol{\theta})+C_{t-1}(\boldsymbol{\theta})\alpha^{\delta_{\alpha}}_{t-1}(\boldsymbol{\theta})\log\lambda^{\beta}_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\beta\delta_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\alpha_{t-1}^{\phi_{\alpha}}(\boldsymbol{\theta})(y_{t-1}-\lambda_{t-1}(\boldsymbol{\theta}))\lambda^{-1}_{t-1}(\boldsymbol{\theta})+C_{t-1}(\boldsymbol{\theta})\alpha^{\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})\log\lambda^{\omega}_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\beta\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\alpha_{t-1}^{\kappa_{\alpha}}(\boldsymbol{\theta})(y_{t-1}-\lambda_{t-1}(\boldsymbol{\theta}))\lambda^{-1}_{t-1}(\boldsymbol{\theta})+C_{t-1}(\boldsymbol{\theta})\alpha^{\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\log\lambda^{\beta}_{t-1}(\boldsymbol{\theta})+A_{t-1}(\boldsymbol{\theta})\log\lambda^{\beta\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\end{bmatrix},$

with

$\displaystyle
C_{t}(\boldsymbol{\theta})=-y_{t}\lambda_{t}(\boldsymbol{\theta}).$

Finally, we have
$\text{vech}\log\lambda^{\boldsymbol{\psi\psi}}_{t}(\boldsymbol{\theta})=\text{vech}\,\alpha_{t}^{\boldsymbol{\psi\psi}}(\boldsymbol{\theta})(y_{t}-\lambda_{t}(\boldsymbol{\theta}))\lambda^{-1}_{t}(\boldsymbol{\theta})$,
where

$\displaystyle\text{vech}\,\alpha_{t}^{\boldsymbol{\psi\psi}}(\boldsymbol{\theta})$
$\displaystyle=\text{vech}\begin{bmatrix}\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\delta_{\alpha}^{2}}&\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\delta_{\alpha}\partial\phi_{\alpha}}&\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\delta_{\alpha}\partial\kappa_{\alpha}}\\\
\cdot&\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\phi^{2}_{\alpha}}&\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\phi_{\alpha}\partial\kappa_{\alpha}}\\\
\cdot&\cdot&\frac{\partial^{2}\alpha_{t}(\boldsymbol{\theta})}{\partial\kappa^{2}_{\alpha}}\end{bmatrix}$
$\displaystyle:=$
$\displaystyle\begin{bmatrix}\alpha^{\delta_{\alpha}\delta_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\alpha^{\delta_{\alpha}\phi_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\alpha^{\delta_{\alpha}\kappa_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\alpha^{\phi_{\alpha}\phi_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\alpha^{\phi_{\alpha}\kappa_{\alpha}}_{t}(\boldsymbol{\theta})\\\
\alpha^{\kappa_{\alpha}\kappa_{\alpha}}_{t}(\boldsymbol{\theta})\end{bmatrix}=\begin{bmatrix}\phi_{\alpha}\alpha^{\delta_{\alpha}\delta_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\alpha^{\delta_{\alpha}}_{t-1}(\boldsymbol{\theta})+\phi_{\alpha}\alpha^{\delta_{\alpha}\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\phi_{\alpha}\alpha^{\delta_{\alpha}\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
(1+\phi_{\alpha})\alpha^{\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})+\phi_{\alpha}\alpha^{\phi_{\alpha}\phi_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\alpha^{\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})+\phi_{\alpha}\alpha^{\phi_{\alpha}\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\\\
\phi_{\alpha}\alpha^{\kappa_{\alpha}\kappa_{\alpha}}_{t-1}(\boldsymbol{\theta})\end{bmatrix}.$

Analogously to the proof of Lemma A.1, to verify the uniform convergence
$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\frac{\partial^{2}\hat{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}-\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\bigg{\|}\xrightarrow[]{e.a.s.}0$
we need to check the condition in Theorem 2.6.4. Straumann, (2005), which, in
addition to the previous proof, it results in proving that
$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{B}_{t}(\boldsymbol{\theta})-B_{t}(\boldsymbol{\theta})\|\xrightarrow[]{e.a.s.}0$
and
$\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\|\hat{C}_{t}(\boldsymbol{\theta})-C_{t}(\boldsymbol{\theta})\|\xrightarrow[]{e.a.s.}0$
where $\hat{B}_{t}(\boldsymbol{\theta}),\hat{C}_{t}(\boldsymbol{\theta})$ and
${B}_{t}(\boldsymbol{\theta}),{C}_{t}(\boldsymbol{\theta})$, are continuous
function of $\hat{\lambda}_{t}(\boldsymbol{\theta})$ and
${\lambda}_{t}(\boldsymbol{\theta})$ (and so of
$\hat{\alpha}_{t}(\boldsymbol{\theta})$ and
${\alpha}_{t}(\boldsymbol{\theta})$), respectively. As remarked in the Proof
of Lemma A.1, given the results obtained in Proposition 3, we can easily show
the required uniform convergences by repeated applications the mean value
theorem together with Lemma 2.5.4 of Straumann, (2005).

Finally, the moment bound
$\mathbb{E}\Big{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\Big{\|}\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\Big{\|}\Big{]}<1$
follows directly since
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|A_{t}(\boldsymbol{\theta})|^{2}]<\infty$
holds by assumption together with
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|\lambda_{t}(\boldsymbol{\theta})|^{m}]<\infty$
by Proposition 3 and $\mathbb{E}[y_{t}^{2}]<\infty$ by Proposition 2, which
further imply
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|B_{t}(\boldsymbol{\theta})|]<\infty$
and
$\mathbb{E}[\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}|C_{t}(\boldsymbol{\theta})|]<\infty$.
∎

######  Lemma A.3.

Let the assumptions of Theorem 3.2 hold, then the second derivatives of the
$\log$-likelihood function have a uniformly bounded moment over the compact
parameter space, that is

$\displaystyle\mathbb{E}\bigg{[}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial^{2}l_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\bigg{\|}\bigg{]}<\infty.$

###### Proof.

By straightforward calculations, it can be seen that

$\displaystyle\frac{\partial^{2}l_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}=\bigg{(}\frac{y_{t}}{\lambda_{t}(\boldsymbol{\theta})}-1\bigg{)}\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}-\frac{y_{t}}{\lambda^{2}_{t}(\boldsymbol{\theta})}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^{\prime}}.$

Thus, as in the proof of Lemma 3.1, we have

$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial^{2}l_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\bigg{\|}\leq$
$\displaystyle\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{y_{t}}{\lambda_{t}(\boldsymbol{\theta})}-1\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial^{2}{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\prime}}\bigg{\|}+\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{y_{t}}{\lambda^{2}_{t}(\boldsymbol{\theta})}\bigg{\|}\sup_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\bigg{\|}\frac{\partial{\lambda}_{t}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
c_{1}\bigg{\|}\frac{y_{t}}{e^{\ell}}-1\bigg{\|}+c_{2}\bigg{\|}\frac{y_{t}}{e^{2\ell}}\bigg{\|}<\infty,$

since $e^{\ell}$ is bounded away from zero and $\mathbb{E}[y_{t}]<\infty$
follows by Proposition 2. Hence, the claimed moment bound follows by an
application of Lemmas A.1 and A.2. ∎