Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Extending causality-related interdependence
measures
Yuzo Hosoya
Tohoku University(emeritus professor)

Waseda International Symposium
23rd October, 2018

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Outline1

Introduction

2

Measures of one-way eﬀect, reciprocity and association

3

The Szegö approxination theory and stationary processes in the plane

4

Locally stationary processes and approximate measures ofinterdependence

5

Extension to reducible nonstationary linear processes

6

Concluding remarks

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theTime series interdependence
Simple relation: Relation focused on a pair of series x(t) and y(t)regardless of third series.
Partial relation: Relation between x(t) and y(t) where the presenceof a third series z(t) is explicitly taken into consideration; the one
way eﬀect component of z(t) is eliminated from x(t) and y(t).

x

-



I



y

x

-



I

y


R

z

z0,0,−1

Figure 1. Characterizing causal interdependence quantitatively.
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Extraction of the one-way
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Introduction to empirical causal analysis
The Wiener-Granger causality of a series {y(t)} to another {x(t)}is defined by the improvement of the prediction error by addition of{y(t)} as predictor.
Sims (1980)’s innovation accounting: Eliciting the variation componetof {x(t)} due to the intrinsic innovation of {y(t)} in the frameworkof the vector MA representation.
Geweke (1982): Prediction improvement by addition ofH{y(t − j), j ∈ Z+ }.
Hosoya (1991): H{y0,−1 (t − j), j ∈ Z+ } wherey0,−1 (t) denotes the projection residual of y(t) onto
H{x(t + 1 − j), y(t − j), j ∈ Z+ } .
All of them characterize time series causality by quantifying what
constitutes the additional information of {y(t)} for prediction
improvement of {x(t)}?
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

The Szegö condition for multivariate spectrum
Let {x(t), y(t), z(t), t ∈ Z} be a zero mean jointly
covariance-stationary process, where x, y, z are p1 , p2 , p3 -vectors.
The process is assumed to have p × p spectral density matrix f (λ),
where p = p1 + p2 + p3 .
∫π
−π ln det f (λ)dλ > −∞ ⇐⇒ f (λ) satisfying the Szegö
condition.
If the condition is satisfied, the density is canonically factorizablef (λ) =12π

Λ(e−iλ )Λ(e−iλ )∗ ,where Λ(z) is analytic and of full rank inside the unit disc, having
the expansion
Λ(z) =∞∑

Λ[j]z jj=0
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in thePredicition-error formula (1)Denote the covariance matrix of the one step ahead prediction errorof the process {x(t), y(t), z(t)} by Σ† whereΣ† = Cov{x−1,−1,−1 (t), y−1,−1,−1 (t), z−1,−1,−1 (t)}and x−1,−1,−1 (t) indicates the projection residual of x(t) onto thepast x(t), y(t), z(t); namely, onto
H{x(t − j), y(t − j), z(t − j), j ∈ Z+ }.
If the Szegö condition holds, Σ† is positive definite. item

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in thePredection-error formula (2)Λ(z) is canonical if and only ifdet{Λ(0)Λ(0)∗ } = det Σ†= (2π)p exp{

1
2π

∫ π
log det f (λ)dλ}.
−π

For related expositions, see Szegö(1939), Helson and Lowdenslager (1958),
Rozanov (1967), Hannan (1970), Brockwell and Davis (1987).

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Monograph by Hosaya et al.

Hosoya, Y., Oya, K., Takimoto, T. and Kinoshita, R. (2017):

Characterizing Interdependencies of Multiple Time Series:
Theory and Application, JSS Research Series in Statistics,
Springer.
Chapter 1:
Chapter 2:
Chapter 3:
Chapter 4:
Chapter 5:
Appendix:

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Introduction
The Meaures of One-way Eﬀect, Reciprocity and Association
Representation of Partial Meaures
Inference Baed on the Vector Autoregressive andMoving Average Model
Inference on Changes in Interdependence Measures
Technical Supplements.

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

The partial measures of interdependence

Section 3 of Hosoya et al. (2017) is for exposition of the overall as well as
frequency-wise measures of one-way eﬀect, reciprocity and association:
The equality holds between the partial OMO and FMO:∫ π
1
P My→x:z (λ)dλ;
P My→x:z =2π −πIn view of the definitions of the respective measures and thecorresponding equality for P Mx→y:z , we haveP Mx,y:z (λ) = P Mx→y:z (λ) + P Mx.y:z (λ) + P My→x:z (λ),
P Mx,y:z = P Mx→y:z + P Mx.y:z + P My→x:z .

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theStationary processes in the plane (1)12

The Szegö theory is extensible to doubly parametrized p-vector
stationary processes {w(s, t), s, t ∈ Z}.
The extension provides the prediction error formula for linear
prediciton of w(0, 0) by H{w(s, t), s, t ∈ S} where S is a halfplane.

References:
Helson and Lowdenslager (1958). “Prediction theory and Fourier series inseveral variables” , Acta Math. (1961), 106, 175-213.
Szegö, B.G. (1939). Orthogonal Polynomials, Colloquium publications,
American Mathematical Society, 23, New York.
Wiener, N. (1949). The Extrapolation, Interpolation and Smoothing of
Stationary Time Series, Wiley, New York.
Whittle, P. (1954). On Stationary Processes in the Plane, Biometrika, 41,434-449.
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Example of half plane in Z2 (the lattice points)
3
2
1
0
−1
−2
−3
−3−2−1 0 1 2 3
Figure 2: Half planes: Each of the sets of the large dots (•) andthe small dots (·) indicates a half plane respectively.

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theStationary processes in the plane (2)A half plane S ≡ S(0, 0) in the set of lattice points Z2 is formallydefined by
S = {(s, t), s = 0, t ≥ 1} ∪ {(s, t), s ≥ 1, t ∈ Z}.
Half plane introduces a total order structure (s0 , t0 ) < (s1 , t1 ) if
(s1 , t1 ) ∈ S(s0 , t0 ) in the set of lattice points and it is used for defining
prediction and allied concepts.
Suppose that a p-vector stationary series {w(s, t), (s, t) ∈ Z2 } hasa partition w(s, t) = {u(s, t)′ , v(s, t)′ }′ where u and v are p1 andp2 vectors.
Suppse that the spectral density of the process {w(s, t)} is given byh(λ1 , λ2 ) which has the partition:[]h11 (λ1 , λ2 ) h12 (λ1 , λ2 )h(λ1 , λ2 ) =, −π < λ1 , λ2 ≤ π,
h21 (λ1 , λ2 ) h22 (λ1 , λ2 )where h11 (λ) is the p1 × p1 spectral density of {u(s, t)}.
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theStationary processes in the plane (3)Assumption 1:
The spectral density matrix h(λ1 , λ2 ) satisfies the Szegö condition∫ π ∫ πlog det h(λ1 , λ2 )dλ1 dλ2 > −∞.
−π−π

It follows from the assumption that h(λ1 , λ2 ) has a factorizationh(λ1 , λ2 ) =12πΓ† (e−iλ1 , e−iλ2 )Γ† (e−iλ1 , e−iλ2 )∗ ,where Γ† is a function defined on C2 and square-integrable on the torus,
having the expansion by means of the half plane S∑Γ† (y, z) = Γ† [0, 0] +Γ† [i, j]y i z j(i,j)∈S
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theStationary processes in the plane (4)Define the Hilbert subspace byH † {u(s1 , t1 ), v(s2 , t2 )} ≡ H{u(s1 − S, t1 ), v(s2 − S, t2 − S)}
where u(s1 − S, t1 − S) ≡ {u(s1 − i1 , t1 − j1 ); (i1 , j1 ) ∈ S}.
Denote by u−1,−1 (s, t), u−1,. (s, t) and u−1,0 (s, t) respectively the
projection residuals of u(s, t) onto H † {u(s, t − 1), v(s, t − 1)} ,H † {u(s, t − 1)} and H † {u(s, t − 1), v(s, t)}, whereas denote byu′−1,−1 (s, t) the projection residuals of u(s, t) ontoH † {u(s, t − 1), v0,−1 (s, t − 1)}.
We can choose Γ† (y, z) so that it has the maximality property:
det{Γ† (0, 0)Γ† (0, 0)∗ } = det Σ∫ π ∫ π1plog det h(λ1 , λ2 )dλ1 dλ2 };= (2π) exp{(2π)2 −π −πsee Helson and Lowdenslager (1958) Theorem 12 in p.199.
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theStationary processes in the plane (5)Denote the covariance matrix of the one step ahead prediction error of theprocess {u(s, t), v(s, t)} is given in partitioned form by[]Σ11 Σ12Σ=Σ21 Σ22
where Σ = Cov{u−1,−1 (t), v−1,−1 (t)} = Γ† (0, 0)Γ(0, 0)∗ .
Set Σ22:1 ≡ Σ22 − Σ21 Σ−1
11 Σ12 . Denote by h11 (λ1 , λ2 ) the p1 × p2
upper diagonal block of h(λ1 , λ2 ) and denote by h11:2 (λ1 , λ2 ) the
p1 × p1 matrix defined by∗
h11:2 (λ1 , λ2 ) = h11 (λ1 , λ2 ) − h̃12 (λ1 , λ2 )Σ−1
22:1 h̃12 (λ1 , λ2 ) .

where h̃12 (λ1 , λ2 ) =∗h1· (λ1 , λ2 )Γ† (e−iλ1 , e−iλ2 )Γ† (0, 0)∗ (−Σ21 Σ−1
11 , Ip1 ) .
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theStationary processes in the plane (6)Define the (overall) measure of one-way eﬀect in the plane from v to u by†
Mv→u= log[det{Cov(u−1,. (s, t))}/ det{Cov(u′−1,−1 (s, t))}].

Theorem 1:
1

The one-step ahead time-wise prediction is given by
det Cov{w−1 (s, t)}{}∫ π ∫ π1p
= (2π) exp
log det h(λ1 , λ2 )dλ1 dλ2 .
(2π)2 −π −π2The one-way eﬀect is repsesented by][∫ π ∫ πdet h11 (λ1 , λ2 )†Mv→u (s, t) =dλ1 dλ2 .
log
det h11:2 (λ1 , λ2 )−π −πY. Hosoya

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Whittle’s likelihood method (1)For {w(s, t), s = 1, . . . , m; t = 1, . . . , n} a second-order stationary
series in the plane, define the empirical spectral density by
Jm,n (λ1 , λ2 ) ≡ wm,n (λ1 , λ2 )wm,n (λ1 , λ2 )∗ , −π < λ1 , λ2 ≤ π,
where
wm,n (λ1 , λ2 ) ≡Y. Hosoya

12π(mn)1/2

{m n
∑∑

}
w(s, t) exp(isλ1 + itλ2 ) .

s=1 t=1

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Whittle’s likelihood method (2)Suppose that the spectral density of the process {w(s, t)} depends on
the unknown model parameter ψ and is given by h(λ1 , λ2 ; ψ) whereψ ∈ Ψ and the parameter space Ψ is assuned a compact subset of Rs
with non-empty interior. In integral form the Whittle log-likelihood
function is given by∫ π ∫ πL̄mn (ψ) ≡ −mn[log det h(λ1 , λ2 ; ψ)−π −π]+tr{h−1 (λ1 , λ2 ; ψ)Jmn (λ1 , λ2 )} dλ1 dλ2 .
The function L̄n (ψ) is said the whittle log-likelihood function [theapproximation was originally proposed by Whittle (1952, 1953) for scalarand multiple stationary processes respectively. The likelihood L̄mn (ψ)
was given by Whittle (1954) for scalar processes.
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Whittle’s likelihood method (3)Assumption 2:
1
2

∫π ∫π

−π log deth(λ1 , λ2 ; ψ)dλ1 dλ2 is diﬀerentiable;
At almost all point of (λ1 , λ2 ), h(λ1 , λ2 ; ψ)−1 is diﬀerentiable with−πrespect to ψ.
The partial derivatives are denoted, respectively, by∫π ∫π∂ −π −π log deth(λ1 , λ2 ; ψ)dλ1 dλ2
Hj (ψ) =∂ψjandhj (λ1 , λ2 ; ψ) =∂h−1 (λ1 , λ2 ; ψ)∂ψjand hj is assumed to be measurable with respect to ψ a.e. λ1 , λ2 .
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Whittle’s likelihood method (4)1

2

The notations H(ψ) and tr{h(λ1 , λ2 ; ψ)f (λ1 , λ2 )} represent,
respectively, the s-vectors whose j-th elements are Hj (ψ) andtr{hj (λ1 , λ2 ; ψ)f (λ1 , λ2 )}.
Define Snj (ψ) by∫ π ∫ πSnj (ψ) = Hj (ψ) +−π−πtr{hj (λ1 , λ2 ; ψ)In (λ1 , λ2 )}dλ1 dλ2 ,j = 1, · · · , sand let Sn (ψ) be the vector {Snj (ψ)}.
3

A value ψ̂n satisfying Sn (ψ̂n ) = 0 is said to be a maximum Whittle
likelihood (MWL) estimate of ψ.

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (1)Szegö ’s prediction error formula holds approximately to the locallystationary processes introduced by Dahlhaus (1996). His approach is able
to be adaptably employed for evaluating the measures of interdependence
for the multivariate locally stationary processes.
The locally stationary p-vector process {w(t, T )} is represented by∫ πw(t, T ) =exp(iλt)Γt,T (λ)dξ for t = 1, 2, . . . , T,−πwhere w(t, T ) = 0 for t ≤ 0.

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (2)Assumption 3:
1 ξ(λ), −π < λ ≤ π, is a p-vector random process with cumulants
cumulant{dξα1 (λ1 ), . . . , dξαk (λk )} =
(∑
)
k
ν
λ
j=1 j hα1 ,...,αk (λ1 , . . . , λk )dλ1 · · · λk23where ν(λ) is the delta function (mod 2π).
hα1 (λ1 ) = 0, and hα1 ,α2 (λ1 , λ2 ) = 1(α1 = α2 ), and|hα1 ,...,αk (λ1 , . . . , λk )| is bounded.
Γt,T (λ) is a p × p matrix for which there exists a constant K and a
2π periodic p × p matrix function such that for all T
sup |Γt,T (λ) − Γ(t/T, λ)| ≤ KT −1t,λand Γ(τ, λ) is continuous in τ .
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (3)Denote the time-varying spectral density matrix for each τ, 0 ≤ τ ≤ 1, byh(τ, λ) =1
2πΓ(τ, λ)Γ(τ, λ)∗ .

Suppose the p-vector process w is partitioned asw(t|T ) = {u(t|T )′ , v(t|T )′ }′ where u and v are p1 and p2 vectors(p = p1 + p2 ). Let the Hilbert subspace HT {u(t|T ), v(s|T )} definedby
HT {u(t|T ), v(s|T )} ≡ H{u(i|T ), 1 ≤ i ≤ t; v(j|T ), 1 ≤ j ≤ s}.
Denote by u−1,−1 (t|T ), u−1 (t|T ) and u−1,0 (t|T ) respectively the
projection residuals of u(t|T ) onto
HT {u(t − 1|T ), v(t − 1|T )}, HT {u(t − 1|T )} andHT {u(t − 1|T ), v(t|T )}, whereas denote by u′−1,−1 (t|T ) the
projection residuals of u(t|T ) onto HT {u(t − 1|T ), v(t − 1|T )}.
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (4)Define the time-varying (overall) measure of one-way eﬀect from v to u by
Mv→u (t|T ) = log[det{Cov(u−1,. (t|T ))}/ det{Cov(u′−1,−1 (t|T ))}]and the average measure by
1 ∑TM̄v→u (T ) ≡Y. Hosoya

T t=1

Mv→u (t, T ).

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (5)Assumption 4:
For each τ, 0 ≤ τ ≤ 1, the spectral density h(τ, λ) satisfies the Szegö
condition and has a canonical factorization for each u such thath(τ, λ) =1
2πΓ̄(τ, e−iλ )Γ̄(τ, e−iλ )∗where Γ̄ is bounded from below and has uniformly bounded derivative∂ 2 Γ̄/∂u∂λ.
Set:Σ(τ ) = Γ̄(τ, 0)Γ̄(τ, 0)∗Σ22:1 (τ ) ≡ Σ22 (τ ) − Σ21 (τ )Σ−1
11 (τ )Σ12 (τ ).
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (6)Denote by h11 (τ, λ) the p1 × p1 upper diagonal block of h(τ, λ).
Denote by h11:2 (τ, λ) the p1 × p1 matrix defined by∗
h11:2 (τ, λ) = h11 (τ, λ) − h̃12 (τ, λ)Σ−1
22:1 (τ )h̃12 (τ, λ)where h̃12 (τ, λ)∗= h1· (τ, λ)Γ(τ, e−iλ )Γ(τ, 0)∗ [−Σ21 (τ )Σ−1
11 (τ ), Ip1 ] .

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Locally stationary processes and asymptotics (7)The first proposition of the following theorem is a multivariate extension of
Dahlhaus (1996, p.149) and the second proposition is a time-dependent
version of the one-way eﬀect measure.

Theorem 2:
1

The one-step ahead time-wise prediction is given by{
det Cov{w−1 (t, T )} =2p(2π) exp

1

}

∫ π

2π −π
+ot (1) + oT (1).

log det h(t/T, λ)dλ

The average measure of one-way eﬀect is given by
1 ∑

lim

T →∞ TY. Hosoya

t=1[∫ 1∫ πT

Mv→u (t, T ) =

log
0

−π

Partial measures

det h11 (τ, λ)det h11:2 (τ, λ)

]
dλdτ.

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Nonstationary reducible processes (1)1Let {w(t), t ∈ Z} be a second-order stationary p-vector process,and let {W (t), t ∈ Z+ } ( W (t) = 0 for t ∈ Z0− ) be anotherp-vector possibly non-stationary process with finite second-ordermoments.

2

Let w−1 (t) be the one-step ahead prediction error of w(t) based on
H{w(t − 1)}; then, the process {W (t)} is said to be reducible to(the generating process) {w(t)} if the projection residual of W (t)onto H{W (t − 1), w(0)} is equal to w−1 (t) for all t ∈ Z+ .

3

If W is reducible to w, then H{W (t), w(0)} = H{w(t)} fort ∈ Z+ .

4

Namely, if a process {W (t)} is reducible to {w(t)}, the respective
prediction errors of w(t) and W (t) are identical if the information
H{w(0)} is supplemented, .
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Nonstationary reducible processes (2)Let {X(t)}, {Y (t)}, {Z(t)} respectively be p1 , p2 , p3 -vector processes
and set W (t) = (X(t)∗ , Y (t)∗ , Z(t)∗ )∗ . Set
A(L) =

a
∑

j

A[j]L

and B(L) =

j=0

b
∑

B[k]Lkk=0

where the A[j] and the B[j] are (p1 + p2 + p3 ) × (p1 + p2 + p3 )matrices with A[0] = B[0] = Ip1 +p2 +p3 .
1 Assume that the zeros of det A(z) and det B(z) are either on oroutside the unit circle, and det A(z) and det B(z) do not sharecommon zeros.
2 Suppose {W (t)} is generated by the VARMA process
A(L)W (t) = B(L)ε(t)(1)

for a white noise process {ε(t)} such that E(ε(t)) = 0 andCov(ε(t)) = Σ† , where rank(Σ† ) = p1 + p2 + p3 .
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Nonstationary reducible processes (3)Denote by A(L)♯ the adjugate matrix of A(L) such that


d(L)
0


..
A(L)♯ A(L) =  ≡ D(L).
0d(L)
where D(L) is the diagonal matrix with det A(L) as the common
diagonal element. Applying the operator A(L)♯ to the members of theequation (4), we have



d(L)
0
X(t)


♯
..
  Y (t)  = A(L) B(L)ε(t)

.
Z(t)
0
d(L)


x(t)
≡  y(t)  ≡ w(t). (2)z(t)
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Nonstationary reducible processes (4)It follows from the construction that {w(t)} is a stationary MA process;moreover, because the zeros of det{A(z)♯ B(z)} are either on or outside
the unit circle, the covariance matrix of the one-step ahead prediction
error of w(t) is equal to Σ† and the joint spectral density matrix f (λ) of{w(t)} has canonical factorizationf (λ) =12π

Λ(e−iλ )Λ(e−iλ )∗ ,

(3)

where
Λ(e−iλ ) = A(e−iλ )♯ B(e−iλ )Σ†
Σ†

1/2

Σ†

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1/2

1/2

= Σ† .

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in the

Nonstationary reducible processes (5)Theorem 3:
1 The partial measures of interdependence of the possibly
non-stationary processes {X(t)}, {Y (t)}, {Z(t)} in the
cointegrated process
A(L)W (t) = B(L)ε(t)are identified with the correspondng measures of the stationary
processes {x(t)}, {y(t)}, and {z(t)} which are given byx(t) y(t)  ≡ w(t) ≡ A(L)♯ B(L)ε(t).
z(t)
2All the partial measures of the series{X(t)}, {Y (t)}, {Z(t)} canbe constructed through the corresponding measures on the basis of
the canonical factor Λ(e−iλ ) in (3) of the stationary processes{x(t)}, {y(t)}, and {z(t)}.
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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theConcluding remarks
Construction of the measure of one-way eﬀect relies crucially on the
Szegö theory of approximization, whereas the theory is flexible enoughto allow generalization in a variety of directions.
The talk was limited to theoretical extension of the measures ofinterdependence originally defiend for stationary vetor time series torandom fields, locally stationary processes and nonstationary reducible
processes. Implementation by application to practical data isnecesssary.
Computational issues of the causal measures were intensively
examined for stationary VARMA model based on simulated and real
data sets; see Hosoya et al. (2017).
Yao and Hosoya (2000) suggested a practical approach of statisticalineference on the measure of one-way eﬀect in the cointegrated AR
model set-up which is a typical reducible non-statioanry process.
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Partial measures

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Introduction Measures of one-way eﬀect, reciprocity and association The Szegö approxination theory and stationary processes in theThank you for listening !

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Partial measures

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