Quasi-maximum likelihood estimation for multiple volatility shifts

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1 Quasi-maximum likelihood estimatio for multiple volatility shifts Moosup Kim, Taewook Lee, Jugsik Noh, ad Chagryog Baek 3 Seoul Natioal Uiversity Hakuk Uiversity of Foreig Studies 3 Sugkyukwa Uiversity August 7, 03 Abstract We propose the Gaussia quasi-maximum likelihood estimator QMLE to detect ad locate multiple volatility shifts. Our Gaussia QMLE is show to be cosistet uder suitable coditios ad the rate of covergece is provided. It is also show that the biary segmetatio procedure provides a cosistet estimatio for the umber of volatility shifts. Itroductio I the aalysis of time-varyig volatility of fiacial time series, high persistece i volatility has bee widely recogized as oe of the most promiet features. To capture such high persistece, the itegrated GARCH models were oce itroduced by Egle ad Bollerslev 986. However, some authors argued that high persistece may occur due to structural breaks i volatility. For relevat refereces, we refer to Lamoureux ad Lastrapes 990, Mikosch ad Stărică 004 ad Hillebrad 005. Besides, it has bee also revealed that the existece of volatility shifts ca result i spurious log rage depedece i volatility see, for example, Klemeš 974, Teverovsky et al. 999, Diebold ad Ioue 00, Mikosch ad Stărică 004 ad refereces therei. Accordigly, AMS subject classificatio. Primary: 6M0, 6G0 Secodary: 60G8. Keywords ad phrases: volatility shifts, chage poit aalysis, quasi-maximum likelihood estimatio. Dept. of Statistics, Seoul Natioal Uiversity, Gwaak-ro, Gwaak-gu, Korea 5-74, moosupkim@gmail.com. Dept of Statistics, Hakuk Uiversity of Foreig Studies, 8 Oedae-ro, Mohyeo-myeo, Cheoi-gu, Yogi-si, Kyuggi-do, Korea , twlee@hufs.ac.kr. Dept. of Statistics, Seoul Natioal Uiversity, Gwaak-ro, Gwaak-gu, Korea 5-74, ohjssuy@gmail.com. Correspodig author: Dept of Statistics, Sugkyukwa Uiversity, 5-, Sugkyukwa-ro, Jogo-gu, Seoul, Korea 0-745, crbaek@skku.edu, Tel: , Fax:

2 it has bee i agreemet that structural breaks i volatility should be take ito cosideratio whe modelig the volatility of fiacial time series. The problem of detectig the chage poits, or structural breaks has bee studied for decades i the cotext of chages i mea. Amog may others, we will refer to Csörgő ad Horváth 997, Perro 006, Aue ad Horváth 03 ad refereces therei for comprehesive review. Recetly, with a remarkable attetio to volatility shifts of fiacial time series, detectig ad locatig structural chages i volatility has attracted may researchers. For example, Kokoszka ad Leipus 000 suggested a CUSUM type chage-poit estimator i ARCH models with a sigle volatility shift. Adreou ad Ghysels 00 coducted a extesive simulatio study to compare the performace of existig methods for various multiple volatility shifts. Davis et al. 008 cosidered multiple breaks detectio for a class of segmeted GARCH processes based o the miimum descriptio legth algorithm. This paper itroduces the Gaussia quasi-maximum likelihood estimator QMLE to fid the locatio of multiple volatility shifts ad the biary segmetatio procedure to estimate the umber of volatility shifts. Our proposed method is very straightforward ad easy to implemet. From a theoretical viewpoit, our Gaussia QMLE is show to be cosistet ad coverges i a order of sample size, amely -cosistet. It is also show that the biary segmetatio procedure provides a cosistet estimatio for the umber of volatility shifts. Mote Carlo simulatios supports that our proposed method performs reasoably well eve for o-gaussia settigs. Estimatig the locatios of multiple volatility shifts We cosider the followig volatility shifts model. For a set of break fractios 0 =: 0 < <... < R < R+ :=, let r t = r i,t, if [ i ] < t [ i+ ], σ i = Er i,0 <, for i = 0,..., R,. where r 0,t,..., r R,t : t Z is strictly statioary ad ergodic with mea zero. It is assumed that each σ i is strictly positive ad σ i σ i for every i =,..., R. I fiacial applicatios, r t usually represets a series of log-returs with R volatility shifts, ad each [ i ] deotes the i-th break poit. Furthermore, we will write,..., R as the true locatios of multiple volatility shifts satisfyig i+ i > ς 0 for each i = 0,..., R, for a sufficietly small umber ς 0 > 0. The true volatility o a iterval [ i ], [ i+ ]] is deoted by := Eri,0, i = 0,..., R. I practice, we observe r,..., r with ukow umber of volatility shifts. Here, we will cosider the estimatio of,..., R whe R is kow. Statistical iferece o the ukow umber of volatility shifts R will be discussed i Sectio 3. The idea behid our proposed estimator is very straightforward. Suppose that r t i. are

3 idepedet observatios from Normal distributio. The, the egative log-likelihood is give by L,..., R, σ 0,..., σ R = [ i+ ] t=[ i ]+ r t σ i + log σ i.. By miimizig L with respect to uisace parameters σ 0,..., σ R, that is, solvig L/ σ i = 0, i = 0,..., R, we obtai that σ i = [ i+ ] [ i ] [ i+ ] t=[ i ]+ By plug-i.3 to., a profile likelihood becomes L,..., R, σ 0,..., σ R = [ i+ ] [ i ] + log [ i+ ] [ i ] r t..3 [ i+ ] t=[ i ]+ r t..4 Therefore, we obtai the Gaussia MLE by miimizig L with respect to,..., R. For depedet observatios r t,. divided by sample size ca be iterpreted as the method of momet estimator of σ i i+ i zd + i+ i log σ i, z = I i < i+. Sice the couterpart is miimized at,..., R =,..., R ad σ 0,..., σ R = v 0,..., v R cf. the proof of Theorem, the estimator miimizig L is reasoable. where Fially, we defie our Gaussia QMLE as ˆ,, ˆ R := argmi M,..., R : D R,.5 M := L i, i+, L i, i+ := i+ i + log i+ i with D R deotes the domai of =,..., R such that [ i+ ] t=[ i ]+ 0 < < < R <, ad i+ i > ς 0, for i = 0,..., R. Note that we scaled.4 by ad applied approximatio [x]/ x for clarity ad otatioal coveiece i the proof. Now, we argue that our proposed estimator i.5 is cosistet ad furthermore the rate of covergece is of a order. Such -cosistecy is the key property i the chage poit aalysis. For example, Bai ad Perro 998 shows that least squares chage poit estimator for mea 3 r t

4 chages is -cosistet, ad Kokoszka ad Leipus 000 proved that the same rate holds for a CUSUM-type chage poit estimator i parametric ARCH models. Uder the regularity coditios listed below, our proposed estimator.5 is -cosistet. We first state the cosistecy i Theorem ad the rate of covergece i Theorem. A r0,t,..., r R,t : t Z is strictly statioary ad ergodic. A Every := Er i,0 is strictly positive ad fiite, ad for i =,..., R. Theorem. Assume that A-A hold. The, ˆ,..., ˆ R P,..., R, as. Proof. The proof cosists of two parts; we first verify the uiform covergece of M to a smooth fuctio. Next, the smooth fuctio achieves the miimum oly at,..., R. Let [ i+ ] Z,..., R = i+ i + log Ert i+ i. Observe that if mi D R 0 i R i+ i [ i+ ] t=[ i ]+ Er t t=[ i ]+ mi ς 0, 0 i R where the last term is strictly positive whe ς 0 >, ad it follows from ergodic theorem that [ max sup i+ ] 0 i R r D R i+ i t Ert t=[ i ]+ [ max ς sup i+ ] 0 0 i R r D R t Er t t=[ i ]+ R [] sup ς 0 r i,t t=[ i ]+ : i, i+] = o P. Thus, we have that = sup i+ i D R Also, we ca easily check that sup M,..., R Z,..., R D R log i+ i [ i+ ] t=[ i ]+ r t log i+ i [ i+ ] t=[ i ]+ Er t = o P. where sup D R Z,..., R Z,..., R = o, z := I i < i+, Z,..., R := i+ i + log i+ i i+ i zd. 4

5 Therefore, sup M,..., R Z,..., R = o P. D R i+ Note that, lettig w i = i+ i i zd, log w i i+ i i+ log zd = i i+ i i+ i z w i log z d 0, w i ad the equality hold oly if z = w i for i, i+, equivaletly, i = j ad i+ = j+ for some j. Thus, we have that Z,..., R + = + i+ log zd = + log zd i 0 i+ i log = Z,..., R, where the equality holds oly if,..., R =,..., R. Hece, the proof is completed. Theorem. Assume that A-A hold. The, max i R ˆ i i = O P, as. Proof. Let K > 0 deote geeric costats that ca be differet from lie to lie. By Theorem, we ca take η > 0 with η 0 as such that each ˆ i lies i i η, i + η with probability tedig to. Without loss of geerality, we assume that η. Thus, it suffices to show that lim M lim sup P R ˆi i > M/ i= ad max i R ˆ i i < η = 0..6 For the compactess of proof, we will oly cosider the case ˆ i i the others ca be treated i similar fashios. The,.6 becomes for each i =,..., R, sice where A M := R i= lim M lim sup P A M = 0,.7 ˆi < i M/ Note from the defiitio of our estimator.5, R i= η < ˆ i i 0. L i, i+ L ˆ i, ˆ i

6 First, we deal with R L i, i+ L i, i+. For brevity, let S i, i+ := The, we have that i+ i [ i+ ] t=[ i ]+ r t, S i, i+ := L i, i+ L i, i+ i+ i [ i+ ] t=[ i ]+ r t Er t. = i+ i+ i + i + log S i, i+ + i+ i log S i, i+ S i, i+. If i η < i i < i+ η < i+ i+, the S i, i+ S i, i+ = i+ i+ i + i S i, i+ i+ + i i S i, i i+ i+ S i+, i i+ i i+ i+ i = i+ i+ i+ i Thus, = i+ i+ i + i i+ i S i, i+ + i i i+ i S i, i i+ i+ i+ i S i+, i+ + i i i+ i S i, i+ S i+, i+ + i i i+ i S i, i S i, i+ + S i, i+ S i, i+ S i, i+ = i+ i+ S i, i+ S i+, i+ i+ i + i i S i, i S i, i+ + + o P i i + o P i+ i+, i+ i. ad S i, i+ S i, i+ S i, i+ = o P i i + o P i+ i+ as sice we have that for each i =,..., R, S i, P i+, sup S, i : i η < i = O P. Therefore, if i η < i i for each i, the, = L i, i+ L i, i+ [ i+ i+ i + i + log S i, i+ + i+ i log S i, i+ ] S i, i+ 6

7 = i+ i+ i + i + log + o P i+ i+ i + i S i, i+ i i S i, i i+ i+ S i+, i+ vi i i + = i i log + + v i S i, i + i, i= o P i i where each i is egligible uiformly i satisfyig i i η, i ] for each i. Note also that log + < 0, because of. Now, we are ready to prove.7. Let B i = sup S, i : i η < i, the B i = O P for each i, ad set ϵ = mi i R vi Deote E m, m N, be the evet such that max sup i R D i m log + + log v i > 0. where sup is take over satisfyig j j η, j ] for each j ad D i m = : i η < < i m. S, i + sup i < ϵ,.9 Remark also that ergodic theorem ad egligibility of sup i implies that Observe that if i m/ i i, the i i m Km log log + max i R lim lim if P E m =..0 m B i + sup i S i, i + sup B i + sup i, i where K > 0 depeds o v 0,..., v R oly. O the other had, if.9 holds with i η i < i m/, the i i log + + S i, i + sup i < m ϵ. I particular, whe i η i < i M/, the lefthad is less tha Mϵ/. 7

8 Therefore, from.8 if evet A M E m, M > m, happes, the we have 0 = L i, i+ L ˆ i, ˆ i+ i ˆ i log i= ϵm + KRm Now, observe for every m N, lim M lim sup lim M P A M lim sup P + lim if + max i R lim M ϵm + KRm P E m = lim if + + B i + sup i. S ˆ i, i + i ˆ lim sup P A M E m + lim if P E m + max B i + sup i 0 i R P E m, where the last equality holds due to KRm + max i R B i + sup i = O P. Hece, by lettig m,.0 completes the proof. 3 A sequetial procedure o the umber of shifts R I this sectio, we preset a sequetial procedure for estimatig the umber volatility shifts based o the so-called biary segmetatio method. The algorithm is recursive ad first set R 0 = 0. Cosider sup L 0, L 0, + L, : ς 0 < < ς The, 3. ca be cosidered as the differece of egative log-likelihood betwee o volatility shift ad oe volatility shift. Therefore, if 3. is sigificatly large, it meas that egative likelihood is sigificatly reduced by addig a volatility shift. As a result, we iclude such locatio ad set ew R 0 =. Oce the first volatility shift at [ ] is detected, we obtai two subsamples o [ˆ 0 ], [ˆ ]] ad [ˆ ], [ˆ ]] with ˆ 0 = 0 ad ˆ =. The, the biary segmetatio method exactly applies above procedure for each subsample. Namely, if there is a sigificat reductio i egative loglikelihood o the subsample, the split the subsample ito two. Otherwise, stop splittig. The etire procedure will be recursively iterated till o further volatility shifts is detected. However, it is more coveiet to fid the locatio of volatility oe by oe. Hece, we are goig to order them by takig the largest reductio i egative log-likelihood across all subsamples. To be more specific, suppose that ˆR = R 0 N at some iteratio stage. We make a statistical iferece o whether a additioal volatility shift is preset. I tur, we perform the testig of H 0 : R = R 0 vs H : R = R

9 The locatios of volatility shifts are estimated as i Sectio from R0 ˆ,..., ˆ R0 = argmi L i, i+ :,..., R0 D R0, 0 = 0, R0 + =. It aturally produces R 0 + subsamples, ad for each i-th subsample o [ˆ i ], [ˆ i+ ]], i = 0,..., R 0 with ˆ 0 = 0, ˆ R0 + =, calculate F i R 0 + R 0 := sup L ˆ i, ˆ i+ L ˆ i, L, ˆ i+ : ˆD i, 3.3 where ˆD i = : ˆ i + ς 0 ˆ i+ ˆ i < < ˆ i+ ς 0 ˆ i+ ˆ i for sufficietly small ς 0 > 0 so that ˆD i is oempty. The test statistic is defied as ˆ FR 0 + R 0 := max F i R 0 + R 0, i R 0 ˆγ i where ˆγ 0,..., ˆγ R0 is the cosistet estimator of the log-ru variace γ i = h= Covr i,0, r i,h, ad ˆv 0,..., ˆv R0 is the cosistet estimator of the volatility v 0,..., v R0. For example, oe ca use popular Bartlett log-ru variace estimator with a adaptive choice of badwidth suggested by Adrews 99 ad a method of momet estimator for volatilities. We establish the asymptotic distributio of the test statistic 3.4 i the followig theorem with a additioal coditio: A3 The ivariace priciple holds for r0,t,..., r R,t : t Z, i.e., there exist a positive defiite matrix Γ such that for every a = a 0,..., a R R R+, [] t= a i r i,t where B stads for a stadard Browia motio. FR 0 + R 0 d max sup 0 i R 0 d a Γa B i D[0, ], Theorem 3. Assume that A-A3 hold. The, uder R = R 0 as, B i u ub i u u : ς 0 < u < ς 0 where B i stads for a sequece of idepedet stadard Browia motios. Proof. We have from Theorem that, S ˆ i, S ˆ i, ˆ i+ = S ˆ i, S ˆ i, ˆ i+ + = ˆ i [] t=[ˆ i ]+ r i,t ˆ i+ ˆ i [ˆ i+ ] t=[ˆ i ]+ r i,t + OP, 9

10 ad S, ˆ i+ S ˆ i, ˆ i+ = S, ˆ i+ S ˆ i, ˆ i+ + = ˆ i+ [ˆ i+ ] t=[]+ r i,t ˆ i+ ˆ i [ˆ i+ ] t=[ˆ i ]+ r i,t + OP where all O P -terms are egligible uiformly i ˆD i. Thus, it follows from A3 that Lˆ i, ˆ i+ Lˆ i, L, ˆ i+ = ˆ i log S ˆ i, S ˆ i, ˆ i+ ˆ i+ log S, ˆ i+ S ˆ i, ˆ i+ ˆ i+ = ˆ i = ˆ S ˆ i, i S ˆ i, ˆ i+ S, ˆ i+ S ˆ i, ˆ i+ S ˆ i, S ˆ i, ˆ i+, + ˆ i S ˆ i, S ˆ i, ˆ i+ + ˆ i+ S, ˆ i+ S ˆ i, ˆ i+ + O P + ˆ i+ S, ˆ i+ S ˆ i, ˆ i+ + O P, where all O P -terms are egligible uiformly i ˆD i. Moreover, the leadig term is rewritte as ˆ i S ˆ i, S ˆ i, ˆ i+ + ˆ i+ ˆ i S ˆ i, ˆ i+ S ˆ i, ˆ i+ = ˆ i+ ˆ i ˆ i ˆ i+ ˆ i+ ˆ i S ˆ i, S ˆ i, ˆ i+ ˆ i+ = ˆ i 3 ˆ i ˆ i+ S ˆ i, ˆ i+ [] t=[ˆ i ]+ ri,t ˆ i ˆ i+ ˆ i [ˆ i+ ] t=[ˆ i ]+ r i,t + O P where the O P -term is egligible uiformly i ˆD i. Hece, due to A3 ad Theorem, the proof is completed by mappig theorem. Fially, we will reject the hypothesis testig i 3. with sigificace level α if The critical value is give as FR 0 + R 0 > q R 0. q R 0 = if x : Gx R 0+ > α,, 0

11 where G represet the distributio fuctio of sup ς0 <u< ς 0 B u ub /u u ad aalytically expressed as Gx = see, for example, Csörgő ad Horváth 997 p. 5. x exp x/ log ς 0 π x ς0 + 4 x + ox By applyig our proposed testig procedure sequetially from R 0 = 0 till o further rejectio, we fially set ˆR be the umber of hypothesis testig rejected. The followig result establishes that above procedure provides a cosistet estimatio o the umber of multiple volatility shifts. Theorem 4. Suppose that the size α is such that α 0 while lim if α > 0, the P ˆR = R as. Proof. With -cosistecy established i Theorem, the proof is idetical to Propositio i Bai 997, hece omitted Some umerical results I this sectio, we examie the performace of our proposed method through Mote Carlo simulatios. For coditioal volatility models, we cosider popular GARCH, models with three differet iovatios. We cosider the stadard Normal distributio, t-distributio with degree of freedom 5 ad skewed asymmetric distributio by geeratig skew-normal distributio with skewess parameter α = 4. The desity fuctio of skew-normal distributio is give by fx = ϕxφαx, ϕx = exp x // π, Φx = αx ϕtdt, 4. where ϕt is the desity fuctio of the stadard Normal distributio. Also, ote that iovatios are stadardized for t ad skew-normal distributios to achieve zero mea ad uit variace. I terms of tuig parameters, we use the data depedet badwidth of Adrews 99 for Bartlett log-ru variace estimator. More specifically, for observatios X,..., X it is give as ˆγ i = h ˆγ h, q + q + h <q where ˆγ h = h j= X j+h X X j X are sample covariaces ad q = [.87ˆρ /3 ˆρ /3 /3 ] with ˆρ = j= X jx j / j= X j. Critical values are calculated from 3.5 by disregardig ox term with ς 0 =.05. All results are based o, 000 replicatios of sample sizes = 500,,000 ad,000 with 5% sigificace level. Figure shows the boxplots o the estimated locatio of true break fractio at =.5. I the simulated model, GARCH, model chages parameters θ = ω, α, β from.,.,.3 to.,.,.6, hece volatility shifts from./..3 = /6 to./..6 = /3. We observe that our

12 GARCH Normal GARCH t5 GARCH SN =500 =000 =000 =500 =000 =000 =500 =000 =000 Figure : Estimatio of locatio. Volatility shifts from /6 to /3 i GARCH, model at midpoit. Iovatios ˆR = 0 ˆR = ˆR N 0, t5 skew-normal4 = =, =, = =, =, = =, =, Table : Estimated umber of shifts from the sequetial method whe R =. Parameters chage from θ =.,.,.3 to θ =.,.,.6 at midpoit i the simulated model. QMLE estimator successfully fids the true locatio i all cases cosidered, ad the performace becomes outstadig as sample size icreases. To be more specific, the root mea square error RMSE is reduced from.045,.6,.064 to.08,.080,.03 i a order of Normal, t ad skew- Normal as sample size icreases from =, 000 to, 000. Note also that heaviess i iovatios has more impact o performace tha asymmetric iovatios. Next, we apply a sequetial method explaied i Sectio 3 to determie the umber of volatility shifts. Table shows the frequecy table o the estimated umber of breaks for the same simulatio model used i Figure. I geeral, the performace of our method is satisfactory. Our method fids the true volatility shift approximately 80% to 90% of chace for moderate sample size. The performace o heavier iovatios is lower tha the other two cases, but the margi is arrowed by icreasig sample size. It is also observed that heavier iovatios ted to estimate less umber of breaks. We guess that it is reasoable because a few outlyig observatios ca mask true volatility

13 shifts. I summary, it is deduced that our QMLE method locates ad determies the umber of shifts quite successfully for GARCH model with various types of iovatios. Besides, ote that our method is easy to implemet ad computatioally feasible. Refereces Adreou, E. ad Ghysels, E. 00. Detectig multiple breaks i fiacial market volatility dyamics. Joural of Applied Ecoometrics, 75: Adrews, D. W. K. 99. Heteroskedasticity ad autocorrelatio cosistet covariace matrix estimatio. Ecoometrica, 593: Aue, A. ad Horváth, L. 03. Structural breaks i time series. Joural of Time Series Aalysis, 34: 6. Bai, J Estimatig multiple breaks oe at a time. Ecoometric Theory, 33: Bai, J. ad Perro, P Ecoometrica, 66: Estimatig ad testig liear models with multiple structural chages. Csörgő, M. ad Horváth, L Limit Theorems i Chage-Poit Aalysis. Wiley Series i Probability ad Statistics. Joh Wiley & Sos Ltd., Chichester. Davis, R. A., Lee, T. C. M., ad Rodriguez-Yam, G. A Break detectio for a class of oliear time series models. Joural of Time Series Aalysis, 9: Diebold, F. X. ad Ioue, A. 00. Log memory ad regime switchig. Joural of Ecoometrics, 05:3 59. Egle, R. F. ad Bollerslev, T Modellig the persistece of coditioal variaces. Ecoometric Reviews, 5: 50. Hillebrad, E Neglectig parameter chages i GARCH models. Joural of Ecoometrics, 9- : 38. Klemeš, V The Hurst pheomeo: a puzzle? Water Resources Research, 04: Kokoszka, P. ad Leipus, R Chage-poit estimatio i ARCH models. Beroulli, 63: Lamoureux, C. G. ad Lastrapes, W. D Persistece i variace, structural chage ad the GARCH model. Joural of Busiess ad Ecoomic Statistics, 8:5 34. Mikosch, T. ad Stărică, C Nostatioarities i fiacial time series, the log-rage depedece, ad the IGARCH effects. The Review of Ecoomics ad Statistics, 86: Perro, P Dealig with structural breaks. I Palgrave Hadbook of Ecoometrics, pages 78 35, Lodo. Palgrave Macmilla. Teverovsky, V., Taqqu, M. S., ad Williger, W A critical look at Lo s modified R/S statistic. Joural of Statistical Plaig ad Iferece, 80-: 7. 3

5 Statistical Inference

5 Statistical Inference 5 Statistical Iferece 5.1 Trasitio from Probability Theory to Statistical Iferece 1. We have ow more or less fiished the probability sectio of the course - we ow tur attetio to statistical iferece. I statistical