# A unified theory for ARMA models with
varying coefficients: One solution fits all111The proofs of the statements are
included in the appendices of this paper and they are available to the reader
upon request.

M. Karanasos†▷, A. Paraskevopoulos‡, T. Magdalinos∗, A. Canepa⋆
†Brunel University London, ‡University of Piraeus,
∗University of Southampton, ⋆ University of Turin

###### Abstract

For the large family of ARMA models with variable coefficients (TV-ARMA),
either deterministic or stochastic, we provide an explicit and computationally
tractable representation based on the general solution of the associated
linear difference equation. Analogous representations are established for the
fundamental properties of such processes, including the Wold-Cramér
decomposition and their covariance structure as well as explicit optimal
linear forecasts based on a finite set of past observations. These results are
grounded on the principal determinant, that is a banded Hessenbergian
representation of a restriction of the Green function involved in the solution
of the linear difference equation associated with TV-ARMA models, built up
solely of the autoregressive coefficients of the model. The $L_{2}$
convergence properties of the model are a consequence of the absolute
summability of the aforementioned Hessenbergian representation, which is in
line with the asymptotic stability and efficiency of such processes. The
invertibility of the model is also a consequence of an analogous condition,
but now the Green function is built up of the moving average coefficients. The
structural asymmetry between constant and deterministically time-varying
coefficient models, that is the backward and forward asymptotic efficiency
differ in an essential manner, is formally demonstrated. An alternative
approach to the Hessenbergian solution representation is described by an
equivalent procedure for manipulating time-varying polynomials. The practical
significance of the theoretical results in this work is illustrated with an
application to U.S. inflation data. The main finding is that inflation
persistence increased after 1976, whereas from 1986 onwards the persistence
declines and stabilizes to even lower levels than the pre-1976 period.

Keywords: ARMA process, Asymptotic efficiency, Asymptotic stability, Green’s
function, Hessenbergians, Invertibility, Skew multiplication, Structural
breaks, time-varying persistence, Variable coefficients, Wold decomposition.
JEL Classification: C13, C22, C32, E17, E31, E58.

▷We would like to thank S. Dafnos, who contributed to the interpretation and
synthesis of the work, which led to an earlier version of the paper, and P.
Koutroumpis, who provided the empirical Section for this earlier version of
the paper.

We gratefully acknowledge the helpful conversations we had with L. Giraitis,
and G. Kapetanios in the preparation of the paper. We would also like to thank
R. Baillie, M. Brennan, L. Bauwens, A. Demos, W. Distaso, D. van Dijk, C.
Francq, P. Fryzlewicz, E. Guerre, C. Gourieroux, M. Guidolin, A. Harvey, C.
Hommes, E. Kyriazidou, S. Leybourne, P. Marsh, P. Minford, C. Robotti, W.
Semmler, R. Smith, T. Teräsvirta, E. Tzavalis, P. Zaffaroni, and J-M Zakoian
for suggestions and comments which greatly improved many aspects of the paper.
We are grateful to seminar participants at Aarhus University (Creates), CREST,
London School of Economics (Department of Statistics), Erasmus University
(Econometric Institute), Imperial College (Tanaka Business School), Queen Mary
University of London (School of Economics and Finance), University of Essex
(Business School), BI Norwegian Business School, University of Copenhagen
(Department of Economics), Birkbeck College, University of London (Department
of Economics, Mathematics and Statistics), University of Nottingham (Granger
Centre for Time Series Econometrics), University of Manchester (Business
School), Cardiff University (Business School), Lancaster University
(Management School), University of Reading (ICMA Centre), King’s College
London (Business School), University of Turin (Department of Economics and
Statistics), Athens University of Economics and Business (Department of
Economics) and University of Piraeus (Department of Economics).

We have also benefited from the comments given by participants at the: SNDE
27th, 26th, 25th, 24th, 22nd and 21st Annual Symposiums (Federal Reserve Bank
of Dallas, March 2019; Keio University, Tokyo, March 2018; ESSEC Business
School, Paris, March 2017; University of Alabama, March 2016; The City
University of New York, April 2014; University of Milano-Bicocca, March 2013,
respectively), 5th and 4th AMEF Conferences (University of Macedonia, April
2019, 2018, respectively), 2nd DSSR Conference (University of Milano-Bicocca,
February 2019), 5th, 3rd, 2nd and 1st IAAE Conferences (University of
Montreal, June 2018; University of Milano-Bicocca, June 2016; University of
Macedonia, Greece, June 2015; Queen Mary University of London, June 2014,
respectively), 12th, 11th, 8th and 7th CFE Conferences (University of Pisa,
December 2018, December 2014; Senate House, University of London, December
2013, December 2017), 17th, 13th and 12th CRETE Conferences (Tinos, Crete and
Naxos, July 2018, 2014 and 2013, respectively), SEM Conference (MIT, Boston,
July 2017), 69th and 67th ESEM European Meetings (University of Geneva, August
2016; Toulouse, France, August 2014, respectively), Final and 1st RastaNews
Annual Conferences (Catholic University, Milan, January 2016; Palazzo Clerici,
Milan, January 2014, respectively), 35th ISF Conference (Erasmus University,
Rotterdam, July 2014), 2nd ISNPS Conference (Cadiz, Spain, June 2014), NSVSM
Conference (University of Paderborn, July 2014), 9th and 8th BMRC-QASS
conferences on Macro and Financial Economics (Brunel University London, May
2014 and 2013, respectively), and 3rd Humboldt-Copenhagen Conference on
Financial Econometrics (Humboldt University, Berlin, March 2013).

##  1 Introduction

Modelling time series processes with variable coefficients has received
considerable attention in recent years in the wake of several financial crises
and high volatility due to frequent abrupt changes in the market.
Justification for the use of such structures can be found in Timmermann and
van Dijk (2013); for example, for the dynamic econometric modelling and
forecasting in the presence of instability see the papers in the corresponding
Journal of Econometrics special issue, i.e., Pesaran et al. (2013). For some
more recent references see, for example, Cavaliere and Taylor (2005, 2008),
Cavaliere and Georgiev (2008), Giraitis et al. (2014), Harvey et al. (2018),
Chambers and Taylor (2020). Time-varying coefficient models are extensively
applied by practitioners, and their importance is widely recognized (see, for
example, Granger, 2007 and 2008, Petrova, 2019, Kapetanios et al., 2019,
2020).222A growing empirical literature in macroeconomics is testimony to
their importance. See, for example, Evans and Honkapohja (2001, 2009). Crucial
advances in both the theory and the empirics for these structures are the
works by Whittle (1965), Abdrabbo and Priestley (1967), Rao (1970), Hallin
(1979, 1986), Kowalski and Szynal (1990, 1991) and Grillenzoni (1993,
2000).333See also Francq and Gautier (2004a, 2004b). We refer to the
introduction of Azrak and Mélard (2006) and Alj et al. (2017) for further
references.

This paper provides a general framework for the study of autoregressive moving
average models with variable coefficients and heteroscedastic errors
(hereafter, TV-ARMA). There are two large classes of stochastic processes: the
ones with deterministically and those with stochastically time-varying
coefficients. Both types have been widely applied in many fields of research,
such as economics, finance and engineering (see for example Barnett et al.,
2014, Aastaveit et al., 2017, Mumtaz and Petrova, 2018), but traditionally
they have been examined separately. The new framework unifies them by showing
that one solution fits all. More specifically, we obtain explicit and
computationally feasible solution representations that generate the
fundamental properties of these models, whereas the scope of the useful tool
which is traditionally used to obtain such representations, that is the
characteristic polynomials, is diminished when time parameter variation is
present (see, for details, Hallin, 1979, and Grillenzoni, 1990).

Following Miller (1968), in a series of papers, Hallin (1979, 1984, 1986,
1987), Singh and Peiris (1987), and Kowalski and Szynal (1990, 1991) employ
the one-sided Green’s function (Green function for short) for the Wold
decomposition of TV-ARMA models and their core properties. This is grounded on
a particular solution of the associated linear difference equation. However,
the Green function involved in this solution, is defined implicitly via a
fundamental set of solutions, the elements of which was not, in the general
case, explicitly specified. As an alternative, recursive methods were employed
to compute it (see, for example, Grillenzoni, 2000, and Azrak and Melard,
2006). An explicit representation of the Green function depends upon the
availability of a fundamental set of solutions (marked also by Hallin, 1979)
whose elements (known as fundamental or linearly independent solutions) must
be explicitly expressed and easily handled, being an ongoing research issue.
Having at our disposal such a fundamental solution set, we also have the
general homogeneous and in turn the general nonhomogeneous solution of the TV-
LDE associated with TV-ARMA models.

In this work we provide such a fundamental solution set yielding a banded
Hessenbergian (determinant of a banded Hessenberg matrix) representation of
the Green function restriction involved in the solution of the associated TV-
LDE (see Paraskevopoulos and Karanasos, 2021), termed here as “principal
determinant” and denoted by $\xi(t,s)$ (see eq. (6)). The entries of
$\xi(t,s)$ are the autoregressive coefficients taken at consecutive time
instances. The first fundamental solution is the principal determinant. The
remaining fundamental solutions are also expressed in terms of the principal
determinant and therefore as banded Hessenbergians, yielding an explicit form
for the entries of the companion matrix product. The advantages of the
principal determinant representation of the Green function are multiple. Using
elementary properties of banded Hessenbergians, we obtain the general
homogeneous solution of the linear difference equation associated with a TV-
ARMA model as a linear combination of the principal determinant times
prescribed initial condition values (see eqs. (9, 11)). The particular
solution, mentioned earlier, is also recovered, but in a more explicit form,
as a linear combination of the principal determinant times consecutive
instances of the forcing term (see eq. (12)). These results lead to an
efficient interpretation of the model, described as a decomposition into four
structurally different parts (see eq. (16)). In this interpretation the
fundamental solutions could be portrayed as autoregressive coefficients of
prescribed random variables.

The Hessenbergian representation of TV-ARMA processes gives rise to an
explicit condition, that is the absolutely summability of terms involving the
principal determinant (see (19)), which leads to easily handled explicit
representations of their fundamental properties in $L_{2}$ spaces. The
convergence to zero of the principal determinant along with the boundedness of
the autoregressive coefficients ensure the asymptotic stability of all bounded
solution processes in $L_{2}$. The absolute summability condition along with
the boundedness of the drift and the moving average coefficients guarantee the
existence and uniqueness of the Wold-Cramér decomposition of asymptotically
stable solution processes associated with finite unconditional first two
moments and autocovariance function. The invertibility of such solution
process is guaranteed by the boundedness of the autoregressive coefficients
and the absolutely summability of terms generated by the principal
determinant, whereas, in this case, its nonzero entries are the moving average
coefficients of the model (see Section 6.1). As a consequence, we derive
explicitly the optimal linear predictor for the asymptotically stable solution
process of TV-ARMA models along with the forecasting error and its associated
mean square error, when an infinite set of data is observed. An alternative
approach, employs the aforementioned explicit representation of the model,
which yields its optimal linear predictor when a finite set of data is
observed. Therefore the problem of obtaining such a predictor is reduced to a
matter of dealing with a linear regression. In a causal environment, it turns
out that both approaches yield identical forecasting and mean square errors
and therefore identical optimal predictors. Moreover, we illustrate
mathematically one of the focal points in Hallin’s (1986) analysis concerning
the asymptotic efficiency of such models. Namely, that in a time-varying
setting two forecasts with identical forecasting horizons, but at different
times, yield different mean squared errors. It turns out that the backward
asymptotic efficiency is, in the general case, different from the forward one
(termed by Hallin Granger-Andersen). In this context, we explicitly provide
sufficient conditions for the forward asymptotic boundedness and uniform
boundedness of the mean square forecasting error. Equally importantly, we
demonstrate how the linear algebra techniques, used to obtain the general
solution, are equivalent to a simple procedure for manipulating polynomials
with variable coefficients (see Section 8). In order to do so we employ the
expression of the principal determinant in conjunction with the so called skew
multiplication operator or symbolic operator (see, for example, Hallin, 1986,
and Mrad and Farag, 2002).

Various processes with stochastic coefficients are treated within our unified
framework in Section 7. We investigate two general models. First, for the
random coefficients model, we show that when the principal determinant
converges almost surely, then the process converges in distribution. Second,
the double stochastic autoregressive model is also employed to formulate
explicitly some of its fundamental properties. For example, the convergence in
$L_{2}$ of the principal determinant is a necessary condition for the
unconditional variance to exist.

Banded Hessenbergians are computationally tractable due to the linear running
time for their calculation (see Paraskevopoulos and Karanasos, 2021). Compact
solution representations of banded Hessenbergians, established in Marrero and
Tomeo (2012, 2017) and Paraskevopoulos and Karanasos (2021), can be applied to
derive analogous explicit representations for the principal determinant. These
results modernize and enhance the explicit representations of time-varying
models and their fundamental properties, by compact representations.

The definition of the principal determinant arises naturally from
Paraskevopoulos (2014), who (by introducing the so called infinite Gauss-
Jordan elimination algorithm) provides banded Hessenbergian representations
for the fundamental solution set associated with a TV-ARMA($p,q$) model (see
Paraskevopoulos and Karanasos 2021). The results of the present paper are
established independently of the infinite elimination algorithm.

To sum up, the unified theory enable us to study linear time series models
with either stochastically or deterministically varying coefficients, thus
allowing us to make theoretical comparisons between these two large classes of
models.

This paper concludes with an empirical application on inflation persistence in
the United States (see Section 10), which employs a time-varying model of
inflation dynamics grounded on statistical theory. In particular, we estimate
an autoregressive process with abrupt structural breaks and we compute an
alternative measure of second-order time dependent persistence, which
distinguishes between changes in the dynamics of inflation and its volatility
and their persistence. Our main conclusion is that persistence increased after
1976, whereas from 1986 onwards it declines and stabilizes to even lower
levels than the pre-1976 period. Our results are in line with those in Cogley
and Sargent (2002), who find that the persistence of inflation in the United
States rose in the 1970s and remained high during this decade, before starting
a gradual decline from the 1980s until the early 2000s.

The outline of the paper is as follows. Section 2 introduces the notation used
in the paper followed by the principal determinant. The next Section presents
the explicit representation for an extensive family of “time-varying” ARMA
models, based on the general solution of the associated TV-LDE. In Section 4,
we obtain a necessary and sufficient condition which guarantees the asymptotic
stability of these processes. Section 5 presents explicit formulas for the
fundamental properties of the model, including the Wold-Cramér decomposition,
the unconditional moments and the autocovariance function. Section 6 deals
with the invertibility and forecasting issues. In Section 7, we examine AR
models in which the drift and the autoregressive coefficients are
stochastically varying. In Section 8, we introduce a simple procedure for
manipulating polynomials with variable coefficients. The next Section gives an
illustrative example with abrupt structural breaks and proposes a new measure
of time-varying persistence. In Section 10 we present an empirical study for
inflation persistence. The final Section of the paper contains some concluding
remarks and future work.

The proofs of the statements concerning L of the current paper are
demonstrated in (Paraskevopoulos and Karanasos 2021), while the proofs of
stochastically oriented statements are included in the Appendix of this paper,
which follows the enumeration structure of the main body of the paper. Further
details are provided in the Online and the Additional Appendices.

##  2 Time-varying ARMA

The aim of this Section is to provide a definition of time-varying ARMA
models, hereafter termed TV-ARMA, along with the main mathematical tool for
their analysis. This is the banded Hessenbergian representation of the Green
function associated with such models, which is also referred to here as the
principal determinant.

###  2.1 The Model

This Subsection introduces suitable notation and defines the basic process.
Throughout the paper we adopt the following notational conventions: The set of
integers (resp. positive and non-negative integers) is denoted by $\mathbb{Z}$
(resp. $\mathbb{Z}_{>0}$ and $\mathbb{Z}_{\geq 0}$). Similarly, the set of
real numbers (resp. positive and non-negative real numbers) is denoted by
$\mathbb{R}$ (resp. $\mathbb{R}_{>0}$ and $\mathbb{R}_{\geq 0}$). Moreover,
$(\Omega,\mathcal{F},P)$ stands for a probability space and
$L_{2}(\Omega,\mathcal{F},P)$ (in short $L_{2}$) stands for the Hilbert space
of random variables with finite first two moments defined on
$(\Omega,\mathcal{F},P)$.

Let $p\in\mathbb{Z}_{\geq 1}$ and $q\in\mathbb{Z}_{\geq 1}$. A TV-ARMA($p,q$)
model generates a stochastic process $\\{y_{t}\\}_{t}\in\mathbb{Z}$ (or simply
$y_{t}$ process), which satisfies

$y_{t}=\varphi(t)+\sum\limits_{m=1}^{p}\phi_{m}(t)y_{t-m}+u_{t},\ \text{for\
all}\ t\in\mathbb{Z}$ (1)

with moving average term $u_{t}$ given by

$u_{t}=\varepsilon_{t}+\sum\limits_{l=1}^{q}\theta_{l}(t)\varepsilon_{t-l},$

where the coefficients $\phi_{m}(t),\theta_{l}(t)$ are either deterministic
(real valued functions) or stochastic (random variables), $\varphi(t)$ is the
time-varying drift generating a deterministic or stochastic process, while
$\\{\varepsilon_{t}\\}_{t}$ is a mean zero random process (that is
$\mathbb{E}(\varepsilon_{t})=0$) such that $\mathbb{E}(\varepsilon_{t}\
\varepsilon_{s})=0$ for $s\not=t$ (uncorrelatedness condition),
$\mathbb{E}(\varepsilon_{t}|y_{s},s<t)=0$ for all $t$ (that is
$\\{\varepsilon_{t}\\}$ is a martingale difference sequence relative to
$\\{y_{s}\\}$) and the time-varying variance
$\mathbb{E}(\varepsilon_{t}^{2})=\sigma^{2}(t)$ is non-zero and bounded, that
is $0<\sigma^{2}(t)\leq M<\infty$ for all $t$ and some $M\in\mathbb{R}_{>0}$.
The above conditions guarantee that $\varepsilon_{t}\in L_{2}$ and
$\varepsilon_{t}\perp\varepsilon_{s}$ ($\varepsilon_{t},\varepsilon_{s}$ are
orthogonal), whenever $s\not=t$.

If the non-constant coefficients $\phi_{m}(t),\theta_{l}(t)$ and the drift
$\varphi(t)$ in eq. (1) are deterministic (resp. stochastic) we shall refer to
it as DTV-ARMA (resp. STV-ARMA).444Notice that in our setting the time-varying
coefficients can depend on the length of the series as well, as in Azrak and
Mélard (2006); see the examples in Section 5.3. Further specifications and
adjustments for STV-ARMA models are presented in Section 7.

The forcing term $\upsilon_{t}$ is assigned to be the time-varying drift plus
the moving average term:

$\upsilon_{t}=\varphi(t)+u_{t}.$ (2)

Proceeding with the notation of eq. (2), the associated TV-LDEs($p$) of eq.
(1) is defined by:

$y_{t}=\sum_{m=1}^{p}\phi_{m}(t)y_{t-m}+\upsilon_{t},\ \text{for\ all}\
t\in\mathbb{Z},$ (3)

where $\phi_{m}(t)$ stand for the autoregressive coefficients of eq. (1) and
$\upsilon_{t}$, $y_{t-m}$ for $1\leq m\leq p$ are considered as realizations
of the homonyms random variables.

In this work, both assumptions of stationarity and homoscedasticity have been
relaxed (see also, among others, Singh and Peiris, 1987, Kowalski and Szynal,
1990, 1991, and Azrak and Mélard, 2006), which is likely to be violated in
practice and we allow $\\{\varepsilon_{t}\\}$ to follow, for example, a
stochastic volatility or a time-varying GARCH type of process (see, for
example, the earlier versions of the current paper: Karanasos et al., 2014c,
and Karanasos et al., 2017) or we allow for abrupt structural breaks in the
variance of $\varepsilon_{t}$ (see the example in Section 9).

The TV-ARMA($p,q$) model nests both the TV-AR$(p)$ as a special case when
$q=0$ and the ARMA$(p,q)$ specification when the drift, the autoregressive and
moving average coefficients, and the variances are all constants, adopting for
this purpose the conventional identifications: $\varphi(t)=\varphi,\
\phi_{m}(t)=\phi_{m},\ \theta_{l}(t)=\theta_{l},\sigma^{2}(t)=\sigma^{2}$ for
all $t$.

The relation between the process under consideration and its innovations is
essentially described by the Wold-Cramér decomposition (see Section 5.1),
which is the main analytical tool for studying the asymptotic efficiency of
the model. In this case, the latest time-point of the observed random
variables, denoted here by $s$, moves to the remote past
($s\rightarrow-\infty$), while the forecast time-point, denoted here by $t$,
is kept fixed. The forward asymptotic efficiency of the model (so-called by
Hallin, 1986, Granger-Andersen) is strongly related to the forecasting
problem. It directs attention to the asymptotic properties of the mean square
forecasting error (MSE for short), as the time $t$ moves to the far future,
while $s$, is kept fixed (see Section 6.3).

One of the goals of this work is to obtain an explicit inverse of the time-
varying autoregressive (AR) polynomial associated with eq. (1) being denoted
by $\Phi_{t}(B)$, where $B$ is the backshift or lag operator (see Section 8).
In a time-varying environment, the usual procedure employs the Green function
instead of the characteristic polynomials, which are used in the time
invariant case. More specifically, we employ the principal determinant,
coupled with the so called multiplication skew operator (see the analysis in
Section 8.1) to obtain the inverse of $\Phi_{t}(B)$.

We should also mention that Kowalski and Szynal (1991) used the product of
companion matrices to obtain the associated Green function. Paraskevopoulos
and Karanasos (2021) capitalized on the connection between the product of
companion matrices and time-varying stochastic difference equations but in the
opposite direction. That is, they went the other way around and by finding an
explicit and compact representation of the fundamental solutions associated
with TV-ARMA models, they obtained an analogous representation for the
elements of the associated companion matrix product.

When the coefficients are deterministic time-varying the process is for sure
nonstationary. When the coefficients are stochastic time-varying then the
process can be stationary.

###  2.2 The Principal Determinant

In this and next Subsection, we provide some results on linear difference
equations of order $p$ with variable coefficients (for short TV-LDEs($p$)) and
their banded Hessenbergian solution representation.

We start with the main mathematical tool of this paper, the principal
determinant, denoted as $\xi(t,s)$. It has been shown in (Paraskevopoulos and
Karanasos 2021), that $\xi(t,s)$ is a solution of the homogeneous linear
difference equation associated with eq. (3), that is

$y_{t}=\sum_{m=1}^{p}\phi_{m}(t)y_{t-m}$ (4)

taking on the initial conditions $y_{s+1-p}=0,...,y_{s-1}=0,y_{s}=1$.

Amongst the various implicit representations of the Green function in terms of
undetermined fundamental sets of solutions (see Section 3.1 and
(Paraskevopoulos and Karanasos 2021) for further details), the principal
determinant provides an explicit and computationally tractable representation
of the Green function restriction involved in the solution of the associated
TV-LDE (see for details Paraskevopoulos and Karanasos 2021). As a consequence,
the main properties of time-varying models acquire analogous representations,
as stated in Sections 4, 5 and 6.

To distinguish scalars from vectors we adopt lower and uppercase boldface
symbols within square brackets for column vectors and matrices respectively:
$\mathbf{x}=[x_{i}]$, $\mathbf{X}=[x_{ij}]$. Row vectors are indicated within
round brackets and usually appear as transpositions of column vectors:
$\mathbf{x}^{\prime}=(x_{i})$.

For every pair $(t,s)\in\mathbb{Z}^{2}$ such that $k=t-s\geq 1$ the principal
matrix associated with the AR part of eq. (1), is defined by

$\mathbf{\Phi}_{t,s}=\left[\begin{array}[]{cccccccc}{\footnotesize\phi}_{1}{\footnotesize(s+1)}&{\footnotesize-1}&&&&&&\\\
{\footnotesize\phi}_{2}{\footnotesize(s+2)}&{\footnotesize\phi}_{1}{\footnotesize(s+2)}&\ddots&&&&&\\\
{\footnotesize\vdots}&{\footnotesize\vdots}&\ddots&\ddots&&&&\\\
{\footnotesize\phi}_{p}{\footnotesize(s+p)}&{\footnotesize\phi}_{p-1}{\footnotesize(s+p)}&\ddots&\ddots&\ddots&&&\\\
&{\footnotesize\phi}_{p}{\footnotesize(s+p+1)}&\ddots&\ddots&\ddots&\ddots&&\\\
&&{\small\ddots}&{\small\ddots}&{\small\ddots}&{\small\ddots}&\ddots&\\\
&&&{\footnotesize\phi}_{p}{\footnotesize(t-1)}&{\footnotesize\phi}_{p-1}{\footnotesize(t-1)}&{\small\cdots}&{\footnotesize\phi}_{1}{\footnotesize(t-1)}&{\footnotesize-1}\\\
&&&&{\footnotesize\phi}_{p}{\footnotesize(t)}&{\small\cdots}&{\footnotesize\phi}_{2}{\footnotesize(t)}&{\footnotesize\phi}_{1}{\footnotesize(t)}\end{array}\right],$
(5)

here and in what follows empty spaces in a matrix have to be replaced by
zeros. $\mathbf{\Phi}_{t,s}$ is a lower Hessenberg matrix of order $k$. It is
also a banded matrix with total bandwidth $p+1$ (the number of its non-zero
diagonals, i.e., the diagonals whose elements are not all identically zero),
upper bandwidth $1$ (the number of its non-zero super-diagonals), and lower
bandwidth $p-1$ (the number of its non-zero sub-diagonals). In particular, the
elements of $\mathbf{\Phi}_{t,s}$ are: $(-1)$ occupying the entries of the
superdiagonal, the values of the first autoregressive coefficient
$\phi_{1}(\cdot)$ (from time $s+1$ to time $t$), occupying the entries of the
main diagonal, the values of the $(1+r)$-th autoregressive coefficient
$\phi_{1+r}(\cdot)$ for $r=1,2,\ldots,p-1$ (from time $s+1+r$ to time $t$),
occupying the entries of the $r$-th sub-diagonal, and zero entries elsewhere.
It is clear that for $p\geq k$, $\mathbf{\Phi}_{t,s}$ is a full lower
Hessenberg matrix.

For every pair $(t,s)\in\mathbb{Z}^{2}$ with $s<t$, the principal determinant
associated with eq. (5) is given by:

$\xi(t,s)=\det(\mathbf{\Phi}_{t,s}).$ (6)

Formally $\xi(t,s)$ is a lower Hessenbergian (determinant of a lower
Hessenberg matrix; for details on Hessenbergians see, for example, the book by
Vein and Dale, 1999). We further extend the definition of $\xi(t,s)$ so as to
be defined over $\mathbb{Z}^{2}$ by assigning the initial conditions:

$\xi(t,s)=\left\\{\begin{array}[]{ccc}1&\text{if}&t=s\\\
0&\text{if}&t<s.\end{array}\right.$ (7)

Under these initial values, for each fixed $s$ the sequence
$\\{\xi(t,s)\\}_{t\geq s+1-p}$ turns out to be a solution sequence of eq. (4)
called principal fundamental sequence (see Paraskevopoulos and Karanasos
2021). It is well known that the restriction of the Green function, often
designated by $H(t,s)$ for $t>s$ and $s$ fixed, solves eq. (4), assuming the
prescribed values $H(s,s)=1$ for $p\geq 1$ and $H(t,s)=0$ for
$t=s+1-p,...,s-1$ and $p\geq 1$ (see for example Agarwal, 2000, p.77, Property
(iii) or Lakshmikantham and Trigiante, 2002, Theorem 3.4.1 p.87 ). Since the
principal determinant $\xi(t,s)$ solves eq. (4) under the same initial values,
just as like with $H(t,s)$, the uniqueness of the solution for an initial
value problem entails that for every arbitrary but fixed $s\in\mathbb{Z}$ the
principal determinant coincides with the Green function, that is
$H(t,s)=\xi(t,s)$ for all $t$ such that $t\geq s+1-p$ and $p\geq 1$ (a proof
from first principles is presented in Paraskevopoulos and Karanasos, 2021,
Proposition 2). Therefore we will make use of two different, although
equivalent terminologies: principal determinant or Green’s function with fixed
$s$. An explicit and compact representation of $\xi(t,s)$, called Leibnizian
representation, is provided in the previously cited reference. An analogous
representation of Hessenbergians, as a nested sum, and their inverses are
established by Marrero and Tomeo (2017, 2012), respectively.

We conclude this Section with an illustrative example, concerning the
ARMA($p,q$) model with constant coefficients.

###### Example 1

The AR polynomial $\Phi(B)=1-\sum_{m=1}^{p}\phi_{m}B^{m}$ associated with eq.
(4), whenever $\phi_{m}(t)=\phi_{m}$ (constant autoregressive coefficients),
is explicitly expressed in terms of the characteristic values as
$\Phi(B)=\prod_{m=1}^{p}(1-\lambda_{m}B)$. The definition in eq. (6) gives the
determinant of a banded Hessenbergian Toeplitz matrix (for details on Toeplitz
matrices see, for example, the book by Gray, 2006)555For the use of Toeplitz
matrices on double-differenced AR(1) models see Han (2007). and satisfies the
identity (usually called Widom’s determinant formula; see Widom, 1958)

$\xi_{k}=\left|\begin{array}[]{ccccccccc}\phi_{1}&-1&&&&&&&\\\
\phi_{2}&\phi_{1}&-1&&&&&&\\\ \phi_{3}&\phi_{2}&\phi_{1}&\ddots&&&&&\\\
\vdots&\vdots&\vdots&\ddots&\ddots&&&&\\\
\phi_{p}&\phi_{p-1}&\phi_{p-2}&\ddots&\ddots&\ddots&&&\\\
&\phi_{p}&\phi_{p-1}&\ddots&\ddots&\ddots&\ddots&&\\\
&&\phi_{p}&\phi_{p-1}&\phi_{p-2}&\cdots&\phi_{1}&-1&\\\
&&&\phi_{p}&\phi_{p-1}&\cdots&\phi_{2}&\phi_{1}&-1\\\
&&&&\phi_{p}&\cdots&\phi_{3}&\phi_{2}&\phi_{1}\end{array}\right|={\displaystyle\sum\limits_{m=1}^{p}}\frac{\lambda_{m}^{k+p-1}}{{\displaystyle\prod\nolimits_{\underset{n\neq
m}{n=1}}^{p}}(\lambda_{m}-\lambda_{n})},$ (8)

where $\xi(t,s)$ is now denoted as $\xi_{k}$, since it depends only on the
forecasting horizon $k$ ($k=t-s$ indicates the order of the matrix too). The
second equality in eq. (8) follows (only if $\lambda_{m}\neq\lambda_{n}$) from
standard results in ARMA models (see, for example, Karanasos, 1998, 2001, and
in particular, eq. (2.6) in his Corollary $2$; see also Hamilton, 1994, pp.
12-13).

##  3 TV-ARMA Representations

In the following Subsections we shall use the principal determinant to
describe explicitly the general homogeneous and particular solutions of TV-
LDE($p$) in eq. (3). The sum of these solutions yields the general solution of
eq. (3),666In linear algebra there have been some isolated attempts to deal
with the problem, which have been criticized on a number of grounds. For
example, Mallik (1998) provides an explicit solution for the aforementioned
equations, but it appears not to be computationally tractable (see also
Mallik, 1997 and 2000). Lim and Dai (2011) point out that “although explicit
solutions for general linear difference equations are given by Mallik (1998),
they appear to be unmotivated and no methods of solution are discussed”. which
lead to an explicit and computationally tractable representation of TV-ARMA
processes (see Theorem 1). A useful decomposition of the innovation part of
the solution is presented in Proposition 1.

###  3.1 Homogeneous Solution

Let $a\in\mathbb{Z}$ and $\mathbb{Z}_{a}\overset{\rm
def}{=}\\{z\in\mathbb{Z}:z\geq a\\}$. Let also $s,t\in\mathbb{Z}$ and $s<t$.
Taking into account that $s,t\in\mathbb{Z}_{s+1-p}$, the general solution of
eq. (4) over $\mathbb{Z}_{s+1-p}$ is denoted by $y_{t,s}^{hom}$. The latter
can be expressed in terms of the principal determinant, the autoregressive
coefficients and the initial conditions $y_{s+1-m}$ for $m=1,2,...,p$, as
follows

$y_{t,s}^{hom}=\displaystyle\sum_{m=1}^{p}\sum_{r=1}^{p+1-m}\phi_{m-1+r}(s+r)\xi(t,s+r)y_{s+1-m},$
(9)

(a proof of eq. (9) is provided in (Paraskevopoulos and Karanasos 2021). The
two variable solution notation $y^{hom}_{t,s}$ is consistent with the
analogous notation used for the principal determinant (or the Green function)
and is essential for the TV-ARMA($p,q$) representation in eq. (16). Moreover,
the principal determinant exactly matches the specific restriction of the
Green function involved in the general homogeneous solution in eq. (9) and, in
turn, of the nonhomogeneous solution of the associated TV-LDE, that is the
$H(t,j)=\xi(t,j)$ for all $t,j$ ($j=s+r$) such that $t\geq s+1-p$ and $s\leq
j\leq t+p-1$ (a proof is presented in Paraskevopoulos and Karanasos, 2021,
Theorem 5 and Corollary 2).

To the extent of our knowledge, the expression of the homogeneous solution in
terms of the obtained in eq. (9) is first recorded in earlier versions of this
paper (see also Paraskevopoulos and Karanasos, 2021). The effects of eq. (9)
on this work are multiple. It makes it possible to construct the general
solution of a TV-LDE($p$) entirely in terms of the Green function and any set
of prescribed random variables $\\{y_{s+1-p},...,y_{s}\\}$ at a time-point $s$
of the recent or remote past. Explicit conditions guaranteeing the zero
convergence of eq. (9) yield the asymptotic stability and the Wold-Cramér
decomposition of the model. Another consequence, is an alternative approach to
the forecasting, presented in Section 6.2. As in the case of processes of
order (1,1), this is a consequence of the explicit representation of TV-
ARMA($p,q$) models in eq. (16).

In all that follows the following notation is used:

$\xi^{(m)}(t,s)\overset{\mathrm{def}}{=}\sum_{r=1}^{p+1-m}\phi_{m-1+r}(s+r)\xi(t,s+r)$
(10)

(superscripts within parentheses or brackets [e.g., $(.)^{(m)}$] designate the
position of the corresponding term [e.g., $m$-th term] in a sequence, so as to
distinguish position indices from power exponents). The principal determinant
is identified with $\xi^{(1)}(t,s)$, that is
$\xi(t,s)\overset{\mathrm{def}}{=}\xi^{(1)}(t,s)$. Applying eq. (10) with
$s=t-1$, on account of $\xi(t,t)=1$ and $\xi(t,t+j)=0$ for $j\geq 1$ (see eq.
(7)), we conclude that: $\xi^{(m)}(t,t-1)=\phi_{m}(t)$.

It turns out (see Paraskevopoulos and Karanasos 2021) that for every $m$ with
$1\leq m\leq p$ the sequence $\\{\xi^{(m)}(t,s)\\}_{t\geq s+1-p}$ is the
solution of eq. (4) under the prescribed initial values: $y_{s+1-m}=1$ and
$y_{s+1-r}=0$ for $1\leq r\leq p$ with $r\not=m$. Applying the expression in
eq. (10) to the right-hand side of eq. (9), the homogeneous solution takes a
more condensed form:

$y_{t,s}^{hom}=\sum_{m=1}^{p}\xi^{(m)}(t,s)y_{s+1-m}.$ (11)

Under the initial values $y_{s}=1$ for $p\geq 1$ and $y_{s-r}=0$ for $1\leq
r\leq p-1$, whenever $p\geq 2$, the right-hand side of eq. (11) turns into
$\xi^{(1)}(t,s)=\xi(t,s)$ and thus the homogeneous solution in eq. (11)
coincides with the principal fundamental sequence $\\{\xi(t,s)\\}_{t\geq
s+1-p}$. In (Paraskevopoulos and Karanasos 2021), it is shown that the set

$\varXi_{s}=\\{\xi^{(1)}(t,s),\xi^{(2)}(t,s),...,\xi^{(p)}(t,s):\ t\geq
s+1-p\\}$

is a fundamental (or linearly independent) set of solutions associated with
eq. (4). Moreover, it is shown there that the $m$-th fundamental solution
$\xi^{(m)}(t,s)$ can be expressed as a single banded Hessenbergian too. The
only difference between any two of these fundamental solutions lies in the
first column (see Paraskevopoulos and Karanasos 2021).

###  3.2 A Particular Solution and its Decomposition

A particular solution to eq. (3) subject to the initial condition values
$y_{s}=y_{s-1}=...=y_{s+1-p}=0$ is given by

$y_{t,s}^{par}=\sum_{r=s+1}^{t}\xi(t,r)[\varphi(r)+u_{r}].$ (12)

A proof of the above formula is demonstrated in (Paraskevopoulos and Karanasos
2021). The solution in eq. (12) depends both on $t$ and $s$. This has to be
compared with the equivalent result presented in Miller (1968, p. 40, eqs.
(2.8) and (2.9)).

In Proposition 1 below, we introduce a decomposition of the innovation part in
the particular solution in eq. (12), which is used throughout this paper. But
first we will introduce the following definition:

###### Definition 1

First, we define the function on $\mathbb{Z}^{2}$:

$\xi_{q}(t,r)\overset{\mathrm{def}}{=}\xi(t,r)+\displaystyle\sum_{l=1}^{q}\xi(t,r+l)\theta_{l}(r+l).$
(13)

Second, for each $s\in\mathbb{Z}$, we define the function on $\mathbb{Z}^{2}$:

$\xi_{s,q}(t,r)\overset{\mathrm{def}}{=}\displaystyle\sum_{l=s+1-r}^{q}\xi(t,r+l)\theta_{l}(r+l).\vspace{-0.05in}$
(14)

As $\xi_{q}(t,r)$ is equal to $\xi(t,r)$ plus a sum of terms consisting of the
first $q$ instances of $\xi(t,r+l)$ multiplied by corresponding moving average
coefficients, it can also be expressed as a banded Hessenbergian (the proof is
deferred to the Online Appendix Subsection F1). The same applies to
$\xi_{s,q}(t,r)$. Therefore, we shall refer to $\xi_{q}(t,r)$ and
$\xi_{s,q}(t,r)$ as banded Hessenbergian coefficients. In view of eq. (13)
$\xi_{q}(t,t)=1$, since $\xi(t,t+j)=0$, whenever $j\geq 1$ and the second term
(i.e., the sum), when applied for $r=t$, vanishes. For the same reason
$\xi_{q}(t,r)=0$, whenever $r\geq t+1$. To summarize, $\xi_{q}(t,r)$ coincides
with the corresponding initial conditions of $\xi(t,r)$, whenever $r\geq t$
(see eq. (7)). In view of eq. (14), if $r<s+1-q$ (or $s+1-r>q$), then the
lower summation limit exceeds the upper limit, whence $\xi_{s,q}(t,r)=0$.
Finally, for a pure AR($p$) model, that is when $q=0$, we have:
$\xi_{0}(t,r)=\xi(t,r)$ and $\xi_{s,0}(t,r)=0$. The banded Hessenbergian
coefficient $\xi_{q}(t,r)$ must be compared with the function defined by
Peiris (1986), see his notation “$\rm g(t,s)$” in eq. (2.2).

###### Proposition 1

The innovation part of the particular solution in eq. (12) can be decomposed
into two parts as follows:

$\sum_{r=s+1}^{t}\xi(t,r)u_{r}=\sum\limits_{r=s+1}^{t}\xi_{q}(t,r)\varepsilon_{r}+\sum\limits_{r=s+1-q}^{s}\xi_{s,q}(t,r)\varepsilon_{r}.\vspace{-0.05in}$
(15)

In the first summation of the right-hand side of eq. (15), the length of the
time interval between $s$ and $t$ is the forecasting horizon ($k=t-s$). As a
consequence, the errors whose index ranges over the integer interval
$\mathopen{\big{[}}s+1\mathrel{{.}\,{.}}\nobreak t\mathclose{\big{]}}$ 777If
$a,b\in\mathbb{Z}$, we adopt the notation:
$\mathopen{\big{[}}a\mathrel{{.}\,{.}}\nobreak
b\mathclose{\big{]}}\overset{\rm def}{=}\\{z\in\mathbb{Z}:a\leq z\leq b\\}$.
are unobservable. In the second one the time interval extends from $s+1-q$ to
$s$ and therefore the errors whose index ranges over
$\mathopen{\big{[}}s+1-q\mathrel{{.}\,{.}}\nobreak s\mathclose{\big{]}}$ are
observable. Notice that $\xi_{q}(t,r)$ for $s+1\leq r\leq t$ can be
equivalently expressed as $\xi_{q}(t,t-r)$ for $0\leq r\leq k-1$. Similarly
$\xi_{s,q}(t,r)$ for $s+1-q\leq r\leq s$ can be equivalently expressed as
$\xi_{s,q}(t,t-r)$ for $k\leq r\leq k-1+q$ (see eq. (48)).

Eq. (15) coupled with eq. (16) (in the following Subsection) will be used to
obtain TV-ARMA($p,q$) forecasts in Subsections 6.2 and 7.4. If $s=t-1$ (or
$k=1$), the first sum in the right-hand side of eq. (15) reduces to
$\varepsilon_{t}$ and the second sum reduces to
$\sum\nolimits_{l=1}^{q}\theta_{l}(t)\varepsilon_{t-l}$ (a formal proof is
provided in the Online Appendix Subsection F2). Notice that the above
mentioned reduction is in line with Remark 1 below.

###  3.3 Explicit Model Representation

The results of the previous Section on linear difference equations, are
applied herein to obtain in the following Theorem an equivalent explicit
representation of the stochastic process $y_{t}$ in eq. (1) in terms of
prescribed random variables: $y_{s},...,y_{s+1-p}$ for any $s\in\mathbb{Z}$
with $s<t$. This is a consequence of the general solution representation of
the TV-LDE($p$) in eq. (3) as a sum of the homogeneous and the particular
solutions in eqs. (11, 12), respectively, that is
$y_{t}=y^{hom}_{t,s}+y^{par}_{t,s}$ for all $t,s$ with $s<t$. Notice that
$y_{t}$ is independent of $s$, while, in context with time-series, both
homogeneous and particular solution functions depend on $t$ and $s$. Applying
eq. (15) to the particular solution $y^{par}_{t,s}$ in eq. (12), Theorem 1
follows.

###### Theorem 1

An equivalent explicit representation of $y_{t}$ in eq. (1) in terms of
prescribed random variables $\\{y_{r}\\}_{s+1-p\leq r\leq s}$ for any
$s\in\mathbb{Z}$ and $t\in\mathbb{Z}_{s+1-p}$ such that $s<t$ is given by:

$y_{t}=\underset{\text{Homogeneous Solution
Part}}{\underbrace{\displaystyle\sum_{m=1}^{p}\xi^{(m)}(t,s)y_{s+1-m}}}+\underset{\underset{\text{{\scriptsize
Drift Part}}}{\text{Particular
Solution:}}}{\underbrace{\sum_{r=s+1}^{t}\xi(t,r)\varphi(r)}+}\underset{\text{Particular
Solution: Innovation
Part}}{\underbrace{\sum\limits_{r=s+1}^{t}\xi_{q}(t,r)\varepsilon_{r}+\sum\limits_{r=s+1-q}^{s}\xi_{s,q}(t,r)\varepsilon_{r}.}}$
(16)

The algebraic part of the proof, concerning TV-LDEs($p$) is provided in
(Paraskevopoulos and Karanasos 2021), the stochastic part is given in the
Appendix. The right-hand side of eq. (16) comprises four summation parts. In
view of Proposition 1, the sum of its last three parts is the particular
solution given by eq. (12). More analytically, the first sum (the homogeneous
solution in eq. (11)) is a product of $m$ fundamental solutions multiplied by
observable random variables. The second sum (the drift part of the particular
solution in eq. (12)) is formed by products involving the principal
determinant $\xi(t,r)$ multiplied by the drift $\varphi(r)$. The terms of the
third sum (the first part of the “MA decomposition”, see eq. (15)) are the
banded Hessenbergian coefficients $\xi_{q}(t,r)$, multiplied by the
unobservable errors. Finally, the terms of the fourth sum (the second part of
the “MA decomposition”) are the banded Hessenbergian coefficients
$\xi_{s,q}(t,r)$, multiplied by observable errors. In the Online Appendix G1,
we apply eq. (16) to an AR(1) model, recovering the well known explicit
representation of the latter.

###### Remark 1

When $s=t-1$ (or $k=1$) the expression in Theorem 1 coincides with eq. (1).
This is a consequence of the following statements: i)
$\xi^{(m)}(t,t-1)=\phi_{m}(t)$ (see the discussion next to eq. (10)) and ii)
$\sum_{r=t}^{t}\xi(t,r)[\varphi(r)+u_{r}]=\varphi(t)+u_{t}$.

###### Remark 2

Replacing the homogeneous solution part in eq. (16) by eq. (9), and its
innovation part with the left-hand side of eq. (15), we can solely express eq.
(16) as a linear combination of the Green function. The expression in eq. (16)
has to be compared with the corresponding result in (Agarwal, 2000 p.77, eq.
(2.11.8)), in which the fundamental solutions, denoted there by “$v_{i}(k)$”,
are not in general explicitly expressed, but only in specific cases (see the
examples in the previously cited reference).

The methodology presented in this Section can be used in the study of infinite
order autoregression models as well as in the case of the fourth order moments
for time-varying GARCH models. In the interest of brevity the detailed
examination of the aforementioned models will be the subject of future papers.
We should also mention that another mathematical tool of constant use in
difference equations is the generalized continuous fraction approach (see, Van
de Cruyssen, 1979). The concept of matrix continued fraction was introduced in
Hallin (1984), whereas Hallin (1986) showed the close connection between the
convergence of matrix continued fractions and the existence of dominated
solutions for multivariate difference equations of order two.

Another advantage of our TV-ARMA representation in eq. (16), is its
generality. That is, in deriving it we do not make any assumptions on the time
dependent coefficients. Therefore, it does not require a case by case
treatment. In other words, we suppose that the law of evolution of the
coefficients is unknown, in particular they may be stochastic (either
stationary or non stationary) or deterministic. Therefore, no restrictions are
imposed on the functional form of the time-varying autoregressive and moving
average coefficients. In the non stochastic case the model allows for known
abrupt changes, smooth changes and mixtures of them. If the changes are smooth
the coefficients can depend on an exogenous variable $x_{t}$ or $t$ or both.
In the case of stochastically varying coefficients the model includes the
generalized random coefficient (GRC) AR specification (see, for example,
Glasserman and Yao, 1995, and Hwang and Basawa, 1998) as a special case or
allows for Markov switching behaviour (see, for example, Hamilton, 1989 and
1994, chapter 22). In both aforementioned cases it allows for periodicity. We
should also mention that the solution includes the case where the variable
coefficients depend on the length of the series (see the example in Section
5.3).

###  3.4 Gegenbauer Functions as Hessenbergians

We conclude this Section with an example. We show how the Gegengbauer
functions can be expressed as Hessenbergians. For a discussion of the
Gegenbauer processes and their applications to economics and finance see
Baillie (1996) and Dissanayake et al. (2018); see also, Giraitis and Leipus
(1995) and Caporale and Gil-Alana (2011).

###### Example 2

The Gegenbauer (or ultraspherical) functions, denoted by $c_{j}^{(d)}(\phi)$
(hereafter, for notational simplicity we use $c_{j}$), are defined to be the
coefficients in the power-series expansion of the following equation:

$(1-2\phi z+z^{2})^{-d}=\sum\limits_{j=0}^{\infty}c_{j}z^{j},$

for $\left|z\right|\leq 1$, $\left|\phi\right|\leq 1$, and $0<d<\frac{1}{2}$.

The easiest way to compute $c_{j}$ (for $j\geq 2$) using computers is based on
the solution of the following second order homogeneous linear difference
equation with coefficients functions of $j$

$c_{j}=2\phi\left(\frac{d-1}{j}+1\right)c_{j-1}-\left(2\frac{d-1}{j}+1\right)c_{j-2}\
\ \mathrm{for}\ j\geq 2,$ (17)

subject to the initial condition values $c_{0}=1$ and $c_{1}=2\phi d$ (see,
for example, Baillie, 1996, Chung, 1996, and the references therein). The
solution $c_{j}$ of eq. (17) is given in the following Proposition. This
follows from eq. (11) (or the homogeneous solution part of Theorem 1) applied
with $t=j$, $s=1$,
$\phi_{1}(j)=2\phi(\frac{d-1}{j}+1),\phi_{2}(j)=-(2\frac{d-1}{j}+1)$ (e.g.
$\phi_{1}(2)=\phi(d+1),\phi_{1}(3)=2\phi\frac{d+2}{3},\phi_{2}(3)=-\frac{2d+1}{3}$).
Besides $y_{j,1}^{hom}$ is identified with $c_{j}$.

###### Proposition 2

The $j$-th (for $j\geq 2$) Gegenbauer coefficient is given by
$c_{j}=\xi(j,1)c_{1}+\xi^{(2)}(j,1)c_{0}$ or equivalently (by replacing
$c_{0}=1$ and $c_{1}=2\phi d$):

$c_{j}=\xi(j,1)2\phi d+\xi^{(2)}(j,1),$ (18)

where $\xi(j,1)$ is a $(j-1)$-th order banded Hessenbergian (in this case
tridiagonal matrix):

$\xi(j,1)=\left|\begin{array}[]{cccccc}\phi(d+1)&-1&&&&\vspace{0.02in}\hfil\\\
-\frac{2d+1}{3}&2\phi\frac{d+2}{3}&-1&&&\vspace{0.07in}\hfil\\\
&-\frac{d+1}{2}&\phi\frac{d+3}{2}&-1&&\\\ &&\ddots&\ddots&\ddots&\\\
&&&-\left(2\frac{d-1}{j-1}+1\right)&2\phi\left(\frac{d-1}{j-1}+1\right)&-1\vspace{0.05in}\\\
&&&&-\left(2\frac{d-1}{j}+1\right)&2\phi\left(\frac{d-1}{j}+1\right)\end{array}\right|.$

Recall that $\xi^{(2)}(j,1)$ differs from $\xi(j,1)$ only in the first two
elements of the first column (see eq. (LABEL:Def1:_Phi^m_k)), which, in this
case, are: $-d$ (or $\phi_{2}(2)$) and $0$, respectively. It follows from eq.
(10) that $\xi^{(2)}(j,1)=-d\xi(j,2)$. Thus eq. (18) can be rewritten as:
$c_{j}=\xi(j,1)2\phi d-d\xi(j,2)$. Next, we apply the multi-linearity property
of determinants along the first column to express $c_{j}$ in eq. (18) as a
single Hessenbergian:

###### Corollary 1

The $j$-th Gegenbauer polynomial can be explicitly expressed as a $(j-1)$-th
order banded Hessenbergian:

$c_{j}=\left|\begin{array}[]{cccccc}d[2\phi^{2}(d+1)-1]&-1&&&&\vspace{0.02in}\hfil\\\
-2\phi d\frac{2d+1}{3}&2\phi\frac{d+2}{3}&-1&&&\vspace{0.05in}\hfil\\\
&-\frac{d+1}{2}&\phi\frac{d+3}{2}&-1&&\\\ &&\ddots&\ddots&\ddots&\\\
&&&-\left(2\frac{d-1}{j-1}+1\right)&2\phi\left(\frac{d-1}{j-1}+1\right)&-1\\\
&&&&-\left(2\frac{d-1}{j}+1\right)&2\phi\left(\frac{d-1}{j}+1\right)\end{array}\right|.$

##  4 Asymptotic Stability

In this Section we apply our result in eq. (16) to provide conditions ensuring
the asymptotic stability888As pointed out by Grillenzoni (2000) stability is a
useful feature of stochastic models because it is a sufficient (although not
necessary) condition for optimal properties of parameter estimates and
forecasts. Since model (1) can be expressed in Markovian form, the stability
condition allows many other stability properties, such as irreducibility,
recurrence, regularity, non evanescence and tightness (see Grillenzoni, 2000
for details). of a family of TV-ARMA($p,q$) processes satisfying eq. (1).

###  4.1 Stability Conditions

The asymptotic stability problem is to provide sufficient conditions such that
a class of stochastic processes solving eq. (1) approaches a solution
independently of the prescribed $p$ random variables (the effect of the
prescribed random variables is gradually dying out) as $s\rightarrow-\infty$,
that is when the homogeneous solution in eq. (16) tends to zero, under a
prescribed type of convergence. The explicit representation of the homogeneous
solution in eq. (11) makes it possible to provide such type of conditions in
Theorem 2 ensuring the $L_{2}$ convergence to zero of the homogeneous
solution, that is $y^{hom}_{t,s}\overset{L_{2}}{\to}0$, as
$s\rightarrow-\infty$, which means that
$\lim_{s\to-\infty}||y^{hom}_{t,s}||_{L_{2}}=0$, or equivalently that
$\lim_{s\rightarrow-\infty}\mathbb{E}\big{(}y^{hom}_{t,s}\big{)}^{2}=0$.

###### Theorem 2

i) Let the autoregressive coefficients $\phi_{m}(t)$ be deterministic. If
$\sup_{t}{|\phi_{m}(t)}|<\infty$ for each $m$ with $1\leq m\leq p$, then a
sufficient condition for an $L_{2}$-bounded stochastic process $y_{t}$ (that
is $\sup_{t}\mathbb{E}(y^{2}_{t})<\infty$) which solves eq. (1) to be
asymptotically stable (in $L_{2}$ sense) is:
$\lim_{s\rightarrow-\infty}\xi(t,s)=0$ for each $t$.
ii) Let the autoregressive coefficients $\phi_{m}(t)$ be stochastic. If
$\sup_{t}\mathbb{E}(\phi_{m}^{2}(t))<\infty$ for each $m$ with $1\leq m\leq
p$, then a sufficient condition for an $L_{2}$-bounded stochastic process
$y_{t}$, which solves eq. (1), to be asymptotically stable (in $L_{2}$ sense)
is $\xi(t,s)\overset{L_{2}}{\rightarrow}0$, as $s\rightarrow-\infty$ for each
$t$ (see, for details, Proposition 9 in Section 7.4).999Goldie and Maller
(2000) offered sufficient conditions for the asymptotic stability of an
AR($1$) model with stochastically varying coefficients, that is the a.s.
convergence of the solution, i.e.
$\sum\nolimits_{t=1}^{\infty}\phi_{1}(1)\phi_{1}(2)\cdots\phi_{1}(t-1)\varepsilon_{t}<\infty$,
a.s. (see also Bougerol and Picard, 1992). Recently, in a multivariate
setting, Erhardsson (2014) showed that the only sufficient condition is:
$\left|\phi_{1}(1)\phi_{1}(2)\cdots\phi_{1}(t)\right|\overset{a.s.}{\rightarrow}0$
as $t\rightarrow\infty$, a result analogous to Theorem 2 (ii) (see also
Section 7 below).

In the next Section, the asymptotic stability plays an essential role for the
existence and uniqueness of TV-ARMA($p,q$) solution processes in $L_{2}$.

The conditions in Theorem 2(ii) include the “bounded random walk” of Giraitis
et al. (2014) also used by Petrova (2019). Properties such as stability
characterize the statistical properties ($\sqrt{T}$ convergence and asymptotic
normality, where $T$ is the sample size) of least squares (LS) and quasi-
maximum likelihood (QML) estimators of the time-varying
coefficients.101010Azrak and Mélard (2006) have considered the asymptotic
properties of quasi maximum likelihood estimators for a large class of ARMA
models with time dependent coefficients and heteroscedastic innovations. The
coefficients and the variance are assumed to be deterministic functions of
time, and depend on a finite number of parameters which need to be estimated.
Other researchers have also considered the statistical properties of maximum
likelihood estimators for very general non stationary models. For example,
Dahlhaus (1997) has obtained asymptotic results for a new class of locally
stationary processes, which includes TV-ARMA processes (see Azrak and Mélard,
2006, and the references therein).

In the time invariant case since $\xi(t,s)$ depends neither on $t$ nor on $s$
but only on their difference, that is the forecasting horizon, $k$, the
stability condition in 2(i) reduces to $\lim_{k\rightarrow\infty}\xi_{k}=0$,
which holds if and only if all the roots $\lambda_{m}$ in eq. (8) lie inside
the unit circle.

###  4.2 Two Illustrative Examples

The first of the following examples concerns the logistic smooth transition
AR($1$) model (see, for example, Teräsvirta, 1994). The second example
concerns periodic AR models.

###### Example 3

In the first model the autoregressive coefficient is given by :
$\phi(t)=\phi_{1}F(t;\gamma,\tau)+[1-F(t;\gamma,\tau)]\phi_{2}$ (we drop the
subscript $1$), where $F(t;\gamma,\tau)=[1+e^{\gamma(t-\tau)}]^{-1}$,
$\gamma\in\mathbb{R}_{\geq 0}$, $\tau\in\mathbb{Z}$, is the first-order
logistic function. Clearly, if $t>\tau$, then $F((t;\gamma,\tau))<0.5$ and
regime $2$ prevails, whereas if $t<\tau$, then $F(t;\gamma,\tau)>0.5$ and
regime $1$ prevails. Let also $t_{2}$ be the value of $t$ for which
$F(t;\gamma,\tau)=0$ if $t\geq t_{2}$ and thus $\phi(t)=\phi_{2}$. Similarly,
let $t_{1}$ be the value of $t$ for which $F(t;\gamma,\tau)=1$ if $t\leq
t_{1}$ and thus $\phi(t)=\phi_{1}$ (clearly $t_{2}>t_{1}$, since $F(\cdot)$ is
a decreasing function of time). For this model, if $t\geq t_{2}$ and $s\leq
t_{1}$ then
$\xi(t,s)=\phi_{1}^{t_{1}-s+1}\prod\nolimits_{r=t_{1}+1}^{t_{2}-1}\phi(r)\phi_{2}^{t-t_{2}+1}$.
Clearly, $\lim_{s\rightarrow-\infty}\xi(t,s)=0$ if and only if
$\left|\phi_{1}\right|<1$, whereas $\lim_{t\rightarrow\infty}\xi(t,s)=0$ if
and only if $\left|\phi_{2}\right|<1$.

###### Example 4

In the periodic AR($1;\ell$) model, $\ell\in\mathbb{Z}_{\geq 1}$ is the number
of seasons (i.e., quarters) and $\phi_{r}$, $r=1,\ldots,\ell$ denote the
periodically varying autoregressive coefficients. Moreover,
$T\in\mathbb{Z}_{\geq 0}$ stands for the number of periods (i.e., years).
Accordingly $t=T\ell+r$ is time expressed in terms of seasons (i.e., if
$\ell=4$, $r=4$ and $T=1$, then $t=8$ quarters). If we want to forecast
$k\ell$ seasons ahead, that is $t-s=k\ell$ or $s=t-k\ell$, then:
$\xi(t,s)=[\prod\nolimits_{r=1}^{\ell}(\phi_{r})]^{k}$. Clearly
$\left|\phi_{r}\right|<1$ for all $r$ is a sufficient but not necessary
condition for $\lim_{s\rightarrow-\infty}\xi(t,s)=0$ (or equivalently
$\lim_{k\rightarrow\infty}\xi(t,s)=0$). The necessary and sufficient condition
is $\left|\prod\nolimits_{r=1}^{\ell}(\phi_{r})\right|<1$.111111For a study of
the periodic stationarity of a random coefficient periodic autoregression
(RCPAR) see, for example, Aknouche and Guerbyenne (2009).

##  5 Solution Representation in $L_{2}$

Having specified a general method for manipulating explicitly the TV-ARMA type
of models, we turn our attention to a consideration of their Wold-Cramér
decomposition followed by the fundamental second order properties of these
models. In this Section we shall restrict ourselves to the treatment of DTV-
ARMA processes (those with deterministic coefficients). In Section 7 we
present STV-AR processes (those with stochastically varying coefficients),
which incorporate the GRC and double stochastic AR models.

###  5.1 Wold-Cramér Decomposition

In Theorem 3, we provide the existence of the Wold-Cramér decomposition (see
Cramér, 1961)121212Since a non-stationary generalization of Wold’s result was
given by Cramér, it is referred to as Wold-Cramér decomposition. and,
therefore, impulse response functions (IRFs), for the DTV version of the model
in eq. (1). In particular, we provide an explicit condition (the absolute
summability of the principal determinant):

$\sum_{r=-\infty}^{t}|\xi(t,r)|<\infty,\ \text{for\ all}\ t,\ \ \ \ \
(\text{absolute\ summability\ condition})$ (19)

which, along with the boundedness of the drift and the moving average
coefficients, ensure the existence of the Wold-Cramér decomposition (Theorem
3) of DTV-ARMA models with finite first two unconditional moments and
autocovariance function (Propositions 3, 4). In Theorem 4 it is shown that
every asymptotically stable stochastic process, which solves eq. (1) coincides
with the Wold-Cramér decomposition in eq. (20).

As the absolute summability condition in (19) also guarantees the absolute
summability of $\xi_{q}(t,r)$ in eq. (13) the following Theorem holds:

######  Theorem 3 (Existence)

Let the absolute summability condition in (19) hold. Let also $\varphi(t)$ and
$\theta_{l}(t)$ be bounded functions in $t$. Then there exists a solution of
eq. (1) in $L_{2}$ of the form

$y_{t}=\sum_{r=-\infty}^{t}\xi(t,r)\varphi(r)+\displaystyle\sum_{r=-\infty}^{t}\xi_{q}(t,r)\varepsilon_{r},$
(20)

that is an explicit representation of the Wold-Cramér solution decomposition.

A direct proof of Theorem 3 is given in the Appendix ensuring that $y_{t}$ in
eq. (20) is second order, that is $y_{t}\in L_{2}$. In particular, show there
that $\sup_{t}\mathbb{E}(\upsilon^{2}_{t})<\infty$ (for the definition of
$\upsilon_{t}$ see eq. (2)), which extends a result established by Peiris
(1986) (see the proof of his Theorem 2.1, provided that $\varphi(t)=0$). In
addition, we deduce the equivalence between
$\sum_{r=-\infty}^{t}\xi^{2}_{q}(t,r)<\infty$ and $y_{t}\in L_{2}$, provided
that $\sum_{r=-\infty}^{t}\xi(t,r)\varphi(r)$ converges in $\mathbb{R}$ (see
Proposition G1, in the Online Appendix G2). This extends a result mentioned in
the above cited reference, that is $y_{t}$ is second order if and only if eq.
“(2.4)” holds true, provided that $\varphi(t)=0$. The solution of eq. (1) in
eq. (20) is decomposed into two orthogonal parts (see for a proof the
Additional Appendix L1), a deterministic part and a mean zero random part,
that is, $\mathbb{E}(y_{t})=\sum_{r=-\infty}^{t}\xi(t,r)\varphi(r)$ is the non
random part (see Proposition 3 below), while
$\lim_{s\to-\infty}\mathbb{FE}_{t,s}=\sum_{r=-\infty}^{t}\xi_{q}(t,r)\varepsilon_{r}$
(i.e., the limit of forecast errors, see eqs. (31) below) is the mean zero
random part of $y_{t}$. As eq. (20) is future independent, we shall also
referred to it as a causal solution of DTV-ARMA models.

Hallin (1979), Singh and Peiris (1987), Kowalski and Szynal (1991),
Grillenzoni (2000), and Azrak and Mélard (2006) evaluate the Wold-Cramér
decomposition through recursion. In sharp contrast, eq. (20) in Theorem 3
provides a direct and explicit representation of the Wold-Cramér decomposition
expressed in terms of banded Hessenbergians with entries the coefficients of
the model in eq. (1). Moreover, in the Appendix we show that the Wold-Cramér
decomposition in eq. (20) can be expressed in a compact form:

$y_{t}=\sum_{r=-\infty}^{t}\xi(t,r)\upsilon_{r}.$

Let $\varphi(r)=0$ and $\\{\varepsilon_{t}\\}_{t}$ be an orthonormal sequence
in $L_{2}$, that is $\mathbb{E}(\varepsilon_{t}\varepsilon_{s})=\delta_{t,s}$
(Kroneker delta). Under these additional assumptions to those considered in
Theorem 3, the latter recovers the following uniqueness result: Eq. (20) is
the unique solution of eq. (1) (see Hallin and Ingenbleek, 1983, and Peiris,
1986). In the Online Appendix G1 we apply eq. (20) to an AR(1) model,
recovering the well known Wold solution representation of the latter.

In the Appendix we show that eq. (20) can be alternatively derived from eq.
(16), when the latter is applied with prescribed random variables

$y_{s+1-m}=\sum_{r=-\infty}^{s+1-m}\xi(s+1-m,r)\upsilon_{r}\ \ \ \text{for}\ \
\ 1\leq m\leq p,$

for any arbitrary $s$ such that $s<t$. This result arises from the formulas

$\left.\begin{array}[]{l}\displaystyle\lim_{s\rightarrow-\infty}\sum_{m=1}^{p}\xi^{(m)}(t,s)\displaystyle\sum_{r=-\infty}^{s+1-m}\xi(s+1-m,r)\upsilon_{r}\overset{L_{2}}{=}0,\\\
\\\
\displaystyle\lim_{s\rightarrow-\infty}\displaystyle\sum_{r=s+1-q}^{s}\xi_{s,q}(t,r)\varepsilon_{r}\overset{L_{2}}{=}0\end{array}\right\\}$
(21)

as follows: Applying eqs. (21) to eq. (16), it follows from eq. (12) that eq.
(20) can be equivalently rewritten as:
$y_{t}=\lim_{s\rightarrow-\infty}y_{t,s}^{par}$. The first of the formulas in
(21) ensures that the Wold-Cramér solution decomposition in eq. (20) is
asymptotically stable. Moreover, in the Appendix we show that every
asymptotically stable stochastic process in $L_{2}$, which solves eq. (1)
coincides almost surely with $y_{t}$ in eq. (20), as demonstrated in the
following Theorem:

######  Theorem 4 (Uniqueness)

Let the conditions of Theorem 3 hold. Then there exists a uniquely determined
asymptotically stable stochastic process in $L_{2}$, which solves eq. (1) that
coincides almost surely with $y_{t}$ in eq. (20).

Assuming in addition to the conditions of Theorem 4 that
$\sup_{t}{|\phi_{m}(t)|}<\infty$ for each $m$ with $1\leq m\leq p$ and taking
into account that the absolute summability condition in (19) implies that
$\lim_{s\to-\infty}\xi(t,s)=0$, we infer from Theorem 2(i) that every bounded
process that solves eq. (1), as being asymptotically stable, also coincides
with the Wold-Cramér decomposition in eq. (20). The existence of such a
bounded process is guaranteed by the boundedness of the absolute summability,
as a function of $t$, that is

$\sum_{r=-\infty}^{t}|\xi(t,r)|\leq C<\infty,\ \text{for\ all}\ t\ \
\text{and\ some}\ \ C\in\mathbb{R}_{>1}.$ (22)

We show in the following Corollary that under the condition (22), the Wold-
Cramér solution decomposition in eq. (20) is bounded. Moreover, as (22)
guarantees the conditions of Theorem 2(i) (see for a proof the Proposition L1
in the Additional Appendix L2), it arises from Theorem 4 that the following
uniqueness result holds:

###### Corollary 2

Let the conditions of Theorem 3 hold. Let also (22) holds. Then there exists a
uniquely determined $L_{2}$-bounded stochastic process, which solves eq. (1)
and coincides almost surely with $y_{t}$ in eq. (20).

A proof of these results is included in Appendix. Corollary 2 must be compared
with an analogous result established by Neimi (Theorem 2.1, 1983) for ARMA
non-stationary processes with constant coefficients coupled with the AR-
regularity condition and zero drift ($\varphi(t)=0$). An extension of this
result covering non-stationary TV-ARMA($p,q$) models coupled with zero drift
and extended regularity conditions was obtained by Kowalski and Szynal (1988).

Kowalski and Szynal (1991) and Grillenzoni (2000) derived sufficient
conditions ensuring that $y_{t}$ is second order, that is $y_{t}\in L_{2}$
(Kowalski and Szynal examined the case of deterministic coefficients with zero
drift, whereas Grillenzoni allowed the coefficients to be stochastic as well,
see also Remark 5 in Section 7.4). These conditions for a TV-AR($p$) model
with zero drift are summarized in the following Remark.

###### Remark 3

The above mentioned sufficient conditions for $y_{t}\in L_{2}$ are as follows:
i) The deterministically varying polynomial
$\Phi_{t}(z^{-1})=1-\sum_{m=1}^{p}\phi_{m}(t)z^{-m}$ is regular. That is,
$\phi_{m}(t)$ are such that there exist the limits
$\lim_{t\rightarrow\infty}\phi_{m}(t)=\phi_{m}$ and
$\sum\nolimits_{r=1}^{\infty}\varrho^{2r}<\infty$, where
$\varrho=\varrho(\Phi)+\epsilon$, $\epsilon>0$,
$\varrho(\Phi)=\max\\{\left|z_{m}\right|,$ $\Phi(z_{m}^{-1})=0\\}$ with
$\Phi(z^{-1})=1-\sum_{m=1}^{p}\phi_{m}z^{-m}$ (see eq. (8) in Kowalski and
Szynal, 1991).131313Kowalski and Szynal (1991) showed that $\varrho(\Phi)$ is
the spectral radius of the matrix
$\mathbf{\Phi=}\left[\begin{array}[]{ccccc}0&0&\cdots&0&-\phi_{m}\\\
1&0&\cdots&0&-\phi_{m-1}\\\ 0&1&\cdots&0&-\phi_{m-2}\\\
\vdots&\vdots&\vdots\vdots\vdots&\vdots&\vdots\\\
0&0&\cdots&1&-\phi_{1}\vspace{-0.1in}\end{array}\right]$
(see page 75 in their paper). ii) The deterministically varying polynomial
$\Phi_{t}(z^{-1})$ should have roots that entirely lie inside the unit circle,
with the exception, at most, of a finite set of points (see Proposition 1 in
Grillenzoni, 2000).

The sufficient conditions in Remark 3 are not, however, necessary, since they
do not cover the case of periodic coefficients, see Grillenzoni (1990) or
Karanasos et al. (2014,a,b). Whereas in the case of TV-AR($p$) models
$\xi_{q}(t,r)=\xi(t,r)$, we infer from the discussion next to Theorem 3 (see
also Proposition G.2 in the Online Appendix G) the equivalence between
$\sum_{r=-\infty}^{t}\xi^{2}(t,r)<\infty$ (square summability) and $y_{t}\in
L_{2}$, provided that $\varphi(t)=0$. As a consequence, the stability
condition $\lim_{s\rightarrow-\infty}\xi(t,s)=0$ is necessary for the square
summability to hold or equivalently for $y_{t}\in L_{2}$ and therefore it is
also necessary for the two conditions in Remark 3, as demonstrated below:

$\\{\text{The Sufficient Conditions in Remark \ref{KS&G}}\\}\implies y_{t}\in
L_{2}\iff\sum_{r=-\infty}^{t}\xi^{2}(t,r)<\infty\implies\lim_{s\rightarrow-\infty}\xi(t,s)=0.$

###  5.2 Unconditional Moments

In this Subsection we present explicit formulae for the first and second
unconditional moments of the Wold-Cramér solution decomposition of the DTV-
ARMA family of processes coupled with sufficient and necessary conditions for
their existence, as demonstrates the following Proposition.

###### Proposition 3

Let the conditions of Theorem 3 hold. Then the unconditional mean of the
process $y_{t}$ in eq. (20) with deterministic coefficients, exists in
$\mathbb{R}$ and is given by

$\mathbb{E}(y_{t})=\sum\limits_{r=-\infty}^{t}\xi(t,r)\varphi(r).$ (23)

The absolute summability condition is also sufficient for the existence of the
unconditional variance of the above mentioned process $y_{t}$, which is given
by

$\mathbb{V}ar(y_{t})=\sum_{r=-\infty}^{t}\xi_{q}^{2}(t,r)\mathbb{\sigma}^{2}(r).$
(24)

Necessary conditions for the $y_{t}$ process to be second order respectively
are:

$\lim_{s\rightarrow-\infty}\xi(t,s)\varphi(s)=0\ \mathrm{and}\
\lim_{s\rightarrow-\infty}\xi_{q}^{2}(t,s)\sigma^{2}(s)=0\ \mathrm{for\ all}\
t.$

Moreover, the stability condition, that is
$\lim_{s\rightarrow-\infty}\xi(t,s)=0$, is sufficient for the above two limits
to exist, due to the boundedness of $\varphi(r)$ and $\sigma^{2}(r)$, while it
is necessary for the absolute summability to hold.

A proof of Proposition 3 is provided in Appendix, noticing that the
unconditional mean $\mathbb{E}(y_{t})$ is the same for both the AR and the
ARMA processes.

The main logical connections between the conditions, described in the above
Proposition, are summarized in the following commutative diagrams:

$\begin{array}[]{ccccc}\displaystyle\sum\limits_{r=-\infty}^{t}\xi(t,r)\varphi(r)\in\mathbb{R}&\Longleftarrow&\displaystyle\sum_{r=-\infty}^{t}|\xi(t,r)|<\infty&\Longrightarrow&\displaystyle\sum_{r=-\infty}^{t}\xi_{q}^{2}(t,r)\mathbb{\sigma}^{2}(r)<\infty\vspace{-0.1in}\\\
\rotatebox{-90.0}{$\Longrightarrow$}&\rotatebox{-135.0}{$\Longrightarrow$}&\rotatebox{-90.0}{$\Longrightarrow$}&\rotatebox{-45.0}{$\Longrightarrow$}&\rotatebox{-90.0}{$\Longrightarrow$}\vspace{0.1in}\\\
\lim\limits_{s\rightarrow-\infty}\xi(t,s)\varphi(s)=0&\Longleftarrow&\lim\limits_{s\rightarrow-\infty}\xi(t,s)=0&\Longrightarrow&\lim\limits_{s\rightarrow-\infty}\xi_{q}^{2}(t,s)\sigma^{2}(s)=0.\end{array}$
(25) Figure 1: Commutative Diagrams

###  5.3 Autocovariance Function

In the following Proposition, we state an explicit expression for the
covariance structure for the Wold-Cramér solution decomposition of the DTV-
ARMA($p,q$) process.

###### Proposition 4

Let the conditions of Theorem 3 hold. Then the time varying $\ell$-order
autocovariance function $\gamma_{t}(\ell)=\mathbb{C}ov(y_{t},y_{t-\ell})$,
$\ell\in\mathbb{Z}_{\geq 0}$, of $y_{t}$ in eq. (20), exists in $\mathbb{R}$
and is given by

$\gamma_{t}(\ell)=\sum_{r=-\infty}^{t-\ell}\xi_{q}(t,r)\xi_{q}(t-\ell,r)\mathbb{\sigma}^{2}(r).$
(26)

A proof of Proposition 4 is found in the Appendix. The time-varying variance
of $y_{t}$ in eq. (24), is recovered by applying $\gamma_{t}(\ell)$ for
$\ell=0$, that is $\gamma_{t}(0)=\mathbb{V}ar(y_{t})$. Moreover, the absolute
summability condition implies absolute summable autocovariances:
$\sum\nolimits_{\ell=0}^{\infty}\left|\gamma_{t}(\ell)\right|<\infty$ for all
$t$. Notice again that for a pure AR process, the autocovariance function
formula in eq. (26) must be applied for $\xi_{0}(t,r)=\xi(t,r)$.

From a computational viewpoint, the covariance structure of $\\{y_{t}\\}_{t}$
can be numerically evaluated by computing the banded Hessenbergian
coefficients, $\xi_{q}(t,r)$ in eq. (13) and substituting these in eq. (26).

The next remark highlights the importance of the existence of finite second
moments.

###### Remark 4

Azrak and Mélard (2006) considered the asymptotic properties of QML estimators
for the DTV-ARMA family of models where the coefficients depend not only on
$t$ but on $T$ as well (see Alj et al., 2017, for the multivariate case). In
their Theorem and Lemma $1$ the existence of finite second moments was
required. They also show that the dependence of the model with respect to $T$
has no substantial effect on their conclusions except that a.s. convergence is
replaced by convergence in probability, since convergence in $L_{2}$ norm
implies convergence in probability (see Lemma $1^{\prime}$ in their paper).

We conclude this Section with two more examples and a discussion of forward
asymptotic stability.

#### Two More Examples:

The following examples, concerning AR($1$) processes with variable
autoregressive coefficients, are taken from Azrak and Mélard (2006).

###### Example 5

In the first example, the autoregressive coefficient is a periodic function of
time defined by

$y_{t}=\phi(t)y_{t-1}+\varepsilon_{t},$

where $\varepsilon_{t}$ is a martingale difference sequence defined on $L_{2}$
with constant variance $\sigma^{2}$. Moreover, the autoregressive coefficient
is given by $\phi(t)=\beta_{t-n[t/n]}$, where $n\in\mathbb{Z}_{\geq 2}$ and
$[x]$ is the larger integer less or equal to $x$ (see also Dahlhaus, 1996). By
specializing the results of Theorem 3 and Proposition 3, the Wold-Crámer
decomposition (if and only if $\left|\beta\right|<1$, where
$\beta=\beta_{0}\cdot\beta_{1}\cdots\cdot\beta_{n-1}$) is given by

$y_{t}=\sum\limits_{r=-\infty}^{t}\xi(t,r)\varepsilon_{r},$

with unconditional variance

$\mathbb{V}ar(y_{t})=\sigma^{2}\sum\limits_{r=-\infty}^{t}\xi^{2}(t,r),$

where

$\xi(t,r)=\beta^{[\frac{t-r}{n}]}\left(\prod\limits_{j=0}^{t-r-1-n[\frac{t-r}{n}]}\beta_{t-j}-n\left[\frac{t-j}{n}\right]\right),$

and, therefore

$\sum\limits_{r=-\infty}^{t}\xi^{2}(t,r)=\frac{1}{1-\beta^{2}}\sum\limits_{r=t-n+1}^{t}\left(\prod\limits_{j=0}^{t-r-1-n[\frac{t-r}{n}]}\beta_{t-j}-n\left[\frac{t-j}{n}\right]\right)$

(see also eq. (4.2) in Azrak and Mélard, 2006).

###### Example 6

In the second example (see Example 2 in Azrak and Mélard, 2006), the
autoregressive coefficient is an exponential function of time given by

$\phi(t)=\left\\{\begin{array}[]{lll}\phi&\text{for}&t\leq 0,\\\
\phi\lambda^{t/T}&\text{for}&t=1,\ldots,T-1,\\\ \phi\lambda&\text{for}&t\geq
T,\end{array}\right.$

where $T\in\mathbb{Z}_{\geq 1}$ is the sample size. For this case (assuming
that $t>T$)

$\xi(t,r)=\left\\{\begin{array}[]{lll}(\phi\lambda)^{t-r}&\text{for}&r\in[T,t],\\\
(\phi\lambda)^{t}\phi^{-r}\lambda^{-\left(\frac{T+1}{2}+\frac{r(r-1)}{2T}\right)}&\text{for}&r=1,\ldots,T-1,\\\
\phi^{1-r}\xi(t,1)&\text{for}&r\leq 0.\end{array}\right.$

The condition $\left|\phi\right|<1$ (necessary and sufficient) entails:

$\sum\limits_{r=-\infty}^{t}\xi^{2}(t,r)=\frac{1}{1-\phi^{2}}\xi^{2}(t,1)+(\phi\lambda)^{2t}\sum\limits_{r=1}^{T-1}\phi^{-2r}\lambda^{-(T+1+\frac{r(r-1)}{T})}+\frac{1-(\phi\lambda)^{2(t-T+1)}}{1-(\phi\lambda)^{2}}.$

As pointed out by Azrak and Mélard (2006) the use of variable coefficients,
which depend on the length of the series, is compatible with the approach of
Dahlhaus (1997).

##  6 Forecasting and Asymptotic Efficiency

In this Section we provide explicitly a sufficient condition for the
invertibility of a causal DTV-ARMA$(p,q)$ process. In conjunction with Theorem
3, the above mentioned condition makes it possible to obtain in Subsection 6.2
an explicit form to the $k$-step-ahead optimal (in $L_{2}$ sense) linear
predictor along with the associated MSE (mean square error). An alternative
optimal forecast is based on a finite sequence of observable random variables,
employing for that the representation given by eq. (16). Both approaches yield
identical forecast and mean square errors. In this context, in Subsection 6.3
we discuss, in a unified scheme, the forward asymptotic efficiency of the
model with deterministic coefficients given by eq. (1).

###  6.1 Invertibility

A DTV-ARMA$(p,q)$ process is invertible if and only if the current value of
the input $\varepsilon_{t}$ can be expressed as a (converging) linear
combination of the present and past random variables $y_{r}$ ($r\leq t$) (see
Brockwell and Davis, 2016, p.76). The main result of this Section is presented
in Theorem 5 (see below).

Eq. (1) can be rewritten as

$\varepsilon_{t}=y_{t}-\varphi(t)-\sum\limits_{l=1}^{q}\theta_{l}(t)\varepsilon_{t-l}-\sum\limits_{m=1}^{p}\phi_{m}(t)y_{t-m}.$
(27)

The principal matrix associated with the moving average part is defined by

$\mathbf{\Theta}(t,s)\\!=\\!\\!\left[\\!\\!\begin{array}[]{cccccccc}-{\footnotesize\theta}_{1}{\footnotesize(s+1)}&{\footnotesize-1}&&&&&&\\\
-{\footnotesize\theta}_{2}{\footnotesize(s+2)}&-{\footnotesize\theta}_{1}{\footnotesize(s+2)}&\ddots&&&&&\\\
{\footnotesize\vdots}&{\footnotesize\vdots}&\ddots&\ddots&&&&\\\
-{\footnotesize\theta}_{q}{\footnotesize(s+q)}&-{\footnotesize\theta}_{q-1}{\footnotesize(s+q)}&\ddots&\ddots&\ddots&&&\\\
&-{\footnotesize\theta}_{q}{\footnotesize(s+q+1)}&\ddots&\ddots&\ddots&\ddots&&\\\
&&{\small\ddots}&{\small\ddots}&{\small\ddots}&{\small\ddots}&\ddots&\\\
&&&-{\footnotesize\theta}_{q}{\footnotesize(t-1)}&{\footnotesize\theta}_{q-1}{\footnotesize(t-1)}&{\small\cdots}&-{\footnotesize\theta}_{1}{\footnotesize(t-1)}&{\footnotesize-1}\\\
&&&&-{\footnotesize\theta}_{q}{\footnotesize(t)}&{\small\cdots}&-{\footnotesize\theta}_{2}{\footnotesize(t)}&-{\footnotesize\theta}_{1}{\footnotesize(t)}\end{array}\\!\\!\right]$
(28)

(for $k=t-s\geq 1$). The matrix $\mathbf{\Theta}(t,s)$ has a similar structure
to the principal matrix associated with the AR operator, $\mathbf{\Phi}(t,s)$,
that is both matrices are banded lower Hessenberg of order $k$. It is clear
that for $q\geq t-s$, $\boldsymbol{\Theta}(t,s)$ is a full lower Hessenberg
matrix.

For every pair $(t,s)\in\mathbb{Z}^{2}$ with $t-s\geq 1$ we define the
principal determinant associated with eq. (28):

$\vartheta(t,s)=\det(\mathbf{\Theta}(t,s)).$

Formally $\vartheta(t,s)$ (and similarly to $\xi(t,s)$), is a banded
Hessenbergian. We further extend the definition of $\vartheta(t,s)$ by
assigning the initial values: $\vartheta(s,s)=1$ and $\vartheta(t,s)=0$ for
$t<s$. Accordingly, $\vartheta(t,s)$ is the Green function associated with the
MA($q$) operator.

In analogy with the definition of $\xi_{q}(t,r)$ (see eq. (13)), we define:

$\vartheta_{p}(t,r)=\vartheta(t,r)-\sum\limits_{m=1}^{p}\vartheta(t,r+m)\phi_{m}(r+m).$

In the following Theorem we give a sufficient condition for a causal DTV-
ARMA($p,q$) process determined by eq. (20) to be invertible.

###### Theorem 5

Let the conditions in Theorem 3 hold and $\phi_{m}(t)$ be bounded in $t$. Let
additionally the following absolute summability condition
$\sum_{r=-\infty}^{t}|\vartheta(t,r)|<\infty$ hold for each $t$. Then the
process in eq. (20) is invertible, that is

$\varepsilon_{t}=\displaystyle\sum_{r=-\infty}^{t}\vartheta_{p}(t,r)y_{r}-\sum_{r=-\infty}^{t}\vartheta(t,r)\varphi(r)$
(29)

and solves eq. (27).

The proof of Theorem 5 essentially repeats the arguments of the proof of
Theorem 3; switching the roles of $y_{r}$ and $\varepsilon_{r}$, and replacing
$\xi(t,r)$ with $\vartheta(t,r)$, $\xi_{q}(t,r)$ with $\vartheta_{p}(t,r)$ and
$\varphi(r)$ with $-\varphi(r)$. This result is fulfilled by showing in the
Online Appendix H, that both terms in the right-hand side of eq. (29) are
processes in $L_{2}$. Eq. (29) recovers the formula devised by Hallin (1979)
in Theorem 3, but this time in a more explicit form.

Following laborious research work, the literature contains a diversity of
“time-varying” specifications of linear form whose main time series properties
either remain unexplored or have not been fully examined. Making progress in
interpreting seemingly different models we have provided in this Section a
common platform for the investigation of their time series properties. With
the help of a few detailed examples, i.e., smooth transition AR processes,
periodic and cyclical formulations, we have demonstrated how to encompass
various time series processes within our unified framework. The significance
of our methodology is almost self-evident from the large number of problems
that it can solve. Our proposed approach allows us to handle “time-varying”
models of infinite order, by introducing unbounded order linear difference
equation of index $p$. This type of equations also yield Hessenbergian
solutions, but involving full lower Hessenberg matrices. The latter enables us
to obtain easily handled explicit solutions for the infinite order case along
with the fundamental properties of corresponding models. An advantage of our
technique is that it can be applied with ease, that is without any major
alterations, in a multivariate setting and provides a solution to the problem
at hand without adding complexity.

###  6.2 Optimal Forecasts

Theorems 3 and 4 ensure the existence of a unique asymptotically stable
solution $y_{t}$ of eq. (1) given by eq. (20). Let $k=t-s$ for $s<t$ be the
forecasting horizon. Let also $\mathcal{M}_{s}$ be the smallest closed
subspace of $L_{2}$ based on the sequence of past observations
$\\{y_{s},y_{s-1},...\\}$, which also contains all constant functions (see
Brockwell and Davis, 1991, p. 62-64). Assuming further the invertibility
conditions in Theorem 5, as the terms of the sequence of errors
$\\{\varepsilon_{s},\varepsilon_{s-1}\ldots\\}$ can be expressed in terms of
$y$’s by eq. (29), they also belong to the space $\mathcal{M}_{s}$. The
optimal predictor of $y_{t}$ in eq. (20), i.e., its conditional expectation on
$\mathcal{M}_{s}$, coincides with its linear projection on the space spanned
by $\\{y_{s},y_{s-1},...\\}\cup\\{1\\}$, whenever $\\{\varepsilon_{r}\\}_{r}$
is a martingale difference sequence (see Brockwell and Davis, 2016, p. 334,
see also Rosenblatt, 2000, p. 83).

The conditional expectation of $y_{t}$ given $\mathcal{M}_{s}$, that is
$\mathbb{E}(y_{t}|\mathcal{M}_{s})$, turns out to be the orthogonal projection
of $y_{t}$ on $\mathcal{M}_{s}$, an explicit form of which is given in the
following Proposition.

###### Proposition 5

The $k$-step-ahead optimal (in $L_{2}$-sense) predictor of $y_{t}$, as stated
above, is linear and it is given by

$\mathbb{E}(y_{t}|\mathcal{M}_{s})=\sum_{r=-\infty}^{t}\xi(t,r)\varphi(r)+\displaystyle\sum_{r=-\infty}^{s}\xi_{q}(t,r)\varepsilon_{r},$
(30)

where $\varepsilon_{r}$ for $-\infty<r\leq s$ are given by eq. (29). The
forecast error associated with $\mathbb{E}(y_{t}|\mathcal{M}_{s})$, that is
$\mathbb{FE}_{t,s}\overset{\mathrm{def}}{=}y_{t}-\mathbb{E}(y_{t}\left|{}\right.\mathcal{M}_{s})$,
and the associated mean square error
$\mathbb{M}\mathbb{SE}_{t,s}\overset{\mathrm{def}}{=}\mathbb{V}ar(\mathbb{FE}_{t,s})$
are given by:

$\mathbb{FE}_{t,s}=\sum_{r=s+1}^{t}\xi_{q}(t,r)\varepsilon_{r},\
\mathbb{MSE}_{t,s}=\sum_{r=s+1}^{t}\xi_{q}^{2}(t,r)\mathbb{\sigma}^{2}(r).$
(31)

Besides, as the expectation of $\mathbb{FE}_{t,s}$ is zero, the forecast is
unbiased.

Theorem 5 entails that $\varepsilon_{r}\in\mathcal{M}_{s}$ for $r\leq s$.
Additionally, as $1\in\mathcal{M}_{s}$, it follows that
$\sum_{r=-\infty}^{t}\xi(t,r)\varphi(r)\in\mathcal{M}_{s}$. Thus, both terms
in the right-hand side of eq. (30) belong to $\mathcal{M}_{s}$. Using the
formulas in eqs. (30 and 31), one can conveniently verify that
$\mathbb{E}\big{(}\mathbb{FE}_{t,s}\
\mathbb{E}(y_{t}|\mathcal{M}_{s})\big{)}=0$ (a detailed proof is provided in
the Additional Appendix N). In other words $\mathbb{FE}_{t,s}$ and
$\mathbb{E}(y_{t}|\mathcal{M}_{s})$ are orthogonal (for short
$\mathbb{FE}_{t,s}\perp\mathbb{E}(y_{t}|\mathcal{M}_{s})$), verifying that the
predictor is optimal.

Singh and Peiris (1987), Kowalski and Szynal (1990, 1991), and Grillenzoni
(1990, 2000) obtained the evaluation of the optimal forecasts similar to that
described in eq. (30), using prediction algorithms and a recursive procedure.
We should also mention that Whittle (1965) showed that in general the linear
least-square predictor obeys a recursion (see his eq. (12)) and gave a
recursive method for obtaining its coefficients. In Section 7 we describe how
we can apply this technique in conjunction with our methodology to derive the
optimal forecasts when the coefficients are varying stochastically. In
particular, we examine the GRC-AR specification and a model with coefficients
that follows AR processes.

An alternative and more realistic approach takes advantage of the explicit
form of the process $y_{t}$, employed in the previous paragraph, but now given
by eq. (16). Let $\mathcal{K}_{s}$ be the smallest closed subspace of $L_{2}$
spanned by the set of $(p+q)$ past observations $\\{y_{s},y_{s-1},\ldots
y_{s+1-p}\\}\cup\\{\varepsilon_{s},\ldots,\varepsilon_{s+1-q}\\}$, which also
contains all constant functions. In a causal environment the $p$ past
observations of the process $y_{t}$ in eq. (20), below and including the time
point $s$, are realizations of the corresponding random variables, which are
given by: $y_{s+1-m}=\sum_{r=-\infty}^{s+1-m}\xi(s+1-m,r)\upsilon_{r}$ for
$1\leq m\leq p$. Applying the conditional expectation operator based on
$\mathcal{K}_{s}$ to eq. (16), the next Proposition follows directly:

###### Proposition 6

The $k$-step-ahead optimal predictor of $y_{t}$ based on $\mathcal{K}_{s}$ is
linear and is given by

$\mathbb{E}(y_{t}\left|\mathcal{K}_{s}\right.)=\sum_{m=1}^{p}\xi^{(m)}(t,s)y_{s+1-m}+\sum_{r=s+1}^{t}\xi(t,r)\varphi(r)+\sum\limits_{r=s+1-q}^{s}\xi_{s,q}(t,r)\varepsilon_{r}.$

The forecast error for the $k$-step-ahead predictor, that is
$\mathbb{FE}_{t,s}=y_{t}-\mathbb{E}(y_{t}\left|\right.\mathcal{K}_{s})$ and
its associated mean square error (its variance), coincide with those
established by eqs. (31), that is:

$\mathbb{FE}_{t,s}=\sum_{r=s+1}^{t}\xi_{q}(t,r)\varepsilon_{r},\
\mathbb{MSE}_{t,s}=\sum_{r=s+1}^{t}\xi_{q}^{2}(t,r)\mathbb{\sigma}^{2}(r).$
(32)

Besides, as the expectation of $\mathbb{FE}_{t,s}$ is zero, the forecast is
unbiased.

In the Additional Appendix N, we show that
$\mathbb{E}\big{(}\mathbb{FE}_{t,s}\
\mathbb{E}(y_{t}|\mathcal{K}_{s})\big{)}=0$ or equivalently that
$\mathbb{FE}_{t,s}\perp\mathbb{E}(y_{t}|\mathcal{K}_{s})$, which is a
prerequisite for the optimal nature of the predictors in $L_{2}$. As the
forecast errors in Propositions 5 and 6 coincide (in $L_{2}$ sense), we
conclude that:
$\mathbb{E}(y_{t}\left|\mathcal{K}_{s}\right.)=\mathbb{E}(y_{t}\left|\mathcal{M}_{s}\right.)$.
Moreover, in both cases the unconditional expectations and variances also
coincide with that found earlier in Proposition 3, that is:

$\lim_{s\rightarrow-\infty}\mathbb{E}(y_{t}\left|\mathcal{M}_{s}\right.)=\lim_{s\rightarrow-\infty}\mathbb{E}(y_{t}\left|\mathcal{K}_{s}\right.)=\sum\limits_{r=-\infty}^{t}\xi(t,r)\varphi(r)=\mathbb{E}(y_{t}),$
$\lim_{s\rightarrow-\infty}\mathbb{MS}\mathbb{E}_{t,s}=\sum_{r=-\infty}^{t}\xi_{q}^{2}(t,r)\sigma^{2}(r)=\mathbb{V}ar(y_{t}).$

We remark that in a causal and invertible environment (the conditions of
Theorem 5 hold) the past $q$ errors $\varepsilon_{r}$ for $s+1-q\leq r\leq s$
can be expressed in terms of $y$’s by eq. (29). In practice, eq. (29) can be
used to provide error estimates for
$\\{\varepsilon_{s},\ldots,\varepsilon_{s+1-q}\\}$ by replacing in eq. (29)
the $p$ realizations of the random variables: $\\{y_{s},y_{s-1},\ldots
y_{s+1-p}\\}$. Furthermore, we remark that the explicit form of the variance
for a Gaussian process is essential for the determination of the confidence
intervals of $\mathbb{E}(y_{t}\left|\mathcal{K}_{s}\right.)$.

Finally, we formulate one of the main arguments made by Hallin (1986), which
states that unlike the time invariant case, in a time-varying setting two MSEs
with the same forecasting horizon, but at different time points, are no longer
equal. With this in mind, consider two pairs of time points, say
$(t_{1},s_{1})$ and $(t_{2},s_{2})$, such that $t_{1}-s_{1}=t_{2}-s_{2}=k$.
The MSEs associated with these two time points are:

$\mathbb{MSE}_{t_{1},s_{1}}=\sum_{r=s_{1}+1}^{t_{1}}\xi_{q}^{2}(t_{1},r)\mathbb{\sigma}^{2}(r),\
\ \
\mathbb{MSE}_{t_{2},s_{2}}=\sum_{r=s_{2}+1}^{t_{2}}\xi_{q}^{2}(t_{2},r)\mathbb{\sigma}^{2}(r).$

Shifting the time intervals determined by the summation limits, that is
($[s_{1}+1..t_{1}]$ and $[s_{2}+1..t_{2}]$), both to $[0..k-1]$, we get:

$\mathbb{MSE}_{t_{1},s_{1}}=\sum_{r=0}^{k-1}\xi_{q}^{2}(t_{1},t_{1}-r)\mathbb{\sigma}^{2}(t_{1}-r),\
\ \
\mathbb{MSE}_{t_{2},s_{2}}=\sum_{r=0}^{k-1}\xi_{q}^{2}(t_{2},t_{2}-r)\mathbb{\sigma}^{2}(t_{2}-r).$

In a time-varying environment, a comparison between
$\mathbb{MSE}_{t_{1},s_{1}}$ and $\mathbb{MSE}_{t_{2},s_{2}}$, whenever
$t_{1}-s_{1}=t_{2}-s_{2}$, entails, in the general case, that :
$\mathbb{MSE}_{t_{1},s_{1}}\neq\mathbb{MSE}_{t_{2},s_{2}}$. On the other hand,
in the time invariant case $\xi_{q}(t,t-r)$, which will now be denoted as
$\xi_{r}^{(q)}$, becomes
$\xi_{r}^{(q)}=\xi_{r}+\sum_{l=1}^{q}\xi_{r-l}\theta_{l}$ (see eqs. (8) and
(13)). In this latter case the two MSEs coincide, as being identical to:

$\mathbb{M}\mathbb{SE}_{k}=\sigma^{2}\sum_{r=0}^{k-1}(\xi_{r}^{(q)})^{2}.$

###  6.3 Forward Asymptotic Efficiency

As pointed out by Hallin (1986), if a researcher wants to study the “causal”
properties of the observed process, then he/she should examine the Wold-Crámer
decomposition (see Theorem 3 in Section 5.1). If forecasting is the main
objective, then the forecast obtained by the model should be asymptotically
efficient in some sense. Of course the asymptotic forecasting properties of
the model rely on its behaviour in the far future, whereas its causal
properties involve its remote past only. If the processes is of constant
coefficients, these two issues coincide. In the case of time dependent
coefficients, however, they apparently differ strongly.

To reiterate one of the main purposes in building models for stochastic
processes is to provide convenient forecast procedures. The researcher would
like to minimize (asymptotically) the MSE or in other words to achieve
asymptotic efficiency. The asymptotic efficiency of a forecasting procedure
can be defined in two alternative ways (seemingly, analogous to each other,
but indeed basically different). The first one (termed by Hallin efficiency,
see Definition 5.1 in his paper) is obtained by considering the asymptotic
forecasting performance of a model as $s$ tends to $-\infty$ (see Proposition
3). The Wold-Cramér representation of a DTV-ARMA model in eq. (20) is
efficient if and only if the model is invertible. If it is not an invertible
model, then the mean square forecasting error associated with the above
mentioned solution representation is not, in the general case, bounded.

A more realistic approach to efficiency consists of considering the asymptotic
behaviour of the mean square forecasting error as $t\rightarrow\infty$ for $s$
being arbitrary but fixed. This forward efficiency concept is also called the
Granger-Andersen efficiency (see Definition 5.2 in Hallin, 1986, and the
references therein). As the forecast error and the mean square error for the
predictors given by eqs. (31 and 32), respectively, coincide, we can examine
the forward asymptotic efficiency of DTV-ARMA models within the same
forecasting scheme.

In the following Proposition, we give a weak condition, which guarantees the
forward asymptotic boundedness of the mean square error.

###### Proposition 7

Let $F(t,s)\overset{\mathrm{def}}{=}\sum_{r=s+1}^{t}|\xi(t,r)|$ for $t>s$. If
$\\{F(t,s)\\}_{t}$ is bounded, as a function of $t$ for each fixed $s$, then
the mean square error is also bounded, as a function of $t$. Equivalently, the
boundedness of $\\{F(t,s)\\}_{t}$ entails that for each $s$ either
$\displaystyle\lim_{t\to\infty}\mathbb{MSE}_{t,s}$ exists in $\mathbb{R}_{\geq
0}$ or $\\{\mathbb{MSE}_{t,s}\\}_{t}$ oscillates with oscillation: $\mathit{\
\Omega}(s)=\displaystyle\inf_{t}(\sup_{r\geq t}\mathbb{MSE}_{r,s}-\inf_{r\geq
t}\mathbb{MSE}_{r,s})$.

In the following Corollary, we provide a stronger condition, which guarantees
the forward asymptotic uniform boundedness of the mean square error.

###### Corollary 3

If further the condition (22) holds, that is the function
$F(t)=\sum_{r=-\infty}^{t}|\xi(t,r)|$ is bounded in $t$, then
$\mathbb{MSE}_{t,s}$ is uniformly bounded.

The proofs of the above statements are available in the Appendix.

##  7 Stochastic Coefficients

In Sections 5 and 6, we restricted ourselves to the treatment of DTV models.
In this Section we examine processes with stochastically time varying
coefficients. For simplicity instead of ARMA processes, we will concentrate on
the AR($p$) specification (STV-AR). In particular we will investigate three
models: the random coefficients one, its generalization and the double
stochastic AR process. But first we will express the STV-AR model in a
companion matrix form. The proofs of the present Section are provided in the
Appendix.

###  7.1 Companion Matrix Form

In this Subsection we show how to utilize the principal determinant (Green’s
function) and the $m$-th fundamental solution in order to obtain a compact and
explicit representation of the companion matrix.

The STV-AR($p$) process, can be expressed as

$y_{t}=\phi_{0t}+\boldsymbol{\phi}_{t}^{\prime}\mathbf{y}_{t-1}+\varepsilon_{t},$
(33)

where $\mathbf{y}_{t-1}=(y_{t-1,}$ $y_{t-2,}\ldots,y_{t-p})^{\prime}$ is a
$p\times 1$ vector of preceding random variables of $y_{t}$, and
$\boldsymbol{\phi}_{t}=(\phi_{1t},$ $\phi_{2t},\ \ldots,\phi_{pt})^{\prime}$
is a $p\times 1$ vector of the autoregressive random coefficients. Notice that
we denote the STV coefficients, including the drift $\phi_{0t}$, by
$\phi_{mt}$, $m=0,\ldots,p$, instead of $\phi_{m}(t)$, which was the notation
used for the deterministic ones.

It is well known that the model (33) can be written in a companion matrix
form:

$\mathbf{y}_{t}=\boldsymbol{\phi}_{0t}+\mathbf{\Phi}_{t}\mathbf{y}_{t-1}+\boldsymbol{\varepsilon}_{t},$
(34)

where $\boldsymbol{\phi}_{0t}=(\phi_{0t}$ $0$ $\ldots 0)^{\prime}$,
$\boldsymbol{\varepsilon}_{t}=(\varepsilon_{t}$ $0\ldots 0)^{\prime}$, and the
companion (square) matrix $\mathbf{\Phi}_{t}$ of order $p$ associated to the
vector $\boldsymbol{\phi}_{t}$ is given by

$\mathbf{\Phi}_{t}=\left(\begin{array}[]{ccccc}\phi_{1t}&\phi_{2t}&\ldots&\phi_{p-1,t}&\phi_{pt}\\\
1&0&\ldots&0&0\\\ 0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots\vdots\vdots&\vdots&\vdots\\\
0&0&\ldots&1&0\end{array}\right).$ (35)

That is, the STV-AR($p$) process is converted to a $p$-dimensional vector STV-
AR($1$) model. For any set of $p$ prescribed random variables
$\mathbf{y}_{s}$, iterating eq. (34) yields

$\mathbf{y}_{t}=\mathbf{C}_{t,s}\mathbf{y}_{s}+\sum\limits_{r=s+1}^{t}\mathbf{C}_{t,r}(\boldsymbol{\phi}_{0,r}+\boldsymbol{\varepsilon}_{r}),$
(36)

where $\mathbf{C}_{t,s}=\prod\limits_{r=s+1}^{t}\mathbf{\Phi}_{t,r}$ is the
product of companion matrices with initial square matrix
$\mathbf{C}_{t,t}=\mathbf{I}$ of order $p$. It follows directly from the above
equation and Theorem 1 (see also, for more details, Paraskevopoulos and
Karanasos, 2021) that the $p$-dimensional square matrix $\mathbf{C}_{t,s}$ is
given by

$\mathbf{C}_{t,s}=\left(\begin{array}[]{cccc}\xi^{(1)}(t,s)&\xi^{(2)}(t,s)&\cdots&\xi^{(p)}(t,s)\\\
\xi^{(1)}(t-1,s-1)&\xi^{(2)}(t-1,s)&\cdots&\xi^{(p)}(t-1,s)\\\
\lx@intercol\hfil\vdots\hfil\lx@intercol&\lx@intercol\hfil\vdots\hfil\lx@intercol&\lx@intercol\hfil\vdots\vdots\vdots\hfil\lx@intercol&\lx@intercol\hfil\vdots\hfil\lx@intercol\\\
\xi^{(1)}(t-p+1,s)&\xi^{(2)}(t-p+1,s)&\cdots&\xi^{(p)}(t-p+1,s)\end{array}\right).$

In other words the element occupying the $(n+1,m)$-th entry of the matrix
$\mathbf{C}_{t,s}$ ($n=0,\ldots,p-1$) is the $m$-th fundamental solution
$\xi^{(m)}(t-n,s)$. We recall that $\xi^{(1)}(t,s)$ is given in eqs. (5 and 6)
(where now, $\phi_{m}(t)$ in eq. (5) is replaced by $\phi_{mt}$) and,
similarly, $\xi^{(m)}(t,s)$ is given in eq. (10).

###  7.2 Random Coefficients AR Model

In this Subsection we examine the random coefficient AR($p$) model (with
acronym RC-AR($p$)), which is given by eq. (33), using for this the following
notation and specifications: $\boldsymbol{\phi}_{t}^{\ast}=(\phi_{0t}$
$\boldsymbol{\phi}_{t})^{\prime}$, $t=s+1,s+2\ldots$, is an $i.i.d.$
$(p+1)$-dimensional random vector of the coefficients, and the $i.i.d.$
errors, $\\{\varepsilon_{t},t\geq s+1\\}$ are independent of the random drift
and autoregressive coefficients. Let also $\varepsilon_{s}$ be a random
variable independent of everything else, being the initial state.

Let us call $\left|\cdot\right|$ the Euclidean norm on the space
$\mathbb{R}^{p}$. Let $\mathbb{R}^{p\times p}$ be the space of $p\times p$
matrices with elements in $\mathbb{R}$ and denote by
$\left|\left|\cdot\right|\right|$ the matrix norm induced by
$\left|\cdot\right|$ (this is known as the spectral norm, defined as the
largest singular value of the matrix).

###### Condition 1

$\xi(t,s)\overset{a.s}{\rightarrow}0$ as $t\rightarrow\infty$.

The proof of the next Theorem follows from the fact that
$\xi_{t,s}\overset{a.s}{\rightarrow}0$ as $t\rightarrow\infty$ implies that
$\left|\left|\mathbf{C}_{t,s}\right|\right|\overset{a.s.}{\rightarrow}0$ as
$t\rightarrow\infty$ and, therefore, Theorem 2.1 in Erhardsson (2014) applies.

###### Theorem 6

Consider the RC-AR($p$) model. Under Condition 1 the following are equivalent:

  1. i)

$\mathbf{y}_{t}$ convergences in distribution as $t\to\infty$,

  2. ii)

$\sum\limits_{r=s+1}^{\infty}\left|\mathbf{C}_{r-1,s}\boldsymbol{\varepsilon}_{r}\right|<\infty$
a.s.,

  3. iii)

$\sum\limits_{r=s+1}^{t}\mathbf{C}_{r-1,s}\boldsymbol{\varepsilon}_{r}$
converges a.s., as $t\to\infty$,

  4. iv)

$\mathbf{C}_{t-1,s}\boldsymbol{\varepsilon}_{t}\overset{\rm a.s.}{\to}0$, as
$t\to\infty$,

  5. v)

$\sup_{t\geq
s+1}\left|\mathbf{C}_{t-1,s}\boldsymbol{\varepsilon}_{t}\right|<\infty$ a.s.

As pointed out by Erhardsson (2014) the implications
$(ii)\implies(iii)\implies(iv)\implies(v)$ remain valid even if Condition (1)
does not hold.

###  7.3 The Generalized RC-AR Model

A generalization of the RC-AR process (GRC-AR($p$)) in eq. (33) is discussed
in this Subsection. For this model, we show how the expected value of the
$m$-th fundamental solution can be expressed as a Hessenbergian, which then
can be employed to obtain an explicit representation of the autocovariance
function.

The generalized process extends the simple one to cover contemporaneous
dependence between the vector of the drift and the autoregressive
coefficients, both included in $\boldsymbol{\phi}_{t}^{\ast}$, and the vector
of errors $\varepsilon_{t}$.

The GRC-AR($p$) model also integrates the following AR processes (for details,
see Hwang and Basawa, 1998):

  1. I.

Random coefficient model: $\phi_{mt}=\phi_{m}+\eta_{mt}$, where
$\\{\eta_{mt}\\}$ is a sequence of $i.i.d.$ random variables with
$\mathbb{E}(\eta_{mt})=0$, and $\\{\eta_{mt}\\}$ is independent of
$\\{\varepsilon_{\tau},\tau\in\mathbb{Z}\\}$ for all $m$ and $t\neq\tau$.
Notice also that if we set $\eta_{mt}=0$, for all $m$ and $t$, we get the
ordinary AR($p$) process.

  2. II.

Markovian bilinear model: $\phi_{mt}=\phi_{m}+\vartheta_{m}\varepsilon_{t}$.

  3. III.

Generalized Markovian bilinear model:
$\phi_{mt}=c_{m}+\vartheta_{m}\varepsilon_{t}^{r_{m}}$,
$r_{m}\in\mathbb{Z}_{>0}$, and $\varepsilon_{t}^{r_{m}}$ has finite moments
$\mathbb{E}(\varepsilon_{t}^{r_{m}})$ for all $m$. Note that if we set
$r_{m}=1$, for all $m$, we get the Markovian bilinear model coefficients.

  4. IV.

Random coefficient exponential model:
$\phi_{mt}=c_{m}+(\vartheta_{m1}+\vartheta_{m2}e^{-\vartheta_{m3}\varepsilon_{t}^{2}})\varepsilon_{t}$.

In cases I - IV all the coefficients without time $t$ subscript are constants.
In the next Proposition and Theorem we will make use of the following
notation:

###### Notation 1

$\begin{array}[]{rcl}y&\overset{\rm def}{=}&\mathbb{E}(y_{t}),\
\phi_{m}\overset{\rm def}{=}\mathbb{E}(\phi_{mt}),\
\sigma_{m\varepsilon}\overset{\rm def}{=}\mathbb{E}(\phi_{mt}\varepsilon_{t})\
\text{for}\ m=0,\ldots,p,\\\ \sigma_{mn}&\overset{\rm
def}{=}&\mathbb{C}ov(\phi_{mt},\phi_{nt})=\bar{\phi}_{mn}-\phi_{m}\phi_{n}\
\text{ for }m,n=0,\ldots,p.\end{array}$

###### Condition 2

$\left|\sum\nolimits_{m=1}^{p}\phi_{m}\right|<1.$

###### Proposition 8

Under Condition 2 the expected value of the GRC-AR($p$) model is given by

$y=\frac{\phi_{0}}{1-\sum\limits_{m=1}^{p}\phi_{m}}.$

In what follows we adopt the notation $\mathbf{\Phi}_{t}^{\otimes
2}=\boldsymbol{\Phi}_{t}\otimes\mathbf{\Phi}_{t}$, where
$\boldsymbol{\Phi}_{t}$ is the companion matrix (see eq. (35)), $\otimes$ is
the Kronecker product, and
$\lambda_{\max}[\mathbb{E}(\mathbf{\Phi}_{t}^{\otimes 2})]$ refer to the
modulus of the largest eigenvalue of $\mathbb{E}(\mathbf{\Phi}_{t}^{\otimes
2})$.

###### Condition 3

$\lambda_{\max}[\mathbb{E}(\mathbf{\Phi}_{t}^{\otimes 2})]<1$.

In the next Theorem we shall also make use of the following notation:

###### Notation 2

i) $\varphi_{ij}$ denotes the element occuping the $(i,j)$-th entry of the
matrix $[\mathbf{I}_{p^{2}}-\mathbb{E(}\mathbf{\Phi}_{t}^{\otimes 2})]^{-1}$,

$ii)\text{
}\sigma^{2}=(\sigma_{\varepsilon}^{2}+\sigma_{00}+2\sigma_{0\varepsilon})+2y\sum\limits_{m=1}^{p}(\sigma_{m\varepsilon}+\sigma_{m0})+y^{2}\sum\limits_{n=1}^{p}\sum\limits_{m=1}^{p}\sigma_{mn},$
(37)

where $\sigma_{\varepsilon}^{2}=\mathbb{V}ar(\varepsilon_{t})$. iii)
$\xi_{k}^{(m)}=\mathbb{E}(\xi^{(m)}(t,s))$, recall that $k=t-s$, is the
$k$-dimensional banded Hessenbergian:

$\xi_{k}^{(m)}=\left|\begin{array}[]{cccccccc}{\footnotesize\phi}_{m}&{\footnotesize-1}&&&&&&\\\
{\footnotesize\phi}_{m+1}&{\footnotesize\phi}_{1}&\ddots&&&&&\\\
{\footnotesize\vdots}&{\footnotesize\vdots}&\ddots&\ddots&&&&\\\
{\footnotesize\phi}_{p}&{\footnotesize\phi}_{p-1}&\ddots&\ddots&\ddots&&&\\\
&{\footnotesize\phi}_{p}&\ddots&\ddots&\ddots&\ddots&&\\\
&&{\small\ddots}&{\small\ddots}&{\small\ddots}&{\small\ddots}&\ddots&\\\
&&&{\footnotesize\phi}_{p}&{\footnotesize\phi}_{p-1}&{\small\cdots}&{\footnotesize\phi}_{1}&{\footnotesize-1}\\\
&&&&{\footnotesize\phi}_{p}&{\small\cdots}&{\footnotesize\phi}_{2}&{\footnotesize\phi}_{1}\end{array}\right|$
(38)

(we recall that $\xi^{(1)}(t,s)$ is given in eqs. (5 and 6), where now,
$\phi_{m}(t)$ is replaced by $\phi_{mt}$ and, similarly, $\xi^{(m)}(t,s)$ is
given in eq. (10)).

###### Theorem 7

Under Condition 3 the covariance structure of the GRC-AR($p$) model is given
by

$\gamma(\ell)=\left\\{\begin{array}[]{ccc}\varphi_{\ell+1,1}\sigma^{2}&\text{for}&\ell=0,\ldots,p-1\\\
\sum\limits_{m=1}^{p}\xi_{\ell}^{(m)}\gamma(m-1)&\text{for}&\ell\geq
p,\end{array}\right.$

where $\xi_{\ell}^{(m)}$ is given by eq. (38).

Let us remark that Condition 2 is equivalent to
$\lim_{\ell\rightarrow\infty}\xi_{\ell}^{(1)}=0$, which implies that
$\lim_{\ell\rightarrow\infty}\xi_{\ell}^{(m)}=0$ for all $m$, and, therefore,
ensures that $\lim_{\ell\rightarrow\infty}\gamma(\ell)=0$. As another example
we examine the autoregressive process of order 2.

###### Corollary 4

Consider the GRC-AR($2$) model and let the following condition hold:

$1-\frac{2\bar{\phi}_{12}\phi_{1}}{1-\phi_{2}}>\bar{\phi}_{11}+\bar{\phi}_{22}>0.$

Under the above condition its covariance structure is given by

$\
\gamma(\ell)=\left\\{\begin{array}[]{ccc}\varphi_{\ell+1,1}\sigma^{2}&\text{for}&\ell=0,1\\\
\sum\limits_{m=1}^{2}\xi_{\ell}^{(m)}\gamma(m-1)&\text{for}&\ell\geq
2,\end{array}\right.$

where

$\displaystyle\varphi_{1,1}$ $\displaystyle=$
$\displaystyle\frac{1-\phi_{2}^{2}}{(1-\phi_{2}^{2})(1-\bar{\phi}_{11}-\bar{\phi}_{22})-2\bar{\phi}_{12}\phi_{1}(1+\phi_{2})},$
$\displaystyle\varphi_{2,1}$ $\displaystyle=$
$\displaystyle\varphi_{2,1}=\frac{\phi_{1}(1+\phi_{2})}{(1-\phi_{2}^{2})(1-\bar{\phi}_{11}-\bar{\phi}_{22})-2\bar{\phi}_{12}\phi_{1}(1+\phi_{2})},$

and the tridiagonal matrices of order $\ell$, that is $\xi_{\ell}^{(m)}$,
$m=1,2$, are given by

$\xi_{\ell}^{(1)}=\left[\begin{array}[]{ccccc}{\footnotesize\phi}_{1}&{\footnotesize-1}&&&\\\
{\footnotesize\phi}_{2}&{\footnotesize\phi}_{1}&\ddots&&\\\
&\ddots&\ddots&\ddots&\\\
&&\ddots&{\footnotesize\phi}_{1}&{\footnotesize-1}\\\
&&&{\footnotesize\phi}_{2}&{\footnotesize\phi}_{1}\end{array}\right],\text{ \
}\xi_{\ell}^{(2)}={\footnotesize\phi}_{2}\xi_{\ell-1}^{(1)}.$

###  7.4 Double Stochastic AR Models

In this Subsection we examine the more general case where the autoregressive
coefficients follow AR processes. We show that for this model the
unconditional variance exists in $\mathbb{R}_{>0}$ provided that the
associated Green function convergences in $L_{2}$, a result which is in line
with Theorem 2(ii). In other words, we investigate the double stochastic AR
model, hereafter termed DS-AR (for double stochastic processes, and in
particular ARMA processes with ARMA coefficients see, for example,
Grillenzoni, 1993, and the references therein). The DS-AR model is defined by
eq. (33) but in this case the autoregressive coefficients, $\phi_{mt}$ for
$m=1,\ldots,p$, follow AR processes:

$\phi_{mt}=\beta_{m0}+\sum\limits_{l=1}^{p_{m}}\beta_{ml}\phi_{m,t-l}+e_{mt},$
(39)

where $\beta_{m0}$ and $\beta_{ml}$ are constant coefficients and
$p_{m}\in\mathbb{Z}_{>0}$ for all $m:1\leq m\leq p$. $\\{e_{mt}\\}$ are
martingale difference sequences defined on $L_{2}$, where $e_{mt}$ and
$\varepsilon_{t\pm b}$, $b\in\mathbb{Z}$, are independent of each other for
all $m$, and $t\in\mathbb{Z}$. For simplicity, we will assume that the drift
in eq. (33) is time invariant, that is, $\phi_{0t}=\phi_{0}$ for all $t$.

The results in Sections 5 and 6 can be easily modified to cover DS-AR models
by replacing the fundamental solutions with their respective (unconditional
and conditional) expectations. More specifically, we present two Theorems
followed by two Propositions (their proofs essentially repeat the arguments of
the proofs of those in Section 5 and 6).

Sufficient conditions ensuring the Wold-Cramér decomposition of DS-AR($p$)
models, and therefore the existence of the first two unconditional moments,
are the following two:

$\left.\begin{array}[]{lll}\displaystyle\sum_{r=-\infty}^{t}|\mathbb{E}\big{(}\xi(t,r)\big{)}|&<&\infty,\
\text{for\ all}\ t\ \text{(first-order\ absolute\ summability)},\\\
\displaystyle\sum_{r=-\infty}^{t}\mathbb{E}\big{(}\xi^{2}(t,r)\big{)}&<&\infty,\
\text{for\ all}\ t\ \text{ (second-order\ summability).}\end{array}\right\\}$
(40)

###### Remark 5

Generally, it is very difficult to verify if the two summability conditions
are fulfilled. Only some special cases allow to write explicit solutions (see,
Andĕl, 1991, and the references therein). A sufficient condition for the
absolute summability to hold is that $\\{\sum\nolimits_{m=1}^{p}\phi_{mt}\\}$
belongs with probability one to the interval $(-1,1)$, nearly everywhere, that
is, with the exception, at most, of a finite number of $t$ (see, for example,
Grillenzoni, 1993). Similarly, a sufficient condition for the square
summability to hold is that with probability one
$\lambda_{t}^{(\max)}[\mathbf{\Phi}_{t}^{\otimes 2}]<1$, nearly everywhere,
where $\lambda_{t}^{(\max)}[\mathbf{\Phi}_{t}^{\otimes 2}]$ refers to the
modulus of the largest eigenvalue of $\mathbf{\Phi}_{t}^{\otimes 2}$; we
recall that $\mathbf{\Phi}_{t}$ is the companion matrix (see eq. (35)).

###### Theorem 8

Let the two summability conditions in (40) hold. The Wold-Cramér decomposition
(in $L_{2}$ sense) is a solution of the DS-AR($p$) model in eq. (33), where
its autoregressive coefficients are given by eq. (39), being of the form:

$y_{t}=\sum_{r=-\infty}^{t}\xi(t,r)(\phi_{0}+\varepsilon_{r}).$ (41)

###### Theorem 9

If all the autoregressive coefficients, $\phi_{mt}$, $m=1,\ldots,p$, are
strictly stationary, then eq. (33) has a stationary solution of the type (41)
if and only if

$\sum_{r=1}^{\infty}\left|\mathbb{E}\big{(}\xi(r,1)\big{)}\right|<\infty\ \
\text{\rm and}\ \ \
\sum_{r=1}^{\infty}\mathbb{E}\big{(}\xi^{2}(r,1)\big{)}<\infty.$

In what follows, we present explicit formulae for the first and second
unconditional moments for the DS-AR family of processes coupled with
sufficient and necessary conditions for their existence.

###### Proposition 9

Under the two conditions in (40), it follows from Theorem 8 that the
unconditional mean of the DS-AR($p$) process, $y_{t}$, exists in $\mathbb{R}$
and is given by

$\mathbb{E}(y_{t})=\phi_{0}\sum\limits_{r=-\infty}^{t}\mathbb{E}\big{(}\xi(t,r)\big{)}.$
(42)

A necessary condition for the absolute summability to hold is

$\lim_{s\rightarrow-\infty}\mathbb{E}\big{(}\xi(t,s)\big{)}=0.$

Moreover, the unconditional variance of the process exists in
$\mathbb{R}_{>0}$ and it is given by

$\mathbb{V}ar(y_{t})=\phi_{0}^{2}\sum\limits_{r=-\infty}^{t}\mathbb{V}ar\big{(}\xi(t,r)\big{)}+\sigma_{\varepsilon}^{2}\sum_{r=-\infty}^{t}\mathbb{E}\big{(}\xi^{2}(t,r)\big{)}.$

A necessary condition for the second-order summability to hold is

$\lim_{s\rightarrow-\infty}\mathbb{E}\big{(}\xi^{2}(t,s)\big{)}=0.$

We notice that
$\lim_{s\rightarrow-\infty}\mathbb{E}\big{(}\xi^{2}(t,s)\big{)}=0$ is
equivalent to $\lim_{s\to-\infty}||\xi(t,s)||^{2}_{L_{2}}=0$, which, in turn,
is equivalent to $\lim_{s\to-\infty}||\xi(t,s)||_{L_{2}}=0$. In this latter
case, we write $\lim_{s\to-\infty}\xi(t,s)\overset{L_{2}}{=}0$, which is in
line with Theorem 2(ii).

###### Proposition 10

Following the notation of Proposition 6, let $\mathcal{K}_{s}$ be the smallest
closed subspace of $L_{2}$ spanned by the finite observed sequence

$\\{y_{s},...,y_{s+1-p}\\}\cup\bigg{(}\displaystyle\bigcup_{i=1}^{m}\\{\phi_{i,s},\phi_{i,s+1},\
\ldots,\phi_{i,s+1-p_{i}}\\}\bigg{)}.$

The $k$-step-ahead optimal (in $L_{2}$-sense) linear predictor of the DS-
AR($p$) process is:

$\mathbb{E}(y_{t}\left|\mathcal{K}_{s}\right.)=\sum_{m=1}^{p}\mathbb{E}\big{(}\xi^{(m)}(t,s)|\mathcal{K}_{s}\big{)}y_{s+1-m}+\phi_{0}\sum_{r=s+1}^{t}\mathbb{E}\big{(}\xi(t,r)\left|\mathcal{K}_{s}\right.\big{)}.$

We remark that
$\lim_{s\to-\infty}\mathbb{E}(y_{t}\left|\mathcal{K}_{s}\right.)=\mathbb{E}(y_{t})$
is given by eq. (42). In addition, the forecast error for the above $k$-step-
ahead predictor, $\mathbb{FE}_{t,s}$, is given by

$\mathbb{FE}_{t,s}=\phi_{0}\sum_{r=s+1}^{t}\bigg{(}\xi(t,r)-\mathbb{E}(\xi(t,r)\left|\mathcal{K}_{s}\right.)\bigg{)}+\sum_{r=s+1}^{t}\xi(t,r)\varepsilon_{r}+\sum_{m=1}^{p}\big{(}\xi^{(m)}(t,s)-\mathbb{E}(\xi^{(m)}(t,s)|\mathcal{K}_{s})\big{)}y_{s+1-m}.\
$

The conditional variance of $y_{t}$ based on $\mathcal{K}_{s}$, that is
$\mathbb{V}ar(y_{t}\left|\mathcal{K}_{s}\right.)=\mathbb{V}ar(\mathbb{FE}_{t,s}\left|\mathcal{K}_{s}\right.)$
is given by

$\mathbb{V}ar(y_{t}\left|\mathcal{K}_{s}\right.)=\phi_{0}^{2}\sum_{r=s+1}^{t}\mathbb{V}ar\big{(}\xi(t,r)\left|\mathcal{K}_{s}\right.\big{)}+\sigma_{\varepsilon}^{2}\sum_{r=s+1}^{t}\mathbb{E}\big{(}\xi^{2}(t,r)\left|\mathcal{K}_{s}\right.\big{)}+\sum_{m=1}^{p}\mathbb{V}ar\big{(}\xi^{(m)}(t,s)\left|\mathcal{K}_{s}\right.\big{)}y_{s+1-m}^{2}$

(we recall that $\xi(t,r)$ have been introduced in eqs. (5) and (6), where now
$\phi_{m}(t)$ are replaced by $\phi_{mt}$).
We remark that
$\lim_{s\to-\infty}\mathbb{V}ar(y_{t}\left|\mathcal{K}_{s}\right.)=\mathbb{V}ar(y_{t})$,
which is given in Proposition 9.

##  8 Time-varying Polynomials

In Sections 3, 5 and 6, we employed techniques of linear algebra in order to
obtain an explicit representation of the TV-ARMA($p,q$) model and its first
two (conditional and unconditional) moments. The main mathematical tool used
was the Hessenbergian determinant. Now that we have expressed the Green’s
function as a Hessenbergian, we will see how the summation terms in the
various equations in Sections 3 and 5 can be expressed as time-varying
polynomials.

Recall that $B$ denotes the backshift (or lag operator), defined such that
$By_{t}=y_{t-1}$. The time-varying AR and moving average (MA) polynomial
(backshift) operators associated with the TV-ARMA($p,q$) model are denoted as:

$\mathbf{\Phi}_{t}(B)=1-\sum\limits_{m=1}^{p}\phi_{m}(t)B^{m},\text{
}\Theta_{t}(B)=1+\sum\limits_{l=1}^{q}\theta_{l}(t)B^{l}.$ (43)

Under this notation eq. (1) can be written in a more condensed form

$\Phi_{t}(B)y_{t}=\varphi(t)+\Theta_{t}(B)\varepsilon_{t}.$ (44)

In the time invariant case one can employ the roots of the time invariant
polynomial $\Phi(z^{-1})$ to obtain its general time series properties such as
the Wold decomposition and the second moment structure. In a time-varying
environment, according to Grillenzoni (1990), the so called here principal
fundamental sequence $\\{\xi(t,s)\\}_{t\geq s+1-p}$ cannot be obtained as in
stationarity, that is by expanding in Taylor series the rational polynomial
$\Phi_{t}^{-1}(B)$. As an alternative, Hallin (1986) used some results on
difference operators involving the symbolic product of operators, which has
also been termed by researchers, in the field of engineering, as the skew
multiplication operator (see, for example, Mrad and Farag, 2002). Hence, now
that we have at our disposal an explicit and computationally tractable
representation of the Green function as a banded Hessenbergian, coupled with
the use of the time-domain noncommutative multiplication operation -which, as
pointed out by Mrad and Farag (2002), is based on the manipulation of
polynomial operators with time-varying coefficients using operations
restricted to the time domain- we are able to state some important Theorems in
relation to the results in Sections 3 and 5.

###  8.1 The Skew Multiplication Operator

In a time-varying environment, the time-varying polynomial operators in eq.
(43) can be manipulated by using the skew multiplication operator “$\circ$”
defined by

$B^{i}\circ B^{j}=B^{i+j}\text{ \ and \ }B^{i}\circ f(t)=f(t-i)B^{i}\text{,}$
(45)

where $f(t)$ is a function of time. This time-domain multiplication operation
is associative but noncummutative (see Karmen, 1988, Bouthellier and Ghosh,
1988, and Mrad and Farag, 2002). Using the properties of “$\circ$”, from eq.
(44), under the necessary and sufficient conditions in Proposition 3, we can
obtain the unique inverse of $\Phi_{t}(B)$, that is
$\Phi^{-1}_{t}(B)\circ\Phi_{t}(B)=1$, where $\Phi^{-1}_{t}(B)$ is provided in
eq. (51) below.

###  8.2 Polynomial Operators

Next, and equally important, we will provide a critical and essential further
link between the linear algebra techniques used in Section 3 (to obtain the
explicit representation of the TV-ARMA model in eq. (16)) and the time-varying
polynomial approach, in which we make use of the skew multiplication operator.
Certainly, from an operational point of view, both are equally satisfying and
recommendable. First, let us define the two time-varying polynomial
(backshift) operators associated with eq. (16).

###### Definition 2

I) Let $\Xi_{t,p}^{[k]}(B)$ be defined as follows:

$\Xi_{t,p}^{[k]}(B)=1-\sum\limits_{m=1}^{p}\xi^{(m)}(t,s)B^{k-1+m}.$ (46)

IIa) Let $\Xi_{t}^{(k)}(B)$ be defined as follows:

$\Xi_{t}^{(k)}(B)=\sum_{r=s+1}^{t}\xi(t,r)B^{t-r}\overset{\mathrm{or}}{=}\sum_{r=0}^{k-1}\xi(t,t-r)B^{r}.$
(47)

IIb) The limit of $\Xi_{t}^{(k)}(B)$ as $k\rightarrow\infty$ is denoted by
$\Xi_{t}(B)$.

Notice that we have employed the notation $(.)^{[k]}$ ($k$ inside brackets)
for the polynomial operator defined in eq. (46) in order to be distinguished
from the one defined in eq. (47).

###### Remark 6

$\Xi_{t,p}^{[k]}(B)$ in eq. (46) is a polynomial of order $k-1+p$ associated
with the homogeneous solution in eq. (11), and it is expressed in terms of the
$m$ fundamental solutions, defined in eq. (10).
Notice that: i) $\Xi_{t,p}^{[1]}(B)=\Phi_{t}(B)$, since
$\xi^{(m)}(t,t-1)=\phi_{m}(t)$ (see the discussion next to eq. (10)), and
ii) Under the stability condition in Theorem 2(i), that is for the
deterministic case, $\lim_{k\rightarrow\infty}\Xi_{t,p}^{[k]}(B)=1$, since
$\lim_{s\rightarrow-\infty}\xi^{(m)}(t,s)=0$.

###### Remark 7

$\Xi_{t}^{(k)}(B)$ is a polynomial of order $k-1$ associated with the
particular solution in eq. (12), and it is expressed in terms of the first $k$
instances of the principal determinant, $\xi(t,s+r)$ for $1\leq r\leq k$.
Notice also that $\Xi_{t}^{(1)}(B)=1$.

Next we define two additional time-varying polynomial operators associated
with the innovation part of the particular solution (see eq. 15) and the Wold-
Cramér decomposition (see eq. 20) respectively.

###### Definition 3

I) Let $\Xi_{t,q}^{(k)}(B)$ be defined as follows

$\Xi_{t,q}^{(k)}(B)=\sum_{r=0}^{k-1}\xi_{q}(t,t-r)B^{r}+\sum_{r=k}^{k-1+q}\xi_{s,q}(t,t-r)B^{r}.$
(48)

II) Let $\Xi_{t,q}(B)$ be defined as follows

$\Xi_{t,q}(B)=\sum_{r=0}^{\infty}\xi_{q}(t,t-r)B^{r},$

that is: $\lim_{k\to\infty}\Xi_{t,q}^{(k)}(B)=\Xi_{t,q}(B)$.

###### Remark 8

$\Xi_{t,q}^{(k)}(B)$ is a polynomial of order $k-1+q$ associated with the
innovation part of the particular solution, and is expressed in terms of the
first $k$ instances of $\xi_{q}(t,s+r)$ for $1\leq r\leq k$ and the first $q$
instances of $\xi_{s,q}(t,s+r-q)$ for $1\leq r\leq q$, which have been
introduced in Definition 1.
Notice that, i) for the pure AR model $\Xi_{t,0}^{(k)}(B)=\Xi_{t}^{(k)}(B)$,
since $\xi_{0}(t,t-r)=\xi(t,t-r)$ and the second summation in eq. (48)
(adopting the convention $\sum_{r=k}^{k-1}a_{r}=0$) vanishes. ii)
$\Xi_{t,q}^{(1)}(B)=\Theta_{t}(B)$, since the first summation is equal to
$\xi(t,t)=1$, and the second summation is equal to $1-\Theta_{t}(B)$ (see the
discussion next to Proposition 1).

###  8.3 Polynomial Representations

The next Proposition is an alternative to Proposition 1, expressed in terms of
polynomial operators. Proposition 11 and Theorems 10, 11 below can be deduced
by applying the properties of the skew multiplication operator “$\circ$”(see
eq. (45)). By analogy to the decomposition in eq. (15), we deduce the
following two part-decomposition:

###### Proposition 11

$\Xi_{t}^{(k)}(B)\circ u_{t}$ takes the following alternative expressions:

$\Xi_{t}^{(k)}(B)\circ
u_{t}=\Xi_{t,q}^{(k)}(B)\varepsilon_{t}\overset{\mathrm{or}}{=}\sum_{r=0}^{k-1}\xi_{q}(t,t-r)\varepsilon_{t-r}+\sum_{r=k}^{k-1+q}\xi_{s,q}(t,t-r)\varepsilon_{t-r}.$
(49)

A proof of the above result is provided in Appendix. The first of the
following Theorems (Theorem 10) is equivalent to Theorem 1.

###### Theorem 10

The explicit representation of eq. (1) given by eq. (16), can be equivalently
expressed, in terms of the polynomial operators given in Definitions 2 and 3,
as:

$\Xi_{t,p}^{[k]}(B)y_{t}=\Xi_{t}^{(k)}(B)\circ[\varphi(t)+u_{t}]=\Xi_{t}^{(k)}(B)\circ\varphi(t)+\Xi_{t,q}^{(k)}(B)\varepsilon_{t}.$
(50)

Clearly, the results for the pure AR model are obtained by setting
$u_{t}=\varepsilon_{t}$ in the first equality of eq. (50) or by noticing that
$\Xi_{t}^{(k)}(B)=\Xi_{t,0}^{(k)}(B)$ in the second equality of eq. (50) (see
Definition 3 and Remark 8). Notice also that Proposition 6 can be expressed in
terms of the time-varying polynomial operators as well (results not reported
here).

In what follows we will make use of the infinite order polynomials,
$\Xi_{t}(B)$ and $\Xi_{t,q}(B)$, which have been introduced in Definitions
2(IIb) and 3(II), respectively. The next Theorem shows that $\Xi_{t,q}(B)$ is
the time-varying Wold-Cramér polynomial operator associated with the DTV-
ARMA($p,q$) model.

###### Theorem 11

Under the absolute summability condition in (19), the Wold-Cramér operators
are given by

$\Xi_{t}(B)=\Phi_{t}^{-1}(B)\ \ \text{\rm and }\ \
\Xi_{t,q}(B)=\Phi_{t}^{-1}(B)\circ\Theta_{t}(B)\ \ \ \ \mathrm{w.r.t}\ \
``\circ".$ (51)

The Wold-Cramér decomposition in Theorem 3 can be written in terms of the
Wold-Cramér operator as

$y_{t}=\Xi_{t}(B)\circ\varphi(t)+\Xi_{t,q}(B)\varepsilon_{t},$ (52)

which of course implies that

$\mathbb{E(}y_{t})=\Xi_{t}(B)\circ\varphi(t).$

Appendix Section LABEL:APPENDTVARPOLYNOM contains the proofs of both Theorems
10 and 11. In the Online Appendix I, we show how the time-varying techniques,
introduced here, that is, polynomials with the time dependent coefficients
(time-varying polynomials for short) expressed as Hessenbergians (coupled with
the usage of the skew multiplication operator), incorporate as a special case
the standard approach to time series analysis which is based on characteristic
polynomials. In the Online Appendix J, we present a summary of the time-
varying polynomials results.

##  9 An Example

In this Section we will consider the AR($2$) process with $2$ deterministic
abrupt breaks [DAB-AR($2;2$)] at fixed points of time $t_{1}$ and $t_{2}$,
where $t_{1}>t_{2}$.

###### Example 7

The process is defined by

$y_{t}=\left\\{\begin{array}[]{ccc}\varphi_{1}+\phi_{1,1}y_{t-1}+\phi_{2,1}y_{t-2}+\sigma_{1}e_{t}&\text{for}&t>t_{1},\\\
\varphi_{2}+\phi_{1,2}y_{t-1}+\phi_{2,2}y_{t-2}+\sigma_{2}e_{t}&\text{for}&t_{2}<t\leq
t_{1},\\\
\varphi_{3}+\phi_{1,3}y_{t-1}+\phi_{2,3}y_{t-2}+\sigma_{3}e_{t}&\text{for}&t\leq
t_{2},\end{array}\right.$ (53)

where $e_{t}\sim$ i.i.d. $(0,1)$ for all $t$ and $0<\sigma_{i}^{2}\leq M$,
$i=1,2,3$. Applying the results of Theorem 1 the following Corollary provides
the solution representation (at time $t_{1}+l$, $l\in\mathbb{Z}_{\geq 0}$) of
eq. (53). But first, we define the following matrices.

###### Definition 4

Let the two tridiagonal matrices of order $r\in\mathbb{Z}_{\geq 1}$, denoted
by $\Phi_{r}^{(j)}$, $j=1,2$, be defined as

$\mathbf{\Phi}_{r}^{(j)}=\left[\begin{array}[]{cccccc}\phi_{1,j}&-1&&&&\\\
\phi_{2,j}&\phi_{1,j}&-1&&&\\\ &\phi_{2,j}&\phi_{1,j}&-1&&\\\
&&\ddots&\ddots&\ddots&\\\ &&&\phi_{2,j}&\phi_{1,j}&-1\\\
&&&&\phi_{2,j}&\phi_{1,j}\end{array}\right],$

###### Definition 5

i) The tridiagonal matrix (of order $l-r$) $\mathbf{\Phi}_{t_{1}+l,t_{1}+r}$,
for $r=1,\ldots,l-1$ and $l\geq 1$, is defined as:

$\mathbf{\Phi}_{t_{1}+l,t_{1}+r}=\mathbf{\Phi}_{l-r}^{(1)},$

where its determinant is
$\xi(t_{1}+l,t_{1}+r)=\left|\mathbf{\Phi}_{t_{1}+l,t_{1}+r}\right|$ with
initial values $\xi(t_{1}+l,t_{1}+l)=1$ and $\xi(t_{1},t_{1}+r)=0$.
ii) The Hessenberg matrix $\mathbf{\Phi}_{t_{1}+l,t_{1}-r}$, for
$r=0,\ldots,t_{1}-t_{2}$ and $r+l\geq 1$, is defined as:

$\mathbf{\Phi}_{t_{1}+l,t_{1}-r}=\left[\begin{array}[]{cc}\mathbf{\Phi}_{r}^{(2)}&\bar{\mathbf{0}}\\\
\widetilde{\mathbf{0}}&\mathbf{\Phi}_{l}^{(1)}\end{array}\right],$

where (for $r,l\neq 0$) $\bar{\mathbf{0}}$ is an $r\times l$ matrix of zeros
except for $-1$ in its $r\times 1$ entry, and $\widetilde{\mathbf{0}}$ is an
$l\times r$ matrix of zeros except for $\phi_{2,1}$ in its $1\times r$ entry.
Notice that $\mathbf{\Phi}_{t_{1}+l,t_{1}-r}$ is a block square matrix of
order $l+r$. Its determinant is
$\xi(t_{1}+l,t_{1}-r)=\left|\mathbf{\Phi}_{t_{1}+l,t_{1}-r}\right|$ with
initial value $\xi(t_{1},t_{1})=1$.

Applying Theorem 1 to the DAB-AR($2;2$) model we obtain the following
Corollary.

###### Corollary 5

The explicit representation of $y_{t_{1}+l}$ in eq. (53) in terms of the
prescribed random variables $y_{t_{2}}$, $y_{t_{2}-1}$, is given by

$y_{t_{1}+l}=\sum_{r=t_{2}+1}^{t_{1}+l}\xi(t_{1}+l,r)(\varphi(r)+\varepsilon_{r})+\xi(t_{1}+l,t_{2})y_{t_{2}}+\phi_{2,1}\xi(t_{1}+l,t_{2}+1)y_{t_{2}-1},$
(54)

where $\varphi(r)=\left\\{\begin{array}[]{ll}\varphi_{2}&\text{if}\ \ r\leq
t_{1}\\\ \varphi_{1}&\text{if}\ \ r>t_{1}\end{array}\right.$.

###  9.1 Second Moment Structure

In this Section we will examine the second moment structure of the DAB-AR
($2;2$) model. To obtain the time-varying variance of $y_{t_{1}+l}$, we will
directly apply Corollary 5.

First, let
$1-\phi_{1,i}B-\phi_{2,i}B^{2}=(1-\lambda_{1,i}B)(1-\lambda_{2,i}B)$, for
$i=1,2,3$.

Assumption 1 (Second-Order Stationarity): $\left|\lambda_{m,i}\right|<1$,
$m=1,2$, for $i=1,3$.
The above condition implies that the DAB-AR($2;2$)-process is second-order
stationary.

The following Proposition states expressions for the time-varying variance of
$y_{t_{1}+l}$ in eq. (54).

###### Proposition 12

Consider the model in eq. (53). Under Assumption 1, the
$\mathbb{V}ar(y_{t_{1}+l})$ is given by

$\mathbb{V}ar(y_{t_{1}+l})=A_{t_{1}+l}\sigma_{1}^{2}+B_{t_{1}+l}\sigma_{2}^{2}+C_{t_{1}+l}\sigma_{3}^{2},$
(55)

where

$\displaystyle A_{t_{1}+l}$
$\displaystyle=\sum_{r=1}^{l}\xi^{2}(t_{1}+l,t_{1}+r),\text{
}B_{t_{1}+l}=\sum_{r=0}^{t_{1}-t_{2}-1}\xi^{2}(t_{1}+l,t_{1}-r)\text{,}$
$\displaystyle C_{t_{1}+l}$
$\displaystyle=\frac{\left[(1-\phi_{2,3})\left(\xi^{2}(t_{1}+l,t_{2})+\phi_{2,1}^{2}\xi^{2}(t_{1}+l,t_{2}+1)\right)+2\phi_{1,3}\xi(t_{1}+l,t_{2})\phi_{2,1}\xi(t_{1}+l,t_{2}+1)\right]}{(1+\phi_{2,3})[(1-\phi_{2,3})^{2}-\phi_{1,3}^{2}]}\text{.}$

Further, if in the above expression we set: $t_{1}=t_{2}$, and therefore
$\phi_{m,1}=\phi_{m,2}$ for $m=1,2$, and $\sigma_{1}=\sigma_{2}$, then we
obtain the $\mathbb{V}ar(y_{t_{2}+l})$, which is equivalent to the case of one
break (notice that in this case $B_{t_{2}+l}=0$):

$\mathbb{V}ar(y_{t_{2}+l})=A_{t_{2}+l}\sigma_{2}^{2}+C_{t_{2}+l}\sigma_{3}^{2}.$

Finally, if in addition we set $l=0$ then we obtain the
$\mathbb{V}ar(y_{t_{2}})$, which (since $A_{t_{2}}=0$, $\xi_{t_{2},t_{2}}=1$,
$\xi_{t_{2},t_{2}+1}=0$) is the well known formula for the time invariant
AR($2$) model:

$\mathbb{V}ar(y_{t_{2}})=\frac{(1-\phi_{2,3})\sigma_{3}^{2}}{(1+\phi_{2,3})[(1-\phi_{2,3})^{2}-\phi_{1,3}^{2}}.$

In the next Section we will show how the above results can be used to derive a
time-varying second-order measure of persistence.

###  9.2 Time-varying Persistence

The most often applied time invariant measures of first-order (or mean)
persistence are the largest autoregressive root (LAR), and the sum of the
autoregressive coefficients (SUM); see, e.g., Pivetta and Reis (2007). As
pointed out by Pivetta and Reis in relation to the issue of recidivism by
monetary policy its occurrence depends very much on the model used to test the
natural rate hypothesis, i.e., the hypothesis that the SUM or the LAR for
inflation data is equal to one. Obviously, if both measures ignore the
presence of breaks then they will potentially under or over estimate the
persistence in the levels. The LAR has been used to measure persistence in the
context of testing for the presence of unit roots (see, for details, Pivetta
and Reis, 2007).

In the following, we suggest a time-varying second-order (or variance)
persistence measure that is able to take into account the presence of breaks
not only in the mean but in the variance as well. Fiorentini and Sentana
(1998) argue that any reasonable measure of shock persistence should be based
on the IRF’s. For a univariate process $x_{t}$ with i.i.d. errors, $e_{t}$,
they define the persistence of a shock $e_{t}$ on $x_{t}$ as
$P(x_{t}\left|e_{t}\right.)\overset{\mathrm{def}}{=}\displaystyle\frac{\mathbb{V}ar(x_{t})}{\mathbb{V}ar(e_{t})}$.
Clearly $P(x_{t}\left|e_{t}\right.)$ will take its minimum value of one if
$x_{t}$ is white noise and it will not exist (will be infinite) for an I($1$)
process. It follows directly from eq. (55) that
$P(y_{t_{1}+l}\left|\varepsilon_{t_{1}+l}\right.)=\displaystyle\frac{\mathbb{V}ar(y_{t_{1}+l})}{\mathbb{\sigma}_{1}^{2}}$,
is given by

$P(y_{t_{1}+l}\left|\varepsilon_{t_{1}+l}\right.)=A_{t_{1}+l}+B_{t_{1}+l}\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}+C_{t_{1}+l}\frac{\sigma_{3}^{2}}{\sigma_{1}^{2}}.\vspace{-0.1in}$
(56)

If Assumption 1 is violated then conditional measures of second-order
persistence can be constructed using the variance of the forecast error
instead of the unconditional variance (results not reported but are available
upon request).

Having derived explicit formulas for time-varying second-order (or variance)
persistence measures, in the next Section we show the empirical relevance of
these results using U.S. inflation data.141414Cogley and Sargent (2002)
measured persistence by the spectrum at frequency zero, $S_{0}$. As an
example, for the time invariant AR($2$) model this will be given by:
$S_{0}=\displaystyle\frac{\sigma_{\varepsilon}^{2}}{2\pi(1-\phi_{1}-\phi_{2})^{2}}$.

##  10  Inflation Data

In this Section we directly link econometric theory with empirical evidence.
In our empirical application we consider the possible presence of structural
breaks in inflation for the United States. We use quarterly data on the GDP
deflator as the measure of price level. The data set consists of observations
from 1963Q4 to 2018Q1. Inflation is calculated as the quarterly change of
price level at an annualized rate calculated as
$\pi_{t}=400(\ln(P_{t}/P_{t-1}).$

In terms of inflation modelling, the period under consideration is of
particular interest as it covers the boom-time inflation of the late 1960s,
the stagflation in the 1970s, and the double-digit inflation of the early
1980s. During this period substantial shifts in monetary policy occurred, most
notably the Fed’s radical step of switching policy from targeting interest
rates to targeting the money supply in the early 1980s. Therefore when
modelling inflation it is important to allow for time-varying parameters.

###  10.1 Unit Root Tests

Although we allow for regime shifts, we are particularly interested in
modelling changes in inflation persistence. In the related literature