This document summarizes a presentation on testing for volatility transmission between international markets using high frequency data. It discusses using realized volatility to estimate true latent volatility processes while controlling for jumps and microstructure noise. The presentation focuses on testing for transmission of only extreme or large volatility values between markets. A quantile model is used to define extreme periods, and cross-covariances are computed to test for non-causality between markets' extreme periods using Ljung-Box statistics. Simulations are performed based on a three-regime smooth-transition model to assess the test in finite samples.

1. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
Transmission between International Markets & Causality
A. Charpentier (Université de Rennes 1)
Discussion of
Testing for Extreme Volatility Transmission
with Realized Volatility Measures
By Sessi Tokpavi,
with Christophe Boucher, Gilles de Truchis and Elena Dumitrescu
IMAC3, International MACroeconomics Workshop, December 2016.
http://freakonometrics.hypotheses.org
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2. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
Granger Causality
Causality 2→ 1,
P[X
(1)
t+1 ∈ A | F
(1,2)
t ] = P[X
(1)
t+1 ∈ A | F
(1)
t ]
see Granger (1969), or more conveniently
f(x
(1)
t+1|x
(1)
t ) = f(x
(1)
t+1|x
(1)
t , x
(2)
t ).
Classical interpretation on the first moment
E[X
(1)
t+1 | F
(1,2)
t ] = E[X
(1)
t+1 | F
(1)
t ]
with a linear model (VAR) it is a nullity test, i.e.
X
(1)
t+1 = φ11X
(1)
t + φ12
?
=0
X
(2)
t + εt+1
See also Taamouti et al. (2012) or Bouezmarni et al. (2009).
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3. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
Volatility transmission between international markets
Volatility transmission with High Frequency Data
Use of realized volatility to estimate the true latent process of volatility
But as mentioned in Corradi, Distaso & Fernandes (2012) it is necessary to control
for the effect of jumps and microstructure noise in the log-price process when
testing for spillover effects in the integrated variance.
Here volatility transmission test focuses only on extreme or large values of the
volatility process.
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4. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
The Model
Underlying continous time model
dp(t) = µ(t)dt + Σ(t)dW t + ξdNt
with J(t) = ξ(t)dNt.
From an econometrician perspective, we have discrete time observed return
T = number of days and M = ∆−1
= number of intraday returns,
rt+j∆ = pt+j∆ − pt+(j−1)∆ where pt+j∆ = pt+j∆ + t+j∆
Realized variance (see Barndorff-Nielsen & Shephard (2001)) is
RV t,M =
M
j=1
rt−1+j∆rT
t−1+j∆
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5. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
Inference
Quadratic variation is QV t+1 =
t+1
t
Σ(τ)dτ
IV t+1
+
t+1
τ=t
J(τ)J(τ)T
dτ
JV t+1
A consistent
nonparametric estimator of QV t+1 (see Barndorff-Nielsen & Shephard (2002)) is
RV t,M , since RV t,M
P
→ Qt as M → ∞.
A consistent nonparametric estimator of IV t+1 (see Barndorff-Nielsen & Shephard
(2002)) is BV t,M (bipolar variation), with
BV t,M =
π
2
M
j=2
|rt−1+j∆||rT
t−1+(j−1)∆|
IV t = integrated variance, with cdf Ft(·), and maginal quantile functions qi,t(·).
Extrem periods are defined as zi,t(α) = 1(IVi,t > qi,t(α)).
For non causality 2 → 1, test here H0 : E[z1,t(α)|F
(1,2)
t−1 ] = E[z1,t(α)|F
(1)
t−1].
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6. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
A model for qt(α)
Consider a heterogeneous autoregressive quantile model (HARQ), see Zikes &
Barunik (2014)
qt(α) = θ1 + θ2IV t−1 + θ3V
(5)
t + θ4V
(22)
t +εt
with V
(J)
t = 1
J
J
j=1 IV t−j.
Using Koenker & Bassett Jr (1978), estimate θi by solving
argmin
θ∈R4
1
T
T
t=2
(α − 1(ui,t < 0)) · ui,t where ui,t = IV i,t − qi,t(α, θ)
and then set zi,t(α) = 1(IV i,t > qi,t(α)).
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7. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
Cross-covariance at lag-order h
Then compute cross-covariances
ρ
(1,2)
h =
C
(1,2)
h
α(1 − α)
with C
(1,2)
h =
1
T−j
T
t=1+j
[z1,t(α) − (1 − α)][z2,t(α) − (1 − α)]
and consider Ljung & Box (1978)’s statistics,
Q
(1,2)
H = T(T + 2)
H
h=1
[ρ
(1,2)
h ]2
T − h
Under H0, Q
(1,2)
H ∼ ξ2
(H) as T, M, → ∞.
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8. Arthur CHARPENTIER - IMAC3 - International MACroeconomics Workshop
From asymptotics to finite sample
M = 23, 400 (1 second sampling frequency and a trading day of 6.5 hours)
Simulations based on log-volatility follows a stationary three-regime
smooth-transition heterogeneous autoregressive (HARST) model
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