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

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Rustam Ibragimova;b, Rasmus Pedersenc, Anton Skrobotovd;b
a Imperial College Business School
b SPBU
c University of Copenhagen
d RANEPA
Eesti Pank 2019

Published in: Economy & Finance
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New Approaches to Robust Inference on Market (Non-)Effciency, Volatility Clustering and Nonlinear Dependence

  1. 1. 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
  2. 2. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets 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, ..., Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 2 / 50
  3. 3. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets 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 @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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 3 / 50
  4. 4. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets 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 @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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 4 / 50
  5. 5. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets Stylized facts of nancial markets €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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 5 / 50
  6. 6. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets Stylized facts of nancial markets 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 6 / 50
  7. 7. Volatility Clustering Heavy Tails in GARCH
  8. 8. 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
  9. 9. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets Stylized facts of nancial markets 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 7 / 50
  10. 10. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Stylized facts of nancial markets Stylized facts of nancial markets 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 8 / 50
  11. 11. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 9 / 50
  12. 12. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 10 / 50
  13. 13. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 11 / 50
  14. 14. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 12 / 50
  15. 15. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 ‚—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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 13 / 50
  16. 16. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Measures of market (non-)eciency, nonlinear dependence and volatility clustering based on absolute returns and their powers 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 14 / 50
  17. 17. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Measures of market (non-)eciency, nonlinear dependence and volatility clustering based on absolute returns and their powers 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 15 / 50
  18. 18. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 16 / 50
  19. 19. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Autocovariance and autocorrelation functions of powers of absolute values of returns Autocovariance and autocorrelation functions of powers of absolute values of returns 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. Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 17 / 50
  20. 20. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Autocovariance and autocorrelation functions of powers of absolute values of returns Autocovariance and autocorrelation functions of powers of absolute values of returns 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. Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 18 / 50
  21. 21. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Autocovariance and autocorrelation functions of powers of absolute values of returns Autocovariance and autocorrelation functions of powers of absolute values of returns 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 19 / 50
  22. 22. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Autocovariance and autocorrelation functions of powers of absolute values of returns Autocovariance and autocorrelation functions of powers of absolute values of returns 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. Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 20 / 50
  23. 23. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 21 / 50
  24. 24. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 22 / 50
  25. 25. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 23 / 50
  26. 26. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 24 / 50
  27. 27. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 25 / 50
  28. 28. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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—nd˜ook 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 ˜ounded —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 —˜out 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 26 / 50
  29. 29. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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). Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 27 / 50
  30. 30. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 28 / 50
  31. 31. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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. Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 29 / 50
  32. 32. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations Asymptotic normality of sample autocovariance and 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 30 / 50
  33. 33. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations 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). Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 31 / 50
  34. 34. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Asymptotic normality of sample autocovariance and autocorrelations HAC estimator „he ™ov—ri—n™e m—trix AΓ† A of the s—mple ™orrel—tion fun™tion ™—n ˜e estim—ted ˜y ˆAˆΓ† ˆA with ˆΓ† given in @ITA —nd ˆA 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 32 / 50
  35. 35. 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)
  36. 36. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 33 / 50
  37. 37. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches Robust t−statistic inference approaches „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). Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 34 / 50
  38. 38. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches Robust t−statistic inference approaches 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 35 / 50
  39. 39. 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
  40. 40. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches Robust t−statistic inference approaches 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 36 / 50
  41. 41. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches Robust t−statistic inference approaches „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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 37 / 50
  42. 42. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches Robust t−statistic inference approaches „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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 38 / 50
  43. 43. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Robust t−statistic inference approaches Robust t−statistic inference approaches „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. Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 39 / 50
  44. 44. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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. Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 40 / 50
  45. 45. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for linear dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 41 / 50
  46. 46. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for linear dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 42 / 50
  47. 47. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for linear dependence ▯▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ ▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ○○○○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 ϕ Figure: Size-adjusted power for ARCH(1), symmetric noice Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 43 / 50
  48. 48. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for linear dependence ▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△△△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △△ △△△△△△△△△△△△△△△△△△△△△△△△△ ▽▽▽▽▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 ϕ Figure: Size-adjusted power for ARCH(1), asymmetric noice Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 44 / 50
  49. 49. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for linear dependence ▯▯▯▯▯▯ ▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△△△ ▽▽▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ○○○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 ϕ Figure: Size-adjusted power for GJR-GARCH Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 45 / 50
  50. 50. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 46 / 50
  51. 51. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for the presence of nonlinear dependence and volatility clustering 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 : Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 47 / 50
  52. 52. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for linear dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 48 / 50
  53. 53. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for the presence of nonlinear dependence and volatility clustering ▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△△ △△ △ △ △ △ △ △△ △ △ △ △ △ △△ △ △△ △△ △△△△△△△△△△△△△△△△△△△△△△△ ▽▽▽▽▽▽▽▽▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽▽▽ ▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ○○○○○○○○○○○○ ○○ ○○ ○○○ ○○ ○○ ○ ○ ○○ ○ ○○ ○ ○○ ○○○○○○○○○○○○○○○○○○ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 ϕ Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 49 / 50
  54. 54. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence Numerical results Numerical results: Testing for the presence of nonlinear dependence and volatility clustering ▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯ △△△△△△ △ △△ △ △ △ △ △ △ △ △ △ △ △△ △ △ △△△△ △△△△△△△△△△△△△△△△△△△△△△△△ ▽▽▽▽▽▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽▽▽ ▽▽ ▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ○○○○○○○○○○○○○ ○○○ ○ ○ ○○ ○ ○ ○ ○○○○ ○○ ○ ○○ ○○ ○ ○ ○○○ ○○ ○○○○○○○○○○ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 ϕ Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 50 / 50
  55. 55. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 50 / 50
  56. 56. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 IVS–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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 50 / 50
  57. 57. New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering and Nonlinear Dependence 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 Anton Skrobotov New Approaches to Robust Inference on Market (Non-)Eciency, Volatility Clustering andOctober 10, 2019 50 / 50

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