Slides Paris 6 Jussieu Slides on Causality

Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7

Causality with Non-Gaussian Time Series

Arthur Charpentier (Université de Rennes 1 & UQàM)

Université Paris 7 Diderot, May 2016.

http://freakonometrics.hypotheses.org

@freakonometrics 1


Motivation (Earthquakes)

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Time before and after a major eathquake (magnitude >6.5) in days

Num

ber

of e

arth

quak

es (

mag

nitu

de >

2) p

er 1

5 se

c., a

vera

ge b

efor

e=10

0

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

−15 −10 −5 0 5 10 15

020

040

060

080

010

00

Same techtonic plate as major oneDifferent techtonic plate as major one

see Boudreault & C. (2011) on contagion among tectonic plates

@freakonometrics 2

http://arxiv.org/abs/1112.0929


Motivation (Onsite vs. Online)

onsite protestors, camped-out, arrests and injuries

vs. online #indignados, #occupy and #vinegar on Twitter & Facebook

see Bastos, Mercea & C. (2015)

@freakonometrics 3

http://sci-hub.io/10.1111/jcom.12145


Multivariate Stationary Time Series

Definition A time series (Xt = (X1,t, · · · , Xd,t))t∈Z with values in Rd is called aVAR(1) process if

X1,t = φ1,1X1,t−1 + φ1,2X2,t−1 + · · ·+ φ1,dXd,t−1 + ε1,t

X2,t = φ2,1X1,t−1 + φ2,2X2,t−1 + · · ·+ φ2,dXd,t−1 + ε2,t

· · ·Xd,t = φd,1X1,t−1 + φd,2X2,t−1 + · · ·+ φd,dXd,t−1 + εd,t

(1)

or equivalentlyX1,t

X2,t...

Xd,t

︸︷︷︸

Xt

=

φ1,1 φ1,2 · · · φ1,d

φ2,1 φ2,2 · · · φ2,d...

......

φd,1 φd,2 · · · φd,d

︸︷︷︸

Φ

X1,t−1

X2,t−1...

Xd,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t...εd,t

︸︷︷︸

εt

@freakonometrics 4



For some real-valued d× d matrix Φ, and some i.i.d. random vectors εt withvalues in Rd.

Assume that εt is a Gaussian white noise N (0,Σ), with density

f(ε) = 1√(2π)d|det Σ|

exp(−ε

TΣ−1ε

2

), ∀ε ∈ Rd.

Assume also that εt is independent of Xt−1 = σ({Xt−1,Xt−2, · · · , }). : (εt)t∈Zis the innovation process.

Definition A time series (Xt)t∈N is said to be (weakly) stationary if

• E(Xt) is independent of t (=: µ)

• cov(Xt,Xt−h) is independent of t (=: γ(h)), called autocovariance matrix

@freakonometrics 5



Define the autocorrelation matrix,

ρ(h) := ∆−1γ(h)∆−1, where ∆ :=√

diag (γ(0)).

(Xt)t∈N a stationary AR(1) time series, Xt = ΦXt−1 + εtProposition (Xt)t∈N is a stationary AR(1) time series if and only if the deigenvalues of Φ should have a norm lower than 1.

Proposition If (Xt)t∈N is a stationary VAR(1) time series,

ρ(h) = Φh, h ∈ N.

@freakonometrics 6


Causality, in dimension 2

Two stationary time series (Xt, Yt)t∈Z. Heuristics on independence,

f(xt, yt|Xt−1, Y t−1) = f(xt|Xt−1) · f(yt|Y t−1)

Write (with X for Xt−1)

f(xt, yt|X,Y )f(xt|X) · f(yt|Y )︸︷︷︸

(X,Y )

= f(xt|X,Y )f(xt|X)︸︷︷︸X→Y

· f(yt|X,Y )f(yt|Y )︸︷︷︸X←Y

· f(xt, yt|X,Y )f(xt|X,Y ) · f(yt|X,Y )︸︷︷︸

X⇔Y

Gouriéroux, Monfort & Renault (1987) define the following Kullback-measures

C(X,Y ) = E[log f(Xt, Yt|X,Y )

f(Xt|X) · f(Yt|Y )

]

@freakonometrics 7

https://annals.ensae.fr/wp-content/uploads/pdf/n0607/vol67-17.pdf



C(X → Y ) = E[log f(Xt|X,Y )

f(Xt|X)

]

C(Y → X) = E[log f(Yt|X,Y )

f(Yt|Y )

]

C(X ⇔ Y ) = E[log f(Xt, Yt|X,Y )

f(Xt|X,Y ) · f(Yt|X,Y )

]so that C(X,Y ) = C(X → Y ) + C(X ← Y ) + C(X ⇔ Y ).

From Granger (1969)

(X) causes (Y ) at time t if L(yt|Xt−1, Y t−1) 6= L(yt|Y t−1)

(X) causes (Y ) instantaneously at time t if L(yt|Xt, Y t−1) 6= L(yt|Xt−1, Y t−1)

@freakonometrics 8

http://sci-hub.io/10.2307/1912791


Causality, in dimension 2, for VAR(1) time series

Xt

Yt

︸︷︷︸

Xt

=

φ1,1 φ1,2

φ2,1 φ2,2

︸︷︷︸

Φ

Xt−1

Yt−1

︸︷︷︸

Xt−1

+

utvt

︸︷︷︸

εt

, with Var

utvt

=

σ2u σuv

σuv σ2v

From Granger (1969) (see also Toda & Phillips (1994))

(X) causes (Y ) at time t, X → Y , if φ2,1 6= 0

(Y ) causes (X) at time t, Y → X, if φ1,2 6= 0

(X) causes (Y ) instantaneously at time t, X ⇔ X, if σu,v 6= 0

@freakonometrics 9

http://sci-hub.io/10.2307/1912791

http://www.tandfonline.com/doi/abs/10.1080/07474939408800286?journalCode=lecr20


Testing Causality, in dimension d

For lagged causality, we test

H0 : Φ ∈ P against H1 : Φ /∈ P,

where P is a set of constrained shaped matrix, e.g. P is the set of d× d diagonalmatrices for lagged independence, or a set of block triangular matrices for laggedcausality.

Proposition Let Φ denote the conditional maximum likelihood estimate of Φ inthe non-constrained MINAR(1) model, and Φ

cdenote the conditional maximum

likelihood estimate of Φ in the constrained model, then under suitable conditions,

2[logL(X, Φ|X0)− logL(X, Φc|X0)] L→ χ2(d2−dim(P)), as T →∞, under H0.

Example Testing (X1,t)←(X2,t) is testing whether φ1,2 = 0, or not.

@freakonometrics 10


Modeling Counts Processes

Steutel & van Harn (1979) defined a thinning operator as follows

Definition Define operator ◦ as

p ◦N =N∑i=1

Yi = Y1 + · · ·+ YN if N 6= 0, and 0 otherwise,

where N is a random variable with values in N, p ∈ [0, 1], and Y1, Y2, · · · are i.i.d.Bernoulli variables, independent of N , with P(Yi = 1) = p = 1− P(Yi = 0). Thusp ◦N is a compound sum of i.i.d. Bernoulli variables.

Hence, given N , p ◦N has a binomial distribution B(N, p).

Note that p ◦ (q ◦N) L= [pq] ◦N for all p, q ∈ [0, 1].

Further

E (p ◦N) = pE(N) and Var (p ◦N) = p2Var(N) + p(1− p)E(N).

@freakonometrics 11

http://projecteuclid.org/download/pdf_1/euclid.aop/1176994950


(Poisson) Integer AutoRegressive processes INAR(1)

Based on that thinning operator, Al-Osh & Alzaid (1987) and McKenzie (1985)defined the integer autoregressive process of order 1:

Definition A time series (Xt)t∈N with values in R is called an INAR(1) process if

Xt = p ◦Xt−1 + εt, (2)

where (εt) is a sequence of i.i.d. integer valued random variables, i.e.

Xt =Xt−1∑i=1

Yi + εt, where Y ′i s are i.i.d. B(p).

Such a process can be related to Galton-Watson processes.

@freakonometrics 12

http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9892.1987.tb00438.x/epdf

http://sci-hub.io/10.1111/j.1752-1688.1985.tb05379.x


INAR(1) & Galton-Watson

Xt+1 =Xt∑i=1

Yi + εt+1, where Y ′i s are i.i.d. B(p)

@freakonometrics 13


Proposition E (Xt) = E(εt)1− p , Var (Xt) = γ(0) = pE(εt) + Var(εt)

1− p2 and

γ(h) = cov(Xt, Xt−h) = ph.

It is common to assume that εt are independent variables, with a Poissondistribution P(λ), with probability function

P(εt = k) = e−λλk

k! , k ∈ N.

Proposition If (εt) are Poisson random variables, then (Xt) will also be asequence of Poisson random variables.

Note that we assume also that εt is independent of Xt−1, i.e. past observationsX0, X1, · · · , Xt−1. Thus, (εt)t∈N is called the innovation process.

Proposition (Xt)t∈N is a stationary INAR(1) time series if and only if p ∈ [0, 1).

Proposition If (Xt)t∈N is a stationary INAR(1) time series, (Xt)t∈N is anhomogeneous Markov chain.

@freakonometrics 14


Markov Property of INAR(1) Time Series

π(xt, xt−1) = P(Xt = xt|Xt−1 = xt−1) =xt∑k=0

P

(xt−1∑i=1

Yi = xt − k

)︸︷︷︸

Binomial

·P(ε = k)︸︷︷︸Poisson

.

@freakonometrics 15


Inference of INAR(1) Processes

Consider a Poisson INAR(1) process, then the likelihood is

L(p, λ;X0,X) =[n∏t=1

ft(Xt)]· λX0

(1− p)X0X0! exp(− λ

1− p

)where

ft(y) = exp(−λ)min{Xt,Xt−1}∑

i=0

λy−i

(y − i)!

(Yt−1

i

)pi(1− p)Yt−1−y, for t = 1, · · · , n.

Maximum likelihood estimators are

(p, λ) ∈ argmax {logL(p, λ; (X0,X ))}

@freakonometrics 16


Multivariate Integer Autoregressive processes MINAR(1)

Let Xt := (X1,t, · · · , Xd,t), denote a multivariate vector of counts.

Definition Let P := [pi,j ] be a d× d matrix with entries in [0, 1]. IfX = (X1, · · · , Xd) is a random vector with values in Nd, then P ◦X is ad-dimensional random vector, with i-th component

[P ◦X]i =d∑j=1

pi,j ◦Xj ,

for all i = 1, · · · , d, where all counting variates Y in pi,j ◦Xj ’s are assumed to beindependent.

Note that P ◦ (Q ◦X) L= [PQ] ◦X.

Further, E (P ◦X) = PE(X), and

E((P ◦X)(P ◦X)T) = PE(XXT)P T + ∆,

with ∆ := diag(V E(X)) where V is the d× d matrix with entries pi,j(1− pi,j).

@freakonometrics 17


Multivariate Integer Autoregressive processes MINAR(1)

Definition A time series (Xt) with values in Nd is called a d-variate MINAR(1)process if

Xt = P ◦Xt−1 + εt (3)

for all t, for some d× d matrix P with entries in [0, 1], and some i.i.d. randomvectors εt with values in Nd.

(Xt) is a Markov chain with states in Nd with transition probabilities

π(xt,xt−1) = P(Xt = xt|Xt−1 = xt−1) (4)

satisfying

π(xt,xt−1) =xt∑

k=0P(P ◦ xt−1 = xt − k) · P(ε = k).

@freakonometrics 18


Inference for MINAR(1)

Proposition Let (Xt) be a d-variate MINAR(1) process satisfying stationaryconditions, as well as technical assumptions (called C1-C6 in Franke & Subba Rao(1993)), then the conditional maximum likelihood estimate θ of θ = (P ,Λ) isasymptotically normal,

√n(θ − θ) L→ N (0,Σ−1(θ)), as n→∞.

Further,

2[logL(N , θ|N0)− logL(N ,θ|N0)] L→ χ2(d2 + dim(λ)), as n→∞.

@freakonometrics 19

https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/734/file/gruen_95.pdf

https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/734/file/gruen_95.pdf


Granger causality with BINAR(1)

(X1,t) and (X2,t) are instantaneously related if ε is a noncorrelated noise,g gg g g g g g g g g g g g

X1,t

X2,t

︸︷︷︸

Xt

=

p1,1 p1,2

p2,1 p2,2

︸︷︷︸

P

◦

X1,t−1

X2,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t

︸︷︷︸

εt

, with Var

ε1,t

ε2,t

=

λ1 ϕ

ϕ λ2

@freakonometrics 20



1. (X1) and (X2) are instantaneously related if ε is a noncorrelated noise, g g gg g g g g g g g g g g

X1,t

X2,t

︸︷︷︸

Xt

=

p1,1 p1,2

p2,1 p2,2

︸︷︷︸

P

◦

X1,t−1

X2,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t

︸︷︷︸

εt

, with Var

ε1,t

ε2,t

=

λ1 ?

? λ2

@freakonometrics 21



2. (X1) and (X2) are independent, (X1)⊥(X2) if P is diagonal, i.e.p1,2 = p2,1 = 0, and ε1 and ε2 are independent,

X1,t

X2,t

︸︷︷︸

Xt

=

p1,1 00 p2,2

︸︷︷︸

P

◦

X1,t−1

X2,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t

︸︷︷︸

εt

, with Var

ε1,t

ε2,t

=

λ1 00 λ2

@freakonometrics 22



3. (N1) causes (N2) but (N2) does not cause (X1), (X1)→(X2), if P is a lowertriangle matrix, i.e. p2,1 6= 0 while p1,2 = 0,

X1,t

X2,t

︸︷︷︸

Xt

=

p1,1 0? p2,2

︸︷︷︸

P

◦

X1,t−1

X2,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t

︸︷︷︸

εt

, with Var

ε1,t

ε2,t

=

λ1 ϕ

ϕ λ2

@freakonometrics 23



4. (N2) causes (N1) but (N1,t) does not cause (N2), (N1)←(N2,t), if P is aupper triangle matrix, i.e. p1,2 6= 0 while p2,1 = 0,

X1,t

X2,t

︸︷︷︸

Xt

=

p1,1 ?

0 p2,2

︸︷︷︸

P

◦

X1,t−1

X2,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t

︸︷︷︸

εt

, with Var

ε1,t

ε2,t

=

λ1 ϕ

ϕ λ2

@freakonometrics 24



5. (N1) causes (N2) and conversely, i.e. a feedback effect (N1)↔(N2), if P is afull matrix, i.e. p1,2, p2,1 6= 0

X1,t

X2,t

︸︷︷︸

Xt

=

p1,1 ?

? p2,2

︸︷︷︸

P

◦

X1,t−1

X2,t−1

︸︷︷︸

Xt−1

+

ε1,t

ε2,t

︸︷︷︸

εt

, with Var

ε1,t

ε2,t

=

λ1 ϕ

ϕ λ2

@freakonometrics 25


Bivariate Poisson BINAR(1)

A classical distribution for εt is the bivariate Poisson distribution, with onecommon shock, i.e. ε1,t = M1,t +M0,t

ε2,t = M2,t +M0,t

where M1,t, M2,t and M0,t are independent Poisson variates, with parametersλ1 − ϕ, λ2 − ϕ and ϕ, respectively. In that case, εt := (ε1,t, ε2,t) has jointprobability function

e−[λ1+λ2−ϕ] (λ1 − ϕ)k1

k1!(λ2 − ϕ)k2

k2!

min{k1,k2}∑i=0

(k1

i

)(k2

i

)i!(

ϕ

[λ1 − ϕ][λ2 − ϕ]

)with λ1, λ2 > 0, ϕ ∈ [0,min{λ1, λ2}].

λ =

λ1

λ2

and Λ =

λ1 ϕ

ϕ λ2

@freakonometrics 26


Bivariate Poisson BINAR(1) and Granger causality

For instantaneous causality, we test

H0 : ϕ = 0 against H1 : ϕ 6= 0

Proposition Let λ denote the conditional maximum likelihood estimate ofλ = (λ1, λ2, ϕ) in the non-constrained MINAR(1) model, and λ⊥ denote theconditional maximum likelihood estimate of λ⊥ = (λ1, λ2, 0) in the constrainedmodel (when innovation has independent margins), then under suitableconditions,

2[logL(X, λ|X0)− logL(X, λ⊥|X0)] L→ χ2(1), as n→∞, under H0.

@freakonometrics 27


Bivariate Poisson BINAR(1) and Granger causality

For lagged causality, we test

H0 : P ∈ P against H1 : P /∈ P,

where P is a set of constrained shaped matrix, e.g. P is the set of d× d diagonalmatrices for lagged independence, or a set of block triangular matrices for laggedcausality.

Proposition Let P denote the conditional maximum likelihood estimate of P inthe non-constrained MINAR(1) model, and P

cdenote the conditional maximum

likelihood estimate of P in the constrained model, then under suitable conditions,

2[logL(X, P |X0)− logL(X, Pc|X0)] L→ χ2(d2−dim(P)), as n→∞, under H0.

Example Testing (X1,t)←(X2,t) is testing whether p1,2 = 0, or not.

@freakonometrics 28


Autocorrelation of MINAR(1) processes

Proposition Consider a MINAR(1) process with representationXt = P ◦Xt−1 + εt, where (εt) is the innovation process, with λ := E(εt) andΛ := Var(εt). Let µ := E(Xt) and γ(h) := cov(Xt,Xt−h). Thenµ = [I− P ]−1λ and for all h ∈ Z, γ(h) = P hγ(0) with γ(0) solution ofγ(0) = Pγ(0)P T + (∆ + Λ).

See Boudreault & C. (2011) for additional properties

@freakonometrics 29

http://arxiv.org/abs/1112.0929


Granger causality X1 → X2 or X1 ← X2

1. North American Plate, 2. Eurasian Plate,3. Okhotsk Plate, 4. Pacific Plate (East), 5. Pacific Plate (West), 6.

Amur Plate, 7. Indo-Australian Plate, 8. African Plate, 9. Indo-Chinese Plate, 10. Arabian Plate, 11. Philippine

Plate, 12. Coca Plate, 13. Caribbean Plate, 14. Somali Plate, 15. South American Plate, 16. Nasca Plate, 17.

Antarctic Plate

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@freakonometrics 30






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@freakonometrics 31






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@freakonometrics 32


Using Ranks for Time Series

Haugh (1976) suggested to use ranks to test for independence.

Set Rt denote the rank of Xt within {X1, · · · , XT }, and set

Ut = RtT

= 1T

T∑s=1

1Xt≤Xs= FX(Xt)

and similarly

Vt = StT

= 1T

T∑s=1

1Yt≤Ys= FY (Yt)

See also Dufour(1981) for rank tests for serial dependence.

@freakonometrics 33

http://sci-hub.io/10.1080/01621459.1976.10480353

http://sci-hub.io/10.1111/j.1467-9892.1981.tb00317.x



From Taamouti, Bouezmarni & El Ghouch (2014), consider some copula basedcausality approach:

C(X → Y ) = E[log f(Xt|X,Y )

f(Xt|X)

]can be written, for Markov 1 processes

C(X → Y ) = E[log f(Xt|Xt−1, Yt−1)

f(Xt|Xt−1)

]= E

[log f(Xt, Xt−1, Yt−1) · f(Xt−1)

f(Xt, Xt−1) · f(Xt−1, Yt−1)

]i.e.

C(X → Y ) = E[log c(FX(Xt), FX(Xt−1), FY (Yt−1))

c(FX(Xt), FX(Xt−1)) · c(FX(Xt−1), FY (Yt−1))

]

@freakonometrics 34

http://sci-hub.io/10.1016/j.jeconom.2014.03.001


Using a Probit-type Transformation

Following Geenens, C. & Paindaveine (2014), consider some Probit-typetransformation, for stationary time series

Xt = Φ−1(Ut) = Φ−1(FX(Xt))

Yt = Φ−1(Vt) = Φ−1(FY (Yt))

Application in Bastos, Mercea & C. (2015)

@freakonometrics 35

http://www.e-publications.org/ims/submission/BEJ/user/submissionFile/20785?confirm=a7adf80a



Online vs. Onsite Causality

For #occupy and #indignados

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Application in Bastos, Mercea & C. (2015)

@freakonometrics 36


Science

Slides Paris 6 Jussieu Slides on Causality