Matching a Distribution by Matching Quantiles Estimation

Transcription

1 Jural f the America Statistical Assciati ISSN: (Prit) X (Olie) Jural hmepage: Matchig a Distributi by Matchig Quatiles Estimati Niklas Sgurpuls, Qiwei Ya & Claudia Yastremiz T cite this article: Niklas Sgurpuls, Qiwei Ya & Claudia Yastremiz (2015) Matchig a Distributi by Matchig Quatiles Estimati, Jural f the America Statistical Assciati, 110:510, , DOI: / T lik t this article: The Authr(s). Published with licese by Taylr & Fracis Grup, LLC Duglas S Ramsay, Stephe C Wds, ad Karl J Kaiyala Accepted authr versi psted lie: 12 Ju Submit yur article t this jural Article views: 796 View related articles View Crssmark data Full Terms & Cditis f access ad use ca be fud at Dwlad by: [ ] Date: 28 Octber 2015, At: 08:08

2 Niklas SGOUROPOULOS, Qiwei YAO, ad Claudia YASTREMIZ Matchig a Distributi by Matchig Quatiles Estimati Mtivated by the prblem f selectig represetative prtflis fr backtestig cuterparty credit risks, we prpse a matchig quatiles estimati (MQE) methd fr matchig a target distributi by that f a liear cmbiati f a set f radm variables. A iterative prcedure based the rdiary least-squares estimati (OLS) is prpsed t cmpute MQE. MQE ca be easily mdified by addig a LASSO pealty term if a sparse represetati is desired, r by restrictig the matchig withi certai rage f quatiles t match a part f the target distributi. The cvergece f the algrithm ad the asympttic prperties f the estimati, bth with r withut LASSO, are established. A measure ad a assciated statistical test are prpsed t assess the gdess-f-match. The fiite sample prperties are illustrated by simulati. A applicati i selectig a cuterparty represetative prtfli with a real dataset is reprted. The prpsed MQE als fids applicatis i prtfli trackig, which demstrates the usefuless f cmbiig MQE with LASSO. KEY WORDS: Gdess-f-match; LASSO; Ordiary least-squares estimati; Prtfli trackig; Represetative prtfli; Sample quatile. Dwladed by [ ] at 08:08 28 Octber INTRODUCTION Basel III is a glbal regulatry stadard bak capital adequacy, stress testig ad market liquidity risk put frward by the Basel Cmmittee Bakig Supervisi i , i respse t the deficiecies i risk maagemet revealed by the late-2000s fiacial crisis. Oe f the madated requiremets uder Basel III is a extesi f the backtestig f iteral cuterparty credit risk (CCR) mdels. Backtestig tests the perfrmace f CCR measuremet, t determie the eed fr recalibrati f the simulati ad/r pricig mdels ad readjustmet f capital charges. Sice the umber f the trades betwee tw majr baks culd easily be i the rder f tes f thusads r mre, Basel III allws baks t backtest represetative prtflis fr each cuterparty, which csist f subsets f the trades. Hwever, the selected represetative prtflis shuld represet the varius characteristics f the ttal cuterparty prtfli icludig risk expsures, sesitivity t the risk factrs, etc. We prpse i this article a ew methd fr cstructig such a represetative prtfli. The basic idea is t match the distributi f ttal cuterparty prtfli by that f a selected prtfli. Hwever, we d t match the tw distributi fuctis directly. Istead we chse the represetative prtfli t miimize the mea squared differece betwee the quatiles f the tw distributis acrss all levels. This leads Niklas Sgurpuls, Qiwei Ya, Claudia Yastremiz. This is a Ope Access article distributed uder the terms f the Creative Cmms Attributi Licese ( which permits urestricted use, distributi, ad reprducti i ay medium, prvided the rigial wrk is prperly cited. The mral rights f the amed authr(s) have bee asserted. Niklas Sgurpuls is Quatitative Aalyst, QA Expsure Aalytics, Barclays, Ld, UK ( iklas.sgurpuls@barclays.cm). Qiwei Ya is Prfessr, Departmet f Statistics, The Ld Schl f Ecmics ad Plitical Sciece, Hught Street, Ld, WC2A 2AE, UK; Guaghua Schl f Maagemet, Pekig Uiversity, Chia ( q.ya@lse.ac.uk). Claudia Yastremiz is Seir Techical Specialist, Market ad Cuterparty Credit Risk Team, Prudetial Regulati Authrity, Bak f Eglad, Ld, UK ( claudia.yastremiz@bakfeglad.c.uk). Partially supprted by the EPSRC research grats EP/G026874/1 ad EP/L01226X/1. The views expressed i this article are thse f the authrs, ad t ecessarily thse f the Bak f Eglad r members f the PRA Bard. Clr versis f e r mre f the figures i the article ca be fud lie at t the matchig quatiles estimati (MQE) fr the purpse f matchig a target distributi. T the best f ur kwledge, MQE has t bee used i this particular ctext, thugh the idea f matchig quatiles has bee explred i ther ctexts; see, fr example, Karia ad Dudewicz (1999), Small ad McLeish (1994), ad Dmiicy ad Veredas (2013). Furthermre, ur iferece prcedure is differet frm thse i the afremetied papers due t the differet ature f ur prblem. Frmally, the prpsed MQE bears sme similarities t the rdiary least squares estimati (OLS) fr regressi mdels. Hwever, the fudametal differece is that MQE is fr matchig (ucditial) distributi fuctis, while OLS is fr estimatig cditial mea fuctis. Ulike OLS, MQE seldm admits a explicit expressi. We prpse a iterative algrithm applyig least-squares estimati repeatedly t the recursively srted data. We shw that the algrithm cverges as the mea squared differece f the tw-sample quatiles decreases mtically. Sme asympttic prperties f MQE are established based the Bahadur-Kiefer buds fr the empirical quatile prcesses. MQE methd facilitates sme variatis aturally. First, it ca be perfrmed by matchig the quatiles betwee levels α 1 ad α 2 ly, where 0 α 1 <α 2 1. The resultig estimatr matches ly a part f the target distributi. This culd be attractive if we are ly iterested i mimickig, fr example, the behavir at the lwer ed f the target distributi. Secd, MQE ca als be perfrmed with a LASSO-pealty, leadig t a sparser represetati. Thugh MQE was mtivated by the prblem f estimatig represetative prtflis, its ptetial usefuless is wider. We illustrate hw it ca be used i a prtfli trackig prblem. Sice MQE des t require the data beig paired tgether, it ca als be used fr aalyzig asychrus measuremets which arise frm varius applicatis icludig atmspheric scieces (He et al. 2012), space physics, ad ther areas (O Brie et al. 2001). Published with licese by Taylr ad Fracis Jural f the America Statistical Assciati Jue 2015, Vl. 110, N. 510, Thery ad Methds DOI: /

3 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 743 Dwladed by [ ] at 08:08 28 Octber 2015 MQE is a estimati methd fr matchig ucditial distributi fuctis. It is differet frm the ppular quatile regressi which refers t the estimati fr cditial quatile fuctis. See Keker (2005), ad refereces therei. It als differs frm the ucditial quatile regressi f Firp et al. (2009) which deals with the estimati fr the impact f explaatry variables quatiles f the ucditial distributi f a utcme variable. Fr rmal mdels, sample quatiles have bee used fr differet iferece purpses. Fr example, Ksrk (1999) used quatiles fr parametric twsample tests. Geitig (2011) argued that quatiles shuld be used as the ptimal pit frecasts uder sme circumstaces. MQE als differs frm the statistical asychrus regressi (SAR) methd itrduced by O Brie et al. (2001), althugh it ca prvide a alterative way t establish a regressilike relatiship based upaired data. See Remark 1(v) i Secti 2. The rest f the article is rgaized as fllws. The MQE methdlgy icludig a iterative algrithm is preseted i Secti 2. The cvergece f the algrithm is established i Secti 3. Secti 4 presets sme asympttic prperties f MQE. T assess the gdess-f-match, a measure ad a assciated statistical test are prpsed i Secti 5. The fiite sample prperties f MQE are examied i simulati i Secti 6. We illustrate i Secti 7 hw the prpsed methdlgy ca be used t select a represetative prtfli fr CCR backtestig with a real dataset. Secti 8 deals with the applicati f MQE t a differet fiacial prblem trackig prtflis. It als illustrates the usefuless f cmbiig MQE ad LASSO tgether. 2. METHODOLOGY Let Y be a radm variable, ad X = (X 1,...,X p ) be a cllecti f p radm variables. The gal is t fid a liear cmbiati β X = β 1 X 1 + +β p X p (2.1) such that its distributi matches the distributi f Y. Weprpse t search fr β such that the fllwig itegrated squared differece f the tw quatile fuctis is miimized 1 0 Q Y (α) Q β X(α)} 2 dα, (2.2) where Q ξ (α) detes the αth quatile f radm variable ξ, that is, P ξ Q ξ (α)} =α, fr α [0, 1]. I fact (2.2) is a squared Mallws metric itrduced by Mallws (1972) ad Taaka (1973). It is als kw as L 2 - Wasserstei distace (del Barri et al. 1999). See als Secti 8 f Bickel ad Freedma (1981) fr a mathematical accut f the Mallws metrics. Give the gal is t match the tw distributis, e may adpt the appraches f matchig the tw distributi fuctis r desity fuctis directly. Hwever, ur apprach f matchig quatiles prvides the better fittig at the tails f the distributis, which is imprtat fr risk maagemet; see Remark 1(iv) belw. Furthermre, it turs ut that the methd f matchig quatiles is easier tha that fr matchig distributi fuctis r desity fuctis directly. Suppse the availability f radm samples Y 1,...,Y } ad X 1,...,X } draw respectively frm the distributis f Y ad X. LetY (1) Y () be the rder statistics f Y 1,...,Y. The Y (j) is the j/th sample quatile. T fid the sample cuterpart f the miimizer f (2.2), we defie the estimatr β = arg mi β Y (j) (β X) (j) } 2, (2.3) where (β X) (1) (β X) () are the rder statistics f β X 1,...,β X. We call β the matchig quatiles estimatr (MQE), as it tries t match the quatiles at all pssible levels betwee 0 ad 1. Ufrtuately β des t admit a explicit sluti. We defie belw a iterative algrithm t evaluate its values. We will shw that the algrithm cverges. T this ed, we itrduce sme tati first. Suppse that β (k) is the kth iterated value, let X (k) (j) } be a permutati f X j } such that (β (k) ) X (k) (1) (β(k) ) X (k) (). (2.4) Step 1. Set a iitial value β (0). Step 2. Fr k 1, let β (k) = arg mi β R k (β), where R k (β) = 1 ( Y(j) β X (k 1) ) 2, (j) (2.5) where X (k 1) (j) } is defied as i (2.4). We stp the iterati whe R k (β (k) ) R k 1 (β (k 1) ) is smaller tha a prescribed small psitive cstat. We the defie β = β k. I the abve algrithm, we may take the rdiary least squares estimatr (OLS) β as a iitial estimatr β (0), where β arg mi (Y j β X j ) 2 = (X X ) 1 X Y. (2.6) β ad Y = (Y 1,...,Y ), X is a p matrix with X j as its jth rw. Hwever we stress that OLS β is a estimatr fr the miimizer f the mea squared errr E(Y β X) 2 }, (2.7) which is differet frm the miimizer f (2.2) i geeral. Hece, OLS β ad MQE β are tw estimatrs fr tw differet parameters, althugh the MQE is btaied by applyig least squares estimati repeatedly t the recursively srted data; see Step 2 abve. T gai sme ituitive appreciati f MQE ad the differece frm OLS, we reprt belw sme simulati results with tw ty mdels. Example 1. Csider a simple sceari Y = X + Z, (2.8) where X ad Z are idepedet ad N(0, 1), ad Z is ubservable. Nw p = 1, the miimizer f (2.7) isβ (1) = 1. Nte that L(Y ) = N(0, 2) = L(1.414X). Thus, (2.2) admits a miimizer β (2) = We geerate 1000 samples frm (2.8) with each sample f size = 100. Fr each sample, we calculate MQE

4 744 Jural f the America Statistical Assciati, Jue 2015 Dwladed by [ ] at 08:08 28 Octber ~ β Figure 1. Bxplts f OLS β fr the true value 1, ad MQE β fr the true value fr mdel (2.8). β usig the iterative algrithm abve with OLS β as the iitial value. Figure 1 presets the bxplts f the 1000 estimates. It is clear that bth OLS β ad MQE β prvide accurate estimates fr β (1) ad β (2), respectively. I fact, the mea squared estimati errrs ver the 1000 replicatis is, respectively, fr β ad fr β. The algrithm fr cmputig β ly tk tw iteratis t reach the cvergece i all the 1000 replicatis. Example 2. Nw we repeat the exercise i Example 1 abve fr the mdel Y = X 1 + X Z, (2.9) where X 1,X 2, ad Z are idepedet ad N(0, 1), ad Z is ubservable. The bxplts f the estimates are displayed i Figure 2. Nwp = 2, the miimizer f (2.7) is(β (1) 1,β(1) 2 ) = (1, 1). Sice L(Y ) = N(0, 4), there are ifiite umbers f miimizers f (2.2). I fact ay (β 1,β 2 ) satisfyig the cditi β1 2 + β2 2 = 2 is a miimizer f (2.2), as the L(β 1 X 1 + β 2 X 2 ) = N(0, β1 2 + β2 2 ) = N(0, 4). Oe such miimizer is (β (2) 1,β(2) 2 ) = (1.414, 1.414). It is clear frm Figure 2 that ver the 1000 replicatis, OLS ( β 1, β 2 ) Figure 2. Bxplts f OLS ( β 1, β 2 ) fr the true value (1, 1), MQE ( β 1, β 2 ), ad β β 2 2 } 1 2 fr the true value 2 fr mdel (2.9). β^ are cetered at the miimizer (β (1) 1,β(1) 2 )f(2.7). While MQE ( β 1, β 2 ) are cetered arud e miimizer (β (2) 1,β(2) 2 )f(2.2), their variatis ver 1000 replicatis are sigificatly larger. O the ther had, the values f β β 2 2} 1 2 are cetered arud its uique true value 2 with the variati cmparable t thse f the OLS β 1 ad β 2. I fact, the mea squared estimati errrs f β 1, β 2, ad β β 2 2}1/2 are, respectively, , , ad The mea squared differeces betwee β 1 ad β (2) 1, ad betwee β 2 ad β (2) 2 are ad , respectively. All these clearly idicate that i the 1000 replicatis, MQE may estimate differet miimizers f (2.2). Hwever, the edprduct, that is, the estimati fr the distributi f Y is very accurate, measured by the mea squared errr fr estimatig β1 2 + β2 2 }1/2. The iterative algrithm fr calculatig the MQE always cverges quickly i the 1000 replicatis. The average umber f iteratis is 5.15 with the stadard deviati Like i Example 1, we used the OLS as the iitial values fr calculatig the MQE. We repeated the exercise with the tw iitial values geerated radmly frm U[ 2, 2]. The bxplts fr β 1 ad β 2, t preseted here t save space, are w cetered at 0 with abut [ 1.5, 1.5] as their iter-half rages. But remarkably the bxplt fr ( β 1 ) 2 + ( β 2 ) 2 } 1/2 remais abut the same. The mea ad the stadard deviati fr the umber f iteratis required i calculatig the MQE are 7.83 ad We cclude this secti with sme remarks. Remark 1. (i) Whe there exist mre tha e miimizer f (2.2), β may estimate differet values i differet istaces. Hwever, the gdess f the resultig apprximatis fr the distributi f Y is abut the same, guarateed by the least squares prperty. See als Therem 2 i Secti 4. (ii) If we are iterested ly i matchig a part f distributi f Y, say, that betwee the α 1 th quatile ad the α 2 th quatile, 0 α 1 <α 2 1, we may replace (2.5) by 1 R k (β; α 1,α 2 ) = 2 1 j= 1 +1 ( Y (j) β X (k 1) ) 2, (j) (2.10) where i = [α i ], where [x] detes the iteger part f x. (iii) T btai a sparse MQE, we chage R k (β) istep2f the iterati t R k (β) = 1 ( Y(j) β X (k 1) (j) ) 2 + λ β i,(2.11) where λ>0isacstat ctrllig the pealty the L 1 rm f β. This is a LASSO estimati, which ca be equivaletly represeted as the prblem f miimizig R k (β)i(2.5) subject t β i C 0, (2.12) where C 0 > 0 is a cstat. The LARS LASSO algrithm due t Efr et al. (2004) prvides the sluti

5 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 745 Dwladed by [ ] at 08:08 28 Octber 2015 path fr the OLS LASSO ptimizati prblem fr all psitive values f C 0. (iv) Sice ur gal is t match the distributi f Y by that f β X, a atural apprach is t estimate β which miimizes, fr example, mi F Y (x) F β x X(x)} 2, where F ξ ( ) detes the distributi fucti f radm variable ξ. Hwever, such a β is predmiatly determied by the ceter parts f the distributis as bth the distributis are clse t 1 fr extremely large values f x, ad are clse t 0 fr extremely egatively large values f x. Fr risk maagemet, thse extreme values are clearly imprtat. (v) MQE des t require that Y j ad X j are paired tgether. It ca be used t recver the early perfect liear relatiship Y β X based upaired bservatis Y j } ad X j }, as the L(Y ) L(β X), where L(ξ) detes the distributi f radm variable ξ. It als applies whe the distributi f Y is kw ad we have ly the bservatis X. I this case, the methdlgy described abve is still valid with Y (j) replaced by the true j/th quatile f L(Y )frj = 1,...,. (vi) Whe Y j ad X j are paired tgether, as i may applicatis, the pairig is igred i the MQE estimati (2.3). Hece, the crrelati betwee Y ad β X may be smaller tha that betwee Y ad β X. Ituitively, the lss i the crrelati shuld t be substatial uless the rati f ise-t-sigal is large, which is cfirmed by ur umerical experimets with bth simulated ad real data. See Table 3 i Secti 6 ad als Secti 7 belw. 3. CONVERGENCE OF THE ALGORITHMS We will shw i this secti that the iterative algrithm prpsed i Secti 2 abve fr cmputig MQE cverges a prperty remiiscet f the cvergece f the EM algrithm (Wu 1983). We itrduce a lemma first. Lemma 1. Let a 1,...,a ad b 1,...,b be ay tw sequeces f real umbers. The (a (i) b (i) ) 2 (a i b i ) 2, (3.1) where a (i) } ad b (i) } are, respectively, the rder statistics f a i } ad b i }. Prf. We prceed by the mathematical iducti. Whe = 2, we ly eed t shw that (a (1) b (1) ) 2 + (a (2) b (2) ) 2 (a (1) b (2) ) 2 + (a (2) b (1) ) 2, which is equivalet t 0 a (1) (b (1) b (2) ) + a (2) (b (2) b (1) ) = (a (2) a (1) )(b (2) b (1) ). This is true. Assumig the lemma is true fr all = k, we shw belw that it is als true fr = k + 1. Withut lss f geerality, we may assume that a k+1 = a (1) ad b l = b (1).Ifl = k + 1, (3.1) hlds fr k + 1 w. Whe l k + 1, it fllws the prf abve fr the case f = 2, (a (1) b (1) ) 2 + (a l b k+1 ) 2 (a l b l ) 2 + (a k+1 b k+1 ) 2. Csequetly, k+1 (a i b i ) 2 (a (1) b (1) ) 2 + (a l b k+1 ) i k,i l (a i b i ) 2 k+1 (a (1) b (1) ) 2 + (a (i) b (i) ) 2. The last iequality fllws frm the iducti assumpti fr = k. This cmpletes the prf. Therem 1. Fr R k ( ) defied i (2.5) r(2.11), ad β (k) = arg mi β R k (β), it hlds that R k (β (k) ) c as k, where c 0 is a cstat. Prf. We shw that the LASSO estimati with R k defied i (2.11) cverges. Whe λ = 0, (2.11) reduces t (2.5). We ly eed t shw that R k+1 (β (k+1) ) R k (β (k) ) fr k = 1, 2,...This is true because R k+1 (β (k+1) ) = 1 1 ( Y (j) β (k+1) X (k) (j) 1 ( Y (j) β (k) X (k) (j) ( Y (j) β (k) X (k 1) (j) ) 2 + λ i=2 ) 2 + λ ) 2 + λ β (k+1) i β (k) i (3.2) β (k) i =R k (β (k) ). (3.3) I the abve expressi, the first iequality fllws frm the defiiti f β (k+1) ad the secd iequality is guarateed by Lemma 1. Remark 2. (i) Therem 1 shws that the iteratis i Step 2 f the algrithm i Secti 2 abve cverge. But it des t guaratee that they will cverge t the glbal miimum. I practice, e may start with multiple iitial values selected, fr example, radmly, ad take the miimum amg the cverged values frm the differet iitial values. If ecessary, e may als treat the algrithm as a fucti f the iitial value ad apply, fr example, simulated aealig t search fr the glbal miimizer. (ii) I practice, we may search fr β X t match a part f distributi f Y ly, that is, we use R k ( ; α 1,α 2 ) defied i (2.10) istead f R k ( )i(2.5). Nte that X (k) (j), 1 < j 2 } may be a differet subset f X j,j= 1,...,} fr differet k, see(2.4). Hece Therem 1 lger hlds. Our umerical experimets idicate that the algrithm still cverges as lg as p is small i relati t (e.g., p 4). See Figure 6 ad Table 4 i Secti 6. (iii) Lemma 1 abve ca be deduced frm Lemmas 8.1 ad 8.2 f Bickel ad Freedma (1981) i a implicit maer, while the prf preseted here is simpler ad mre direct. 4. ASYMPTOTIC PROPERTIES OF THE ESTIMATION We preset the asympttic prperties fr a mre geeral settig i which MQE is cmbied with LASSO, ad the estima-

6 746 Jural f the America Statistical Assciati, Jue 2015 ti is defied t match a part f the distributi betwee the α 1 th quatile ad the α 2 th quatile, where 0 α 1 <α 2 1are fixed. Obviusly matchig the whle distributi is a special case with α 1 = 0 ad α 2 = 1. Furthermre whe λ = 0i(4.1) ad (4.3), it reduces t the MQE withut LASSO. Fr λ 0, let β 0 = arg mi S(β), S(β) S(β; α 1,α 2 ) β = α2 α 1 Q Y (α) Q β X(α)} 2 dα + λ β j. (4.1) Cditi B. (i) Let Y j } be a radm sample frm the distributi f Y ad X j } be a radm sample frm the distributi f X. Bthf Y ( ) ad f X ( )exist. (ii) (The Kiefer cditi.) It hlds fr ay fixed β that sup f β X (Q β X(α)) <, α 1 α α 2 if f β X(Q β X(α)) > 0. (4.5) α 1 α α 2 Furthermre Dwladed by [ ] at 08:08 28 Octber 2015 Ituitively β 0 culd be regarded as the true value t be estimated. Hwever, it is likely that β 0 s defied is t uique. Such a sceari may ccur whe, fr example, tw cmpets f X are idetically distributed. Furthermre it is cceivable that thse differet β 0 may lead t differet distributis L(β 0 X) which prvide a equally gd apprximati t L(Y )ithe sese that S(β 0 ) takes the same value fr thse differet β 0. Similar t (2.3), the MQE fr matchig a part f the distributi is defied as β = arg mi S (β), (4.2) β where S (β) S (β; α 1,α 2 ) = 1 = 1 j= j= 1 +1 Y(j) (β X) (j) } 2 + λ Q,Y (j/) Q,β X β j (j/) } 2 λ β j, (4.3) i = [α i ], (β X) (1) (β X) () are the rder statistics f β X 1,...,β X, Q,Y ( ) is the quatile fucti crrespdig t the empirical distributi f Y j }, that is, Q,Y (α) = ify : F,Y (y) α}, α (0, 1). I the abve expressi, F,Y (y) = 1 1 j I(Y j y). F,β X ad Q,β X are defied i the same maer. Similar t its theretical cuterpart β 0 i (4.1), the estimatr β defied i (4.2) may t be uique either, see Example 2 ad Remark 1(i) abve. Hece, we shw belw that S ( β) cverges t S(β 0 ). This implies that the distributi f β X prvides a ptimal apprximati t the distributi f Y i the sese that the mea square residuals S ( β) cverge t the miimum f S(β), althugh L( β X) may t cverge t a fixed distributi. Furthermre, we als shw that β is csistet i the sese that d( β, B 0 ) mi β B0 β β cverges t 0, where detes the Euclidea rm fr vectrs, ad B 0 is the set csistig f all the miimizers f S( ) defied i (4.1), that is, B 0 =β : S(β) = S(β 0 )}, (4.4) We itrduce sme regularity cditis first. We dete by, respectively, F ξ ( ) ad f ξ ( ) the distributi fucti ad the prbability desity fucti f a radm variable ξ. sup f Y (Q Y (α)) <, α 1 α α 2 (iii) X has buded supprt. Remark 3. if f Y (Q Y (α)) > 0. (4.6) α 1 α α 2 (i) Cditi B (ii) is the Kiefer cditi. It esures the uifrm Bahadur Kiefer buds fr empirical quatile prcesses fr iid samples. Mre precisely, (4.5) implies that sup fβ X(Q β X(α))Q,β X(α) Q β X(α)} α 1 α α 2 + F,β X(Q β X(α)) α} ( = O P 1/4 (lg ) 1/2 (lg lg ) 1/4), (4.7) ad (4.6) implies that sup fy (Q Y (α))q,y (α) Q Y (α)} α 1 α α 2 + F,Y (Q Y (α)) α} ( = O P 1/4 (lg ) 1/2 (lg lg ) 1/4). (4.8) See Kiefer (1970), ad als Kulik (2007). (ii) The assumpti f idepedet samples i Cditi B(i) is impsed fr simplicity f the techical prfs. I fact, Therem 2 still hlds fr sme weakly depedet prcesses, as the Bahadur-Kiefer buds (4.7) ad (4.8) may be established based the results i Kulik (2007). (iii) The requiremet fr X havig a buded supprt is fr techical cveiece. Whe α 1 = 0 ad α 2 = 1, it is implied by Cditi B(ii), as (4.5) etails that β X has a buded supprt fr ay β. Therem 2. Let Cditi B hld ad λ i (4.1) ad (4.3) be a egative cstat. The as, S ( β) S(β 0 )i prbability, ad d( β, B 0 ) 0 i prbability. We preset the prf f Therem 2 i Appedix I. 5. GOODNESS OF MATCH The gal f MQE is t match the distributi f Y by that f a selected liear cmbiati β X. We itrduce belw a measure fr the gdess f match, ad als a statistical test fr the hypthesis H 0 : L(Y ) = L(β X). (5.1)

7 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 747 Dwladed by [ ] at 08:08 28 Octber A Measure fr the Matchig Gdess Let F ( ) be the distributi fucti f Y. Letg( ) bethe prbability desity fucti f the radm variable F (β X). Whe Y ad β X have the same distributi, F (β X) is a radm variable uifrmly distributed the iterval [0, 1], ad g(x) 1frx [0, 1]. We defie a measure fr the gdess f match as fllws: ρ = g(x) 1 dx. (5.2) 2 0 It is easy t see that ρ [0, 1], ad ρ = 1 if ad ly if the matchig is perfect i the sese that L(Y ) = L(β X). Whe the differece betwee g( ) ad 1 (i.e., the desity fucti f U[0, 1]) icreases, ρ decreases. Hece the larger the differece betwee the distributis f Y ad β X, the smaller the value f ρ. Fr example, ρ = 0.5 ify U[0, 1] ad β X U[0, 0.5], ad ρ = 1/m if Y U[0, 1] ad β X U[0, 1/m] fr ay m 1. With the give bservatis (Y i, X i )},let U i = F (β X i ), where F (x) = 1 A atural estimatr fr ρ defied i (5.2)is ρ = C j = 1 [/k] Cj k/, I(Y j x). where ( ) (j 1)k I <U i jk. (5.3) I the abve expressi, k 1 is a iteger, [x] detes the iteger part f x. It als hlds that ρ [0, 1]. Furthermre, ρ = 1 if ad ly if /k is a iteger ad each f the /k itervals ( (j 1)k, jk )(j = 1,...,/k) ctais exactly k pits frm U 1,...,U. This als idicates that we shuld chse k large eugh such that there are eugh sample pits each f thse [/k] itervals ad, hece, the relative frequecy each iterval is a reasable estimate fr its crrespdig prbability. Remark 4. Frmula (5.2) ly applies whe the distributi f F (β X) is ctiuus. If this is t the case, the radm variable F (β X) has zer prbability masses at 0 r/ad 1, ad (5.2) shuld be writte i a mre geeral frm ρ = dg dx, where G( ) detes the prbability measure f F (β X). It is clear w that ρ = 0 if ad ly if the supprts f L(Y ) ad L(β X) d t verlap. Nte that the estimatr ρ defied i (5.3) still applies. 5.2 A Gdess-f-Match Test There exist several gdess-f-fit tests fr the hypthesis H 0 defied i (5.1); see, fr example, Secti 2.1 f Serflig (1980). We prpse a test statistic T belw, which is clsely assciated with the gdess-f-match measure ρ i (5.3) ad is remiiscet f the Cramér-v Mises gdess-f-fit statistic. Uder the hypthesis H 0, U 1,...,U behave like a sample frm U[0, 1] fr large. Hece based the relative cuts C j } defied i (5.3), we may defie the fllwig gdess-f-match test statistic fr testig hypthesis H 0. T = [/k] C j k/. (5.4) By Prpsiti 1, the distributi f T uder H 0 is distributifree. The critical values listed belw was evaluated frm a simulati with 50,000 replicatis, = 1000, ad bth ξ i } ad η i } draw idepedetly frm U[0, 1]. Sigificace level k/ = k/ = k/ = The chages i the critical values led by differet sample sizes, as lg as 300, are smaller tha 0.05 whe k/ 0.05, ad are smaller tha 0.1 whe k/ = Prpsiti 1. Let ξ 1,...,ξ } ad η 1,...,η } be tw idepedet radm samples frm tw distributis F ad G, ad F be a ctiuus distributi. Let F (x) = 1 i I(ξ i x) ad U i = F (η i ). Let C j be defied as i (5.3) ad T as i (5.4). The, the distributi T is idepedet f F ad G prvided F ( ) G( ). This prpsiti fllws immediately frm the fact that U i = 1 IF (ξ j ) F (η i )} almst surely, ad F (ξ i )} ad F (η i )} are tw idepedet samples frm U[0, 1] whe F ( ) G( ). 6. SIMULATION T illustrate the fiite-sample prperties, we cduct simulatis uder the settig Y j = β X j + Z j = β 1 X j1 + +β p X jp + Z j, j = 1,...,, (6.5) t check the perfrmace f MQE fr β = (β 1,...,β p ), where X j = (X j1,...,x jp ) represet p bserved variables, ad Z j represets cllectively the ubserved factrs. We let X j be defied by a factr mdel X j = AU j + ε j, where A is a p 3 cstat factr ladig matrix, the cmpets f U j are three idepedetly liear AR(1) prcesses defied with psitive r egative cetered lg-n(0, 1) ivatis, the cmpets f ε j are all idepedet ad t-distributed with 4 degrees f freedm. Hece, the cmpets f X j are crrelated with each ther with skewed ad heavy tailed distributis. We let Z j i (6.1) be idepedet N(0,σ 2 ). Fr each sample, the cefficiets β j are draw idepedetly frm U[ 0.5, 0.5], the elemets f the factr ladig matrix A are draw idepedetly frm U[ 1, 1], ad the three autregressive cefficiets i the three AR(1) factr prcesses are draw idepedetly frm U[ 0.95, 0.95]. Fr this example, liear cmbiatis f X j ca prvide a perfect match fr the distributi f Y j. Fr cmparis purpses, we als cmpute OLS β defied i (2.6). Fr cmputig MQE β,weuse β as the iitial value, ad

8 748 Jural f the America Statistical Assciati, Jue 2015 Dwladed by [ ] at 08:08 28 Octber 2015 Table 1. The meas ad stadard deviatis (STD) f the umber f iteratis required fr cmputig MQE β i a simulati with 1000 replicatis p r Mea STD Mea STD let β = β k ad whe rmse( β) =R k (β (k) )} 1/2 (6.2) Rk (β (k) )} 1/2 R k 1 (β (k 1) )} 1/2 < 0.001, (6.3) where R k ( ) is defied i (2.5). The reas t use square-rt f R k istead f R k i the abve is that R k itself ca be very small. We set the sample size = 300 r 800, the dimesi p = 50, 100, r 200, the rati STD(Z j ) r = 0.5, 1, r 2. STD(β 1 X j1 + +β p X jp ) Fr the simplicity, we call r the ise-t-sigal rati, which represets the rati f the ubserved sigal t the bserved sigal. Fr each settig, we draw 1000 samples ad calculate bth β ad β fr each sample. Figure 3 displays the bxplts f the rmse( β) defied i (6.2). It idicates that the apprximati with = 800 is mre accurate tha that with = 300. Whe the ise-t-sigal rati r icreases frm 0.5, 1, t 2, the values ad als the variati f rmse( β) icrease. Figure 3 shws that rmse( β) is rightskewed, idicatig that the algrithm may be stuck at a lcal miimum. This prblem ca be sigificatly alleviated by usig multiple iitial values geerated radmly, which was cfirmed i a experimet t reprted here. Table 1 list the meas ad stadard deviatis f the umber f iteratis required i calculatig MQE β, ctrlled by (6.3), ver the 1000 replicatis. Over all tested settigs, the algrithm cverges fast. The umber f iteratis teds t decrease whe the dimesi p icreases. This may be because there are mre true values f β whe p is larger, r simply whe p becmes really large. With each draw sample, we als geerate a pst-sample f size 300 deted by (y j, x j ),i = 1,...,300}. We measure the matchig pwer fr the distributi Y by rmme( β) frmqe, ad by rmme( β) fr OLS, where the rt mea matchig errr rmme is defied as ( ) 300 1/2 1 rmme(β) = y(j) (β } 2 x) (j), (6.4) 300 where y (1) y (300) are the rder statistics f y j }, ad (β x) (1) (β x) (300) are the rder statistics f β x j }. Figure 4 presets the scatterplts f rmme( β) agaist rmme( β) with sample size = 800. The dashed diagal lies mark the psitis y = x. Sice mst the dts are belw the diagals, the matchig errr fr the distributi Y based MQE β is smaller tha the crrespdig matchig errr based OLS β i mst cases. Whe the ise-t-sigal rati r is as small as 0.5, the differece betwee the tw methds is relatively small, as the the miimizers f (2.2) d t differ that much frm the miimizer f (2.7). Hwever whe the rati icreases t 1 ad 2, the matchig based the MQE is verwhelmigly better. This cfirms that MQE shuld be used whe the gal is t match the distributi f Y. The same plts with sample size = 300 are preseted i Figure 5. Whe the dimesi p is small such as p = 50 r 100, MQE still prvides a better matchig perfrmace verall, althugh the matchig errrs are greater tha thse whe = 800. Whe dimesi p = 200 ad sample size = 300, we step it verfittig territry. While the i-sample fittig is fie (see the tp pael i Figure 3 ad the bttm-left part f Table 3 belw), the pst-sample matchig pwer f bth OLS ad MQE is pr ad MQE perfrms eve wrse tha the wrg methd OLS. T assess the gdess-f-match, we als calculate the measure ρ defied i (5.3) with k = 20. The mea ad stadard deviati f ρ ver 1000 replicatis are reprted fr i Table 2. We lie up side by side the results calculated usig bth the sample used fr estimatig β ad the pst-sample. Except the verfittig cases (i.e., = 300 ad p = 200), the values f ρ with MQE are greater (r much greater whe r = 2r 1) tha thse with OLS, tig the small stadard deviatis acrss all the settigs. With MQE, ρ 0.92 fr the i-sample Table 2. The meas ad stadard deviatis (i paretheses) f estimated gdess-f-match measure ρ defied i (5.3) iasimulatiwith 1000 replicatis, calculated fr bth the sample used fr estimatig β ad the pst-sample OLS, = 300 MQE, = 300 OLS, = 800 MQE, = 800 p r i-sample pst-sample i-sample pst-sample i-sample pst-sample i-sample pst-sample (0.02) 0.89 (0.02) 0.95 (0.01) 0.89 (0.02) 0.88 (0.01) 0.89 (0.02) 0.92 (0.01) 0.89 (0.02) (0.03) 0.85 (0.03) 0.95 (0.01) 0.89 (0.02) 0.83 (0.02) 0.84 (0.03) 0.92 (0.01) 0.89 (0.02) (0.04) 0.77 (0.05) 0.95 (0.01) 0.88 (0.02) 0.71 (0.03) 0.72 (0.04) 0.93 (0.01) 0.88 (0.02) (0.02) 0.87 (0.02) 0.96 (0.01) 0.89 (0.02) 0.86 (0.01) 0.87 (0.02) 0.96 (0.01) 0.89 (0.02) (0.02) 0.85 (0.03) 0.96 (0.01) 0.88 (0.02) 0.83 (0.02) 0.84 (0.03) 0.96 (0.01) 0.88 (0.02) (0.03) 0.81 (0.03) 0.96 (0.01) 0.87 (0.03) 0.74 (0.03) 0.75 (0.04) 0.94 (0.01) 0.88 (0.02) (0.02) 0.86 (0.02) 0.97 (0.01) 0.88 (0.02) 0.86 (0.01) 0.87 (0.02) 0.96 (0.01) 0.89 (0.02) (0.02) 0.86 (0.03) 0.97 (0.01) 0.84 (0.04) 0.83 (0.01) 0.84 (0.02) 0.96 (0.01) 0.88 (0.02) (0.02) 0.82 (0.04) 0.97 (0.01) 0.78 (0.04) 0.78 (0.02) 0.79 (0.04) 0.96 (0.01) 0.88 (0.02)

9 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 749 rmse f MQE, = Dwladed by [ ] at 08:08 28 Octber p=100 p=100 p=100 rmse f MQE, = 800 p=200 p=200 p=200 p=100 p=100 p=100 p=200 p=200 p=200 Figure 3. Bxplts f rmse( β) defiedi(6.2) with sample size = 300 r 800, dimesi p = 50, 100, r 200, ad the ise-t-sigal rati r = 0.5, 1, r 2. matchig, ad ρ 0.87 fr the pst-sample matchig (except whe = 300 ad p = 200). With OLS, the miimum value f ρ is 0.71 fr the i-sample matchig, ad is 0.72 fr the pst-sample matchig. Oe side-effect f MQE β is the disregard f the pairig f (Y j, X j ); see (2.3). Hece we expect that the sample crrelati betwee Y ad β X will be smaller tha that betwee Y ad β X. Table 3 lists the meas ad stadard deviatis f the sample crrelati cefficiets betwee Y ad β X, ad f thse betwee Y ad β X i ur simulati. Over all differet settigs, the mea sample crrelati cefficiet fr bth i-samples ad pst-samples betwee Y ad β X is always greater tha that betwee Y ad β X. Hwever the differece is small. I fact if we take the differece f the tw meas, deted as D, as the estimatr fr the true differece ad treat the tw meas idepedetly f each ther, the (abslute) value f D is always smaller tha its stadard errr ver all the settigs.

10 Dwladed by [ ] at 08:08 28 Octber Jural f the America Statistical Assciati, Jue 2015 Figure 4. Scatterplts f rmme( β) agaist rmme( β) with sample size = 800 i a simulati with 1000 replicatis. The dashed lies mark the diagal y = x. Fially we ivestigate the perfrmace f MQE i matchig ly a part f distributi. T this ed, we repeat the abve exercise but usig R k (β) = R k (β, 0, 0.3) defied i (2.10) istead, that is, the MQE is sught t match the lwer 30% f the distributi f Y. Figure 6 presets the bxplts f rmse( β). Cmparig it with Figure 3, there are etries fr = 300 ad p = 100 r 200, fr which the algrithm did t cverge after 500 iteratis. See Remark 2(ii). Fr the cases preseted i Figure 6, rmse( β) are smaller tha the crrespdig etries i Figure 3. This is because the matchig w is easier, as the MQE is sught such that the lwer 30% f L( β X) matches the cuterpart f L(Y ). But there are ay cstraits the upper 70% f L( β X). Table 4 list the meas ad stadard deviatis f the umber f iteratis required i calculatig MQE ver the 1000 replicatis. Cmparig it with Table 1,the algrithm cverges faster fr matchig a part f L(Y ) tha fr matchig the whle L(Y ). 7. A REAL-DATA EXAMPLE I the ctext f selectig a represetative prtfli fr backtestig cuterparty credit risks, Y is the ttal prtfli f a cuterparty, ad X = (X 1,...,X p )arethep mark-t-market values f the trades. The gal is t fid a liear cmbiati

11 Dwladed by [ ] at 08:08 28 Octber 2015 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 751 Figure 5. Scatterplts f rmme( β) agaist rmme( β) with sample size = 300 i a simulati with 1000 replicatis. The dashed lies mark the diagal y = x. β X which prvides a adequate apprximati fr the ttal prtfli Y. Sice Basel III requires that a represetative prtfli matches varius characteristics f the ttal prtfli, we use the prpsed methdlgy t select β X t match the whle distributi f Y. We illustrate belw hw this ca be de usig the recrds fr a real prtfli. The data ctais 1000 recrded ttal prtflis at e mth ter (i.e., e mth stppig perid) ad the crrespdig mark-t-market values f 146 trades (i.e., p = 146). Thse 146 trades were selected frm ver 2000 trades acrss differet ters (i.e., frm 3 days t 25 years) by the stepwise regressi methd f A et al. (2008). The data has bee rescaled. As sme trades are heavily skewed t the left while the ttal prtfli data are very symmetric fr this particular dataset, we trucate thse trades at μ 6 σ, where μ ad σ dete, respectively, the sample mea ad the sample stadard deviati f the trade ccered. The absece f the heavy left tail i the ttal prtfli data is because there exist highly crrelated trades i ppsite directis (i.e., sales i ctrast t buys) which were elimiated at the iitial stage by the methd f A et al. (2008). We estimate bth OLS β ad MQE β usig the first 700 (i.e., = 700) f the 1000 available bservatis. The algrithm fr cmputig MQE tk 7 iteratis t cverge. We cmpare Y with β X ad β X usig the last 300 bservatis. The i-sample

12 752 Jural f the America Statistical Assciati, Jue 2015 Table 3. The meas ad stadard deviatis (i paretheses) f the sample crrelati cefficiets betwee Y ad β X, ad betwee Y ad β X i a simulati with 1000 replicatis, calculated fr bth the sample used fr estimatig β ad the pst-sample OLS, = 300 MQE, = 300 OLS, = 800 MQE, = 800 p r i-sample pst-sample i-sample pst-sample i-sample pst-sample i-sample pst-sample (0.02) 0.93 (0.02) 0.95 (0.02) 0.92 (0.03) 0.95 (0.02) 0.94 (0.02) 0.94 (0.02) 0.93 (0.02) (0.04) 0.79 (0.06) 0.84 (0.04) 0.76 (0.06) 0.84 (0.04) 0.81 (0.06) 0.81 (0.04) 0.78 (0.06) (0.06) 0.51 (0.10) 0.65 (0.06) 0.47 (0.10) 0.63 (0.06) 0.56 (0.09) 0.58 (0.05) 0.50 (0.08) (0.01) 0.92 (0.03) 0.96 (0.01) 0.91 (0.03) 0.95 (0.02) 0.94 (0.02) 0.95 (0.02) 0.93 (0.02) (0.03) 0.74 (0.07) 0.88 (0.03) 0.72 (0.08) 0.85 (0.04) 0.80 (0.06) 0.83 (0.04) 0.77 (0.06) (0.05) 0.43 (0.10) 0.74 (0.05) 0.40 (0.10) 0.66 (0.06) 0.53 (0.09) 0.63 (0.05) 0.48 (0.09) (0.01) 0.85 (0.05) 0.98 (0.01) 0.84 (0.05) 0.96 (0.01) 0.92 (0.03) 0.95 (0.01) 0.92 (0.03) (0.02) 0.60 (0.10) 0.94 (0.02) 0.59 (0.10) 0.87 (0.03) 0.76 (0.07) 0.86 (0.03) 0.74 (0.07) (0.02) 0.28 (0.10) 0.88 (0.02) 0.28 (0.10) 0.72 (0.04) 0.46 (0.10) 0.71 (0.04) 0.44 (0.10) Dwladed by [ ] at 08:08 28 Octber 2015 ad pst-sample crrelatis betwee Y ad β X are ad The i-sample ad pst-sample crrelatis betwee Y ad β X are ad Oce agai the lss f crrelati with MQE is mir. Settig k/ = 0.05 i (5.3), the i-sample ad pst-sample gdess f fit measures ρ are ad with MQE, ad are ad with OLS. This idicates that MQE prvides a much better matchig tha OLS. The gdess-fmatch test preseted i Secti 5.2 reifrces this asserti. The test statistic T defied i (5.4), whe applied t the 300 pst-sample pits, is equal t fr the MQE matchig, ad is fr the OLS matchig. Cmparig t the critical values listed i Secti 5.2, we reject the OLS matchig at the 0.5% sigificace level, but we cat reject the MQE matchig eve at the 10% level. Nte that we d t apply the test t the i-sample data as the same data pits were used i estimatig β (thugh the cclusis wuld be the same). T further shwcase the imprvemet f MQE matchig ver OLS, Figure 7 plts the sample quatiles f the represetative prtflis β X ad β X agaist the sample quatiles f the ttal cuterparty prtfli Y, based the 300 pst-sample pits. It shws clearly that the distributi f the represetative prtfli based MQE β prvides much mre accurate apprximati fr the distributi f the ttal cuterparty prtfli tha that based the OLS β. Fr the latter, the discrepacy is alarmigly large at the tw tails f the distributi, where matter mst fr risk maagemet. 8. PORTFOLIO TRACKING Prtfli trackig refers t a prtfli assembled with securities which mirrrs a bechmark idex, such as S&P500 r FTSE100 (Jase ad va Dijk 2002, ad Dse ad Cictti 2005). Trackig prtflis ca be used as the strategies fr ivestmet, hedgig ad risk maagemet fr ivestmet, r as macrecmic frecastig (Lamt 2001). Let Y be the retur f a idex t be tracked, X 1,...,X p be the returs f the p securities t be used fr trackig Y. Oe way t chse a trackig prtfli is t select weights w i } t miimize subject t w i = 1 E ( Y ) 2 w i X i (8.1) ad w i c, (8.2) where c 1 is a cstat. See, fr example, Secti 3.2 f Fa et al. (2012). I the abve expressi, w i is the prprti f the capital ivested the ith security X i, ad w i < 0 idicates a shrt sale X i. It fllws frm (8.2) that w i >0 w i 1 + c 2, w i <0 w i c 1 2. (8.3) Hece, the cstat c ctrls the expsure t shrt sales. Whe c = 1, shrt sales are t permitted. Istead f usig the cstraied OLS as i abve, e alterative i selectig the trackig prtfli is t match the whle (r a part) f distributi f Y. This leads t a cstraied MQE, subject t the cstraits i (8.2). Give a set f histrical returs (Y j,x j1,...,x jp ),j= 1,...,}, we use the iterative algrithm i Secti 2 t calculate MQE β subject t the cstrait Table 4. The meas ad stadard deviatis (STD) f the umber f iteratis required fr cmputig MQE β fr matchig the lwer 30% f the distributi f Y (, p) (300, 50) (800, 50) (800, 100) (800, 200) r Mea STD β j δ β (0) i, (8.4) ad β (0) = ( β (0) (0) 1,..., β p ) is the ucstraied MQE fr β, ad δ (0, 1) is a cstat which ctrls, idirectly, the ttal expsure t shrt-sales. This is the stadard MQE-LASSO; see (2.12) i Remark 1(iii) i Secti 2. Frδ 1, β = β (0). We trasfrm the cstraied MQE β = ( β 1,..., β p ) t the

13 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 753 rmse f MQE, = 300 rmse f MQE, = Dwladed by [ ] at 08:08 28 Octber 2015 Figure 6. Bxplts f rmse( β) fr matchig the lwer 30% f the distributi f Y,where is sample size, p is the dimesi f X,adr is the ise-t-sigal rati. estimates fr the prprti weights as fllws: ŵ i = β i / 1 j β j, i = 1,...,p. The ŵ i } fulfill the cstraits i (8.2) with ay c satisfyig the fllwig cditi: c δ i β (0) i / j β j. (8.5) p=100 p=100 p=100 Such a c is always greater tha 1 as δ i β (0) i / j β j δ p=200 i β (0) i p=200 p=200 / β j 1, see (8.4). Nte that the LARS-LASSO algrithm gives the whle sluti path fr all psitive values f δ. Hece fr a give value c i (8.2), we ca always fid the largest pssible value δ frm the sluti path fr which (8.5) hlds. Remark 5. Oe wuld be tempted t absrb the cstrait cditi j w j = 1 i the estimati directly by lettig, fr j OLS represetative prtfli **** ***** *** * **** * * *** * * MQE represetative prtfli ** **** ***** ** * * ***** * Ttal cuterparty prtfli Ttal cuterparty prtfli Figure 7. The plts f the sample quatiles f the represetative prtflis based OLS (the left pael) ad MQE (the right pael) agaist the sample quatiles f the ttal cuterparty prtfli. The straight lies mark the diagal y = x which the tw quatiles are equal. All the quatiles are calculated based the 300 pst-sample pits.

14 754 Jural f the America Statistical Assciati, Jue 2015 Table 5. The mea, maximum ad miimum daily lg returs (i percetages) f FTSE100 ad the estimated track prtflis i The estimati was based the data i Als icluded i the table are the umber f stcks preset i each prtfli, the stadard deviatis (STD) ad the egative mea (NM) f the daily returs, ad the percetages (f the capital) fr shrt sales Prtfli N. f Retur Shrt stcks Mea Max Mi STD NM sales FTSE OLS MQE OLS-lass (δ = 0.7) MQE-lass (δ = 0.7) OLS-lass (δ = 0.5) MQE-lass (δ = 0.5) MQE-lass (δ = 0.7, α 1 = 0, α 2 = 0.5) MQE-lass (δ = 0.7, α 1 = 0.25, α 2 = 0.75) MQE-lass (δ = 0.7, α 1 = 0.5, α 2 = 1) MQE-lass (δ = 0.5, α 1 = 0,α 2 = 0.5) MQE-lass (δ = 0.5, α 1 = 0.25, α 2 = 0.75) MQE-lass (δ = 0.5, α 1 = 0.5, α 2 = 1) Dwladed by [ ] at 08:08 28 Octber Ja Mar May Jul Sep Nv Ja Ja Mar May Jul Sep Nv Ja Figure 8. The plts f the daily lg returs f FTSE100 idex (thick black cycles), the MQE-LASSO prtfli with δ = 0.7 (thi red cycle i the tp pael), ad the MQE-LASSO prtfli with δ = 0.5ad(α 1,α 2 ) = (0, 0.5) (thi blue cycles i the bttm pael).

15 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 755 Mea aual returs Stadard deviatis f returs Dwladed by [ ] at 08:08 28 Octber 2015 Mea returs f prtflis FTSE100 OLS MQE OLS lass MQE lass STD f returs FTSE100 OLS MQE OLS lass MQE lass Year Year Figure 9. The plts f the aual meas ad stadard deviatis (STD) f daily lg returs f FTSE100 idex, the OLS prtfli, the MQE prtfli, the OLS-LASSO prtfli ad the MQE-LASSO prtfli i the perid f example, Y = Y X p, X i = X i X p fr 1 i<p. The, e culd estimate w 1,...,w p 1 directly by regressig Y X 1,...,X p 1. Hwever, this puts the pth security X p a equal ftig as the ther p 1 securities, which may lead t a adverse effect. We illustrate ur prpsal by trackig FTSE100 usig 30 actively traded stcks icluded i FTSE100. The cmpay ames ad the symbls f thse 30 stcks are listed i Appedix II. We use the lg returs (i percetages) calculated usig the adjusted daily clse prices i ( = 758) t estimate the trackig prtflis by MQE with r withut the LASSO, ad cmpare their perfrmace with the returs f FTSE100 i 2007 (i ttal 253 tradig days). We als iclude i the cmparis the prtflis estimated by OLS. The market is verall bullish i the perid The data were dwladed frm Yah!Fiace. Table 5 list sme summary statistics f the daily lg-returs i 2007 f FTSE100 ad the varius trackig prtflis. Bth the OLS ad the MQE track well the FTSE100 idex with almst idetical daily mea 0.014%. I additi t the stadard deviatis (STD), we als iclude i the table the egative mea (NM) as a risk measure, which is defied as the mea value f all the egative returs. Accrdig t bth STD ad NM, bth the OLS ad the MQE are slightly less risky tha FTSE100 i 2007.

16 756 Jural f the America Statistical Assciati, Jue 2015 Mea aual returs Stadard deviatis f returs Dwladed by [ ] at 08:08 28 Octber 2015 Mea returs f prtflis FTSE100 Lwer half Middle half Upper half STD f returs FTSE100 Lwer half Middle half Upper half Year Year Figure 10. The plts f the aual meas ad stadard deviatis (STD) f daily lg returs f FTSE100 idex, ad the prtflis based the MQE-LASSO matchig the lwer half, the middle half, ad the upper half f distributi i the perid f We als frm the prtflis based OLS-LASSO ad MQE- LASSO with the trucated parameter δ = 0.7 ad 0.5; see (8.4). Nw all the fur prtflis yield ticeably greater average daily returs tha that f FTSE100 with ticeably greater risks. Furthermre, the perfrmaces f OLS ad MQE part frm each ther with MQE prducig substatially larger returs with larger risks. Fr example, the MQE-LASSO prtfli with δ = 0.5 yields average daily retur f 0.119% ad NM 1.336% while the OLS-LASSO yields average daily retur f 0.045% ad NM 1.119%. The umber f stcks selected i prtfli is 10 by MQE, ad 14 by OLS. We ctiue the experimet by usig the MQE matchig the lwer half, the middle half ad the upper half f the distributi ly; see Remark 1(ii). With δ = 0.7, the prtflis resulted frm matchig either the lwer r the upper half f the distributi icur excessive shrt sales f, respectively, 38.4% ad 885% f the iitial capital, ad are therefre t risky. By usig δ = 0.5, shrt sales are reduced t 1.4% ad 22% respectively. Especially matchig the lwer half distributi with δ = 0.5 leads t a prtfli with average daily retur 0.223%, the STD 2.33%, the NA 1.56% ad shrt sales 1.4%. Figure 8 plts the daily returs f FTSE100 tgether with the tw prtflis estimated by the MQE-LASSO with δ = 0.7, ad δ = 0.5, (α 0,α 1 ) = (0, 0.5), respectively. Bth the prtflis track well the idex with icreased vlatility. Especially the prtfli pltted i blue is btaied by matchig the lwer

17 Sgurpuls, Ya, ad Yastremiz: Matchig a Distributi by Matchig Quatiles Estimati 757 Dwladed by [ ] at 08:08 28 Octber 2015 half distributi ly. Cmparig with FTSE100, the icrease f the STD is 1.23% while the icrease f the NM is merely 0.654%. The icrease f the retur fr this prtfli is resulted frm mimickig the lss f FTSE100 ad freeig the tp half distributi. Nw we apply the abve apprach with a rllig widw t the data i Mre precisely, fr each caledar year withi the perid, we use the data i its previus three years fr estimati t frm the differet prtflis. We the calculate the meas ad stadard deviatis fr the daily returs i that year based each f the prtflis. (The data fr 2013 were ly up t 10 September whe this exercise was cducted.) The results fr the prtflis based OLS, MQE with ad withut LASSO are pltted i Figure 9. Wesetδ = 0.5 iall the LASSO estimatis. Figure 9 shws that the MQE-LASSO prtfli geerated greater average returs i the 5 ut f 7 years tha the ther fur prtflis. But it als led t greater lsses tha FTSE100 idex i bth 2008 ad Judgig by the stadard deviatis it is the mst risky strategy amg the five prtflis reprted i Figure 9. Nte that bth the OLS ad MQE prtflis icur small icreases i stadard deviati while the gais i average returs i 2011 ad 2013 are ticeable. This shws that it is pssible t match the verall perfrmace f the idex by tradig much fewer stcks. Figure 10 cmpares the three prtflis based the MQE- LASSO matchig, respectively, the lwer half, the middle half ad the upper half f the distributis fr the returs f FTSE100 idex. The first pael i the figure suggests that matchig the upper-half distributis leads t very vlatile average returs which are wrse tha the returs f FTSE100 idex verall. I ctrast, matchig the lwer half r the middle half f the distributis prvide better retur tha the idex i the 6 ut f 7 years durig the perid. The risks f thse prtflis, measured by the stadard deviatis, are higher that thse f the idex; see the secd pael i the figure. Overall the MQE-LASSO prtflis ted t versht at bth the peaks ad the trughs. Therefre they ted t utperfrm FTSE100 idex whe the market is bullish, ad they may als d wrse tha the idex whe the market is bearish (such as 2008 ad 2011). APPENDIX I: PROOF OF THEOREM 2 We split the prf f Therem 2 it several lemmas. Lemma A.1. Uder Cditis B(i) ad (ii), τ S (β) S(β)} 0 i prbability fr ay fixed β ad τ<1/2. Prf. Put W = β X.By(4.7) ad(4.8), 1 j= = Q,Y (j/) Q,W (j/)} 2 j= 1 +1 j= 1 +1 j= 1 +1 Q Y (j/) Q W (j/)} 2 } F,Y (Q Y (α)) α 2 f Y (Q Y (α)) F,W (Q W (α)) α f W (Q W (α)) } 2 + 2R 3/2 j= 1 +1 Q Y (j/) Q W (j/) + F,Y (Q Y (α)) α f Y (Q Y (α)) F,W (Q W (α)) α f W (Q W (α)) } + O P (R 2 /), (A.1) where R = O P ( 1/4 (lg ) 1/2 (lg lg ) 1/4 ) = P (1). By the Dvretzky-Kiefer-Wlfwitz iequality (Massart 1990), it hlds fr ay cstat C>0ad ay iteger 1that P P sup 0 α 1 sup 0 α 1 F,Y (Q Y (α)) α >C F,W (Q W (α)) α >C } 2e 2C2, } 2e 2C2. Let C = τ 1 fr sme τ 1 (τ/2, 1/4), ad } A = sup F,Y (Q Y (α)) α C 0 α 1 } F,W (Q W (α)) α C. sup 0 α 1 The by (A.2), P (A ) 1 4e 2C2 1, ad the set A, τ 1 1 j= 1 +1 j= 1 +1 Q,Y (j/) Q,W (j/)} 2 (A.2) Q Y (j/) Q W (j/)} 2 } = P (1), (A.3) which is guarateed by Cditi B(ii) ad the fact that 1 j= 1 +1 } Q Y (j/) Q W (j/) α2 α 1 Q Y (α) Q W (α)}dα. Nte that α2 α2 } Q Y (α) Q W (α)}dα Q Y (α) + Q W (α) dα α 1 α 1 = E[ Y IG Y (α 1 ) <Y G Y (α 2 )}] + E W <, as Y IG Y (α 1 ) <Y G Y (α 2 )} is buded uder Cditi B(ii). See als cditi B(iii) ad Remark 3(iii). Uder Cditi B(ii), Q Y (α) Q Y (j/) =f Y (j/) 1 /1 + (1)} fr ay α j/ 1/. Hece 1 j= 1 +1 α2 = α 1 Q Y (j/) Q W (j/)} 2 Q Y (α) Q W (α)} 2 dα + (1/). Cmbiig this with (A.3), we btai the required result. Lemma A.2. Let a 1 a be real umbers. Let b i = a i + δ i fr i = 1,...,,adδ i are real umbers. The max a i b (i) max δ j, (A.4) 1 i 1 j where b (1) b () is a permutati f b 1,...,b }. Prf. We use the mathematical iducti t prve the lemma. Let ɛ = max j δ j. It is easy t see that (A.4) istruefr = 2.Letitbe als true fr = k. We w prve it fr = k + 1. Let c i = b i fr i = 1,...,k. The by the iducti assumpti, max a i c (i) ɛ. (A.5) 1 i k If b k+1 = a k+1 + δ k+1 c (k), the required result hlds. Hwever, if fr sme 1 i<k, c (i) b k+1 <c (i+1),