New Approaches to Robust Inference on Market (Non-)Effciency, Volatility Clustering and Nonlinear Dependence

New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence
New Approaches to Robust Inference on Market
(Non-)Eciency, Volatility Clustering and Nonlinear
Dependence
‚ust—m s˜r—gimova,b
D ‚—smus €edersenc
D enton ƒkro˜otovd,b
a
Imperial College Business School
b
SPBU
c
University of Copenhagen
d
RANEPA
Eesti Pank 2019
Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 1 / 50

Stylized facts of nancial markets
„he following —re three most import—nt stylized f—™ts of (n—n™i—l returns time
series (Rt) th—t mu™h of the empiri™—l liter—ture —grees uponD together with the
st—nd—rd me—nEzero property @E(Rt) = 0A th—t implies the —˜sen™e of system—ti™
g—ins or lossesX
@iA e˜sen™e of line—r dependen™e —nd line—r —uto™orrel—tions th—t provides
the support for the we—k e0™ient m—rket hypothesisD th—t isD for the
m—rting—le di'eren™e property of (n—n™i—l returnsX
Corr(Rt, Rt−h) ≈ 0, @IA
even for sm—ll l—gs h = 1, 2, ...,

g—ins or lossesX
@iiA „he presen™e of nonline—r dependen™e —nd vol—tility ™lusteringD ™—ptured
˜y signi(™—nt positive —uto™orrel—tion in simple nonline—r fun™tions of the
returns —nd di'erent me—sures of vol—tilityD su™h —s squ—red returnsX
Corr(R2
t , R2
t−h) 0, @PA
even for l—rge l—gs h 0. „his property impliesD in p—rti™ul—rD th—t (n—n™i—l
returns —re not iFiFdF —nd thus the strong m—rket e0™ien™y hypothesis does
not holdF

g—ins or lossesX
@iiiA re—vy t—ilsX „he returns distri˜utions —re nonEnorm—l —nd exhi˜it
powerEl—w or €—retoElike t—ilsD
P(|Rt| x) ∼ C/xζ
, @QA
for l—rge positive x sD with C 0 —nd the t—il index ζ 0F

€roperties @iAE@iiA ! m—rket e0™ien™y hypotheses ˜y iF p—m— —nd @qAe‚gr
time series ˜y ‚F ingle —nd gF qr—nger E the ™ontri˜utions re™ognised with
the xo˜el wemori—l €rize in i™onomi™ ƒ™ien™esF
iFgFD @qAe‚gr time series ! —˜sen™e of line—r —uto™orrel—tions —nd the
presen™e of vol—tility ™lustering E stylized f—™ts @iAE@iiA E th—t they ™—pture ˜y
the very de(nition @seeD —mong othersD the reviews in ghF IP in g—mp˜ell
et —lF @IWWUAD ghF R in ghristo'ersen @PHIPA —nd ghF R in w™xeil et —lF
@PHISAAF
re—vyEt—iled power l—ws @QA produ™eD —s is ™on(rmed ˜y m—ny empiri™—l
studiesD — good (t to the distri˜ution of (n—n™i—l returns —nd other import—nt
v—ri—˜les in (n—n™e —nd e™onomi™sF

qe‚gr @qener—lized euto‚egressive gondition—l reterosked—sti™ityA model
of dyn—mi™ v—ri—n™e
sde—X vol—tility depends on the p—st
Rt = σtZt, t ∈ Z,
Zi ∼ i.i.d.N(0, 1)F
e‚gr@IAX
σ2
t = ω + αR2
t−1.
qe‚gr@IDIAX
σ2
t = ω + αR2
t−1 + βσ2
t−1,
α + β 1F

Volatility Clustering Heavy
Tails in GARCH

Stylized Facts of Real-World Returns
28
0. Mean =0 1. No linear autocorrelations: Efficient markets (?)
2. Volatility clustering: Nonlinear dependence
3. Crises, large fluctuations (unconditional) heavy tails

re—vyEt—iled distri˜utions —lso provide — ™onvenient fr—mework for modelling
—nd qu—ntifying @˜y their key p—r—meter E the t—il index ζA the likelihood of
l—rge downf—llsD l—rge )u™tu—tions —nd ™rises in (n—n™i—l —nd e™onomi™
m—rketsF
sn @QAD the sm—ller v—lues of the t—il index p—r—meter ζ ™orrespond to — l—rger
likelihood of ™risesD l—rge downf—lls —nd l—rge )u™tu—tions —'e™ting the
(n—n™i—l returns time series (Rt), —nd vi™e vers— @see the dis™ussion in ghF I
in s˜r—gimov et —lF @PHISAAF
„he t—il index p—r—meter ζ governs existen™e of moments of Rt, withD for
inst—n™eD the v—ri—n™e of Rt ˜eing de(ned —nd (niteX V ar(Rt) ∞ if —nd
only if ζ 2, —ndD more gener—llyD the pth moment E|Rt|p
, p 0, ˜eing
(niteX E|Rt|p
∞ if —nd only if ζ p.
„he most of the empiri™—l liter—ture on he—vyEt—iled distri˜utions —grees th—tD
in the ™—se of developed (n—n™i—l m—rketsD the returns9 t—il indi™es ζ ˜elong to
the interv—l (2, 4), thus implying (nite v—ri—n™es —nd in(nite fourth momentsF
…se of the ™ommon me—sure of he—vyEt—ilednessD iFeF kurtosisD is
in—ppropri—teF

qe‚gr models —lso ™—pture the he—vy t—ils stylized f—™t ! the t—il index ζ
depends on the p—r—meters @—ndD in the gener—l ™—seD on the distri˜ution of
the qe‚gr innov—tionsA of the qe‚gr model vi— uesten9s equ—tion @see
h—vis —nd wikos™h @PHHH—AD wikos™h —nd ƒt¨—ri™¨— @PHHHAAF
sn p—rti™ul—rD even thinEt—iled norm—l distri˜ution of the innov—tions E —
st—tion—ry qe‚gr time series (Rt) would h—ve powerEl—w t—ils @QA with the
t—il index ζ ∈ (2, 4), —s in the ™—se of (n—n™i—l returns in re—lEworld developed
m—rketsD for —n —ppropri—te ™hoi™e of qe‚gr p—r—meters s—tisfying simple
™onditions implied ˜y uesten9s equ—tionF

Measures of market (non-)eciency, nonlinear dependence and volatility clustering based on absolute returns and their powers
Measures of market (non-)eciency, nonlinear dependence
and volatility clustering based on absolute returns and their
powers
„he st—nd—rd —ppro—™h to testing properties @iAE@iiA ! fullEs—mple estim—tes of
popul—tion —uto™orrel—tion ™oe0™ients @s—mple —uto™orrel—tionsA of the
returns —nd their squ—res —nd —ppe—ling to the ™entr—l limit theorem for themF
roweverD —s the t—il index ζ is sm—ller th—n RX ζ 4D thus implying in(nite
fourth momentsD the popul—tion —uto™orrel—tions Corr(R2
t , R2
t−h) of squ—red
returns —re not even de(nedD m—king ™ondition @PA me—ninglessF „heir s—mple
—n—loguesD the s—mple —uto™orrel—tions of R2
t , —re not ™onsistentD implying
r—ndom )u™tu—tions even in l—rge s—mplesF
ƒimil—rlyD the popul—tion line—r —uto™orrel—tions Corr(Rt, Rt−h) —re not
de(ned in the ™—se of t—il indi™es ζ sm—ller th—n PX ζ 2 —nd in(nite
v—ri—n™es th—t —re often o˜served for (n—n™i—l returns in emerging —nd
developing m—rketsD thus m—king me—ningless ™ondition @IA —nd its testing
using s—mple line—r —uto™orrel—tions of the returns th—t lose ™onsisten™y in
this ™—seF

powers
euto™orrel—tions of —˜solute returns Corr(|Rt|, |Rt−h|) —nd their powers
Corr(|Rt|p
, |Rt−h|p
), p 0, with most of the studies —nd empiri™—l
—ppli™—tions fo™using on the ™—se p = 1 th—t ™orresponds to the returns9
—˜solute v—lues @see hing —nd qr—nger @IWWTAD hing et —lF @IWWQAAF
hing —nd qr—nger @IWWTAD hing et —lF @IWWQAD qr—nger —nd hing @IWWSAD —nd
gont @PHHIA indi™—te th—tD for — given l—g h, the —uto™orrel—tion
Corr(|Rt|p
, |Rt−h|p
) —ppe—rs to ˜e m—ximised for p = 1 thus implying th—t
—˜solute returns tend to ˜e more predi™t—˜le th—n other powers of returnsF
fut the —n—lysis of —uto™orrel—tions of powers of —˜solute returns in the
liter—ture did not rely on e™onometri™—lly justi(ed or ro˜ust inferen™eD whi™h
is the m—in fo™us in this studyF

powers
sn view of unde(ned —uto™orrel—tions Corr(R2
t , R2
t−h) —nd in™onsisten™y of
their s—mple —n—logues in the ™—se of (n—n™i—l returns time series (Rt) with
in(nite fourth momentsD these ™ontri˜utions m—ke it n—tur—l to ™onsider the
modi(™—tion of vol—tility ™lustering stylized f—™t Q in terms of —uto™orrel—tions
of —˜solute returnsX Corr(|Rt|, |Rt−h|) 0, even for l—rge l—gs h 0.
€ro˜lemX in the ™—se of qe‚gr models for (Rt), —symptoti™ norm—lity of
s—mple line—r —uto™orrel—tions of Rt —nd the —˜solute v—lues |Rt| holds only
in the ™—se of t—il indi™es ζ gre—ter th—n RX ζ 4 —nd (nite fourth moments
@see h—vis —nd wikos™h @PHHH—AD wikos™h —nd ƒt¨—ri™¨— @PHHHAAF

powers
sntuitionX v—ri—n™e V ar(RtRt−1) of the summ—nds RtRt−1 th—t —ppe—rs in
s—mple (rstEorder —uto™ov—ri—n™es —nd —uto™orrel—tions of — returns time
series (Rt) th—t follows qe‚gr@ID IA pro™ess is given ˜y
V ar(RtRt−1) = E(R2
t R2
t−1) = E(σ2
t σ2
t−1) = E((ω+αR2
t−1+βσ2
t−1)σ2
t−1) =
ωE(σ2
t−1) + (α + β)E(σ4
t−1) = ωV ar(Rt) + (α + β)E(R4
t )/E(z4
t ) —nd is
thus (nite if —nd only if the fourth moment of Rt is (niteX E(R4
t ) ∞.
qe‚gr modelsX (Rt), with t—il indi™es 2 ζ 4 —s is typi™—l for (n—n™i—l
returns in developed m—rketsD the s—mple —uto™ov—ri—n™es —nd
—uto™orrel—tions of Rt —nd |Rt|, —l˜eit ™onsistentD ™onverge in distri˜ution to
nonEnorm—l limits given ˜y st—˜le rFvF9s or their r—tiosF
smpossi˜le to use these we—k ™onvergen™e results for testing for the stylized
f—™ts @iA —nd @iiA dire™tlyF „his is ˜e™—use the r—te of ™onvergen™e of the
s—mple —uto™ov—ri—n™es —nd —uto™orrel—tions of Rt —nd |Rt| depends on the
unknown t—il index ζ of the qe‚gr pro™ess (Rt)F

powers
‚—te of ™onvergen™e of s—mple —uto™ov—ri—n™es —nd —uto™orrel—tions is slower
th—n
√
n —nd thus the true ™on(den™e ˜—nds for the —uto™ov—ri—n™es —nd
—uto™orrel—tions of Rt —nd |Rt| —re wider th—n those implied ˜y the ™entr—l
limit theoremF
ƒimil—r ™on™lusions hold for the we—k ™onvergen™e of s—mple —uto™ov—ri—n™es
—nd —uto™orrel—tions of squ—res R2
t of qe‚gr time series (R2
t ) : „heir
—symptoti™ norm—lity requires ζ 8 —nd thus (nite eighth moments of (Rt).
sn the ™—se 4 ζ 8, the s—mple —uto™ov—ri—n™es —nd —uto™orrel—tions of
R2
t —re ™onsistent ˜ut ™onverge to nonEnorm—l limits given ˜y st—˜le rFvF9s or
their r—tiosD with the r—te of ™onvergen™e th—t is slower th—n
√
n —nd —lso
dependsD —s do the limiting distri˜utionsD on the unknown t—il index v—lue ζF
„he ™on(den™e ˜—nds for —uto™orrel—tions of squ—red qe‚gr time series R2
t
—re thus wider th—n those implied ˜y the ™entr—l limit theoremF

Contributions and key results
‡e provide — support for the use of —uto™ov—ri—n™es Cov(|Rt|p
, |Rt−h|p
) of
sm—ll order powers of (n—n™i—l returns @with p 0.5 for (n—n™i—l returns (Rt)
with the t—il index ζ 2 —nd (nite v—ri—n™es —nd p 0.25 for the returns
with the t—il index ζ 1 —nd (nite (rst momentsA —s me—sures of nonline—r
dependen™e —nd vol—tility in (n—n™i—l m—rketsD with the —n—logues of property
@iiA given ˜y
(ii) Cov(|Rt|p
, |Rt−h|p
) 0, even for large lags h 0.
‡e propose the —uto™ov—ri—n™es Cov(Rt, |Rt−h|q
sign(Rt−h)), of 9signed9
powers of —˜solute returns —s me—sures of m—rket @nonEAe0™ien™y in
—n—logues of property @iAX
(i') Cov(Rt, |Rt−h|q
sign(Rt−h) ≈ 0, even for small lags h = 1, 2, ...
@property @i9A ™oin™ides with un™orrel—tedness property P in the ™—se q = 1.
„he ™hoi™e of sm—ll powers q is justi(ed under empiri™—lly o˜served
he—vyEt—iledness in re—lEworld (n—n™i—l m—rketsD with the —ppropri—te ™hoi™e
of power v—lues q ˜eing q ζ/2 − 1 for the returns time series with the t—il
index ζ ∈ (2, 4)F

Contributions and key results
‡e ™onsider —uto™orrel—tions of gener—l fun™tions of (n—n™i—l returnsD
in™luding the ™—se of power fun™tions used in de(nitions @i9A —nd @iiA —nd
est—˜lish the results on —symptoti™ norm—lity of s—mple —n—logues of these
me—suresF
‡e fo™us on the development of ro˜ust inferen™e —ppro—™hes on them th—t
do not require di0™ult estim—tion of limiting v—ri—n™es of the me—sures9
estim—tesF
„he —ppro—™hes —re ˜—sed on ro˜ust inferen™e methods exploiting
™onserv—tiveness properties of t−st—tisti™s @s˜r—gimov —nd w¤ullerD PHIHD
PHITA —nd sever—l new results on their —ppli™—˜ility in the settings ™onsideredF

Autocovariance and autocorrelation functions of powers of absolute values of returns
Autocovariance and autocorrelation functions of powers of
absolute values of returns
vet Z = {..., −2, −1, 0, 1, 2, ...}. gonsider — qe‚gr@ID IA pro™ess (Rt)t∈Z,
de(nedD given nonneg—tive p—r—meters ω, α, β, ˜y
Rt = σtZt, t ∈ Z, @RA
where (Zt)t∈Z is sequen™e of iFiFdF r—ndom v—ri—˜les @rFvF9sA with me—n zero
—nd unit v—ri—n™eX E(Zt) = 0, V ar(Zt) = 1, —nd (σ2
t ) is — vol—tility pro™ess
σ2
t = ω + αR2
t−1 + βσ2
t−1. @SA
„he vol—tility pro™ess (σ2
t ) of — qe‚gr@ID IA pro™ess @RAE@SA h—s — st—tion—ry
version if ω 0 —nd E[log(α1Z2
+ β1)] 0. elsoD in this ™—seD st—tion—rity of
(σ2
t ) implies st—tion—rity of the qe‚gr@ID IA pro™ess (Rt).
fy uesten9s theoremD the distri˜ution of st—tion—ry qe‚gr@ID IA pro™ess
(Rt) h—s power l—w t—ils @QA with the t—il index ζ 0 th—t the unique positive
solution to the equ—tion
E(αZ2
+ β)ζ/2
= 1. @TA

vet κZ = E(Z4
) denote the kurtosis of innov—tions Z of — st—tion—ry pro™ess
@RAE@SAF
prom uesten9s equ—tion @TAD the t—il index ζ of st—tion—ry qe‚gr@ID IA
pro™ess (Rt) is gre—ter th—n PX ζ 2, —nd thus the @un™ondition—lA v—ri—n™e
of Rt is (niteX V ar(Rt) ∞ if —nd only if α + β 1.
„he t—il index is sm—ller th—n RX ζ 4 —ndD thusD the fourth moment of Rt is
in(niteX E(Rt)4
∞ if —nd only if α2
κZ + 2αβ + β2
1, th—t isD if —nd
only if (α + β)2
1 − (κZ − 1)α2
.
„hereforeD the t—il index ζ of — st—tion—ry qe‚gr@ID IA pro™ess (Rt) ˜elongs
to the interv—l (2, 4) : ζ ∈ (2, 4), —s in the ™—se of re—lEworld developed
m—rkets if —nd only if 1 − (κZ − 1)α2
(α + β)2
1.
sn the ™—se of the ™ommonly used st—nd—rd norm—l innov—tions Z ∼ N(0, 1),
the ™onditions for ζ ∈ (2, 4) —re 1 − 2α2
(α + β)2
1.

ƒimil—rlyD the se™ond moment of — st—tion—ry qe‚gr@ID IA pro™ess (Rt) in
@RAE@SA is in(niteX E(Rt)2
= ∞, —s is often o˜served for (n—n™i—l returns in
emerging —nd developing m—rketsD if —nd only if α + β ≥ 1.
sn the ™—se of —n e‚gr@IA pro™ess (Rt) with β = 0 in @RAE@SAD the ™onditions
for ζ ∈ (2, 4) ˜e™ome 1/
√
κZ α 1. sn the ™—se of e‚gr@IA with
st—nd—rd norm—l innov—tions Z ∼ N(0, 1) the ™ondition ζ ∈ (2, 4) holds if
—nd only if 1/
√
3 α 1.

por 0 p ζ/4, —nd h = 0, 1, 2, ..., denote ˜y γ|R|p (h) —nd ρ|R|p (h) the
@deterministi™D popul—tionA —uto™ov—ri—n™e —nd —uto™orrel—tion fun™tions
@eg†p —nd egpA of the p−th power |Rt|p
of the —˜solute v—lues |Rt| :
γ|R|p (h) = Cov(|Rt|p
, |Rt−h|p
), ρ|R|p (h) = γ|R|p (h)/γ|R|p (0)F
sn p—rti™ul—rD for p = 2 —nd ζ 8, one gets the tr—dition—l eg†p —nd egp
γR2 (h) = Cov(R2
t , R2
t−h), —nd ρR2 (h) = γR2 (h)/γR2 (0), where
γR2 (0) = σ2
R2 , of the squ—red returns R2
t in st—nd—rd vol—tility ™lustering
de(nition @PAF
sn the ™—se p = 1 —nd ζ 4, one gets the eg†p —nd egp
γ|R|(h) = Cov(|Rt|, |Rt−h|), ρ|R|(h) = γ|R|(h)/γ|R|(0), where
γ|R|(0) = σ2
|R| of —˜solute v—lues |Rt| of the returnsF

por ζ 2, 0 q ζ/2 − 1, we —lso denote ˜y γR,|R|qsign(R)(h) —nd
ρR,|R|qsign(R)(h) the me—sures of m—rket @nonEAe0™ien™y given ˜y 9signed9
powers of —˜solute returnsX
γR,|R|qsign(R)(h) = Cov(Rt, |Rt−h|q
sign(Rt−h)),
ρR,|R|qsign(R)(h) = γR,|R|qsign(R)(h)/(σRσ|R|q )F
sn the ™—se q = 1, the —uto™ov—ri—n™es —nd —uto™orrel—tions γR,|R|qsign(R)(h)
—nd ρR,|R|qsign(R)(h) ˜e™ome the st—nd—rd line—r —uto™ov—ri—n™es —nd
—uto™orrel—tions γR(h) = Cov(Rt, Rt−h), ρR(h) = Corr(Rt, Rt−h) of the
pro™ess (Rt)F
sn the —˜ove not—tionD —n—logues of stylized f—™ts @iAD @iiA on —˜sen™e of line—r
—uto™orrel—tions —nd presen™e of nonline—r dependen™e —nd vol—tility
™lustering in (n—n™i—l returns ˜e™ome
(i') γR,|R|qsign(R)(h), ρR,|R|qsign(R)(h) ≈ 0, even for small lags h = 1, 2, ...
(ii) γ|R|p (h), ρ|R|p (h) 0, even for large lags h 0.

Asymptotic normality of sample autocovariance and autocorrelations
Asymptotic normality of sample autocovariance and
autocorrelations
henote ˜y ˆµR = 1
T
T
t=1 Rt the s—mple me—n of Rt, —ndD for p 0, denote
˜y ˆµ|R|p = 1
T
T
t=1 |Rt|p
the s—mple me—n of p−th powers |Rt|p
of —˜solute
v—lues |Rt|. por q 0, we —lso denote ˜y
ˆµ|R|qsign(R) = 1
T
T
t=1 |Rt|q
sign(Rt) the s—mple me—n of signed q−th
powers of the —˜solute v—lues |Rt|.
por p 0 —nd h = 0, 1, 2, ..., denote ˜y ˆγ|R|p (h), ˆρ|R|p (h) the usu—l
@fullEs—mpleA estim—tors of the —˜ove popul—tion eg†p —nd egp9s γ|R|p (h)
—nd ρ|R|p (h), th—t is the s—mple eg†p —nd egp9s
ˆγ|R|p (h) = 1
T
T
t=h+1(|Rt|p
− ˆµ|R|p )(|Rt−h|p
− ˆµ|R|p ),
ˆρ|R|p (h) = ˆγ|R|p (h)/ˆγ|R|p (0).
sn p—rti™ul—rD in the ™—ses p = 2 —nd p = 1, the estim—tes ˆγR2 (h), ˆρR2 (h) —nd
ˆγ|R|(h), ˆρ|R|(h) ˜e™ome the s—mple eg†p —nd egp9s of the squ—red returns
—nd the —˜solute v—lues of the returnsD respe™tivelyF

autocorrelations
ƒimil—rlyD denote ˜y ˆγ R,|R|qsign(R)(h), ˆρR,|R|qsign(R)(h) the usu—l
@fullEs—mpleA estim—tors of the popul—tion eg†p —nd egp9s of signed powers
γR,|R|qsign(R)(h) —nd ρR,|R|qsign(R)(h), th—t is the s—mple eg†p —nd egp9s
ˆγR,|R|psign(R)(h) = 1
T
T
t=h+1(Rt − ˆµR)(|Rt−h|q
sign(Rt−h) − ˆµ|R|qsign(R)),
ˆρR,|R|qsign(R)(h) = ˆγR,|R|qsign(R)(h)/ˆγR,|R|qsign(R)(0).
sn the ™—se q = 1, the estim—tors ˆγR,|R|qsign(R)(h) —nd ˆρR,|R|qsign(R)(h)
˜e™ome the usu—l s—mple line—r —uto™ov—ri—n™es —nd —uto™orrel—tions

autocorrelations
„he following result provides — ˜—sis for —symptoti™ inferen™e on property @iiA E
—n—logue of properties @iiA E on the presen™e of nonline—r dependen™e —nd vol—tility
™lustering in (n—n™i—l returnsF
Theorem
sf 0 p ζ/4, then
√
T(ˆγ|R|p (h) − γ|R|p (h))h=0,1,...,m →d (Gh,p)h=0,1,...,m. @UA
sf 0 p ζ/8, then
√
T(ˆρ|R|p (h) − ρ|R|p (h))h=0,1,...,m →d (Hh,p)h=0,1,...,m, @VA
where the limits —re multiv—ri—te q—ussi—n with me—n zeroF

autocorrelations
„he following theorem provides — ˜—sis for testing —nd —symptoti™ inferen™e on
—n—logue @i9A of stylized f—™t @iA on the —˜sen™e of line—r —uto™orrel—tions in
(n—n™i—l returnsF
Theorem
sf 0 q ζ/2 − 1, then
√
T( ˆγ R,|R|psign(R)(h) − γ R,|R|psign(R)(h))h=0,1,...,m →d (Gh,p)h=0,1,...,m. @WA
sf q ζ/4 − 1, then
√
T(ˆρ R,|R|psign(R)(h) − ρ R,|R|psign(R)(h))h=0,1,...,m →d (Hh,p)h=0,1,...,m, @IHA
where the limits —re multiv—ri—te q—ussi—n with me—n zeroF

autocorrelations
ƒimil—r to the results in h—vis —nd wikos™h @PHHH—AD wikos™h —nd ƒt¨—ri™¨— @PHHHAD
the limiting distri˜utions in „heorems RFI —nd RFP ˜e™ome slower th—n
√
T —nd
the limits ˜e™ome nonEnorm—l in the ™—se ζ/4 p ζ/2 in RFI —nd
ζ/2 − 1 p ζ − 2 in RFPF sn su™h ™—sesD the limits G —nd G of s—mple
—uto™ov—ri—n™es in the theorems ˜e™ome st—˜le rFvF9s with the index of st—˜ility
@the t—il indexA α given ˜y α = ζ/(2p) 2 —nd α = ζ/(1 + p) 2, respe™tively
@see s˜r—gimov et —lF @PHISA —nd referen™es therein for — review of properties of
st—˜le distri˜utionsAF ƒimil—rlyD if p ζ/4 in RFI —nd p ζ/2 − 1 in RFID the limits
H —nd H of s—mple —uto™orrel—tions in the theorems —re r—tions of st—˜le rFvF9s
with indi™es of st—˜ility th—t depend on ˜oth p —nd the unknown ζ. ƒimil—r to
€edersen @PHIWAD one ™—n show th—tD under the —˜ove ™onditions on p —nd ζ, the
st—˜le limits G —nd G of s—mple —uto™ov—ri—n™es in the theorems —re symmetri™ if
the distri˜ution of qe‚gr innov—tions (Zt) is symmetri™F

autocorrelations
„he limiting distri˜ution for s—mple —uto™ov—r—in™es —nd —ut™orrel—tions for
qe‚gr pro™esses @in™luding their squ—resA h—ve ˜een studied ˜y f—sr—k et
—lF @PHHPD ƒ€eA —nd vindner @PHHWD in r—ndõok of pin—n™i—l „ime ƒeriesAF
ƒuppose th—t E[log(α1Z2
+ β1)] 0 —nd th—t Z h—s — ve˜esgue density
supported on R —nd õunded —w—y from zero on ™omp—™t su˜sets of RF
„hen it follows ˜y „heorem Q of weitz —nd ƒ—ikkonen @PHHVA th—t the
w—rkov ™h—in ((Rt, σ2
t ) : t = 0, 1, . . .) is geometri™—lly ergodi™ —nd the
—sso™i—ted stri™tly st—tion—ry solution ((Rt, σ2
t ) : t ∈ Z) is βEmixing with
geometri™ de™—yF
„his implies th—t (xt) is strongly mixing with geometri™ de™—yF por some
(xed h, s 0 let f : Rh+1
→ Rs
˜e me—sur—˜leF
ƒin™e mixing is — property —õut the σE(eld gener—ted ˜y (Rt : t ∈ Z)D it
holds th—t (f(Rt, . . . , Rt−h) : t ∈ Z) is strongly mixing with geometri™ de™—yF
„he ide— is then to —pply — gv„ for strongly mixing pro™essesD su™h —s
„heorem IVFSFQ of s˜r—gimov —nd vinnik @IWUIAF

autocorrelations
Theorem
Let f : R → R and g : R → R be measurable functions. Consider the sample auto cross
covariance function of f(Rt) and g(Rt) for some order h ≥ 0 and the population
analogues [given that they exist].
Assume that (Rt : t ∈ Z) is strongly mixing with geometric decay. Suppose that there
exists a δ 0 such that max{E[|f(Rt))|2+δ
], E[|g(Rt))|2+δ
]} ∞ and
maxh=0,...,m{E[|f(Rt)g(Rt−h)|2+δ
]} ∞. Then
√
T(ˆγT,f(R),g(R)(h) − γf(R),g(R)(h))h=0,...,m
d
→ (Gh,f(R),g(R))h=0,...,m, (11)
where (Gh,f(R),g(R))h=0,...,m is an (m + 1)-dimensional Gaussian vector with zero mean
and covariance matrix given by Γ = Var(Y0) + 2 ∞
k=1 Cov(Y0, Yk),
where Yt = (Yt,h)h=0,...,m,
Yt,h = (f(Rt) − E[f(Rt)])(g(Rt−h) − E[g(Rt−h)]) − γf(R),g(R)(h).

autocorrelations
Theorem
ƒuppose th—t in —ddition there exists — δ 0 su™h th—t
max{E[|f(Rt)|4+δ
], E[|g(Rt)|4+δ
]} ∞F „hen
√
T(ˆρT,f(R),g(R)(h) − ρf(R),g(R)(h))h=0,...,m
d
→ ( ˜Gh,f(R),g(R))h=0,...,m, @IPA
where ( ˜Gh,f(R),g(R))h=0,...,m is — q—ussi—n ve™tor with me—n zero —nd ™ov—ri—n™e
given ˜y AΓ†
A D where A is ™onst—nt m—trix de(ned in @ISA —nd
Γ†
= Var(Y †
0 ) + 2
∞
k=1
Cov(Y †
0 , Y †
k ), @IQA
with Y †
t = (Yt , Vt,1, Vt,2) D Vt,1 = (f(Rt) − E[f(Rt)])2
− γf(R),f(R)(0)D
Vt,2 = (g(Rt) − E[g(Rt)])2
− γg(R),g(R)(0)F

autocorrelations
Consider the limiting distribution of the sample correlations. Using that
max{E[|f(Rt)|4+δ
], E[|g(Rt)|4+δ
]} ∞ for some δ 0 it holds arguments similar
to the ones given above that
√
T


(ˆγT,f(R),g(R)(h) − γf(R),g(R)(h))h=0,...,m
ˆγT,f(R),f(R)(0) − γf(R),f(R)(0)
ˆγT,g(R),g(R)(0) − γg(R),g(R)(0)

 d
→ G†
, (14)
where G†
is an (m + 3)-dimensional Gaussian vector with mean zero and covariance
given by Γ†
dened in (13). Let x = (x1, . . . , xm+3) ∈ Rm+3
and dene the
function ˜g : Rm+3
→ Rm+1
as
˜g(x) = (
x1
√
xm+2xm+3
, . . . ,
xm+1
√
xm+2xm+3
) .
Dene the matrix
A =
∂˜g(x)
∂x x=γ†
, γ†
= ((γf(R),g(R)(h))h=0,...,m, γf(R),f(R)(0), γg(R),g(R)(0)) .(15)
The convergence in (12) is then obtained by an application of the ∆-method.

autocorrelations
ƒuppose th—t f(R) = R —nd g(R) = |R|p
for some p 0F fy —n —ppli™—tion
of r¤older9s inequ—lity we h—ve th—t whenever E[|R|2(p+1)+
] ∞ for some
0D the limit result for —uto™ov—ri—n™es in @IIA —ppliesF por inst—n™eD if
p = 1/2 we need (nite moments of —t le—st order QF
sf we w—nt to —pply the limit result for —uto™orrel—tions in @IPA we need th—t
Rt h—s (nite moments of —t le—st order RF

HAC estimator
Note that the limiting distributions in Theorem 2 are not particularly useful in practice since the
asymptotic covariance matrices are unknown.
We note that under suitable conditions the matrices may be estimated by HAC-type estimators
(Newey and West, 1987).
In particular, assuming that E[ Y †
t
4+
] ∞ for some 0, an application of Newey and West
(1987, Theorem 2) yields that the matrix Γ†
dened in (13) may be estimated by
ˆΓ
†
= ˆΩ
†
0 +
NT
j=1
wj (NT )ˆΩ
†
j , ˆΩ
†
j =
1
T
T
t=1
ˆY
†
t
ˆY
†
t−j , (16)
where wj (NT ) are suitable weights and NT is an increasing sequence (depending on T ), and
ˆY †
t = ( ˆYt , ˆVt,1, ˆVt,2) with ˆYt = ( ˆYt,h)h=0,...,m,
ˆYt,h = (f(Rt) −
1
T
T
t=1
f(Rt))(g(Rt−h) −
1
T
T
t=1
g(Rt)) − ˆγT,f(R),g(R)(h)
ˆVt,1 = (f(Rt) −
1
T
T
t=1
f(Rt))
2
− ˆγT,f(R),f(R)(0),
ˆVt,2 = (g(Rt) −
1
T
T
t=1
g(Rt))
2
− ˆγT,g(R),g(R)(0).

HAC estimator
„he ™ov—ri—n™e m—trix AΓ†
A of the s—mple ™orrel—tion fun™tion ™—n ˜e
estim—ted ˜y ÂˆΓ† Â with ˆΓ†
given in @ITA —nd Â given ˜y @ISA with γ†
repl—™ed ˜y
ˆγ†
= ((ˆγT,f(R),g(R)(h))h=0,...,m, ˆγT,f(R),f(R)(0), ˆγT,g(R),g(R)(0))
xote th—t the @su0™ientA moment requirement E[ Y †
t
4+
] ∞ is quite
restri™tiveF por inst—n™eD if f(R) = R we need th—t Rt h—s (nite moments of
—t le—st order VF

t−statistic based robust inference: small samples
• Bakirov and Sz´ekely (2005), Ibragimov and M¨uller (2006): Usual small
sample t−test of level α ≤ 5% : conservative for independent hetero-
geneous Gaussian observations (not α = 10%)
• Xj ∼ N(µ, σ2
j), j = 1, · · · , q : H0 : µ = 0 against H1 : µ = 0
t-statistic t =
√
q
¯X
sX
¯X = q−1 q
j=1 Xj, s2
X = (q − 1)−1 q
j=1(Xj − ¯X)2
cvq(α) = critical value of Tq−1 : P(|Tq−1| cvq(α)) = α
• P(|t| cv(α)|H0) ≤ P(|t| cv(α)|H0, σ2
1 = ... = σ2
q) =
P(|Tq−1| cv(α)) = α
• Holds under heavy tails: mixtures of normals (stable, Student-t)

Robust t−statistic inference approaches
ƒuppose we w—nt to ™ondu™t inferen™e —˜out some s™—l—r p—r—meter β of —
—uto™orrel—tedD heterogenous —nd possi˜ly he—vyEt—iled time series (Xt) using
— l—rge d—t— set of T o˜serv—tions X1, X2, ..., XT .
por — wide r—nge of time series models —nd estim—tors ˆβ of β, it is known
th—t the distri˜ution of ˆβ is —pproxim—tely norm—l in l—rge s—mplesD th—t isD√
T(ˆβ − β) →d N(0, σ2
) —s T → ∞.
sf the —uto™orrel—tions in (Xt) —re perv—sive —nd pronoun™ed enoughD then it
will ˜e ™h—llenging to ™onsistently estim—te the limiting v—ri—n™e σ2
, eFgFD ˜y
reg —ppro—™hesD —nd inferen™e pro™edures for β th—t ignore the s—mple
v—ri—˜ility of — ™—ndid—te ™onsistent estim—tor ˆσ2
F

„he results in s˜r—gimov —nd w¤uller @PHIHD PHITA provide the following
gener—l —ppro—™h to ro˜ust inferen™e —˜out —n —r˜itr—ry p—r—meter β of —
time series or other e™onomi™ or (n—n™i—l models under heterogeneityD
™orrel—tion —nd he—vyEt—iledness of — l—rgely unknown formF
gonsider — p—rtition of the origin—l d—t— s—mple X1, X2, ..., XT into — (xed
num˜er q ≥ 2 of groups of ™onse™utive o˜serv—tions Xs with
(j − 1)T/q s ≤ jT/q}.
henote ˜y ˆβj the estim—tor of β using o˜serv—tions in group j only
ƒuppose th—t the group estim—tors ˆβj —re —symptoti™—lly norm—lX√
T(ˆβj − β) →d N(0, σ2
j ) —ndD —lsoD
√
T(ˆβi − β) —nd
√
T(ˆβj − β) —re
—symptoti™—lly independent for i = j
„he ™ondition of —symptoti™ independen™e of ˆβi —nd ˆβj is — ™ondition on the
degree of @we—kA dependen™e in time series (Xt).

s˜r—gimov —nd w¤uller @PHIHAX one ™—n perform —n —symptoti™—lly v—lid test of
level α, α ≤ 0.05 of H0 : β = β0 —g—inst β = β0 ˜e reje™ting H0 when |tβ|
ex™eeds the (1 − α/2) per™entile of the ƒtudentEt distri˜ution with q − 1
degrees of freedomD where tβ is the usu—l t−st—tisti™ in group estim—tors ˆβj,
j = 1, 2, ..., q :
tβ =
√
q
ˆβ − β0
sˆβ
@IUA
with ˆβ = q−1 q
j=1
ˆβj —nd s2
ˆβ
= (q − 1)−1 q
j=1(ˆβj − ˆβ)2
, respe™tively the
s—mple me—n —nd s—mple v—ri—n™e of ˆβj, j = 1, ..., q.
es dis™ussed in s˜r—gimov —nd w¤uller @PHIHAD the t−st—tisti™ —ppro—™h to
ro˜ust inferen™e ™—n —lso ˜e used for test levels α2Φ(
√
3) ≈ 0.08326..., where
Φ(x) is the st—nd—rd norm—l ™dfF

Monte Carlo Results
Same design as in Andrews (1991): Linear Regression, 5 regressors, 4
nonconstant regressors are independent draws from stationary Gaussian
AR(1), as are the disturbances, + heteroskedasticity. T = 128, 5% level
test about coeﬃcient of one nonconstant regressor.
t-statistic (q) ˆω2
QA ˆω2
PW ˆω2
BT (b)
2 4 8 0.05 0.1 0.3 1
ρ Size
0 4.9 4.7 4.6 7.1 8.1 6.7 6.6 6.0 6.2
0.5 4.8 4.6 4.6 10.4 9.9 9.4 8.4 7.5 7.0
0.8 4.8 4.9 5.4 19.1 17.3 18.6 15.6 12.8 11.9
0.9 4.9 5.1 6.1 28.9 25.4 29.9 24.9 20.5 18.8
ρ Size Adjusted Power
0 15.1 38.4 53.7 62.7 60.6 60.7 58.6 51.9 47.2
0.5 14.5 38.2 55.9 57.0 56.2 56.0 53.5 48.4 44.2
0.8 15.4 45.1 66.0 52.9 51.7 54.0 52.6 46.9 42.4
0.9 17.2 56.7 77.6 57.5 54.6 58.7 57.5 51.4 46.6
18

por inferen™e on he—vyEt—iled time series gener—ted ˜y qe‚grEtype modelsD
the t−st—tisti™ ro˜ust inferen™e —ppro—™h rem—ins v—lid —s long —s the group
estim—tors ˆβj, j = 1, 2, ..., q, —re —symptoti™—lly independent —nd ™onvergen™e
@—t —n —r˜itr—ry r—teA to he—vyEt—iled s™—le mixtures of norm—lsF
x—melyD the —ppro—™h is —symptoti™ v—lid if mT (ˆβj − β)q
j=1 →d ZjVj,
j = 1, 2, ..., q, for some re—l sequen™e mT , where Zj ∼ i.i.d.N(0, 1), the
r—ndom ve™tor {Vj}q
j=1 is independent of the ve™tor {Zj}q
j=1 —nd
maxj |Vj| 0 —lmost surelyF
„he ™l—ss of limiting s™—le mixtures of norm—ls is — r—ther l—rge ™l—ss of
distri˜utions th—t in™ludes —ll symmetri™ st—˜le distri˜utions @see ƒe™tion
PFIFP in s˜r—gimov et —lF @PHISAA th—t —rise —s distri˜ution—l limits of
estim—tors in e™onometri™ models under he—vyEt—ilednessF
iFgFD symmetri™ st—˜le distri˜utions —riseD under symmetri™ innov—tions Z, —s
limits of s—mple line—r —uto™ov—ri—n™es ˆγR(h) of st—tion—ry qe‚grEtype
pro™esses @eFgFD qe‚gr@ID IAA with t—il indi™es ζ 4 @see €edersen @PHIWAA
—nd their power —n—logues ˆγ R,|R|psign(R)(h), p 0, in „heorem RFP in the
™—se of qe‚grEtype pro™esses with t—il indi™es ζ 4pF

„he t−st—tisti™ ro˜ust —ppro—™h is used for inferen™e on the p—r—meter β of —
st—tion—ry qe‚gr@ID IA pro™ess (Rt) given ˜y the popul—tion
—uto™ov—ri—n™es β = γR,|R|p (h), γR,|R|psign(R)(h), p 0,F „he p—r—meters of
interest —re the popul—tion —uto™ov—ri—n™es β = γR2 (h) of squ—red returns
R2
t in the ™—se p = 2, —nd the popul—tion —uto™ov—ri—n™es β = γ|R|(h) of
—˜solute v—lues |Rt| —nd line—r —uto™ov—ri—n™es β = γR(h) of Rt for p = 1F
vet R1, R2, ..., RT ˜e — l—rge s—mple of o˜serv—tions on the qe‚gr@ID IA
pro™ess ™onsideredF pollowing the t−st—tisti™ —ppro—™hD one p—rtitions the
s—mple into — (xed num˜er q ≥ 2 of groups of ™onse™utive o˜serv—tions Rs
with (j − 1)T/q s ≤ jT/q}. „he inferen™e on the p—r—meter β given ˜y
the —˜ove —uto™ov—ri—n™es is ™ondu™ted using group estim—tors ˆβj,
j = 1, 2, ..., q, given ˜y s—mple —uto™ov—ri—n™es ˆγj,|R|p (h) —nd
ˆγj,|R|psign(R)(h) th—t —re ™—l™ul—ted using the o˜serv—tions in group j onlyX
ˆγj,|R|p (h) =
q
T
jT/q
s=(j−1)T/q+h+1
(|Rs|p
− ˆµj,|R|p )(|Rt−h|p
− ˆµj,|R|p ), @IVA

„he null hypothesis H0 : β = β0, eFgFD equ—lity to zero @with β0 = 0A of
—uto™ov—ri—n™es γ|R|p (h), γ|R|psign(R)(h) is reje™ted —t level α ≤ 0.05 if the
—˜solute v—lue |tβ| of the t−st—tisti™ tβ in the group estim—tors ˆβj,
j = 1, 2, ..., q @group s—mple —uto™ov—ri—n™es ˆγj,|R|p (h) —nd ˆγj,|R|psign(R)(h)A
in IU ex™eeds the (1 − α/2) per™entile of the ƒtudentEt distri˜ution with
q − 1 degrees of freedomF
prom the gener—l results it follows th—t the —˜ove t−st—tisti™ —ppro—™h to
ro˜ust inferen™e on —uto™ov—ri—n™es γ|R|p (h), γ|R|psign(R)(h) —nd ro˜ust
tests —re —symptoti™—lly v—lid under the —symptoti™ norm—lity —nd —symptoti™
independen™e of the group s—mple —uto™ov—ri—n™es ˆγj,|R|p (h) —nd
ˆγj,|R|psign(R)(h) @group estim—tors ˆβj in this ™ontextAF
esymptoti™ norm—lity of the —˜ove group s—mple —uto™ov—ri—n™es hold —s
long it holds for the full s—mple —uto™ov—ri—n™es ˆγ|R|p (h) —nd ˆγ|R|psign(R)(h)F

„he following lemm— est—˜lishes —symptoti™ independen™e of the group s—mple
—uto™ov—ri—n™es under the s—me ™onditions —s „heorems RFI —nd RFP —nd thus
™ompletes justi(™—tion of —ppli™—˜ility of the ro˜ust t−st—tisti™ —ppro—™hes in the
settings ™onsideredF
Lemma
„he group s—mple —uto™ov—ri—n™es ˆγi,|R|p (h) —nd ˆγj,|R|p (h) —re —symptoti™—lly
independent for i, j = 1, 2, ..., q, with i = j, if 0 p ζ/2. „he group s—mple
—uto™ov—ri—n™es ˆγi,|R|psign(R)(h) —nd ˆγj,|R|psign(R)(h) —re —symptoti™—lly
independent for i, j = 1, 2, ..., q, with i = j, if 0 p ζ − 2.

Numerical results
Numerical results: Testing for linear dependence
Rt = φRt−1 + εt, (20)
εt = σtZt, t = 2, . . . , T, (21)
ARCH(1), symmetric noise:
εt = σtZt, t = 2, . . . , T, (22)
σ2
t = 0.1 + (π1/3
/2)R2
t−1 (23)
where {Zt}T
t=2 is sequence of i.i.d. N(0,1) r.v.'s.
ARCH(1), asymmetric noise:
σ2
t = 0.1 + (π1/3
/2)R2
t−1 (24)
where {Zt}T
t=2 is sequence of i.i.d. Student-t r.v.'s with 3 d.o.f. and skewness
parameter 0.5.
GJR-GARCH(1,1,1), symmetric noise:
σ2
t = 0.1 + 0.9(|Rt−1| − 0.1Rt−1)2
+ 0.8σ2
t−1 (25)
where {Zt}T
t=2 is sequence of i.i.d. N(0,1) r.v.'s.

Numerical results
por (rst qe‚gr pro™esses α = π1/3
/2 whi™h ™orresponds t—il index ζ = 3 in
RF „hereforeD in order to h—ve st—nd—rd norm—l limit for ˆγ|R|qsign(R)(h)D q
should ˜e lower th—n HFS in this ™—seF
…nder the null hypothesis φ = 0 in @PHAF por the power we simul—te our hq€
with φ in r—nge from H to HFSF
ˆρR(h)X D ˆρ|R|0.5sign(R)(h)X · ˆρ|R|0.25sign(R)(h)X D
ˆρ|R|0.1sign(R)(h)X D ˆρR(h)D q = 12 : D ˆρ|R|0.5sign(R)(h)D
q = 12 : D ˆρ|R|0.25sign(R)(h)D q = 12 : D ˆρ|R|0.1sign(R)(h)D
q = 12 : D

Numerical results
Table: Sizes of tests for autocorrelation
ˆρR(h) ˆρ|R|0.5sign(R)(h) ˆρ|R|0.25sign(R)(h) ˆρ|R|0.1sign(R)(h)
ARCH(1), symm. 10.7 7.9 7.0 6.6
ARCH(1), asymm. 19.7 11.0 9.6 8.9
GJR-GARCH 20.0 12.3 10.2 9.3
Table: Sizes of tests for autocorrelation (Robust t-tests)
ˆρR(h) ˆρ|R|0.5sign(R)(h) ˆρ|R|0.25sign(R)(h) ˆρ|R|0.1sign(R)(h)
q 8 12 16 8 12 16 8 12 16 8 12 16
5.1 5.4 6.2 5.3 5.7 6.6 5.4 5.9 6.7 5.5 5.9 6.4
9.3 11.4 14.5 7.2 9.3 11.2 6.6 8.0 10.1 6.3 7.6 9.1
5.0 5.2 5.8 5.2 5.7 6.2 5.3 5.8 6.4 5.4 5.9 6.6

Numerical results
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Figure: Size-adjusted power for ARCH(1), symmetric noice

Numerical results
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Figure: Size-adjusted power for ARCH(1), asymmetric noice

Numerical results
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Figure: Size-adjusted power for GJR-GARCH

Numerical results
Numerical results: Testing for the presence of nonlinear
dependence and volatility clustering
Rt = σtZt, t = 2, . . . , T, @PTA
so th—t Rt follows qe‚gr pro™ess of the following typesX
e‚gr@IAD symmetri™ noiseX
εt = σtZt, t = 2, . . . , T, @PUA
σ2
t = 0.1 + αR2
t−1 @PVA
where {Zt}T
t=2 is sequen™e of iFiFdF ƒtudentEt rFvF9s with Q dFoFfF
e‚gr@IAD —symmetri™ noiseX
σ2
t = 0.1 + αR2
t−1 @PWA
where (Zt)t∈Z is sequen™e of iFiFdF ƒtudentEt rFvF9s with Q dFoFfF —nd skewness
p—r—meter HFSF

Numerical results
por (rst qe‚gr pro™esses t—il index is equ—l to QF „hereforeD in order to h—ve
st—nd—rd norm—l limit for ˆγj,|R|p (h)D p should ˜e lower th—n HFUS in this ™—seF
…nder the null hypothesis φ = 0 in @PTAF por the power we simul—te our hq€
with α in r—nge from H to HFSF
ˆρR2 (h)X D ˆρ|R|(h)X D ˆρ|R|0.5 (h)X · ˆρ|R|0.25 (h)X D
ˆρ|R|0.1 (h)X D ˆρR2 (h)D q = 12 : D ˆρ|R|(h)D q = 12 : D ˆρ|R|0.5 (h)D
q = 12 : D ˆρ|R|0.25 (h)D q = 12 : D ˆρ|R|0.1sign(R)(h)D q = 12 :

Numerical results
Table: Sizes of tests for nonlinear dependence (null of no GARCH)
ˆρR2(h)
ˆρ|R|(h)
ˆρ|R|0.5(h)
ˆρ|R|0.25(h)
ˆρ|R|0.1(h)
ARCH(1), symm. 28.1 8.8 6.0 5.8 5.8
ARCH(1), asymm. 34.4 10.1 6.2 5.6 5.7
Table: Sizes of tests for nonlinear dependence (null of no GARCH)
ˆρR2 (h) ˆρ|R|(h) ˆρ|R|0.5 (h) ˆρ|R|0.25 (h) ˆρ|R|0.1 (h)
q 8 12 16 8 12 16 8 12 16 8 12 16 8 12 16
14.8 16.0 18.5 6.9 8.5 10.1 6.0 6.8 8.4 5.6 6.9 7.9 5.5 6.7 7.9
18.5 19.3 20.4 7.1 8.7 10.3 5.9 6.5 7.8 5.7 6.5 7.4 5.7 6.7 7.8

Numerical results
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Numerical results
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References
g—mp˜ellD tF ‰FD voD eF ‡FD —nd w—™uinl—yD eF gF @IWWUAF „he e™onometri™s of
(n—n™i—l m—rkets @ƒe™ond ednAF €rin™eton …niversity €ressD €rin™etonF
ghristo'ersenD €F pF @PHIPAF ilements of pin—n™i—l ‚isk w—n—gement @Pnd ednAF
e™—demi™ €ressD xew ‰orkF
gontD ‚—m— @PHHIAF impiri™—l properties of —sset returnsX ƒtylized f—™ts —nd
st—tisti™—l issuesF u—ntit—tive pin—n™eD 1@PAD PPQ!PQTF
h—visD ‚i™h—rd eF —nd wikos™hD „hom—s @IWWVAF „he s—mple —uto™orrel—tions of
he—vyEt—iled pro™esses with —ppli™—tions to e‚grF enn—ls of ƒt—tisti™sD 26F
h—visD ‚i™h—rd eF —nd wikos™hD „hom—s @PHHH—AF „he s—mple —uto™orrel—tions of
(n—n™i—l time series modelsF sn xonline—r —nd nonst—tion—ry sign—l pro™essing
@g—m˜ridgeD IWWVAD ppF PRU!PURF g—m˜ridge …nivF €ressD g—m˜ridgeF
h—visD ‚F eF —nd wikos™hD „F @PHHH˜AF „he s—mple —uto™orrel—tions of (n—n™i—l
time series modelsF sn xonline—r —nd xonst—tion—ry ƒign—l €ro™essing @edF
‡F tF pitzger—ldD ‚F vF ƒmithD eF „F ‡—ldenD —nd €F ‰oungAD ppF PRU!PURF
g—m˜ridge …niversity €ressF
h—visD ‚F eF —nd wikos™hD „F @PHHWAF ixtreme v—lue theory for g—r™h pro™essesF sn
r—nd˜ook of pin—n™i—l „ime ƒeries @edF „F qF endersenD ‚F eF h—visD tFE€F
ureissD —nd „F wikos™hAD ppF IVU!PHHF ƒpringerF

References
hingD F —nd qr—ngerD gF ‡F tF @IWWTAF wodelling vol—tility persisten™e of
spe™ul—tive returnsX — new —ppro—™hF tourn—l of i™onometri™sD 73D IVSâ€“PITF
hingD FD qr—ngerD gF ‡F tFD —nd inglerD ‚F pF @IWWQAF e long memory property of
sto™k m—rket returns —nd — new modelF tourn—l of impiri™—l i™onometri™sD 83D
I!VQF
qr—ngerD gF ‡F tF —nd hingD F @IWWSAF ƒome properties of —˜solute returnX en
—ltern—tive me—sure of riskF enn—les d9¡i™onomie et de ƒt—tistiqueD 40D TU!WIF
s˜r—gimovD sF eF —nd vinnikD ‰F †F @IWUIAF sndependent —nd ƒt—tion—ry ƒequen™es
of ‚—ndom †—ri—˜lesF ‡oltersExoordho' ƒeries of wonogr—phs —nd „ext˜ooks
on €ure —nd epplied w—them—ti™sF ‡oltersExoordho'F
s˜r—gimovD wFD s˜r—gimovD ‚FD —nd ‡—ldenD tF @PHISAF re—vyEt—iled distri˜utions
—nd ro˜ustness in e™onomi™s —nd pin—n™eF †olume PIRD ve™ture xotes in
ƒt—tisti™sF ƒpringerF
s˜r—gimovD ‚ust—m —nd w¤ullerD …lri™h uF @PHIHAF tEst—tisti™ ˜—sed ™orrel—tion —nd
heterogeneity ro˜ust inferen™eF tourn—l of fusiness 8 i™onomi™ ƒt—tisti™sD 28D
RSQ!RTVF
s˜r—gimovD ‚ust—m —nd w¤ullerD …lri™h uF @PHITAF snferen™e with few
heterogeneous ™lustersF ‚eview of i™onomi™s —nd ƒt—tisti™sD 98D VQ!WTF

References
w™xeilD eF tFD preyD ‚FD —nd im˜re™htsD €F @PHISAF u—ntit—tive ‚isk w—n—gement
gon™eptsD „e™hniques —nd „ools @‚evised ednAF €rin™eton …niversity €ressF
weitzD wik— —nd ƒ—ikkonenD €entti @PHHVD WAF irgodi™ityD mixingD —nd existen™e of
moments of — ™l—ss of m—rkov models with —ppli™—tions to g—r™h —nd —™d
modelsF i™onometri™ „heoryD 24D IPWI!IQPHF
wikos™hD „hom—s —nd ƒt¨—ri™¨—D g¨—t¨—lin @PHHHAF vimit theory for the s—mple
—uto™orrel—tions —nd extremes of — qe‚gr (1, 1) pro™essF enn—ls of
ƒt—tisti™sD 28D IRPU!IRSIF
€edersenD ‚F ƒF @PHIWAF ‚o˜ust inferen™e in ™ondition—lly heterosked—sti™
—utoregressionsF porth™oming in the i™onometri™ ‚eviewsF

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New Approaches to Robust Inference on Market (Non-)Effciency, Volatility Clustering and Nonlinear Dependence