INFERENCE FOR EXTREMAL CONDITIONAL QUANTILE MODELS (EXTREME VALUE INFERENCE FOR QUANTILE REGRESSION)

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1 INFERENCE FOR EXTREMAL CONDITIONAL QUANTILE MODELS (EXTREME VALUE INFERENCE FOR QUANTILE REGRESSION) VICTOR CHERNOZHUKOV Abstract. Quantile regressin is a basic tl fr estimatin f cnditinal quantiles f a respnse variable given a vectr f regressrs. It can be used t measure the effect f cvariates nt nly in the center f a distributin, but als in the upper and lwer tails. Quantile regressin applied t the tails, r simply extremal quantile regressin is f interest in numerus ecnmic and financial applicatins. Fr example, it can be emplyed t measure cnditinal valueatrisk, prductinefficiency, adjustment bands in the (S,s) mdels, and cst functins f mst efficient bidders in auctins. In rder t facilitate the applicatins, this paper prvides feasible inference tls that rely upn extreme value apprximatins f the distributin f selfnrmalized extremal quantile regressin statistics. The methds are simple t implement in practice and are f independent interest even in the nnregressin case. The value f the methds is explred in the analysis f extremely lw percentiles f live infant birthweights (in the ranges between 250 and 1500 grams) and in the study f factrs f extreme fluctuatins f a stck return. Key Wrds: Quantile Regressin JEL: C13, C14, C21, C41, C51, C53 Data and sftware in R available by request Date: May, 2002, revised January, This is still a preliminary revisin.
2 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 1 1. Intrductin This paper prvides feasible inference methds fr extremal quantile regressin (QR) mdels, which are based n extreme value thery. The mdels have many vital applicatins t extremal phenmena arising in ecnmics and ther sciences, ranging frm extreme risks t extreme birthweights. We prvide feasible inference methds fr these mdels and illustrate them with an analysis f ecnmic determinants f extremely lw birthweights, in the range between 250 and 1500 grams, and the analysis f ecnmic factrs f extreme fluctuatins f a stck return. The inferential methds f this paper are based n the extreme value thery fr QR, namely n the sampling thery fr QR that we develp specifically fr the tails. Frmally the tail situatin ccurs when the quantile index τ (0, 1) is either small, clse t zer, r large, clse t 1. Withut lss f generality we assume the frmer, and, by clse t 0 mean that τt k > 0 as the sample size T. Under these cnditins, the central limit therem fails t hld, but different laws f extreme events arise. This apprach, underlying als the classical extreme value thery in univariate settings, turns ut t prvide distributinal apprximatins fr QR that are f better quality in the tails than cnventinal nrmal apprximatins. Hwever, there are fundamental difficulties that have previusly made the inference fr QR based n this apprach bth elusive and infeasible. Main inference methds break dwn fr several reasns. First, the basic inference apprach using limit distributins f Wald statistics is nt feasible, because extreme value apprximatins, bth in the classical univariate case and in the regressin case, rely n cannical nrmalizing cnstants used t achieve nndegenerate asympttic distributins. Cnsistent estimatin f these cnstants is nt always feasible withut additinal strng assumptins. This is the first difficulty encuntered here as well as in the classical univariate case (Bertail, Haefke, Plitis, and White (2004)). Secnd, basic resampling methds als fail t deliver prper inference. The cnventinal btstrap fails due t the nnstandard behavir f extremal quantile statistics (as shwn by Bickel and Freedman (1981) in the univariate case). Cnventinal subsampling (Plitis, Rman, and Wlf 1999) is als incnsistent, at least in in the unbunded supprt case where QR statistics diverge. Mrever, cnventinal subsampling suffers frm the first difficulty as well, requiring cnsistent estimates f the cannical nrmalizing cnstants. In this paper we develp tw types f inference appraches that vercme these difficulties, a resampling apprach and an analytical apprach, with the frmer apprach being the principal ne. Bth appraches are based n selfnrmalized QR (SNQR) statistics that emply randm nrmalizatin (instead f generally infeasible nrmalizatin by cannical
3 2 VICTOR CHERNOZHUKOV cnstants). The use f SNQR allws us t derive feasible limit distributins, which underlie either f ur inference appraches. Mrever, ur resampling apprach is a mdified subsampling methd applied t SNQR statistics. The apprach entirely avids nt nly estimating the cannical nrmalizing cnstants but als all ther nuisance tail parameters knwn t be very hard t estimate. The mdified subsampling methd explres the special relatinship between the rates f cnvergence f extremal and intermediate QR statistics, which allws fr suitably estimating the centering cnstants in subsamples. This paper als prvides inferential methds fr cannicallynrmalized QR (CNQR), fr cmpleteness, but als shws that much strnger assumptins are required fr their feasibility. This paper cntributes t the existing literature by prviding fr the first time the general feasible inference fr extremal quantile regressin. This addresses a prblem that was first asked in the cntext f estimating prbabilistic frntier functins (e.g., Timmer (1971) and Aigner, Amemiya, and Pirier (1976).) The inferential methds prpsed in this paper partially rely n limit results in Chernzhukv (2000b) wh studied the limits f cannicallynrmalized QR under the extreme rder cnditin τt k > 0. Hwever, the limit thery in Chernzhukv (2000b) was mre theretical in nature and prvided limit results that are infeasible fr purpses f inference, in the sense explained earlier. Related limit results fr the linear prgramming estimatr, that crrespnds t taking τ = τt = 0, were cnsidered by Feigin and Resnick (1994), Prtny and Jure ckvá (1999), Knight (2001), and Chernzhukv (1998), all in different cntexts and at varius levels f generality. The linear prgramming estimatr is mst suited t the prblem f estimating the finite bundary f the data, e.g. as in image prcessing and ther technmetric applicatins alike, whereas the current apprach f taking τt k > 0 is mre suited t ecnmetric applicatins, where interest fcuses n the usual quantiles lcated near the minimum, r maximum, and where the bundaries may be unlimited. Hwever, sme f ur theretical develpments are mtivated and build upn this previus literature. The paper is rganized as fllws. Sectin 2 mtivates the analysis and frmulates the ecnmetric prblem. Sectin 3 describes the mdel and regularity cnditins. Sectin 4 describes the limit behavir f selfnrmalized QR statistics. Sectin 5 describes the inference methds. Sectin 6 describes the MnteCarl wrk. Sectin 7 presents empirical applicatins, and the Appendix cllects prfs and figures. 2. Ecnmic Prblems and Ecnmetric Prblems 2.1. Extremal Cnditinal Quantiles in Ecnmic Analysis. There are many useful applicatins f inferential methds develped in this paper. Fr purpses f discussin in
4 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 3 this sectin, let Q Y (τ X) dente the τquantile f a respnse variable Y given regressrs X, that is the τ cnditinal quantile functin. Example 1. Determinants f ValueatRisk. Valueatrisk analysis seeks t frecast r explain very lw quantiles f an institutin s prtfli return Y using tday s available infrmatin W. This type f analysis is a basic, every day activity that banking and ther financial institutins perfrm, as required by the Securities and Exchange Cmmissin and the Basle Cmmittee n Banking Supervisin. The new inferential tls shuld be pertinent there, as the valueatrisk prblem naturally lends itself t the quantile regressin dmain, e.g. Chernzhukv and Umantsev (2001) and Engle and Manganelli (2001). As a risk measure valueatrisk is mtivated by the safetyfirst decisin principle (Ry 1952), in which ne makes ptimal decisins subject t the cnstraint that a disastrus event ccurs with a small prbability. This and similar measures are ften used in reallife financial management, insurance, and actuarial science (Embrechts, Klüppelberg, and Miksch 1997). Example 2. Determinants f Very Lw Birthweights. In the analysis f infant birth weights, we may be interested in hw smking, absence f prenatal care, and ther types f maternal behavir affect varius quantiles f birth weights, cf. Abreveya (2001). Of special interest, hwever, are the very lw quantiles, since lw birth weights have been linked t subsequent health prblems. This paper prvides inference tls fr statistical investigatin f this questin using quantile regressins. Example 3. Prbabilistic Frntiers fr Prducers. An imprtant frm f efficiency analysis in industrial rganizatin and ecnmics f regulatin is the determinatin f prductin frntiers (e.g. Timmer, 1971). Given prductin factrs W, high percentiles f prductin levels Y, called prbabilistic frntiers, namely Q Y (τ W ), fr τ [1 ɛ, 1] and ɛ > 0 are attained by an ɛfractin f mst highly efficient firms. The inference n ɛfrntiers (??) will be facilitated by the results f this paper. Example 4. (S,s)Rules and Other Apprximate Reservatin Rules in Ecnmic Decisins. A related example is that f (S, s)adjustment mdels, which arise as ptimal plicies in many ecnmic mdels (Arrw, Harris, and Marschak 1951). Fr example, the capital Z is adjusted up t the level S nce it has depreciated t the level s. In terms f an ecnmetric specificatin, the bserved capital satisfies Z i = s(w i ) + v i, where W i are cvariates and v i is a disturbance that is psitive mst f the time, i.e. P (v i 0) is clse t 1. Once capital reaches the critical level, Z i s(w i ) 0, it is adjusted in the next perid. When the disturbance v i is negative, it is assumed t capture unbserved hetergeneity and small decisin mistakes which are independent f bserved cvariates W i, as in Caballer and Engel (1999). In a given crsssectin r time series, adjustment will ccur infrequently, s in fact data at r belw the lwer adjustment band s(x i ) will be bserved with a small
5 4 VICTOR CHERNOZHUKOV prbability P (v i 0), hence Q Z (τ W ) = s(w ) + Q v (τ) fr τ (0, P (v 0)]. The lwer band functin s(w ) therefre cincides with the lwer cnditinal quantile functin up t an additive cnstant. A similar argument can be made fr the upper band functin S(W ). Example 5. Structural Auctin Mdels. In the standard specificatin f the firstprice prcurement auctin where bidders hld independent valuatins, the winning bids B i satisfy: B i = c(x i )β(n i )+ɛ i, ɛ i 0, where c(x i ) is the efficient cst functin, and β(n i ) 1 is a markup that appraches 1 as the number f bidders n i appraches infinity (Dnald and Paarsch 2002). By cnstructin, c(x i )β(n i ) is the extreme cnditinal quantile functin. In the empirical analysis, it is realistic t let the disturbance ɛ i take n sme small negative value, s that when negative these disturbances capture small decisin mistakes that are independent f included explanatry variables. In this case the quantile functin satisfies Q B (τ X, n) = C(X)β(n) + Q ɛ (τ), fr τ (0, P [ɛ 0]], hence inference n c(x) and β(n) can be made using the extremal QR, cmplementing the apprach f Dnald and Paarsch (2002) The Ecnmetric Prblem. The linearinparameters QR mdel specifies a τth quantile f interest f a respnse variable Y given regressrs X as X β(τ). Suppse we have T = 200 bservatins and run QR, btain β(τ), and prceed with inference n β(τ) fr τ = 5%. Fr inference purpses, we may use the cnventinal central nrmal limit thery which states T ( β(τ) β(τ)) d N (0, Ω(τ)), fr fixed τ (0, 1), as T, (2.1) fr Ω(τ) = τ(1 τ)j(τ) 1 E[XX ]J(τ) 1, J(τ) = E[f U (F 1 U (τ X) X)XX ], and U = Y X β fr sme β R d. In this paper we develp an inference methd based n a limit thery btained under the extreme rder cnditins n the quantile index τt k > 0 as T. The idea is that this fits ur example with τt = 10 and ther similar cases better than the nrmal apprach arising frm impsing τ T. Indeed, the cannically nrmalized QR (CNQR) satisfies: 1 A T ( β(τ) β(τ)) d Ẑ (k), fr A T = F 1 as τt k, T, (2.2) U (1/T ), where Ẑ (k) is an extremetype variate specified in Sectin 3 that depends n the tail thickness parameter ξ f variable U. The nrmalizatin by A T is cannical in the sense 1 In the apprach f Dnald and Paarsch (2002) ne first estimates a saturated bundary mdel using cellbycell extreme rder statistics and then prjects the estimated bundary nt a lwer dimensinal nnlinear mdel using the minimumdistance methd. One can prceed likewise using quantile regressin t estimate the first stage and then prject it nt further structural mdel.
6 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 5 that the same nrmalizatin is used in the classical univariate thery f extreme rder statistics. Figure 1 shws that the extremal apprximatin (2.2) may ptentially prvide an imprved basis fr inference in the tails. It explains the finite sample distributin f β(τ) in the tails cnsiderably better than the nrmal distributin des. Hwever, a majr bstacle t this rute and t adaptatin f this apprximatin in practice is the feasibility: we need t have accurate estimates f the tail index parameter ξ and f the nrmalizatin factr 1/F 1 U (1/n). The scaling 1/F 1 U (1/T ) is a reciprcal f an extreme ppulatin quantile which is hard r infeasible t estimate cnsistently in the heavytailed cases. Under a minimal assumptin that disturbances belng t the dmain f minimum attractin, we have that 1/F 1 U (1/T ) = L(1/T ) T ξ (1 + (1)) where L( ) is a nnparametric, slwly varying functin, which is t weak a restrictin t be useful (Bertail, Haefke, Plitis, and White 2004). Estimatin f L(1/T ) becmes feasible when parametric assumptins n L( ) are made; and sme will be given later. Estimatin f tail parameters such as the extreme value index ξ is anther knwn difficult prblem, as the cmmn estimatrs behave well nly when the slwlyvarying cmpnent L(1/T ) is nt present ((Embrechts, Klüppelberg, and Miksch 1997)). As neither the nrmalizatin 1/F 1 U (1/T ) nr the tail parameters ζ are f primary interest, it is reasnable t attempt t bypass the estimatin f these quantities. In rder t d s, ur inference will be based n the SelfNrmalized QR statistic (SNQR) A T ( β(τ) β(τ) ) d Z (k), fr A T := τt, (2.3) X ( β(mτ) β(τ)) where Z (k) is anther extreme type variate, t be specified later. The randm scaling A T is functin f data nly, and it is nt an estimate f A T. Using this result we will prvide tw inference appraches: an analytical apprach and a resampling apprach based n mdified subsampling. The analytical apprach will directly rely n the (simulated) limit distributin, but will require estimatin f the tail parameter ξ. Hwever, ur resampling apprach eliminates the need t estimate ξ. Our subsampling apprach is different frm cnventinal subsampling in its use f recentering terms and f randm nrmalizatin. The cnventinal subsampling that uses recentering by the full sample estimate β(τ) is nt cnsistent when that estimate is diverging; and here we indeed have A T 0 when ξ > 0. Instead, we will emply recentering by intermediate rder QR estimates, which will be cnsistent enugh t estimate the limit distributin f SNQR cnsistently. Thus, the mdified subsampling apprach explres the special relatinship between the rates f cnvergence/divergence f extremal and intermediate QR statistics, and shuld be f independent interest even in a nnregressin setting.
7 6 VICTOR CHERNOZHUKOV 3. The Set Up and the Mdel 3.1. Basic Set Up. Suppse Y is the respnse variable in R, and W is a vectr f the cnditining variables. Dente the cnditinal distributin f Y given W = w, F Y ( w). The τth cnditinal quantile functin F 1 Y (τ w) is the inverse f F Y ( w) evaluated at prbability index τ. We use the fllwing flexible functinal frm fr the extremal cnditinal quantile functin f Y given W : F 1 Y (τ X) := X β(τ), ( τ I = (0, η), η > 0), (3.4) where X = B(W ) is a d 1 vectr f apprximating functins that may include pwer functins, splines, and ther transfrmatins f riginal variables W. As nted in Hedricks and Kenker (1992) and Newey (1997), ecnmic thery rarely prescribes exact functinal frms mtivating the use f flexible functinal frms t prvide parsimnius apprximatins f unknwn ecnmic structures and the cmputatinal cnvenience. Given T bservatins {Y t, X t, t = 1,..., T }, the quantile regressin (QR) slves: β(τ) arg min β R d T ( ρ τ Yt X tβ ), (3.5) t=1 where ρ τ (u) = (τ 1(u < 0))u. The median τ = 1/2 case f (3.5) was intrduced by Laplace (1818) and the general quantile frmulatin (3.5) by Kenker and Bassett (1978). Viewing the sample regressin quantiles as rder statistics in the regressin setting, we will refer t τt as the rder r rank f QR. The sequence f quantile indexsample size pairs (τ T, T ) will be said t be an extreme rder sequence, if τ T 0 and τ T T k > 0, an intermediate rder sequence, if τ T 0 and τ T T, and a central rder sequence, if τ is fixed and T. Our inference analysis is based n extremerder sequences that prvide nnnrmal apprximatins t the distributin f QR statistics The Extremal Quantile Regressin Mdel. In develping ur inference methds, we assume that the respnse variable Y has Parettype tails. The Parettype tails are regularly varying tails, and are prevalent in ecnmic data, as discvered by Vilfred Paret in The Parettype tails encmpass r apprximate a rich variety f tail behavir, including that f thicktailed and thintailed distributins, having either bunded r unbunded supprt. 2 Paret called the tails f this frm Distributin Curve fr Wealth and Incmes. Further empirical substantiatin has been given by Sen (1973), Zipf (1949), Mandelbrt (1963), and Fama (1965), amng thers. The mathematical thery f regular variatin in cnnectin t extreme value thery has been develped by Gnedenk and de Haan.
8 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 7 Cnsider a randm variable u with the distributin functin F u and the lwer endpint s = r s = 0. F u is said t be regularly varying at s, if F u (lv) lim l s F u (l) = v 1/ξ, ( v > 0), ( ξ > 0 r ξ < 0). (3.6) Regular variatin equivalently means that the distributin functin F u and its quantile functin F 1 u exhibit apprximately pwerlaw behavir in the tails: 3 F u (t) L(l) l 1/ξ, as l s, (3.7) F 1 u (τ) L(τ) τ ξ, as τ 0, (3.8) where L(l) is a nnparametric, slwlyvarying functin at s and L(τ) is a nnparametric slwly varying functin at 0. A functin l L(l) is slwly varying at s if lim l s [L(l)/L(ml)] = 1 fr any m > 0, fr example, L(l) = L, L(l) = L lg( l), etc. The number ξ is called the extreme value (EV) index and its abslute value ξ measures heavytailedness f distributins. A distributin F u with regularly varying tails necessarily has a finite lwer supprt pint if ξ < 0 and infinite lwer supprt pint if ξ > 0. Distributins with ξ > 0 include stable distributins, Paret distributins, t distributins, and thers. Fr instance, the tdistributin with ν degrees f freedm has the EV index ξ = 1/ν, where ξ = 1 yields the Cauchy distributin that has heavy tails, and ξ < 1/30 clsely apprximates the nrmal distributin that has light tails. On the ther hand, distributins with ξ < 0 include the unifrm, expnential, and Weibull distributins, amng thers. Our main assumptin is that the respnse variable Y, transfrmed by sme auxiliary regressin line, has regularly varying tails at s = and 0 with EV index ξ. C1. In additin t (3.4), suppse there exists an auxiliary extremal regressin parameter β e such that U Y X β e has endpint s = 0 r s = a.s. and satisfies as τ 0 F 1 U (τ x) x c Fu 1 (τ), unifrmly in x X, where Fu 1 (τ) RV ξ (0). (3.9) Since this assumptin nly affects the tails, it allws cvariates t affect the extremal quantile and the central quantiles very differently. Mrever, the lcal effect f cvariates in the tail is apprximately given by β e + cfu 1 (τ), which allws fr a differential impact f cvariates acrss varius extremal quantiles. C 2. In additin t C1, F 1 U τ 0, unifrmly in x X, where F 1 (τ x)/ τ exists and F 1 u (τ)/ τ RV ξ 1 (0). U (τ x)/ τ x c Fu 1 (τ)/ τ as This assumptin strengthens C1 by impsing regular variatin n the quantile density functin, which is imprtant fr btaining the main inferential results. 3 The ntatin a b means that a/b 1 as apprpriate limits are taken.
9 8 VICTOR CHERNOZHUKOV 3.3. Sampling Cnditins. The fllwing sampling cnditins will be impsed. C3. The regressr X is f the frm (1, Z ), has cmpact supprt X, mean µ X = (1, 0,...) and EXX > 0, and satisfies the nnlattice cnditin stated in the mathematical appendix (satisfied, fr instance, if Z is abslutely cntinuus). Cmpactness is a necessary technical assumptin needed t insure the cntinuity and rbustness f the mapping frm extreme events in Y t the extremal QR statistics. Even if X is nt cmpact, we can select nly that data fr which X belngs t a cmpact regin. The nndegeneracy cnditin is a standard cnditin that requires invertibility f EXX. The nnlattice cnditin is required fr existence f finitesample density f QR ficients and is needed here even asympttically, since the asympttic distributin thery clsely resembles the finitesample thery, which is nt a surprise given the rare nature f events that have the prbability f rder 1/T. Data is als assumed t satisfy the cannical iid sampling. C4. The sequence f variables {(Y t, X t ), t 1} are independent and identically distributed. Mre generally, data can be a time series with the fllwing restrictin. C5. The sequence f vectrs {(Y t, X t ), t 1} frms a statinary, αmixing prcess with gemetric mixing rate and the extreme events satisfying a nnclustering cnditin: P t (U t K, U t+j K) CP t (U t K) 2 fr all sufficiently small K s, all j 1, all t 1, and sme cnstant C > 0, where P t dentes the prbability cnditinal n the infrmatin set up t time t (that is, the σalgebra generated by {Y t, X t, t = t 1, t 2,...}). The assumptin f mixing is a standard assumptin made in ecnmetric analysis, e.g. White. The nnclustering cnditin f the Meyer (1973)type states that the pssibility f tw extreme events cccuring at fixed dates is much lwer than prbability f just ne extreme event. Fr example, it assumes that a large market crash is nt likely t be immediately fllwed by anther large crash. This assumptin leads t the limit distributins f QRs as if independent sampling had taken place. The plausibility f the nnclustering assumptin is an empirical matter; we cnjecture that ur primary inference methd based n subsampling is rbust t a vilatin f this assumptin. 4. Asympttic Extreme Value Apprximatins: Feasible and Infeasible This sectin establishes imprtant preliminary results that will underlie ur inferential prcedures.
10 INFERENCE FOR EXTREMAL QUANTILE REGRESSION Weak Limits f Extreme Order QR and SelfNrmalized QR statistics. The asympttic distributins are given under the extreme rder sequence τt k > 0 as T 0. Define the selfnrmalized QR (SNQR) statistic as: ) Z T (k) := A T ( β(τ) β(τ) fr A T := τt, (4.10) X ( β(mτ) β(τ)) where τt (m 1) > d, and the cannicallynrmalized QR statistic (CNQR) as: Ẑ T (k) := A T ( β(τ) β(τ) ) fr A T := 1/F 1 u ( ) 1. (4.11) T The first statistic uses a randm nrmalizatin, while the secnd uses an infeasible cannical nrmalizatin. Therem 1 (Weak Limits). Suppse cnditins C1C5 hld. Define Ẑ (k) arg min z R d [ kµ X(z k ξ c) + i=1 [Γ ξ i X i c + X i (z k ξ c)] + ], (4.12) where {Γ 1, Γ 2,...} {E 1, E 1 + E 2,...}, {E 1, E 2,...} is an iid sequence f expnential variables, which is independent f {X 1, X 2,...}, an iid sequence with distributin F X. Then, 1. The SNQR statistics f rder k behave as fllws, fr any m such that k(m 1) > d: as τt k > 0 and T, we have that Z T (k) d Z (k), where Z (k) := k Ẑ (k) µ X (Ẑ (mk) Ẑ (k)). (4.13) + (m ξ 1)k ξ 2. The CNQR statistics f rder k behave as fllws: as τt k > 0 and T, Ẑ T (k) d Ẑ (k). Cmment 4.1. The cnditin that k(m 1) > d insures that A T is well defined, thanks t Therem 3.2 in Bassett and Kenker (1982). Result 1 n SNQR statistics is the main new result that we will need fr inference. Result 2 n CNQR statistics is needed primarily fr auxiliary purpses. Chernzhukv (2000b) presents sme extensins f Result 2. Previusly, Feigin and Resnick (1994), Prtny and Jure ckvá (1999), and Knight (1999) investigated cannically nrmalized statistics under the limit case τt 0 + (knwn as the linear prgramming estimatr), under varius degrees f generality. The nearextreme case τ T k > 0 cnsidered here is mtivated primarily by ecnmic applicatins listed in Sectin 2.
11 10 VICTOR CHERNOZHUKOV 4.2. Discussin. Few basic bservatins are in rder. Recall that in the classical nnregressin case, when X = 1, and U t are sampled iid accrding t F u, we have that A T ( β( T k ) β( T k )), k = 1,..., J, cnverge in distributin t Γ ξ k k ξ, k = 1,..., J, where β( T k ) := U (k) is the kth rder statistic and the k/t th sample quantile. These basic variables als appear in the limiting expressins fr SNQR and CNQR statistics with a prefactr X i c that makes them crrelated with regressrs X i. The limit distributins are nt nrmal; they have n finite mments f any rder exceeding 1/ξ when ξ > 0; they are nt centered at zer, and significant first rder asympttic median biases may exist. It is therefre desirable t cnstruct statistics that are asympttically medianunbiased. In cntrast t limit thery fr SNQR statistics, Result 1, the limit thery f CNQR statistics, Result 2, is generally infeasible fr purpses f inference, since it requires knwledge r estimability f the scaling cnstant A T = 1/Fu 1 ( 1 T ), the reciprcal f the extremal quantile f variable U defined in C1. That is, ne requires ÂT such that ÂT /A T p 1, which is nt generally feasible unless further strng parametric restrictins n the tail are made. Even thugh these restrictins facilitate estimatin f A T and hence inference n CNQR, as described later, this will nt be ur primary inferential methd. The last pint is imprtant. By assumptin, A T = 1/F 1 u ( 1 T ) = L( 1 T )T ξ, where L( 1 T ) is a nnparametric slwlyvarying functin. The class f slwly varying functins is quite brad: by Karamata s Representatin L(τ) = l(τ) exp( 1/τ 1 t 1 ε(t)dt), where lim t 0 l(t) = l (0, 1), lim t 0 ε(t) = 0, and l(t) and ε(t) are measurable, nnnegative functins, cf. Resnick (1987). Nnparametric estimatin f L(τ) at τ = 1/T under these restrictins and using the data sample f size T appears t be an infeasible prblem (e.g. Bertail, Haefke, Plitis, and White (2004)). Our main prpsal fr inference is based n SNQR and Result 1, and it entirely avids estimating A T. We will estimate the distributin f SNQR statistics A T ( β(τ) β(τ)) instead, using either a subsampling methd r an analytical methd. An imprtant ingredient here is the feasible nrmalizing variable A T that is randmly prprtinal t the cannical nrmalizatin A T, in the sense that A T /A T is a randm variable in the limit. 4 An advantage f the subsampling methd ver analytical methds is that des nt require estimatin f nuisance parameters ξ and c. 4 The idea f feasible randm nrmalizatin has been used in many ther cntexts (e.g. tstatistics). In extreme value thery, Dekkers and de Haan (1989) applied a similar randm nrmalizatin idea t the extraplated quantile estimatrs in the nnregressin setting, precisely t prduce limit distributins that can be easily used fr inference.
12 INFERENCE FOR EXTREMAL QUANTILE REGRESSION Generic Inference and MedianUnbiased Estimatin. Next we utline prcedures fr cnducting inference and cnstructing asympttically median unbiased estimates f linear functins ψ β(τ) f the ficient vectr β(τ) fr extremal values f τ. Inference Using SNQR. By Therem 1 fr any nnzer ψ, ψ A T ( β(τ) β(τ)) d ψ Z (k). Given a cnsistent estimate ĉ α f the αquantile c α f ψ Z (k), the asympttically medianunbiased estimatr and the α%cnfidence interval can be cnstructed as ψ β(τ) c1/2 /A T and [ψ β(τ) cα/2 /A T, ψ β(τ) c1 α/2 /A T ], where the biascrrectin term c 1/2 /A T and cmpnents c α/2 /A T and c 1 α/2 /A T f the cnfidence interval depend n the randm scaling A T. Cnsistent estimates ĉ α are prvided in the next sectin. Therem 2 (Inference and medianunbiased estimatin using SNQR). Under cnditins f Therem 1, suppse we have ĉ α such that ĉ α p c α, where c α is the αquantile f ψ Z (k). Then, lim T P {ψ β(τ) ĉ1/2 /A T ψ β(τ)} = 1/2 and lim T P {ψ β(τ) ĉα/2 /A T ψ β(τ) ψ β(τ) ĉ1 α/2 /A T } = α. Inference Using CNQR. Under the cnditins stated abve, fr any nnzer ψ, ψ A T ( β(τ) β(τ)) d ψ Ẑ (k). Given a cnsistent estimate ÂT and a cnsistent estimate ĉ α f the αquantile c α f ψ Ẑ (k), we culd cnstruct the asympttically medianunbiased estimatr and α%cnfidence intervals as, respectively: ψ β(τ) + ĉ 1/2 /ÂT and [ψ β(τ) + ĉ α/2 /ÂT, ψ β(τ) + ĉ 1 α/2 /ÂT ]. Cnstructin f cnsistent estimates f A T requires additinal strng, nearly parametric restrictins n the underlying mdel. Fr example, suppse that the nnparametric slwly varying cmpnent L(τ) f A T is replaced by a cnstant L, i.e. 1/F 1 u (τ) = L τ ξ (1 + δ(τ)), δ(τ) = (1), L R, as τ 0. (4.14) Then, ne can estimate the cnstants L and ξ via Pickandstype prcedures: ξ = 1 ln 2 ln X ( β(4τ T ) β(τ T )) X ( β(2τ T ) β(τ T )) and L = X ( β(2τ T ) β(τ T )), (4.15) (2 ξ 1) T ξ where τ T is chsen t f an intermediate rder, τ T T and τ T 0. By Therem 4 in Chernzhukv (2000b), these estimates are cnsistent under C1C5, and under further
13 12 VICTOR CHERNOZHUKOV cnditins n the sequence (δ(τ T ), τ T ), 5 ne has that ξ = ξ + (1/ ln T ), which prduces a cnsistent estimate ÂT = L(1/T ) ξ such that ÂT /A T p 1. Cnsistent estimates f c α are prvided in the next sectin. Therem 3 (Inference and MedianUnbiased Estimatin using CNQR). Assume cnditins f Therem 1 hld. Suppse that we have ÂT such that ÂT /A T p 1 and ĉ α such that ĉ α p c α, where c α is the αquantile f ψ Ẑ (k). Then, lim T P {ψ β(τ) ĉ 1/2 /ÂT ψ β(τ)} = 1/2 and lim T P {ψ β(τ) ĉ α/2 /ÂT ψ β(τ) ψ β(τ) ĉ1 α/2 /ÂT } = α. 5. Estimatin f Critical Values 5.1. SubsamplingBased Estimatin f Critical Values. The methd develped belw uses subsamples t estimate the sampling distributin f SNQR, just as in standard subsampling. Hwever, by using SNQR, the methd bypasses estimatin f the unknwn cnvergence rate A T, required in standard subsampling. The methd als emplys recentering terms that avid the incnsistency f the standard subsampling caused by divergence f the QR estimatr when ξ > 0. The methd is as fllws. First, cnsider all subsets f data {W t = (Y t, X t ), t = 1,..., T } f size b. If {W t } is a time series, cnsider B T = T b + 1 subsets f size b f the frm {W i,..., W i+b 1 }. Then cmpute the analgs f SNQR statistics in subsamples, dented V i,b, and defined belw in equatin (5.17), fr each ith subsample, fr i = 1,..., B T. Secnd, estimate c α by ĉ α defined as the sample αquantile f { V i,b,t, i = 1,..., B T }. In practice, a smaller number B T f randmly chsen subsets can be used, prvided B T as T, see Sec. 2.5 in Plitis et. al. (1999). Subsample size b can be chsen accrding t Plitis et. al. (1999) and Bertail et. al. (2004). Recall that the SNQR statistic cmputed in the full sample f size T is: V T := A T ψ ( β T (τ T ) β(τ T )), A T := τ T T / X ( β(mτt ) β(τ T ) ), (5.16) where m will be set equal In this sectin we write τ T t emphasize the theretical dependence f the quantile f interest τ n the sample size. In each ith subsample f size b, we cmpute the fllwing analg f the abve statistic: V i,b,t := A i,b,t ψ ( β i,b,t (τ b ) β(τ b )), A i,b,t := τ b b / X ( βi,b,t (mτ b ) β i,b,t (τ b ) ), (5.17) where β(τ T ) is the τ T regressin quantile ficient cmputed using the full sample; β(τ b ) is the τ b regressin quantile ficient cmputed using the full sample; βi,b,t (τ b ) is the 5 The rate cnvergence f ξ is max[ 1, ln δ(τt )], which gives the fllwing cnditin n the sequence τt T (δ(τ T ), τ T ) : max[ 1 τt T, ln δ(τt )] = (1/ ln T ). 6 Such chice, namely m = 2, is ften used in the nnregressin cases, cf. Dekkers and de Haan (1989).
14 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 13 τ b regressin quantile ficient cmputed using ith subsample; βi,b,t (mτ b ) is the mτ b  regressin quantile ficient cmputed using ith subsample; and τ b := k T /b := (τ T T ) /b. 7 Nte that determinatin f τ b is a critical decisin that sets apart the central rder apprximatin frm the extreme rder apprximatin. In the frmer case case ne sets τ b = τ T in subsamples. Under additinal parametric assumptins n the tail behavir stated earlier, ne can als estimate the quantiles f the limit dsitributin f CNQR. The resampling prcedure can be dne in tw steps: First, create subsamples i = 1,..., B T as befre, cmpute in each subsample: V i,b,t := Âbψ ( β i,b,t (τ b ) β(τ b )), where Âb is any cnsistent estimate f A b. Estimate c α by ĉ α defined as the αquantile f {V i,b,t, i = 1,..., B T }. Fr example, under the parametric restrictins specified in (4.14), set Âb = Lb ξ fr L and ξ specified in (4.15). The fllwing therems establish cnsistency f ĉ α and ĉ α. Therem 4 (Critical Values fr SNQR by Resampling). Given the assumptins f Therems 1 and 2 and that b/n 0, b, n, B n, we have that ĉ α p c α. Therem 5 (Critical Values fr CNQR by Resampling). Suppse that the assumptins f Therems 1 and 2 hld and b/n 0, b, n, B n, and the estimate Âb is such that Âb/A b 1. Then ĉ α p c α. Cmment 5.1. The methd prduces cnsistent inference in the QR case and might be f independent interest in the nnregressin case. The methd differs frm cnventinal subsampling in several ways: First, ur methd als uses randm nrmalizatin term A T. Secnd, QR statistics are diverging when ξ > 0. In this case, the standard subsampling that emplys the recentering by the full sample estimate β(τ T ) des nt apply. In rder t use this recentering ne requires A b /A T 0 (Plitis and Rman 1994), but instead A b /A T. Our apprach instead uses β(τ b ) fr recentering. This statistic itself may be diverging, but the speed f its divergence is strictly slwer than A T, because it is an intermediate rder QR statistic. Hence this methd f recentering explits the special structure f rder statistics Analytical Estimatin f Critical Values. Analytical inference uses the quantiles f the limit distributins fund in Therem 1, which can nly be dne numerically via simulatin prcedures. This apprach is much mre demanding than the previus subsampling 7 In practice, it is reasnable t use the fllwing finitesample adjustment t τb : τ b = min[(τ T T ) /b,.2] if τ T <.2, and τ b = τ T if τ T.2. The idea is that if τ T >.2, it is judged t be nnextremal, and the subsampling prcedure resrts t the central rder inference. If τ T <.2, then τ b = min[(τ T T ) /b,.2]. The truncatin f τ b by.2 is a finitesample adjustment that restricts the key statistics V i,b,t t be extremal in subsamples. Adjustments f this kind d nt affect the asympttic arguments.
15 14 VICTOR CHERNOZHUKOV methd. The methd develped belw is als f independent interest in situatins where the limit distributins invlve Pisssn prcesses with unknwn nuisance parameters, as, fr example, in Chernzhukv and Hng (2004). Define the fllwing randm vectr Ẑ (k) = arg min z R d[ k µ X(z k ξc) + i=1 (Γ ξ i X i ĉ + X i (z k ξc)) + ] (5.18) fr sme estimates ξ and ĉ, fr example thse stated belw, µ X = X, {Γ 1, Γ 2,...} = {E 1, E 1 + E 2,...}, where {E 1, E 2,...} is an i.i.d. sequence f unitexpnential variables, and {X 1, X 2,...} is an iid sequence with distributin functin F X, where F X is a smthed cnsistent estimate f F X. Als, let Z (k) = kẑ (k)/[ X (Ẑ (mk) Ẑ (k)) + m ξ 1 k ξ]. Estimates ĉ α and ĉ α are btained by taking αquantiles f variables ψ Ẑ (k) and ψ Z (k). In practice, these quantiles can nly be evaluated numerically as described belw. The analytical inference prcedure requires cnsistent estimates f ξ and c. The cnsistent estimates based n Pickands type prcedures are prvided in Chernzhukv ( (2000b), Therem 5) under cnditins implied by Assumptin 1: ξ = 1 ln 2 ln X ( β(4τ T ) β(τ T )) X ( β(2τ T ) β(τ T )) and ĉ = β(2τ T ) β(τ T ) X ( β(2τ T ) β(τ T )), (5.19) where τ T T and τ T 0. Als, the empirical distributin functin F X is a unifrmly cnsistent estimate f F X via the GlivenkCantelli Therem. Therem 6 (Critical Values fr CNQR by Analytical Methd). Assume cnditins f Therem 1 hld. Then, fr any cnsistent estimates f nuisance parameters such that ξ p ξ, ĉ p c, such as the nes prvided abve, ĉ α = c α + p (1). Therem 7 (Critical Values fr SNQR by Analytical Methd). Assume cnditins f Therem 1. Then fr any cnsistent estimates f nuisance parameters such that ξ p ξ, ĉ p c, such as the nes prvided abve, ĉ α = c α + p (1). Cmment 5.2. Since the distributins f the variables Ẑ (k) and Z (k) are nt in clsed frm, except in very special cases, we can nly btain ĉ α and ĉ α numerically via a Mnte Carl prcedure. Fr instance, fr each i = 1,..., B the prcedure Ẑ i, (k) using the frmula ( ) abve except truncating the sum at sme value L. We find that chsing L 200 and setting B 100 prvides numerically accurate estimates (see Chernzhukv (2000a)). 6. Extreme Value vs. Nrmal Inference: Cmparisns 6.1. Prperties f Cnfidence Intervals with Unknwn Nuisance Parameters. In this sectin we asses the cvering perfrmance f nrmal and extremal CI using the mdel
16 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 15 Y t = X tβ + U t, t = 1,..., 500, β = 1, where disturbances {U t } are i.i.d. and fllw either (1) a tdistributin with ν {1, 3, 30} degrees f freedm, r, (2) a Weibull distributin with the shape parameter α {.5, 1, 4}. These distributins have the extreme value index ξ = 1/ν {1, 1/3, 1/20} and ξ = 1/α { 2, 1, 1/4}, respectively. Regressrs are drawn with replacement frm the empirical applicatin in Sectin 7.1 in rder t match a real situatin as clsely as pssible. 8 The design f the first type crrespnds t tail prperties f financial data, including returns and trade vlumes, and the design f the secnd type crrespnds t tail prperties f micrecnmic data, including birthweights, wages, and bids. Figures 2 and 3 plt cverage prperties f CI based n ur primary extremal methd and n the nrmal inference methd suggested by Pwell (1986). The figures are based n QR estimates fr τ {0.005,.02,.05,.1}, i.e. τt {2.5, 10, 25, 50}. When disturbances U are tdistributed, extremal CI have gd cverage prperties, whereas nrmal CI typically undercver with the perfrmance that deterirates in the degree f heavytailedness and imprves in the index τt. In heavytailed cases, ξ {1, 1/3}, nrmal CI substantially undercver, as might be expected frm the nrmal distributin failing t capture heavy tails f the actual distributin f QR statistics. In thintailed cases, ξ = 1/30, nrmal CI have an imprved cverage f 70 80%. In all cases extremal CI d better cnsistently, giving cverage in a gd neighbrhd f 90%. When disturbances U are Weibulldistributed, extremal CI als have gd cverage prperties, whereas nrmal CI typically vercver with the perfrmance that deterirates in the degree f heavytailedness and imprves in the index τ T. In heavytailed cases, ξ { 2, 1}, nrmal CI strngly vercver, which results frm the verdispersin f the nrmal distributin relative t the actual distributin f QR statistics. In thintailed cases, α = 4, nrmal CI cver at the right rate, even thugh frmally they are nt valid in the tails. In all cases, extremal CI d better r as well as nrmal CI, giving a cverage in a gd neigbrhd f 90%. We have als cmpared frecasting prperties f rdinary QR estimatrs and medianbiascrrected QR estimatrs f the true intercept and slpe ficients, using the median abslute deviatin and median bias as measures f the perfrmance (ther measures may be infinite). We have fund that gains t biascrrecting appear t be very small, except in the finitesupprt case with disturbances heavytailed near the bundary. Tabulatins f these results are therefre mitted A RulefThumb fr Applicatin f Extremal Inference. Equipped with bth simulatin experiments and practical experiences, we wuld like t find a simple rule fr the applicatin f the extremal inference. T this end, nte that the rder τt f the QR 8 These data can be dwnladed at
17 16 VICTOR CHERNOZHUKOV statistic β(τ) shuld bviusly play the crucial rule, as ne needs τt t have nrmal laws apply. The number f regressrs may als play a crucial rle. Cnsider the case where all d regressrs, ther than the intercept, are indicatrs that equally divide the sample f size T int subsamples f size T/d. Then the rder f QR in each f these subsamples will be τt/d, and τt/d is then the effective rder. Next, ne requires a sample f nrmal length, say 50, t be belw the fitted quantiles fr the nrmal laws t start t apply. This gives the simple rule τt/d 50 fr the use f the extremal inference. The rule may r may nt be cnservative. Fr example, when regressrs are cntinuus, ur cmputatinal experiments indicate that the nrmal inference catches up with the extremal inference when τt/d On the ther hand, imagine having an indicatr variable that picks ut a 1/50th fractin f the entire sample, as in the birthweight applicatin, then the number f bservatins belw the fitted quantile fr this subsample will be τt/50, mtivating a far mre cnservative rule τt/50 50 fr this case. Overall, it seems prudent t use bth extremal and nrmal inference at the same time in mst cases, with the idea that the discrepancies between the tw can indicate extreme situatins. 7. Empirical Illustratins 7.1. Extremal Risk f a Stck. We cnsider the prblem f finding factrs that affect valueatrisk f the Occidental Petrleum daily stck return, a prblem interesting fr ecnmic analysis and realwrld activities f financial firms. 9 Our dataset cnsists f 1000 daily bservatins n the neday return Y n the Occidental Petrleum stck, cvering a perid in , and regressrs, cnsisting f the lagged return n the spt price f il, dented X 1, the lagged neday return f Dw Jnes Industrials index, X 2, and the lagged wn return, X 3. We begin by stating verall estimatin results fr the basic predictive linear mdel. A detailed specificatin and gdnessffit analysis f this mdel had been given in Chernzhukv and Umantsev (2001), and here we fcus n the extremal analysis in rder t illustrate new inferential tls. Figure 4 plts QR estimates β(τ) = ( β j (τ), j = 0,..., 3), and the shaded area represents pintwise 90% cnfidence intervals (CI). Nrmal CI are used fr central quantiles,.15 < τ <.85, and extremal CI are used fr extremal quantiles, τ.15 and τ.85. The ficient n the spt price f il is psitive and increasing in the right tail f the distributin. The ficient is large but nly marginally significant at the cnventinal 90% level in the far right tail. Hwever, extremal CI indicate that the distributin f the 9 See Chernzhukv and Umantsev (2001), Christffersen, Hahn, and Inue (1999), Engle and Manganelli (1999), Diebld, Schuermann, and Strughair (2000).
18 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 17 QR statistic is asymmetric, with verwhelming part f this distributin psitive. Therefre, the ecnmic effect f this determinant is ptentially quite strng in the upper tail. The ficient n the DJ index is significantly psitive fr the entire middle part f the distributin. The ficient n lagged wn return Y t 1 is significantly negative fr nearly the entire distributin except fr the tails, which suggests a reversin effect in the center f the distributin and n reversin effects in the tails. The lag is thus mre imprtant fr the determinatin f intermediate risks. We wuld als like t cmpare the extremal inference with the nrmal inference. This empirical example is clsely matched by the Mnte Carl design with heavytailed t(3) disturbances and regressrs having the same distributin by cnstructin. Hence we might expect that nrmal CI wuld understate the estimatin uncertainty f QR and be cnsiderably mre narrw than extremal CI in the tails. As shwn in Figures 5 and 6, nrmal CI are indeed much mre narrw than extremal CI at τ < Extremal Birthweights. In this sectin, we investigate the impact f varius demgraphic characteristics and maternal behavir n extremely lw quantiles f birthweights f live infants brn t black mthers f ages between 18 and 45, in the United States, using the June 1997 Detailed Natality Data published by the Natinal Center fr Health Statistics. Previus studies f these data by Abreveya (2001) and Kenker and Hallck (2001) fcused the analysis n nrmal birthweights, in a range between 2000 and 4500 grams. In cntrast, equipped with extremal inference, we nw can and we will venture far in the tails and study extremely lw birthweight quantiles, in the range between 250 and 1500 grams. Sme f ur findings will sharply differ frm the previus results fr nrmal quantiles. We fcus ur analysis n black mthers nly, mtivated by Figure 7 that shws a trubling heavy tail f lw birthweigts t black mthers. In ur analysis we adpt the regressin specificatin similar t that in Kenker and Hallck (2001), which has been tested fr misspecificatin and rigrusly validated fr the nrmal quantiles by He et. al.. The specificatin we estimate is linear in parameters, with the respnse variable being the birthweight recrded in grams, and the fllwing regressrs: By and Married are indicatrs f infant gender and f whether the mther was married r nt. N Prenatal, Prenatal Secnd, and Prenatal Third are indicatr variables that divide the sample int 4 categries: thse with n prenatal visit (less than 1% f the sample), thse whse first prenatal visit was in the secnd trimester, and thse whse first prenatal visit was in the third trimester. The cntrl categry is mthers with a first visit in the first trimester, which cnstitute 83% f the sample. Smker and Cigarettes/Day are an indicatr f whether the mther smked during pregnancy and the mther s reprted average number f cigarettes smked per day. Weight Gain and Weight Gain 2 are the mther s reprted weight gain during pregnancy
19 18 VICTOR CHERNOZHUKOV (in punds) minus the mean mther s weight gain and the mther s weight gain squared minus the square f the mean f the mther s weight gain. Educatin is a categrical variable taking a value f 0, if the mther had less than a highschl educatin, 1, if cmpleted high schl educatin, 2, if btained sme cllege educatin, and 3, if graduated frm cllege. Age and Age 2 are the mther s age minus the mean f the mther s age and the mther s age squared minus the square f the mean f the mther s age. Thus the cntrl grup cnsists f mthers f average age that had their first prenatal visit during the first trimester, that had attained educatin level f less than high schl, that did nt smke, and that had average weight gain during pregnancy. The intercept in the estimated quantile regressin mdel will measure quantiles in this grup, and will therefre be called the centercept. Figure 8 reprts estimatin results fr extremal quantiles, and Figure 9 fr the nrmal quantiles. These figures shw pint estimates, extremal CI, and nrmal CI, at the cnfidence level f 90%. Nte that the centercept in Figure 8 varies frm 250 t abut 1500 grams, indicating the apprximate range f birthweights that ur extremal analysis applies t. In what fllws, we fcus the discussin nly n key cvariates and n differences between extremal and central inference. Estimatin results fr sme cvariates, e.g. squared age, are nt pltted t save space. While density f birthweights, shwn in Figure 7, has a finite lwer supprt pint, it has little prbability mass near that bundary. This puts us in the situatin similar t the MnteCarl design with Weibull(4) disturbances, fr which differences between central and extremal inference were nt large. This is what we als bserve in this empirical example. Fr the mst part, nrmal CI tend t be nly 0 10% mre narrw than extremal CI, with the exceptin f the ficient n N Prenatal, fr which nrmal CI are twice as narrw as extremal CI. Since nly 1.9% f the mthers had n prenatal care, the sample size used t estimate this ficient is nly 635. The effective rder f τquantile fr this subsample is τ 635, which suggests that there shuld be little r n discrepancy between extremal CI and central CI fr the ficient n N prenatal nce τ 3 5%. As Figure 9 shws, there is indeed little difference between extremal CI and nrmal CI nce τ 3%. The analysis f extremal birthweights, shwn in Figure 8, reveals several surprises that we d nt see in the results n nrmal birthweights, shwn in Figure 9. Mst surprisingly, smking appears t have n negative impact n extremal quantiles, in cntrast t strngly negative impact at the nrmal quantiles. The absence f the effect culd be due t pssible selectin, where nly mthers cnfident f gd utcmes smke, r it culd be indeed due smking having n r little causal effect n very extreme utcmes. This finding mtivates further analysis, pssibly using data sets that enable instrumentatin strategies.
20 INFERENCE FOR EXTREMAL QUANTILE REGRESSION 19 The prenatal medical care has a strng impact n extremal quantiles and relatively little impact n nrmal quantiles, especially in the middle f the distributin. The effect f Nprenatal is nnpsitive and may be strngly negative in the tails, anywhere between values f 0 and 800 cntained in the extremal CI. The impact f Prenatal Secnd and Prenatal Third is psitive and very strng in the tails in cntrast t the relatively little impact fund in the middle f the distributin. This culd be due t mthers cnfident f gd utcmes chsing t have a late first prenatal visit. Alternatively, it culd be that substantial examinatins, that accmpany the first visit, during the secnd and especially third trimesters help intervene t reduce the prbability f extreme utcmes. Finally, the mther s weight gain seems quite imprtant fr imprving extreme birthweight quantiles, much mre s than fr nrmal quantiles. 8. Cnclusin The empirical sectin reprted several findings, which are interesting in their wn right. The new tls give us the ability t maintain that the findings hld with a given statistical cnfidence within the pstulated mdel.
21 20 VICTOR CHERNOZHUKOV Appendix A. Prf f Therem 1 The prf will be given fr the case when ξ < 0. The case with ξ > 0 fllws very similarly. The prf will emply the Cnvexity Lemma, cf. Geyer (1996) and Knight (1999). 10 Step 1. Cnsider the pint prcess N defined by N(A) := T t=1 1{(A T U t, X t ) A} fr Brel subsets A f E := [0, + ) R d. The pint prcess N cnverges weakly in the metric space f pint measure M p (E), that is equipped with the metric induced by the tplgy f vague cnvergence. The limit prcess is a Pissn pint prcess. This cnvergence fllws by nting that fr any set F defined as intersectin a bunded rectangle with E, ne has that (a) lim T E N(F ) = m(f ) := F (x c) 1/ξ u 1/ξ dudf X (x), where ne uses regular variatin f F u and that (b) lim T P [ N(F ) = 0] = e m(f ), where the latter fllws by Meyer s (1973) therem after bserving that we have T T/k j=2 P ((A T U 1, X 1 ) F, (A T U j, X j ) F ) O(T T/k P ((A T U 1, X 1 ) F ) 2 ) = O(1/k) by C5 and have gemetric strng mixing by C5. Cnsequently, (a) and (b) imply by Kallenberg s therem that the weak limit is a Pissn pint prcess N with intensity measure m. The pints f N can be represented as in the statement f Therem 1: (J i = (X i c) Γ ξ i, X i ), fr i = 1, 2,... Step 2. Observe that Z T T (k) := A T ( β(τ) β r ) = arg min z R d t=1 ρ ( τ AT U t X tz ) [write z := A T (β β r ).] Rearranging terms gives T t=1 ρ ( τ AT U t X tz ) τt X z T t=1 1(A T U t X t z) ( A T U t X tz ) + T t=1 τa T U t. Subtract T t=1 τa T U t that des nt depend n z and des nt affect ptimizatin, and define Q T (z, k) := T t=1 l(a T U t, X tz) = l(u, x)dn(u, x), where l(u, v) := 1(u v)(v u). Since l is cntinuus and vanishes utside a cmpact subset f E, Q T (z, k) is a cntinuus mapping f N, an element f M p (E), t the real line. N N therefre implies by the Cntinuus Mapping Therem that the finitedimensinal limit f Q T (z, k) is given by Q (z, k) := kµ X z + E l(j, x z)dn(j, x) := kµ X z + i=1 l(j i, X i z). Then by Cnvexity lemma we have that Z T (k) arg min Q z R d T (z, k) cnverges in distributin t Z (k) = arg min Q z R d (z, k). Uniqueness and abslute cntinuity f Z (k) are shwn in Lemma 1, Appendix H. Step 3. By C1 A T (β(τ) β r ) k ξ c. Thus A T ( β(τ) β(τ)) d Ẑ (k) := Z (k) + k ξ c. Then Ẑ (k) = Z (k) + k ξ c = arg min z R d[ kµ X (z k ξ c) + i=1 l(j i, X i (z k ξ c))]. Claim 2 is thus prven. Step 4. Similarly t step 2 it fllws that ( ZT (mk), Z T (k) ) argmin (z 1,z 2 ) R 2d Q T (z 1, mk) + Q T (z 2, k) weakly cnverges t ( Z (mk), Z (k)) = argmin (z 1,z 2 ) R 2d Q (z 1, mk) + Q (z 2, k). Step 5. By Lemma 1 in Appendix H, µ X ( Z (mk) Z (k)) = Z (mk) 1 Z (k) 1 0 a.s. prvided mk k > d. It fllws by the Extended Cntinuus Mapping Therem that Z T (k) = k Ẑ T (k)/( X ( Z T (mk) Z T (k))) d Z (k) = Ẑ (k)/(µ X ( Z (mk) Z (k))). Claim 1 fllws by using relatins Ẑ (k) = Z (k) + k ξ c and Ẑ (mk) = Z (mk) + (mk) ξ c and using µ X c = 1 by C1. 10 The lemma states: Suppse (i) a sequence f cnvex lwersemicntinus functin Q T : R d R cnverges in finitedimensinal sense t Q : R d R ver a dense subset f R d, (ii) Q is finite ver a nnempty pen set Z 0, and (iii) Q is uniquely minimized at a randm vectr Z. Then any argmin f Q T, dented ẐT, cnverges in distributin t Z.
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