STX FACULTY WORKING PAPER NO EC*0BB4. Roger W. Koenker. and Related Empirical Processes. Strong Consistency of Regression Quantiles

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3 J30 $385 lo COPY 2 STX FACULTY WORKING PAPER NO Strog Cosistecy of Regressio Quatiles ad Related Empirical Processes Gilbert W. Bassett Roger W. Koeker JHE LIBRARY oeihe 0EC*0BB4 UNWE atu; College of Commerca ad Busiess Admiistratio Bureau of Ecoomic ad Busiess Research Uiversity of Illiois, Urbaa-Champaig

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5 BEBR FACULTY WORKING PAPER NO College of Commerce ad Busiess Admiistratio Uiversity of Illiois at Urb aa- Champaig November, 1984 Strog Cosistecy of Regressio Quatiles ad Related Empirical Processes Gilbert W. Bassett Uiversity of Illiois at Chicago Roger W. Koeker, Professor Departmet of Ecoomics The support of the NSF is gratefully ackowledged.

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7 Abstract The strog cosistecy of regressio quatile statistics (Koeker ad Bassett (1978)) i liear models with iid errors is established. Mild regularity coditios o the regressio desig sequece ad the error distributio are required. Strog cosistecy of the associated empirical quatile process (itroduced i Bassett ad Koeker (1982)) is also established uder aalogous coditios. However, for the proposed estimate of the coditioal distributio fuctio of Y, o regularity coditios o the error distributio are required for uiform strog covergece, thus establishig a Gliveko- Catelli-type theorem for this estimator.

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9 1. Itroductio I several recet papers, Koeker ad Bassett (1978, 1982) ad Bassett ad Koeker (1982), we have explored the problem of estimatig liear models for coditioal quatile fuctios of related radom variables. This approach complemets classical least-squares methods for liear models as well as recet robust methods which focus exclusively o estimatio of coditioal cetral tedecy. th We might hypothesize that the coditioal quatile fuctio of Y,..., Y is a liear fuctio of a p- vector of exogeous variables x, i.e., Q Y (8 x)=*0, (1.1) Give this hypothesis, oe may ask uder what coditios ca we cosistetly estimate the parameter vector 3? I Koeker ad Bassett (1978) we showed that for the liear model with iid errors ay sequece of solutios {3 fl } t the problem b R'.,» =1 where p(-) is the check fuctio. mi E p 9 (y,- -*,-*) (i.2) = ' e«u s= o Pa(«) ie - 1)«u. <o had the property that, uder mild coditios o the sequece of desigs ad the assumptio that F had a positive desity i a eighborhood of Q(9) = F (8), v (ft _ f$) coverged i law to a /:-variate Gaussia

10 distributio with mea vector = (Q (8), 0,..., 0)' R. Thus, uder the foregoig coditios p fl is weakly cosistet for B fl = B + I Koeker ad Bassett (1982) we showed that similar asymptotic behavior prevailed i sequeces of liear models with heteroscedasticity of order 0(I/V ). Here we maitai the hypothesis of iid errors while relaxig our previous smoothess coditios o the error distributio F. We begi by treatig the behavior of the p -dimesioal regressio quatiles ad coclude by treatig the associated empirical processes itroduced i Bassett ad Koeker (1982). Almost sure covergece results are established uder mild regularity coditios o the desig ad the distributio fuctio of the errors. I the case of our proposed estimate of the error distributio o regularity coditios are required o F, thus providig a atural extesio of the Gliveko-Catelli theorem to the realm of liear models. The latter result provides a itriguig alterative to methods based o residuals for assessig distributioal features of liear models with iid errors. Applicatios to bootstrappig ad other model diagostics immediately suggest themselves. 2. Strog Cosistecy of Regressio Quatiles We will assume throughout that we have data geerated from the model,

11 I (I) = Y. = x. 8 + u., (2.1) The errors u. are assumed to be idepedetly ad idetically distributed with distributio fuctio F. About F we will assume that it is a proper, right-cotiuous distributio fuctio. Its "iverse" will be deoted by Q(0) =if{u \F(u)^*} (2.2) so Q is left cotiuous o [0, lj. The parameter (3 is a ukow p -vector. The sequece {x. } of fixed desig vectors is assumed to "cotai a itercept," that is, x. = 1 for all i ad to satisfy the followig regularity coditios: lim if d = lim if if E \ -* ] (i) = 1 x. <u = d > (Ti\\ -1 2 lim if D = lim if sup E(x. a>) =D < <» (T)^ 71 -* We may ow state: 1 Theorem 1. If Dl ad D2 hold ad F has a uique 8 quatile the ay sequece of solutios {p (0)} to problem (1.3) satisfies 3 (0) -» (3 + (L almost surely. Proof. FLx 8 ad cosider, *, (8) r. (8) (23) t =1

12 s 18-1 = [p9 (a. - x. 8 - Q (8)) - p,(«. - Q (6))] t' =1 Sice R (0) = 0, ad R (8) is a sum of covex fuctios ad therefore covex, it suffices to show that for ay A > I lim if if R (8) > a.s. -x I I =A (2.4) We begi by establishig that R (8) - E R (8) a.s. (2.5) Sice r. (8) < j. 8, Kolmogorov's criterio yields o o J7 Var(r. (8))/t * (*. 8) / (2. 6) t =1 s s I'p'rti.sr/trfx.s 2 which is coverget so (2.5) follows. This may be stregtheed to uiform covergece o compacts by otig that for ay 8 R, we have. s - 6 : sup R «(8) - R (8 Q ) < sup{ A E \ x. (8-8 Q ) (2.7) sup E J I leal I -1 X. 0) I 1/2 So it remais to show that ER (8) is bouded away from 0, for ay

13 8 I I > 0. For a > 0, g(a)=e [p e (ti - a - Q (6)) - p 9 ( U - Q (6)] (2.8) Q + a * Q + a = / (1 -e)a<zf(u) - / a6rff(it)- / (u-q)df(u) -a g + a Q ad itegratig by parts gives, Q + a <7(a) = / (F(u) -Q)du. (2.9) Q For a < 0, the sig ad limits of itegratio are simply reversed. The fuctio g is covex, g (0) =0, ad g (a) > for ay a # by the uiqueess of the 6 quatile. Now, let h (a) deote the covex hull of g (a) ad g (-a) the we have, ^ (8)> _1 * (*,») (2.10) > ~ l Eh( \x. 8 ) -i h{~ E x. 8 ) >A(<* 8 ) by the symmetry of A, Jese's iequality ad (D2) respectively. The

14 fuctio h (a) is bouded away from zero for a by the uiqueess of the 8 quatile, thus completig the proof. Reviewig the precedig argumet it is clear that uiqueess of the 8 quatile is eeded oly to argue that ER (8) has a uique miimum at the origi. Whe the 8 quatile is ot uique the ER (8) has a larger miimum set, but it is straightforward to show that solutios to (1.3) coverge almost surely to elemets of this set. See Koeker ad Bassett(1984) for a example of weak but ot strog covergece i this framework. The followig result will be used i subsequet sectios. Theorem 2.2. Suppose F is ay proper, right cotiuous distributio fuctio ad D1-D2 prevail. Let A (A) = {5 R P \ER (8) ^ A}, the A~ level set of ER (8). The ay sequece of solutios to (1.3), {p (6)}, satisfies for all A > 0. P ft (e)-p-s 9.d (4) a.s. (2.11) Proof. Whe Q (8 + 0) = Q (8) this follows immediately from Theorem 1 sice trivially, A (A) for all A. Whe, (8) = Q(8 + 0) -^(8) >0 (2.12) the fuctio g (a) is idetically zero o [0, (0)] ad therefore ER (8) will attai a miimum of zero o a set cotaiig the origi. It suffices to verify that the (ecessarily covex) sets A (A) are bouded sice we may the, by

15 the prior argumet, fiitely cover the boudary of ay such set ad by the uiform covergece of R (8) to ER (3) o compacts we may coclude that ay solutio to (1.3) lies iside this boudary. For ay A. A (A) we have h(d \\\ ) ^ A by the argumet of (2.10). Sice h(-) is covex ad zero oly o the bouded iterval [- (9), (6)]> f r a >' -1 > 0, there exists a M <» such that h (a) ^ A implies a < A/. Thus X < M /d for ay X. A (A), ad this completes the proof. 3. Strog Covergece of Empirical Processes Based o Regressio Quatiles I Bassett ad Koeker (1982) we proposed a estimate of the coditioal quatile fuctio of Y give x, where as above Q Y (B\x) = m!{xb 6 <E 5(6)} (3.1) B (9) = {6 R P 27 p f ( f. - x. 6 ) = mi!}. (3.2) There it is show that at x = x = Ex., the sample paths of the radom fuctio Q r (e) = Q' r (e7) (3.3) are o-decreasig, left cotiuous jump fuctios o (0, 1). However, ulike the ordiary empirical quatile fuctio to which Q Y W specializes whe X = 1 ; Q Y (Q), jumps at irregularly spaced poits o (0, 1). Similarly, oe may show that

16 Q y (e + 0)» lim g y ( Q + ) (3.4) = sup{b \b 5(6)} is a odecreasig, right- cotiuous jump fuctio o (0, 1). It was also show that properly ormalized versios of these processes have fiite dimesioal distributios which coverge to those of the Browia bridge process. Portoy (1983) has recetly stregtheed these results to establish weak covergece of the estimate to the Browia bridge. F Y (y)=sup{q\q Y (Q)^y} (3.5) Here we wish to ivestigate the strog covergece of Q Y {Q) ad F y (Q) usig results from the previous sectio. We may begi by otig that give th the iid error assumptio of model (1.2), the 6 coditioal quatile fuctio of Y give x may be writte as, We will restrict attetio as previously to Q Y (9\z)=zl+Q(9) (3.6) We may ow state: <9 r (e) = g r (ej) = pj + Q(e). (3.7) Theorem 3.1. If Dl ad D2 hold ad Q (0) is cotiuous o a closed iterval 0C(O, 1) the

17 9 sup I e e Q Y (Q) -Q Y (Q)\ -0 a.s. (38) Proof. From Theorem 2.1 we have poitwise covergece of Q Y (Q) ad Q (9 + 0) to Q Y (Q) ad usig the mootoicity ad cotiuity of Q Y (Q) this may be stregtheed to uiform almost sure covergece o 0. See, for example, the argumet i Billigsley (1979, p. 233) for the Gliveko-Catelli theorem. Whe there are jumps i Q Y (') some ew difficulties arise, sice Q Y (Q) may oscillate betwee Q y (6) ad Q Y (Q + 0), but based o Theorem 2.2 we may establish the followig result. Theorem 3.2. If Dl ad D2 hold the a.s. sup \F Y (u) \u ) -F -r [u Y {u)\ y ) -Q -u a.s «u 6 R ^ g j Proof. We will cosider the case of cotiuous F first; discotiuities i F are treated i the Appedix. We will begi by fixig 8 ad establishig the iequalities: limif Q y (Q) > Q y (Q) a.s. - * lim sup Q Y {Q + 0) < Q y (Q + 0) a.s. (3 10) ^ j From Theorem 2.2 we have for av Q Y (0) ^ QY (Q) + if {XI X,l (^)} (3.12) ad

18 10 Q Y (Q + 0) ^ Q Y (B) + sup {\J X A (A)} (3.13) so, by Jese's iequality, usig Dl, {\x X A (A)} = {\x -1 ' g (x. \) < A} (3 14)» =1 C{\x \g{\x) < A}. Sice g ( ) is covex ad zero oly o [0, Q (0 + 0) - Q (8)] we have -o (A) ^ if {a g (a) < 4} (3.15) < sup {a (a) < 4} <; Q(8 + 0) -Q(Q) + o(zi) so the iequalities follow lettig A - 0. It is straightforward to verify that ( ) imply lim sup F Y ( Q y (6)) ^ F y ( y (6)) (3 16) ad lim if F y (<? y (e) -0) a/(<? r (6) -0). (3 17) So by cotiuity of F we have lim if F Y (Q Y (Q)-0)= lim sup F y ( y (6)) = 6. (3 lg) Ad by stadard argumets for the Gliveko-Catelli theorem, agai see, Billigsley (1979, p. 233), this implies (3.9).

19 to 11 Appedix Whe F has discotiuities some ew difficulties arise which are take up i this appedix. Pricipally, we must verify that F coverges to a proper distributio fuctio. See the remark followig the the proof of the lemma. To complete the proof of Theorem 3.2 we will eed: Lemma. Fix 0 (O,1), ad assume, (Q)=F [Q (Q))-F (Q (8)-0) > 0, so th F has strictly positive mass at the quatile. The for ay > 0, there is a such that P{B(e)=p + fl, > # }*!-*. (A.l) Proof. As i (2.3) cosider, # (8) = ~ l E = «r 1 (») i 2>«(«,- - *,- 8 - e e ( )) - p 9K - e (ej) (a.2) ad correspodig directioal derivitives, R '(b,w) = ~ l JJr.'(b w) = ~ l } [\-9 -\sg* (u.,x. w )]x. w (A.3) where sg* (u,v ) = sg (u ) for u ^0 ad sg(v) otherwise. We must establish, lim if if R' (0,w ) > a. s. - * =1 fa4l Followig the approach used i the proof of Theorem 2.1 we begi by showi that for fixed w,

20 to I =1 w 12 R ' (0,w ) -ER' (0,w )-0 a.s. (A.5) Sice r '. (0,tv ) ^ \x. w applicatio of Kolmogorov's criterio yields, X 2..,2,.2 EVar(r'.%w))/i < E(x.w)/i <«(A 6) i =i» =i by (D2). This may be stregtheed to uiform covergece o the sphere \w =1 usig (D2) ad the cotiuity of R ' (0,w ) i w. Hece, we have, Now, lim if if R '(Q,w ) - lim if if ER '(0,w ) a.s. - x I I v ' - * I &'(<>, ) =! to I I =1 [F(Q(6))-e]z. w if 1. 1» a ' ' -[ [F(Q (0) - 0)-6]x. w otherwise (A.7) (A.8) so settig m (6) = mi{f (<? (6))-6,6-F (Q (8)-0)} we have, Er.'(0,w) > m(6) or. to, (A.9) hece, usig (Dl), lim if if ER '(0,w ) > m {Q)d > 0, - x I I =1 (A. 10) which completes the proof. A immediate cosequece of the lemma is that uder the same coditios, P{Q Y 0) = Q Y W = Q r (e+0) = Q r (8+0), > o }> 1 - X(A.ll) which implies,

21 13 limsupf y (C? r (e))a:f r (Q y (e)) - * (A.12) lim if F Y y\^y (Q Y (9-0)) ^ F Y y\^y (Q Y (9-0)) - x (A.13) Which together with ( ) imply that at poits of discotiuity i F we have the required covergece. Sice there are oly coutably may such poits this completes the proof. Remark. The followig example illustrates the ecessity of the lemma. Let Q (9) =0 for 9 (0,1] so that the associated df is F( (0 a <0 1 a >0 (A.14) Cosider, QW \ 6 ((V4] e*(v] (A.15) ad F(u) = Ml u < l -l - <«< (A.16) 1 -l ^u The () satisfies ( ) but ot (A.ll). Q actually coverges to Q uiformly, but \\mf(qw) =F(0) = *# 1 -w so F fails to coverge to F. Note =F(0) (A.17) that sice the poitwise limit of F is

22 14 either right or left cotiuous at the limit fails to be a proper distributio fuctio.

23 15 Refereces Bassett, G. W. ad Koeker R. (1982). A empirical quatile fuctio for liear models with iid errors, Joural of the America Statistical Associatio, Billigsley, P. (1979) Probability ad Measure, (New York: Wiley). Koeker, R. ad Bassett, G. W. (1978) Regressio Quatiles, Ecoometrica, 46, Koeker, R. ad Bassett, G. W. (1982) Robust Tests for Heteroscedasticiiy based o Regressio Quatiles, Ecoometrica, 50, Koeker, R.W. ad Bassett, G.W. (1984) Four (Pathological) Examples i Asymptotic Statistics, The America Statisticia, 38, Portoy, S. (1983) Tightess of the sequece of empiric cdf processes defied from regressio fractiles, Proceedigs of the Coferece o Robust ad No-liear Methods i Time-Series Aalysis, Heidelburg, FRG.

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