Post on 12-Apr-2017
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Motivation (Earthquakes)
●
●●●
●
●
●
●
●
●
●●●●●●●●●●
●●●●●●●●●●
●
●●●●●●●●●●●
●●●
●●●●●●●●●●
●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●
●●
●
●●●●●●●●●●●●
●●●●●
●●●
●
●●●●
●●●
●
●●●●●●●●●●●●
●●
●
●
●●●●
●●●●
●
●●●
●●●●
●
●●●●●●
●●●●●●●
●●
●●●
●●●●●●
●●●●●●●●●●
●●
●
●
●
●●●
●
●●●●
●●●●●●●●
●●●●●●●●●
●●●●
●●●●●●●●●
●●●●●
●●
●
●●●●●
●
●●●●●
●●●●●●●●●●●●●
●
●●●●●●●●●●●
●
●●
●●
●●●●●●
●
●●●
●●●●●●●●●●●●●●
●
●●●●●●●
●
●
●
●●
●●●
●●
●●●●●●●●
●●●●●●●●●
●
●
●●●
●●
●
●●●●●●●●●●
●
●●
●
●
●●●●●●●●●●●
●
●●●●●●●
●●●●●
●
●
●
●●●
●●●●
●●●●●●●●
●●●●●
●●●●●●
●●
●
●
●●●
●●●●●●●●●●●
●●●●
●●●●●●●
●●●●
●
●●
●
●●●
●●●
●
●●●●●●●
●
●●
●●●●●●
●●●●
●●●●●●●●●●●●●●●●
●●●●●●●
●
●●●●●●●●●●●
●
●●●●
●●●
●
●●●●●●●●
●
●●
●●●●●●●●●●●●●
●
●
●●
●●●●●●
●●
●●●
●●●●●●
●
●●●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●
●
●
●●●●●●●●
●●●
●
●
●●●●●●●
●●●●●●●●
●
●●
●
●●●
●
●●
●●●●●●●●●●●
●●●●●
●●●●●
●
●●●●●●●●●●
●
●
●●●●●●●
●●●●●
●
●
●●●●●●●●●●●●●
●
●●●●
●●●●
●●●
●●●●●●
●
●●
●●●●
●●●●●●●
●
●●●●●●●●●●●
●
●●●●
●
●●●●
●●●●●●●
●●●●●●
●
●●
●
●●●
●●●●●●
●●●●●●●●●
●●●●
●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●
●
●●●●●●●●●
●●
●
●
●●●●●●●●●
●●●
●●●
●
●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●
●
●●●●●
●
●
●●●●●●●●●●
●●●●●●●
●
●●●●●●●●●●
●
●●●
●
●●●●●●●
●
●●●●●●●●●
●●●●
●
●●●
●
●●●●●
●●
●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●
●●●●●●●
●●
●●●
●
●
●●●●●
●
●●●●●
●
●●●●●●●●
●●●●
●
●
●●
●
●
●
●
●●●●
●
●●●
●
●●●●●
●●●●●
●
●●●
●●
●●●●●●
●
●●●●
●●●●●●●●●●●●
●●●●●
●
●●●
●●●●●●●●●
●●
●
●●●●●●
●
●●●●●
●
●●●●●●●
●
●
●●
●●●●
●●●●●●
●
●●●●
●●
●●●●●●●●●●●
●●●
●●●●●●●
●●●●●●
●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●
●●●●●
●●●●
●●●
●●●●●●●
●●●●
●●●●●●●
●●●●●●●●●●●●●
●
●●●●●●●●●
●●
●●
●
●●●●●●●●●●●
●●●●●●
●●●
●
●●●●
●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●●
●●
●
●●●●●
●●●●●●
●●
●●●●●●●●●●●●●
●●●●
●●●
●
●
●
●
●●●●
●
●
●●
●
●●
●
●●●
●●●●●●
●●●
●●●
●
●●●●●●
●
●
●
●●
●●
●
●
●
●●
●
●
●
●
●●
●
●
●
●●
●●
●
●●
●●●
●
●
●
●●●
●
●
●
●
●●●●●
●
●
●
●●●
●
●
●●●
●
●
●
●
●●
●
●
●
●
●●
●
●●●●●●●
●●●
●●
●
●●
●
●●●●
●●
●
●
●●
●
●●●●●●●●●
●●●
●●
●
●●●●●●●●
●●
●●●
●
●●●
●●●●●●
●
●●
●●●
●●●
●●●●●●●●●●●
●●●●●
●
●●●
●●●
●
●●●●
●
●
●●●●●
●
●
●
●
●
●
●●●●●●●
●●●●●●
●●●●●
●●
●●●●
●●●●●
●
●●●●●●●
●
●
●●●●●●●
●
●
●●●●●●●
●●
●●●●●
●●●●●●
●
●
●●
●●
●●
●
●●
●●●
●●●
●●
●●●●●
●●●
●
●
●●●●●●●●●●●
●
●●●●●●●
●
●●
●
●●
●
●
●
●●●●
●●●
●●●●●●
●
●
●
●●●●●●
●●
●●●●●●●
●
●●●●●●
●
●●●●●
●
●
●
●●●●●
●●●
●
●●
●●●●●●●●●
●
●
●
●
●
●●●●●●
●●
●●●●
●
●●●●
●●
●●
●●●●
●
●●●
●●●●
●●●●●●●●●●●
●
●●●●●●●
●
●
●●●●●●●●●●
●
●
●●●●
●
●
●●●
●
●●
●
●●●●●
●●●●
●
●●●
●
●●●●●●
●●●●●●
●
●
●
●
●
●●●●●●●●●●●●●●●●
●
●●
●●●●●
●●
●●●
●
●
●●
●●●●●
●●●●●
●
●●
●●
●●
●●●●●●
●●●●●●
●●
●
●●
●●●●●●●●●●●●●
●
●
●
●●
●
●●
●
●
●●●●●●●●●●
●●
●●●●
●
●●
●
●
●●●●
●●●●●●●●●●●●●●●●●●●
●●●●
●
●●●●●●●●●
●●●●●●
●
●
●●●●
●●●●
●
●●
●●
●●●●●●●
●
●
●
●●●
●●●●●●
●
●●●●
●
●
●●●●●
●
●●●●●●
●●●●
●
●●●●●●●●●●●●●●●
●●●●●●
●
●
●
●●●●●●●●●
●
●●●●
●
●●●●●●●
●●
●
●●●●●●●●●●●
●
●
●●●
●●●
●
●●●
●●●
●
●●●●●●●●●
●
●
●●
●●●●●●
●●●
●
●
●
●
●●
●
●●●
●●●●●●●●●●●●●
●●●●
●
●
●●●●●●
●●
●●●●●●●●●●●
●
●●
●
●●●●
●
●●●●●●●●●●●
●
●●
●●●
●
●●●●●
●●●●
●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●
●●
●
●●●●●●●
●●●●●●
●●
●●●●●●●●●●●●●●●●●
●
●●●●●●
●
●
●●●●●●●●●●●●●●
●
●●●
●
●●●●●
●
●●●●
●
●●●●●●●●
●
●●●
●
●●●●●●●
●
●●●●●●
●
●
●●●
●
●●●●
●
●
●●●●●
●●●
●●●●●●●●●●●●●●
●
●●●●●●
●
●●
●
●●●●
●●●●
●●●
●●●●●●
●
●●●●●●●●●●●●
●
●
●●●
●●●●●●●
●●●●●●●
●●●●●●●●●
●●●●●●●
●
●
●●●●●●●●●●●●●●●●●●●
●●●●●●●
●
●●●●●●
●
●
●●●●●●●
●●●●●
●●
●●●●●●●●●●
●●●●
●
●●
●●●●●●
●
●●●●
●●●
●
●●
●●
●
●
●●●●●●●●
●
●●
●●
●
●●●●●●●
●●●●●●●●
●●●●●●●●●
●●●●●
●●●●
●●
●
●●
●●●●●●●●●●●●●●
●
●
●●●
●
●
●
●●●●●●●
●●●●●
●
●●●●●●●●●●
●
●
●●●●
●●●●●●●●●●●●●
●
●
●
●
●●●●
●
●●
●
●●●●●●●●●●
●●●●●
●
●●●●●●●
●●●
●●
●
●●●●●●●
●
●●●●
●
●●
●●●
●
●●●●●●●●●●●●●●●●●●●
●
●
●
●●●●●●●●●●●●●●●
●
●●
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
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
−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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Multivariate Stationary Time Series
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Multivariate Stationary Time Series
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Causality, in dimension 2
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
(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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
INAR(1) & Galton-Watson
Xt+1 =Xt∑i=1
Yi + εt+1, where Y ′i s are i.i.d. B(p)
@freakonometrics 13
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Granger causality with BINAR(1)
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Granger causality with BINAR(1)
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Granger causality with BINAR(1)
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Granger causality with BINAR(1)
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Granger causality with BINAR(1)
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 3 hours
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 6 hours
@freakonometrics 30
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 12 hours
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 24 hours
@freakonometrics 31
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 36 hours
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 48 hours
@freakonometrics 32
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Causality, in dimension 2
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
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
Arthur Charpentier, Causality & (non-Gaussian) Time Series, P7
Online vs. Onsite Causality
For #occupy and #indignados
●
●
●
●
●
●
F
T
Protestors
P
Injuries
I
Arrests
A
●
●
●
●
●
●
F
T
Protestors
P
Camped
C
Arrests
A
Application in Bastos, Mercea & C. (2015)
@freakonometrics 36