On model fitting and estimation of strictly stationary processes
Marko Voutilainen Lauri Viitasaari Pauliina Ilmonen



Research Article




a]M.Marko Voutilainencor1[label=e1]marko.voutilainen@aalto.fi
[type=corresp,id=cor1]Corresponding author.
b]L.Lauri Viitasaari[label=e2]lauri.viitasaari@iki.fi
a]P.Pauliina Ilmonen[label=e3]pauliina.ilmonen@aalto.fi

[a]Department of Mathematics and Systems Analysis

Aalto University School of Science

P.O. Box 11100, FI-00076 Aalto, Finland
[b]Department of Mathematics and Statistics

University of Helsinki

P.O. Box 68, FI-00014 University of Helsinki, Finland






M. Voutilainen et al.On model fitting and estimation of strictly stationary processes


Stationary processes have been extensively studied in the literature.
Their applications
include modeling and forecasting numerous real life phenomena such as natural
disasters, sales and market movements. When stationary processes are
considered, modeling is traditionally based on fitting an
autoregressive moving average (ARMA) process. However, we challenge
this conventional approach. Instead of fitting an ARMA model, we apply
an AR(1) characterization in modeling any strictly stationary
processes. Moreover, we derive consistent and asymptotically normal
estimators of the corresponding model parameter.



AR(1) representation
asymptotic normality
consistency
estimation
strictly stationary processes


[2010]
60G10
62M09
62M10
60G18


13 9 2017

22 11 2017

25 11 2017

22 12 2017




§ INTRODUCTION

Stochastic processes are widely used in modeling and forecasting
numerous real life phenomena such as natural disasters, activity of the
sun, sales of a company and market movements, to mention a few.
When stationary processes are considered, modeling is traditionally
based on fitting an autoregressive moving average (ARMA) process.
However, in this paper, we challenge this conventional approach.
Instead of fitting an ARMA model, we apply the AR(1) characterization
in modeling any strictly stationary processes. Moreover, we derive
consistent and asymptotically normal estimators of the corresponding
model parameter.




One of the reasons why ARMA processes have been in a central role in
modeling of time-series data is that for every autocovariance function
γ(·) vanishing at infinity and for every n∈ℕ
there exists an ARMA process X such that γ(k) = γ_X(k)
for every k = 0,1,..,n. For a general overview of the theory of
stationary ARMA processes and their estimation, the reader may consult
for example <cit.> or <cit.>.

ARMA processes, and their extensions, have been studied extensively in
the literature.
A direct proof of consistency and asymptotic normality of Gaussian
maximum likelihood estimators for causal and invertible ARMA processes
was given in <cit.>. The result was originally
obtained, using asymptotic properties of the Whittle estimator, in
<cit.>. The estimation of the parameters of
strictly stationary ARMA processes with infinite variances was studied
in <cit.>, again, by using Whittle estimators. Portmanteau tests
for ARMA models with stable Paretian errors with infinite variance were
introduced in <cit.>. An efficient method for
evaluating the maximum likelihood function of stationary vector ARMA
models was presented in <cit.>. Fractionally
integrated ARMA models with a GARCH noise process, where the variance
of the error terms is also of ARMA form, was studied in <cit.>. Consistency and asymptotic normality of the
quasi-maximum likelihood estimators of ARMA models with the noise
process driven by a GARCH model was shown in <cit.>.
A least squares approach for ARMA parameter estimation has been studied
at least in <cit.> by contrasting its
efficiency with the maximum likelihood estimation.
Also estimators of autocovariance and their limiting behavior have been
addressed in numerous papers. See for example <cit.> and <cit.>.













Modeling an observed time-series with an ARMA process starts by fixing
the orders of the model. This is often done by an educated guess, but
there also exists methods for estimating the orders, see e.g. <cit.>. After the orders are fixed, the related
parameters can be estimated, for example, by using the maximum
likelihood or least squares estimators. These estimators are expressed
in terms of optimization problems and do not generally admit closed
form representations. The final step is to conduct various diagnostic
tests to determine whether the estimated model is sufficiently good or
not. These tests are often designed to recognize whether the residuals
of the model support the underlying assumptions about the error terms.
Depending on whether one considers strict or weak stationarity, the
error process is usually assumed to be an IID process or white noise,
respectively. If the tests do not support the assumptions about the
noise process, then one has to start all over again. Tests for the
goodness of fit of ARMA models

have been suggested e.g. in <cit.>. 








The approach taken in this paper is based on the discrete version of
the main theorem of <cit.> leading to an AR(1)
characterization for (any) strictly stationary processes. Note that
this approach covers, but is not limited to, strictly stationary ARMA
processes. 


It was stated in <cit.> that a process is strictly
stationary if and only if for every fixed 0<H<1 it can be represented
in the AR(1) form with ϕ= e^-H and a unique, possibly
correlated, noise term. Although the representation is unique only
after H is fixed, we show that in most of the cases, given just one
value of the autocovariance function of the noise, one is able to
determine the AR(1) parameter and, consequently, the entire
autocovariance function of the noise process. It is worth emphasizing
that since the parameter–noise pair in the AR(1) characterization is
not unique, it is natural that some information about the noise has to
be assumed. Note that conventionally, when applying ARMA models,


we have assumptions about the noise process much stronger than being
IID or white noise.
That is, the autocovariance function of the noise is assumed to be
identically zero except at the origin. When founding estimation on the
AR(1) characterization, one does not have to select between different
complicated models. In addition, there is only one parameter left to be
estimated. Yet another advantage over classical ARMA estimation is that
we obtain closed form expressions for the estimators.






The paper is organized as follows. We begin Section <ref> by
introducing some terminology and notation. After that, we give a
characterization of discrete time strictly stationary processes as
AR(1) processes with a possibly correlated noise term together with
some illustrative examples. The AR(1) characterization leads to
Yule–Walker type equations for the AR(1) parameter ϕ. In this
case, due to the correlated noise process, the equations are of
quadratic form in ϕ. For the rest of the section, we study the
quadratic equations and determine ϕ with as little information
about the noise process as possible.
The approach taken in Section <ref> leads to an estimator of the
AR(1) parameter. We consider estimation in detail in Section <ref>.
The end of Section <ref> is dedicated to
testing the assumptions

we make when constructing the estimators.
A simulation study to assess finite sample properties of the estimators
is presented in Section <ref>.
Finally, we end the paper with three appendices containing a technical
proof, detailed discussion on some special cases and tabulated
simulation results.



§ ON AR(1) CHARACTERIZATION IN MODELING STRICTLY STATIONARY PROCESSES



Throughout the paper we consider strictly stationary processes.


Assume that X = (X_t)_t∈ℤ is a stochastic process. If


    (X_t+n_1, X_t+n_2,…,X_t+n_k ) law= (X_n_1, X_n_2,…,X_n_k )


for all k∈ℕ and t, n_1, n_2,…,n_k ∈ℤ, then
X is strictly stationary.



Assume that G = (G_t)_t∈ℤ is a stochastic process and
denote Δ_t G = G_t - G_t-1. If (Δ_t G
)_t∈ℤ is strictly stationary, then the process G is a
strictly stationary increment process.


The following class of stochastic processes was originally introduced
in <cit.>.



Let H > 0 be fixed and let G= (G_t)_t∈ℤ be a
stochastic process. If G is a strictly stationary increment process
with G_0 = 0 and if the limit


    lim_k→-∞∑_t=k^0
    e^tHΔ_t G


exists in probability and defines an almost surely finite random
variable, then G belongs to the class of converging strictly
stationary increment processes, and we denote G∈𝒢_H.


Next, we consider the AR(1) characterization of strictly stationary
processes. The continuous time analogy was proved in <cit.> together with a sketch of a proof for the discrete case.
For the reader's convenience, a detailed proof of the discrete case is
presented in Appendix <ref>.



Let H >0 be fixed and let X = (X_t)_t∈ℤ be a
stochastic process. Then X is strictly stationary if and only if lim_t→-∞e^tHX_t = 0 in probability and


    Δ_t X = (e^-H - 1
    )X_t-1 + Δ_t G


for a unique G∈𝒢_H.



Let H>0 be fixed. Then every discrete time strictly stationary
process (X_t)_t∈ℤ can be represented as


    X_t - ϕ^(H) X_t-1 =
    Z_t^(H),


where ϕ^(H) = e^-H and Z_t^(H) = Δ_t G is another
strictly stationary process.


It is worth to note that the noise Z in Corollary <ref> is
unique only after the parameter H is fixed. The message of this
result is that every strictly stationary process is an AR(1) process
with a strictly stationary noise that may have a non-zero
autocovariance function. The following examples show how some
conventional ARMA processes can be represented as an AR(1) process.



Let X be a strictly stationary AR(1) process defined by


    X_t - φ X_t-1 = ϵ_t,     (
    ϵ_t)∼ IID(0, σ^2)


with φ>0. Then we may simply choose ϕ^(H) = φ and
Z_t^(H) = ϵ_t.




Let X be a strictly stationary ARMA(1,q) process defined by


    X_t - φ X_t-1 = ϵ_t +
    θ_1ϵ_t-1+⋯ +θ_q
    ϵ_t-q,      (ϵ_t )∼ IID(0,
    σ^2)


with φ>0. Then we may set ϕ^(H) = φ, and
Z_t^(H) then equals to the MA(q) process.



Consider a strictly stationary AR(1) process X with φ<0.
Then X admits an MA(∞) representation


    X_t = ∑_k=0^∞φ^k ϵ_t-k.


From this it follows that


    Z_t^(H) = ϵ_t + ∑
    _k=0^∞φ^k (φ-ϕ
    ^(H))ϵ_t-1-k


Hence


    Z_t^(H)   = ∑
    _k=0^∞φ^kϵ_t-k
    - ϕ^(H)∑_k=0^∞φ^k ϵ_t-1-k
       =ϵ_t + ∑_k=0^∞φ^k+1ϵ_t-1-k - ϕ ^(H)∑
    _k=0^∞φ^k ϵ_t-1-k
       =ϵ_t + ∑_k=0^∞φ^k (φ-ϕ^(H))
    ϵ_t-1-k.


Let var(ϵ_t) = σ^2. Then, for t>0 it holds that


    cov(Z_t^(H),
    Z_0^(H))    = φ^t-1 (φ- )
    σ^2
    
         + cov(∑_k=t^∞φ^k (φ- )ϵ_t-1-k, ∑
    _k=0^∞φ^k (φ- )
    ϵ_-1-k).


The covariance on the on right-hand side can be written as


    cov(∑
    _n=-1^-∞φ^t-n-1 (φ- )
    ϵ_n, ∑_n=-1^-∞φ^-n-1 (φ- )ϵ_n ).
    
       = φ^t-2 (φ- )^2σ^2 ∑
    _n=1^∞φ^2n.


Since X was stationary, the sum above converges and we obtain that
and


    cov(Z_t^(H), Z_0^(H)) = φ^t-2 (φ- )σ^2 (φ+
    (φ-)∑_n=1^∞(
    φ^2)^n ).


Hence in the case of an AR(1) process with a negative parameter, the
autocovariance function of the noise Z of the representation (<ref>) is non-zero everywhere.


Next we show how to determine the AR(1) parameter ϕ^(H) in
(<ref>) provided that the observed process X is known. In what
follows, we omit the superindices in (<ref>). We assume that the
second moments of the considered processes are finite and that the
processes are centered. That is, (X_t) = (Z_t) = 0 for every t∈ℤ. Throughout the rest of the paper, we use the notation
cov(X_t, X_t+n) = γ(n) and cov(Z_t,Z_t+n) = r(n) for every
t,n∈ℤ.



Let centered (X_t)_t∈ℤ be of the form (<ref>). Then


    ϕ^2 γ(n) - ϕ(γ(n+1) +
    γ(n-1) ) + γ (n) - r(n) = 0


for every n∈ℤ.




Let n∈ℤ. By multiplying both sides of


    X_n - ϕ X_n-1 = Z_n


with Z_0 = X_0 - ϕ X_-1 and taking expectations, we obtain


    (X_n(X_0- ϕ
    X_-1) )- ϕ(X_n-1(X_0 - ϕ
    X_-1) )
    
         = ϕ^2γ(n) - ϕ(γ(n+1) + γ(n-1))+
    γ(n) = r(n).




    γ(k) - ϕγ(k-1)    =
    (Z_tX_t-k)
    
       = (Z_t(Z_t-k+ϕ X_t-k-1))
    
       = r(k) + ϕ(Z_t(Z_t-k-1 + ϕ X_t-k-2)
    )
    
       =∑_i=0^∞ϕ^i r(i+k).


By using k=n+2 in the equation above and multiplying the both sides
with ϕ, we obtain


    ϕγ(n+2) - ϕ^2γ(n+1) = ∑
    _i=1^∞ϕ^i r(i+n+1).


Let us next subtract the equation above from (<ref>) with k =
n+1 yielding


    ϕ^2 γ(n+1) -ϕ(γ(n+2) + γ(n)) +
    γ(n+1) = r(n+1).






Let centered (X_t)_t∈ℤ be of the form (<ref>) and
let N∈ℕ be fixed.




  (1) If γ(N) ≠0, then either


    ϕ = γ(N + 1)  + γ(N - 1)  + √( (γ(N + 1) + γ
    (N - 1) )^2  -  4γ(N)(γ(N)  -  r(N)))/2γ(N)


or


    ϕ = γ(N + 1)  + γ(N - 1)  - √( (γ(N + 1) + γ
    (N - 1) )^2  -  4γ(N)(γ(N)  -  r(N)))/2γ(N).



  (2) If γ(N) = 0 and r(N) ≠0, then


    ϕ= -r(N)/γ(N+1) + γ(N-1).






Note that if γ(N) = r(N) = 0, then Lemma <ref> yields only γ(N+1) + γ(N-1) = 0 providing
no information about the parameter ϕ. As such, in order to
determine the parameter ϕ, we require that either γ(N) ≠
0 or r(N) ≠0.


If the variance r(0) of the noise is known, then (<ref>) and
(<ref>) reduces to


    ϕ= γ(1) ±√(γ(1)^2 - γ(0) (γ(0)
    - r(0) ))/γ(0).




At first glimpse it seems that Corollary <ref> is
not directly applicable. Indeed, in principle it seems like there could
be complex-valued solutions although representation (<ref>)
together with (<ref>) implies that there exists a solution
ϕ∈(0,1). Furthermore, it is not clear whether the true value is
given by (<ref>) or (<ref>). We next address these
issues. We start by proving that the solutions to (<ref>)
cannot be complex. At the same time we are able to determine which one
of the solutions one should choose.



The discriminants of (<ref>) and (<ref>) are always
non-negative.



Let k∈ℤ. By multiplying both sides of (<ref>) with
X_t-k, taking expectations, and applying (<ref>) repeatedly we obtain


    γ(k) - ϕγ(k-1)    = (Z_tX_t-k) = (Z_t(Z_t-k+ϕ X_t-k-1))
    
       = r(k) + ϕ(Z_tX_t-k-1 )
    
       = r(k) + ϕ(Z_t(Z_t-k-1 + ϕ X_t-k-2)
    )
    
       =r(k) + ϕ r(k+1) + ϕ^2 (Z_t X_t-k-2
    ).


Proceeding as above l times we get


    γ(k) - ϕγ(k-1) = ∑_i=0^l-1ϕ^i r(k+i) + ϕ ^l(Z_t(ϕ
    X_t-k-l-2)).


Letting l approach infinity leads to


    γ(k) - ϕγ(k-1) = ∑_i=0^∞ϕ^i r(k+i),


where the series converges as r(k+i)≤ r(0) and 0<ϕ<1. It now
follows from (<ref>) that


    γ(N)    = ϕγ(N-1) + ∑
    _i=0^∞ϕ^i r(N+i)
    
       =ϕγ(N-1) +r(N)+ϕ∑_i=1^∞ϕ^i-1r(N+i)
    
       =ϕγ(N-1) +r(N)+ϕ∑_i=0^∞ϕ^ir(N+i+1)
    
       = ϕγ(N-1) + ϕ(γ(N+1) - ϕγ(N) )+r(N).


Denote the discrimant of (<ref>) and (<ref>) by D.
That is,


    D = (γ(N-1)+γ(N+1) )^2 - 4γ(N) (γ
    (N)-r(N) ).


By using the equation above we observe that


    D    = (γ(N) + ϕ^2γ(N)-r(N)/ϕ)^2 -
    4γ(N) (γ(N)-r(N)).


Denoting a_N = r(N)/γ(N), multiplying by ϕ
^2/γ(N)^2, and using the identity


    (a+b)^2 -4ab = (a-b)^2


yields


    ϕ^2/γ(N)^2D = (1+ϕ^2-a_N
    )^2 - 4ϕ^2(1-a_N) = (ϕ
    ^2-1+a_N)^2 ≥0.


This concludes the proof.


Note that if r(N) = 0, as ϕ<1, the discriminant is always
positive. Let a_N = r(N)/γ(N). The proof above now gives
us the following identity


    ϕ= 1/2ϕ(1+ϕ^2-a_N±|γ(N)|/γ
    (N)ϕ^2-1+a_N ).


This enables us to consider the choice between (<ref>) and
(<ref>). Assume that γ(N) > 0. If ϕ^2 - 1 + a_N >
0, then ϕ is given by (<ref>) (as ϕ∈(0,1)).
Similarly, if ϕ^2 - 1 +a_N<0, then ϕ is determined by (<ref>). Finally, contrary conclusions hold in the case γ(N)
< 0. In particular, we can always choose between (<ref>) and
(<ref>) provided that either a_N ≤0 or a_N ≥1.


Moreover, from (<ref>) it follows that


    r(N)/γ(N) = r(N+k)/γ(N+k)


if and only if


    γ(N+1)+γ(N-1)/γ(N) = γ(N+1+k)+γ
    (N-1+k)/γ(N+k),


provided that the denominators differ from zero. Since (<ref>)
and (<ref>) can be written as


    ϕ   = γ(N+1)+γ(N-1)/2γ(N)
         ±1/2sgn(γ(N))√((γ(N+1)+γ(N-1)/γ
    (N))^2-4 (1-r(N)/γ(N))),


we observe that one can always rule out one of the solutions (<ref>) and (<ref>) provided that a_N ≠ a_N+k.
Therefore, it always suffices to know two values of the autocovariance
r such that a_N ≠ a_N+k, except the worst case scenario where
a_j = a ∈(0,1) for every j∈ℤ. A detailed analysis of
this particular case is given in Appendix <ref>.


Consider a fixed strictly stationary process X. If we fix one value
of the autocovariance function of the noise such that Corollary <ref> yields an unambiguous AR(1) parameter, then the
quadratic equations (<ref>) will unravel the entire
autocovariance function of the noise process. In comparison,
conventionally, the noise is assumed to be white — meaning that the
entire autocovariance function of the noise is assumed to be known a priori.


We end this section by observing that in the case of vanishing
autocovariance function of the noise, we get the following simplified
form for the AR(1) parameter.



Let centered (X_t)_t∈ℤ be of the form (<ref>) and
let N∈ℕ be fixed. Assume that r(m)=0 for every m≥
N. If γ(N-1)≠0, then for every n≥ N, we have


    ϕ= γ(n)/γ(n-1).


In particular, γ admits an exponential decay for n≥ N.





Let γ(N-1)≠0. It follows directly from (<ref>) and the
assumptions that


    γ(n) = ϕγ(n-1)  for every n≥ N.


The condition γ(N-1)≠0 now implies the claim.


Recall that the representation (<ref>) is unique only after
H is fixed. As a simple corollary for Theorem <ref> we
obtain the following result giving some new information about the
uniqueness of the representation (<ref>).


Let X be a strictly stationary process with a non-vanishing
autocovariance. Then there exists at most one pair (H, G) satisfying
(<ref>) such that the non-zero part of the autocovariance
function of the increment process (Δ_t G)_t∈ℤ is finite.





Assume that there exists H_1, H_2 > 0, and G_1∈𝒢_H_1
and G_2∈𝒢_H_2 such that the pairs (H_1, G_1) and
(H_2, G_2) satisfy (<ref>) and the autocovariances of
(Δ_t G_1)_t∈ℤ and (Δ_t G_2)_t∈ℤ
have cut-off points. From Theorem <ref> it follows that H_1
= H_2 and since for a fixed H the process G in (<ref>) is
unique, we get G_1 = G_2.




§ ESTIMATION

Corollary <ref> gives natural estimators for ϕ provided
that we have been able to choose between (<ref>) and (<ref>), and that a value of r(n) is known. We emphasize that in
our model it is sufficient to know only one (or in some cases two) of
the values r(n), whereas in conventional ARMA modeling much stronger
assumptions are required. (In fact, in conventional ARMA modeling the
noise process is assumed to be white noise.) It is also worth to
mention that, generally, estimators of the parameters of stationary
processes are not expressible in a closed form. For example, this is
the case with the maximum likelihood and least squares estimators of
conventionally modeled ARMA processes, see <cit.>. Within
our method, the model fitting is simpler. 


Finally, it is worth to note that

assumption of one known value of r(n) is a natural one
and cannot be avoided. Indeed, this is a direct consequence of the fact
that the pair (ϕ,Z) in representation (<ref>) is not unique.
In fact, for practitioner, it is not absolutely necessary to know any
values of r(n). The practitioner may make an educated guess and
proceed in estimation. If the obtained estimate then turns out to be
feasible, the practitioner can stop there. If the obtained estimate
turns out to be unreasonable (not on the interval (0,1)), then the
practitioner have to make another educated guess. The process is
similar to selecting p and q in traditional ARMA(p,q) modeling.

Throughout this section, we assume that (X_1, …, X_T) is an
observed series from a centered strictly stationary process that is
modeled using the representation (<ref>). We use (n) to
denote an estimator of the corresponding autocovariance γ(n).
For example, (n) can be given by


    (n) = 1/T∑_t=1^T-n
    X_tX_t+n,


or more generally


    (n) = 1/T∑_t=1^T-n
    (X_t-X̅ ) (X_t+n-X̅ ),


where X̅ is the sample mean of the observations.
For this estimator the corresponding sample covariance (function)
matrix is positive semidefinite. On the other hand, the estimator is
biased while it is asymptotically unbiased. Another option is to use
T-n-1 as a denominator. In this case one has an unbiased estimator,
but the sample covariance (function) matrix is no longer positive
definite. Obviously, both estimators have the same asymptotic
properties. Furthermore, for our purposes it is irrelevant how the
estimators (n) are defined, as long as they are consistent, and
the asymptotic distribution is known.

We next consider estimators of the parameter ϕ arising from
characterization (<ref>). In this context, we pose some
assumptions related to the autocovariance function of the observed
process X. The justification and testing of these assumptions are
discussed in Section <ref>. From a priori
knowledge that ϕ∈(0,1) we enforce also the estimators to the
corresponding closed interval. However, if one prefers to use unbounded
versions of the estimators, one may very well do that. The asymptotic
properties are the same in both cases. We begin by defining an
estimator corresponding to the second part (2) of Corollary <ref>.





For any h∈ℕ the asymptotic joint distribution of


    √(T)((0)-γ(0), (1)-γ(1), …, (h)-γ (h)
    )


is multivariate normal with zero expectation.







The estimator related to <ref> is asymptotically normal with
the rate of convergence of √(T).


Let us denote


    X_T = √(T)(γ̂_T(n+1) -
    γ_T(n+1)),    Y_T = √(T)(
    γ(n) - γ̂_T(n)).


By (**) the asymptotic joint distribution of (X_T, Y_T) is normal and
hence, we write


    (X_T, Y_T) law⟶𝒩( 
    [ 0; 0 ] 
    , 
    [ σ_X^2  σ_XY;  σ_XY σ_Y^2 ])


Now let us study the asymptotic behavior of


    √(T)(γ̂_T(n+1)/(n)- γ
    (n+1)/γ(n))
    
    
    =   1/(n)√(T)((n+1) - γ(n+1)) + √(T)(γ(n+1)/(n) - γ(n+1)/γ(n))
    
    
    =   1/(n)( √(T)((n+1) - γ(n+1)) +
    γ(n+1)/γ(n)√(T)(γ(n)- (n) ) )
    
    
    =   1/(n)(X_T + γ(n+1)/γ(n)Y_T
    ),


where


    (X_T, γ(n+1)/γ(n)Y_T ) law⟶𝒩( 
    [ 0; 0 ]
    
    , 
    [                 σ_X^2       γ(n+1)/γ(n)σ_XY;       γ(n+1)/γ(n)σ_XY γ(n+1)^2/γ
    (n)^2σ_Y^2 ])


and hence


    X_T + γ(n+1)/γ(n)Y_T law⟶𝒩(0, σ_X^2 +
    γ
    (n+1)^2/γ(n)^2σ_Y^2 + 2γ(n+1)/γ(n)σ
    _XY).


By Slutsky's theorem, we may conclude that


    1/(n)(X_T + γ(n+1)/γ(n)Y_T
    ) law⟶𝒩(0,
    σ
    _X^2/γ(n)^2 + γ(n+1)^2/γ(n)^4σ_Y^2 + 2
    γ(n+1)/γ(n)^3σ_XY).









Assume that γ(N) = 0. Then we define


    ϕ̂_T = -r(N)/γ̂_T(N+1) + γ̂_T(N-1)1_(N+1)+(N-1) ≠0


whenever the right-hand side lies on the interval [0,1]. If the
right-hand side is below zero, we set ϕ̂_T = 0 and if the
right-hand side is above one, we set ϕ̂_T = 1.




Assume that γ(N) =0 and r(N)≠0. If the
vector-valued estimator
 [(N+1), (N-1) ]^⊤ is
consistent, then ϕ̂_T is consistent.



Since γ(N) = 0 and r(N) ≠0, Equation (<ref>)
guarantees that γ(N+1) + γ(N-1) ≠0. Therefore
consistency of ϕ̂_T follows directly from the continuous
mapping theorem.




Let ϕ̂_T be given by (<ref>), and assume that
γ(N) = 0 and r(N) ≠0. Set γ =  [γ
(N+1), γ(N-1) ]^⊤ and γ̂_T =  [
(N+1),(N-1) ]^⊤. If


    l(T) (γ̂_T - γ )law⟶𝒩 (0,  )


for some covariance matrix  and some rate function l(T), then


    l(T) (ϕ̂_T - ϕ ) law⟶𝒩(0, ∇ f(γ)^⊤∇
    f(γ) ),






where ∇ f(γ) is given by


    ∇ f(γ) = -r(N)/ (γ(N+1) + γ
    (N-1) )^2·[ 1; 1 ] 
    .





For the simplicity of notation, in the proof we use the unbounded
version of the estimator ϕ̂_T. Since the true value of ϕ
lies strictly between 0 and 1, the very same result holds also for the
bounded estimator of Definition <ref>. Indeed, this is a
simple consequence of the Slutsky's theorem. To begin with, let us
define an auxiliary function f by


    f(x) = f(x_1,x_2) = r(N)/x_1 + x_21_x_1+x_2
    ≠0.


If x_1+x_2≠0, the function f is smooth in a neighborhood of
x. Since γ(N) = 0 together with r(N) ≠0 implies
that γ(N+1) + γ(N-1) ≠0, we may apply the delta method
at x = γ to obtain



    l(T) (ϕ̂_T - ϕ ) = -l(T) (f(γ̂_T) - f(γ) )law⟶𝒩(0, ∇ f(γ)^⊤∇
    f(γ) ),


where ∇ f(γ) is given by (<ref>).
This concludes the proof.



By writing


    = 
    [ σ_X^2  σ_XY;  σ_XY σ_Y^2 ]


the variance of the limiting random variable reads


    r(N)^2/ (γ(N+1) + γ(N-1) )^4(σ_X^2 + 2
    σ_XY + σ_Y^2).





In many cases the convergency rate is the best possible, that is l(T)
= √(T). However, our results are valid with any rate function. One
might, for example in the case of many long memory processes, have
other convergency rates for the estimators (n).


We continue by defining an estimator corresponding to the first part
(1) of the Corollary <ref>. For this we assume that, for
reasons discussed in Section <ref>, we have chosen the solution
(<ref>) (cf. Remark <ref> and Section <ref>).
As above, we show that consistency and asymptotic
normality follow from the same properties of the autocovariance
estimators. In the sequel we use a short notation


    g(x) = g(x_1,x_2,x_3)
    = (x_1+x_3 )^2-4x_2
    (x_2-r(N)).


In addition, we denote


    γ = [γ(N+1), γ(N), γ(N-1) ]^⊤


and


    γ̂_T = [(N+1),(N),(N-1)
    ]^⊤.




Assume that γ(N) ≠0. We define an estimator for ϕ
associated to (<ref>) by


    ϕ̂_T = γ̂_T(N+1) + γ̂_T(N-1) + √(g (γ̂_T ))1_g (_T
    )>0/2γ̂_T(N)1_(N) ≠0


whenever the right-hand side lies on the interval [0,1]. If the
right-hand side is below zero, we set ϕ̂_T = 0 and if the
right-hand side is above one, we set ϕ̂_T = 1.




Assume that γ(N)≠0 and g(γ)>0. Furthermore,
assume that ϕ is given by (<ref>). If γ̂_T is consistent, then ϕ̂_T is consistent.



As g(γ)>0, the result is again a simple consequence of the
continuous mapping theorem.


Before proving the asymptotic normality, we present some short
notation. We set


    C_N = γ(N+1)+γ(N-1)+√(g(γ))/γ(N)


and


    _ϕ   =
    1/4γ(N)^2( (∇√(g(γ)))^⊤∇√(g(γ))+ 2 
    [    1; -C_N;    1 ] 
    ^⊤∇√(g(γ)).
         .+ 
    [    1; -C_N;    1 ] 
    ^⊤[    1; -C_N;    1 ]),


where


    ∇√(g(γ)) = 1/√(g(γ))[ γ(N+1) + γ(N-1); 2(r(N) - 2γ(N)); γ(N+1) + γ(N-1) ] 
    .




Let the assumptions of Theorem <ref> prevail. If


    l(T) (γ̂_T - γ )law⟶𝒩 (0,  )


for some covariance matrix  and some rate function l(T), then
l(T) (ϕ̂_T - ϕ ) is asymptotically normal with
zero mean and variance given by
(<ref>).









The proof follows the same lines as the proof of Theorem <ref>
but for the reader's convenience, we present the details. Furthermore,
as in the proof of Theorem <ref>, since the true value of ϕ
lies strictly between 0 and 1,
for the notational simplicity,
we may and will use the unbounded version of the estimator. Indeed, the
asymptotics for the bounded version then follow directly from the
Slutsky's theorem. We have


    (γ̂_T(N+1)1_(N) ≠0/(N)- γ(N+1)/γ(N))
    
         =1/(N)((N+1)1_(N) ≠0 - γ (N+1)
    ) + (γ(N+1)/(N) - γ
    (N+1)/γ(N))
    
         = 1/(N)( (N+1)1_(N) ≠0 - γ (N+1) -
    γ(N+1)/γ(N)((N)- γ(N) ) ).


Similarly


    (γ̂_T(N-1)1_(N) ≠0/(N)- γ(N-1)/γ(N))
    
         = 1/(N)( (N-1)1_(N) ≠0 - γ (N-1) -
    γ(N-1)/γ(N)((N)- γ(N) ) )


and


    (√(g(γ̂_T))1_g (
    _T )>01_(N) ≠0/(N)- √(g(γ))/γ(N))
    
         = 1/(N)( √(g(γ̂_T))1_g (_T )>01_(N) ≠0 - √(g(γ)) -
    √(g(γ))/γ(N)((N)- γ(N) ) ).


For C_N given in (<ref>) we have


    l(T) (ϕ̂_T-ϕ ) =   l(T)/2γ̂(N)((N+1)- γ(N+1)
    
       + (N-1)- γ(N-1) -C_N ((N)-γ(N) )
    
       + √(g(γ̂_T))- √(g(γ))).


By defining


    h(x) = h(x_1,x_2,x_3) =
    (x_1 + x_3 + √(g(x))1_g(x) >0)1_x_2≠0 - C_N
    x_2


we have


    l(T) (ϕ̂_T-ϕ ) = l(T)/2γ̂_T(N)(h (γ̂_T ) - h(γ) ).


If x_2 ≠0 and g(x) > 0, the function h is smooth in a
neighborhood of x. Therefore we may apply the delta method at
x = γ to obtain


    l(T) (h (γ̂_T ) - h(γ) )
    law⟶𝒩(0, ∇ h(γ)^⊤∇ h(γ) ),


where


    ∇ h(γ)^⊤∇ h(
    γ)    = ( 
    [    1; -C_N;    1 ] 
    + ∇√(g(γ)))^⊤( 
    [    1; -C_N;    1 ] 
    + ∇√(g(γ)))
    
       = (∇√(g(γ)))^⊤∇√(g(γ)) + 2 
    [    1; -C_N;    1 ] 
    ^⊤∇√(g(γ))
          + 
    [    1; -C_N;    1 ] 
    ^⊤[    1; -C_N;    1 ] 
    .


Hence (<ref>) and Slutsky's theorem imply that l(T) (ϕ̂_T - ϕ ) is asymptotically normal with zero mean and variance
given by (<ref>).




One straightforwardly observes the same limiting behavior as in
Theorems <ref> and <ref> for the estimator related to (<ref>). This fact also can be used to determine which one of
Equations (<ref>) and (<ref>) gives the correct ϕ (cf. Section <ref>).




If γ(N) ≠0 and g(γ) = 0 we may define an estimator


    ϕ̂_T = (N+1) + (N-1)/2(N).


Assuming that


    l(T) (γ̂_T - γ )law⟶𝒩 (0,  )


it can be shown similarly as in the proofs of Theorems <ref> and
<ref> that


    l(T) (ϕ̂_T - ϕ ) law⟶𝒩(0, 1/4γ(N)^2[                     1; -γ(N+1) + γ(N-1)/γ(N);                     1 ] 
    ^⊤[                     1; -γ(N+1) + γ(N-1)/γ(N);                     1 ])





The estimator related to Theorem <ref> reads


    ϕ̂_T = γ̂_T(n+1)/γ̂_T(n)1_(n) ≠0,


where we assume that γ(n)≠0.
By using the same techniques as earlier, it can be shown that if


    l(T) (γ̂_T(n+1) - γ(n+1), γ̂_T(n)- γ (n) ) law⟶𝒩( 0, 
    [ σ_X^2  σ_XY;  σ_XY σ_Y^2 ]),


then


    l(T) (ϕ̂_T - ϕ ) law⟶𝒩(0, σ_X^2/γ
    (n)^2 + γ(n+1)^2/γ(n)^4σ_Y^2 - 2γ
    (n+1)/γ(n)^3σ_XY).




Note that the asymptotics given in Remarks <ref> and <ref>
hold also if one forces the corresponding estimators to the interval
[0,1] as we did in Definitions <ref> and <ref>.




The asymptotics of the square root term of the estimator is given by


    √(T)(g(γ̂_T) - g(γ) )
    law⟶𝒩(0, ∇ g(γ)^⊤∑∇ g(γ) )








 §.§ Testing the underlying assumptions


When choosing the estimator that corresponds the situation at hand, we
have to make assumptions related to the values of γ(N) (for some
N) and g(γ). In addition, we have to consider the
question of the choice between (<ref>) and (<ref>).

Let us first discuss how to test the null hypothesis that γ(N) =
0. If the null hypothesis holds, then by asymptotic normality of the
autocovariances, we have that


    l(T)(N)law⟶𝒩(0,σ^2 )


with some σ^2. Hence we may use


    (N) ∼_a 𝒩(0, σ^2/l(T)^2)


as a test statistics. A similar approach can be applied also when
testing the null hypothesis that g(γ) = 0, where g is
defined by (<ref>). The alternative hypothesis is of the form g(γ) > 0. Assuming that the null hypothesis holds, we obtain by
the delta method that


    l(T) (g(γ̂_T) - g(γ) )
    law⟶𝒩(0, σ̃^2 )


for some σ̃^2 justifying the use of


    g(γ̂_T) ∼_a 𝒩(0,
    σ̃^2/l(T)^2)


as a test statistics. If the tests above suggest that γ(N) ≠
0 and g(γ)> 0, then the choice of the sign can be based
on the discussion in Section <ref>. Namely, if for the ratio a_N = r(N)/γ(N) it holds that a_N ≤0 or a_N ≥1, then the
sign is unambiguous. The sign of γ(N) can be deduced from the
previous testing of the null hypothesis γ(N) = 0. By (<ref>), if necessary, one can test the null hypothesis γ(N) =
r(N) using the test statistics








    (N) ∼_a 𝒩(r(N), σ^2/l(T)^2),


where the alternative hypothesis is of the form r(N)/γ(N)
< 1. Finally, assume that one wants to test if the null hypothesis
a_N = a_k holds. By the delta method we obtain that


    l(T) (â_N-â_k - a_N
    +a_k )law⟶𝒩(0,
    σ̅^2 )


for some σ̅^2 suggesting that


    â_N-â_k ∼_a 𝒩(0, σ̅^2/l(T)^2)


could be utilized as a test statistics.



§ SIMULATIONS


We present a simulation study to assess the finite sample performance
of the estimators. In the simulations, we apply the estimator
corresponding to the first part (1) of Corollary <ref>. 






We simulate data from AR(1) processes and ARMA(1,2) processes with
θ_1 = 0.8 and θ_2 = 0.3 as the MA parameters. (Note that
these processes correspond to Examples <ref> and <ref>.) We
assess the effects of the sample size T, AR(1) parameter φ,
and the chosen lag N. We consider the sample sizes T=50, 500, 5000,
50000, lags N=1, 2, 3, …, 10, and the true parameter values
φ= 0.1, 0.2, 0.3, …, 0.9. For each combination, we simulate
1000 draws. The sample means of the obtained estimates are tabulated in
Appendix <ref>.

Histograms given in Figures <ref>, <ref> and <ref> reflect the effects of the sample size T, AR(1) parameter
φ, and the chosen lag N, respectively. In Figure <ref>, the parameter φ= 0.5 and the lag N = 3. In
Figure <ref>, the sample size T = 5000 and the lag N = 3.
In Figure <ref>, the parameter φ= 0.5 and the sample size
T = 5000. The summary statistics corresponding to the data displayed
in the histograms are given in Appendix <ref>.










Figure <ref> exemplifies the rate of convergence of the
estimator as the number of observations grows. One can see that with
the smallest sample size, the lower bound is hit numerous times due to
the large variance of the estimator. In the upper series of the
histograms, the standard deviation reduces from 0.326 to 0.019,
whereas in the lower series it reduces from 0.250 to 0.008. The
faster convergence in the case of ARMA(1,2) can be explained with the
larger value of γ(3) reducing the variance in comparison to the
AR(1) case. The same phenomenon recurs also in the other two figures.

Figure <ref> reflects the effect of the AR(1) parameter on
the value of γ(3) and consequently on the variance of the
estimator. The standard deviation reduces from 0.322 to 0.020 in
the case of AR(1) and from 0.067 to 0.009 in the case of ARMA(1,2).

In Figure <ref> one can see how an increase in the lag increases
the variance of the estimator. In the topmost sequence, the standard
deviation increases from 0.014 to 0.326 and in the bottom sequence
from 0.015 to 0.282.

We wish to emphasize that in general smaller lag does not imply smaller
variance, since the autocovariance function of the observed process is
not necessarily decreasing. In addition, although the autocovariance
γ(N) appears to be the dominant factor when it comes to the
speed of convergence, there are also other possibly significant terms
involved in the limit distribution of Theorem <ref>.

















§ PROOF OF THEOREM <REF>

We provide here a detailed proof of Theorem <ref>. The continuous
time version of the theorem was recently proved in <cit.>
and we loosely follow the same lines in our proof for the discrete time version.


Let H>0. A discrete time stochastic process Y = (Y_e^t)_t∈ℤ with lim_t→-∞ Y_e^t = 0 is H-self-similar if


    (Y_e^t+s )_t∈ℤlaw=(e^sH Y_e^t)_t∈ℤ


for every s∈ℤ in the sense of finite-dimensional distributions.



Let H>0. In addition, let X = (X_t)_t∈ℤ and Y =
(Y_e^t)_t∈ℤ be stochastic processes. We define the
discrete Lamperti transform by


    (ℒ_H X)_e^t = e^tHX_t


and its inverse by


    (ℒ_H^-1 Y)_t =
    e^-tHY_e^t.






If X = (X_t)_t∈ℤ is strictly stationary, then (ℒ_H X)_e^t is H-self-similar. Conversely, if Y= (Y_e^t)_t∈ℤ is H-self-similar, then (ℒ_H^-1 Y)_t is
strictly stationary.






Assume that (X_e^n)_n∈ℤ is H-self-similar and let us
denote Δ_n X_e^n = X_e^n - X_e^n-1. Then, for every n,
l∈ℤ it holds that


    Δ_n-l X_e^n-llaw=
    e^-HlΔ_n X_e^n.



Since there is one-to-one correspondence between H-self-similar and
stationary processes, for some stationary U it holds that


    X_e^n - X_e^n-1   =
    e^HnU_n - e^H(n-1)U_n-1
       law= e^Hn U_n-1 - e^H(n-1)
    U_n-2
       = e^H ( X_e^n-1 - X_e^n-2).


Repeating the step above l times concludes the proof.








Let H>0 and assume that (Y_e^t)_t∈ℤ is
H-self-similar. Let us denote Δ_t Y_e^t = Y_e^t -
Y_e^t-1. Then the process (G_t)_t∈ℤ defined by


    G_t = {[    ∑_k=1^t e^-kHΔ_k Y_e^k,                        t≥1;                         0,                        t=0; -∑_k=t+1^0 e^-kHΔ_k Y_e^k,                       t≤-1 ].


belongs to 𝒢_H.


By studying the cases t≥2, t=1, t=0 and t≤-1 separately, it
is straightforward to see that


    Δ_t G = e^-tHΔ_t
    Y_e^t  for every t∈ℤ.


Now


    lim_k→-∞∑_t = k^0
    e^tHΔ_t G = lim_k→
    -∞∑
    _t=k^0 Δ_t
    Y_e^t = Y_e^0 - lim_k→-∞
    Y_e^k


and since Y is self-similar, we have


    Y_e^k e^kH Y_e^0.


Thus


    lim_k→-∞ Y_e^k = 0


in distribution, and hence also in probability. This implies that


    ∑_t = -∞^0 e^tHΔ_t G


is an almost surely finite random variable. Next we show that G has
strictly stationary increments. For this, assume that t,s,l ∈ℤ with t>s are arbitrary. Then


    G_t - G_s    = ∑
    _k=s+1^t Δ_k G= ∑
    _k=s+1^t e^-kHΔ_k
    Y_e^k = ∑_j = s+l+1^t+l
    e^-(j-l)HΔ_j-l Y_e^j-l
       law=∑_j = s+l+1^t+l
    e^-jHΔ_j Y_e^j = G_t+l -
    G_s+l,


where the equality in law follows from H-self-similarity of
(Y_e^t)_t∈ℤ. Treating n-dimensional vectors
similarly concludes the proof.





Assume first that X is strictly stationary. In this case X clearly
satisfies the limit condition. In addition, there exists a
H-self-similar Y such that


    Δ_t X    = e^-tH
    Y_e^t - e^-(t-1)HY_e^t-1
       = (e^-H - 1 )e^-(t-1)H Y_e^t-1 +
    e^-tH(Y_e^t - Y_e^t-1)
    
       = (e^-H - 1 )X_t-1 + e^-tHΔ_t Y_e^t.


Defining the process G as in Lemma <ref> completes the proof of the
`if' part.
For the proof of the `only if' part, assume that G∈𝒢_H.
From (<ref>) it follows that


    X_t    = e^-HX_t-1 +
    Δ_tG = e^-2HX_t-2 + e^-HΔ_t-1G + Δ_t G
    
       =∑_j=0^n e^-jHΔ_t-j G + e^-(n+1)HX_t-n-1
       = e^-tH(∑_k = t-n^t
    e^kHΔ_k G + e^(t-n-1)HX_t-n-1)


for every n∈ℕ. Since G∈𝒢_H and lim_m→
-∞ e^mH X_m = 0 in probability, we obtain that


    X_t = e^-tH∑
    _k= -∞^t e^kHΔ_k G


for every t∈ℤ. Now, by strictly stationary increments of
G, we have


    e^-tH∑_j=-M^t
    e^jHΔ_j+s G law=
    e^-tH∑_j=-M^t
    e^jHΔ_j G.


for every t, M∈ℤ such that -M ≤ t.
Since the sums above converge as M tends to infinity, we obtain




    X_t+s = e^-(t+s)H∑
    _ j =-∞^t e^(j+s)HΔ_j+s G
    law= e^-tH∑_j=-∞^t
    e^jHΔ_j G = X_t.


Treating multidimensional distributions similarly we thus observe that
X is strictly stationary.
Finally, to prove the uniqueness assume there exist G_1, G_2∈𝒢_H such that


    e^tH X_t = ∑
    _k=-∞^t e^kHΔ_k
    G_1 = ∑_k=-∞^t
    e^kHΔ_k G_2


for every t∈ℤ. Then


    e^tH X_t - e^(t-1)HX_t-1 =
    e^tHΔ_t G_1 = e^tHΔ_t G_2.


Hence Δ_t G_1 = Δ_t G_2 for every t∈ℤ implying
that G_1 = G_2 + c. Since both processes are zero at t=0, it must
hold that c = 0.



Corollary <ref> is almost trivial. However, it is well
motivated by Theorem <ref>. On the other hand, Theorem <ref>
is far away from trivial as it states both sufficient and necessary
conditions. We prove Theorem <ref> using discrete Lamperti
transform. In principle, one could consider proving Theorem <ref>
by starting from Corollary <ref>. However, at this point, we
have not assumed any moment conditions, and thus it is not clear
whether a process G constructed from Z^(H) of Corollary <ref> would satisfy G∈𝒢_H. Indeed, a counter example
is provided in <cit.>. See also <cit.>, where moment conditions are discussed.




§ DISCUSSION ON SPECIAL CASES


In this appendix we take a closer look at “worst case scenario”
processes related to the choice between (<ref>) and (<ref>). These are such processes that, for some 0<a<1, a_j = a
for every j∈ℤ. By (<ref>) this is equivalent to


    γ(j+1)+γ(j-1)/γ(j) = b


for every j∈ℤ, where ϕ<b<ϕ+ 1/ϕ. In
order to study processes of this form, we consider formal power series.


Let


    f(x) = ∑_n=0^∞
    c_nx^n


be a formal power series in x. We now define the coefficient
extractor operator [·]{*} by


    [x^m]{f(x)} = c_m




Setting j=0 in (<ref>) we obtain that γ(1) = b/2γ
(0). This leads to the following recursion.


    γ(n) = bγ(n-1) - γ(n-2)  for n≥2.


It follows immediately from the first step of the recursion that b>2
does not define an autocovariance function of a stationary process.
Note also that for b=2 Equation (<ref>) implies that
γ(n)=γ(0) for every n∈ℤ. This corresponds to
the completely degenerate process X_n = X_0. We next study the case
0 <b<2.
For this, we define a generating function regarded as a formal power
series by


    f(x) = ∑_n=0^∞γ(n) x^n.


Then the coefficients of f(x) satisfy


    [x^n]{f(x)}   = b
    [x^n-1]{f(x)} - [x^n-2]{f(x)}
       =[x^n]{bxf(x)} - [x^n]
    {x^2f(x)}
       = [x^n]{bxf(x)-x^2f(x)}


for n≥2. For simplicity, we assume that γ(0) = 1. By taking
the constant and the first order terms into account we obtain


    f(x) = bxf(x)-x^2f(x)-bx+1+b/2x,


which implies


    f(x) = 1-b/2x/x^2-bx+1.


Since the function above is analytic at x=0, the corresponding power
series expansion is (<ref>). Furthermore, since the
recursion formula is linear, for a general γ(0) it holds that


    γ(n) = γ(0)[x^n] {(1-
    b/2x )∑_n=0^∞(bx-x^2)^n }.











§ TABLES

The simulation results highlighted in Section <ref> are
chosen from a more extensive set of simulations. All the simulation
results are given in a tabulated form in this appendix. The two
processes considered in the simulations are AR(1) and ARMA(1,2).
The used MA parameters are θ_1 = 0.8 and θ_2 = 0.3. The
tables represent the efficiency dependence of the estimator on the
AR(1) parameter φ and the used lag N. We have varied the
column variable AR(1) parameter from 0.1 to 0.9 and the row
variable lag from 1 to 10. The tables display the sample means of
the estimates from 1000 iterations with different sample sizes. At
the end of this appendix, we provide summary statistics tables
corresponding to the histograms presented in Section <ref>.






























18


brockwell


Brockwell, P.J.,
Davis, R.A.:
Time Series: Theory and Methods,
2nd edn.
Springer,
New York
(1991).
mr=1093459






Brockwell, P.J.,
Davis, R.A.:
Time Series: Theory and Methods,
2nd edn.
Springer,
New York
(1991)


structpyb



davis1986limit


Davis, R.,
Resnick, S.:
Limit theory for the sample covariance and correlation
functions of
moving averages.
The Annals of Statistics
14(2),
533–558
(1986).
mr=0840513






Davis, R.,
Resnick, S.:
Limit theory for the sample covariance and correlation
functions of
moving averages.
The Annals of Statistics
14(2),
533–558
(1986)


structpyb



francq2004maximum


Francq, C.,
Zakoïan, J.-M.:
Maximum likelihood estimation of pure GARCH and ARMA-GARCH
processes.
Bernoulli
10(4),
605–637
(2004)






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