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QUANTILE AUTOREGRESSION ROGER KOENKER AND ZHIJIE XIAO Abstract. We cnsider quantile autregressin (QAR) mdels in which the autregressive cefficients can be expressed as mntne functins f a single, scalar randm variable. The mdels can capture systematic influences f cnditining variables n the lcatin, scale and shape f the cnditinal distributin f the respnse, and therefre cnstitute a significant extensin f classical cnstant cefficient linear time series mdels in which the effect f cnditining is cnfined t a lcatin shift. The mdels may be interpreted as a special case f the general randm cefficient autregressin mdel with strngly dependent cefficients. Statistical prperties f the prpsed mdel and assciated estimatrs are studied. The limiting distributins f the autregressin quantile prcess are derived. Quantile autregressin inference methds are als investigated. Empirical applicatins f the mdel t the U.S. unemplyment rate and U.S. gasline prices highlight the ptential f the mdel. 1. Intrductin Cnstant cefficient linear time series mdels have played an enrmusly successful rle in ecnmetrics, and gradually varius frms f randm cefficient time series mdels have als emerged as viable cmpetitrs in particular fields f applicatin. One variant f the latter class f mdels, althugh perhaps nt immediately recgnizable as such, is the linear quantile autregressin mdel. This mdel has received cnsiderable attentin in the theretical literature, and can be easily estimated with the quantile regressin methds prpsed in Kenker and Bassett (1978). Curiusly, hwever, all f the theretical wrk dealing with this mdel (that we are aware f) fcuses exclusively n the iid innvatin case that restricts the autregressive cefficients t be independent f the specified quantiles. In this paper we seek t relax this restrictin and cnsider linear quantile autregressin mdels whse autregressive (slpe) parameters may vary with quantiles τ [0, 1]. We hpe that these mdels might expand the mdeling ptins fr ecnmic time series that display asymmetric dynamics r lcal persistency. Key wrds and phrases. randm cefficient mdel, asymmetric dynamics. JEL Classificatin: C14, C22. Crrespnding authr: Rger Kenker, Department f Ecnmics, University f Illinis, Champaign, Il, 61820. Email: rkenker@uiuc.edu. Versin March 4, 2004. This research was partially supprted by NSF grant SES-02-40781. The authrs wuld like t thank Steve Prtny and Peter Phillips fr valuable discussins regarding this wrk. 1

2 Quantile Autregressin In recent years, cnsiderable research effrt has been devted t mdificatins f traditinal unit rt mdels t incrprate the effect f varius types f shcks. An imprtant mtivatin fr such mdificatins is the intrductin f asymmetries int ecnmic dynamics. It is widely acknwledged that many imprtant ecnmic variables may display asymmetric adjustment paths (e.g. Neftci (1984), Enders and Granger (1998)). The bservatin that firms are mre apt t increase than t reductin in prices is a key feature f many macrecnmic mdels. Beaudry and Kp (1993) shwed that psitive shcks t U.S. GDP are mre persistent than negative shcks, indicating asymmetric business cycle dynamics ver different quantiles f the innvatin prcess. In additin, while it is recgnized that utput fluctuatins are persistent, less persistent results are als fund at lnger hrizns (Beaudry and Kp (1993)), indicating the existence f lcal persistency r tempral persistency in ecnmic time series. See, inter alia, Delng and Summers (1986), Hamiltn (1989), Sichel (1989), Diebld and Rudebusch (1990), Evans and Wachtel (1993), Ptter (1995), Bradley and Jansen (1997), Hess and Iwata (1997), and Kuan and Huang (2001)) amng thers n the study f asymmetric dynamics in ecnmic time series. A related develpment is the grwing literature n threshld autregressin (TAR) with unit rts (e.g. Balke and Fmby (1997); Tsay (1997); Gnzalez and Gnzal (1998); Hansen (1999); and Caner and Hansen (2001)). In particular, Tsay (1997) prpsed a unit rt test when the innvatins fllw a threshld prcess; Gnzalez and Gnzal (1998) studied a TAR(1) mdel that allws fr a unit rt; Caner and Hansen (2001) develps an asympttic thery f inference fr an unrestricted tw regime TAR mdel with a unit rt. We believe that quantile regressin methds can prvide an alternative way t study asymmetric dynamics and persistency in ecnmic time series. Lintn and Whang (2004) have recently prpsed related quantilgram inference methds fr explring linear dependence in time series at varius quantiles. In this paper, we prpse a new quantile autregressin (QAR) mdel whse autregressive cefficient may take different values (pssibly unity) ver different quantiles f the innvatin prcess. We shw that sme frms f the mdel can exhibit unit-rt like tendencies r even temprarily explsive behavir, but with ccasinal episdes f mean reversin sufficient t insure statinarity. The mdels lead t interesting new hyptheses and inference apparatus fr ecnmic time series. The paper is rganized as fllws: We intrduce the mdel and study sme basic statistical prperties f the QAR prcess in Sectin 2. Sectin 3 develps the limiting distributin f the QAR estimatr. Sectin 4 cnsiders sme restrictins impsed n the mdel by the mntnicity requirement n the cnditinal quantile functins. Statistical inference, including testing fr asymmetric dynamics, is explred in Sectin 5. Sectin 6 reprts a Mnte Carl experiment n the sampling perfrmance f the prpsed inference prcedure. Empirical applicatins t the U.S. unemplyment rate

Rger Kenker and Zhijie Xia 3 and the U.S. price f gasline are discussed in Sectin 7. Prfs are prvided in the Appendix. 2. The Mdel There is a substantial theretical literature, including Weiss (1987), Knight (1989), Kul and Saleh(1995), Kul and Mukherjee(1994), Hercé (1996), Hasan and Kenker (1997), Jurečkvá and Hallin (1999) dealing with the linear quantile autregressin mdel. In this mdel the τ-th cnditinal quantile functin f the respnse y t is expressed as a linear functin f lagged values f the respnse. But a striking feature f this literature is that it has fcused exclusively n the case f iid innvatins in which the cnditining variables play their classical rle f shifting the lcatin f the cnditinal density f y t, but they have n effect n cnditinal scale r shape. In this paper we wish t study estimatin and inference in a mre general class f quantile autregressive (QAR) mdels in which all f the autregressive cefficients are allwed t be τ-dependent, and therefre are capable f altering the lcatin, scale and shape f the cnditinal densities. 2.1. The Mdel. Let {U t } be a sequence f iid standard unifrm randm variables, and cnsider the pth rder autregressive prcess, (1) y t = θ 0 (U t ) + θ 1 (U t )y t 1 + + θ p (U t )y t p, where the θ j s are unknwn functins [0, 1] R that we will want t estimate. Prvided that the right hand side f (1) is mntne increasing in U t, it fllws that the τth cnditinal quantile functin f y t can be written as, (2) Q yt (τ y t 1,..., y t p ) = θ 0 (τ) + θ 1 (τ)y t 1 +... + θ p (τ)y t p. r smewhat mre cmpactly as, (3) Q yt (τ F t 1 ) = x t θ(τ). where x t = (1, y t 1,..., y t p ), and F t is the σ-field generated by {y s, s t}. The transitin frm (1) t (2) is an immediate cnsequence f the fact that fr any mntne increasing functin g and standard unifrm randm variable, U, we have Q g(u) (τ) = g(q U (τ)) = g(τ), where Q U (τ) = τ is the quantile functin f U. In the abve mdel, the autregressive cefficients may be τ-dependent and thus can vary ver the quantiles. The cnditining variables nt nly shift the lcatin f the distributin f y t, but als may alter the scale and shape f the cnditinal distributin. We will refer t this mdel as the QAR(p) mdel. We will argue that QAR mdels can play a useful rle in expanding the mdeling territry between classical statinary linear time series mdels and their unit rt alternatives. T illustrate this in the QAR(1) case, cnsider the mdel (4) Q yt (τ F t 1 ) = θ 0 (τ) + θ 1 (τ)y t 1,

4 Quantile Autregressin with θ 0 (τ) = σφ 1 (τ), and θ 1 (τ) = min{γ 0 + γ 1 τ, 1} fr γ 0 (0, 1) and γ 1 > 0. In this mdel if U t > (1 γ 0 )/γ 1 the mdel generates the y t accrding t the standard Gaussian unit rt mdel, but fr smaller realizatins f u t we have a mean reversin tendency. Thus, the mdel exhibits a frm f asymmetric persistence in the sense that sequences f strngly psitive innvatins tend t reinfrce its unit rt like behavir, while ccasinal negative realizatins induce mean reversin and thus undermine the persistence f the prcess. The classical Gaussian AR(1) mdel is btained by setting θ 1 (τ) t a cnstant. The frmulatin in (4) reveals that the mdel may be interpreted as rather unusual frm f randm cefficient autregressive (RCAR) mdel. Such mdels arise naturally in many time series applicatins. Discussins f the rle f RCAR mdels can be fund in, inter alia, Nichlls and Quinn (1982), Tjøstheim(1986), Purahmadi (1986), Brandt (1986), Karlsen(1990), and Tng (1990). In cntrast t mst f the literature n RCAR mdels, in which the cefficients are typically assumed t be stchastically independent f ne anther, the QAR mdel has cefficients that are functinally dependent. Mntnicity f the cnditinal quantile functins impses sme discipline n the frms taken by the θ functins. This discipline essentially requires that the functin Q yt (τ y t 1,..., y t p ) is mntne in τ in sme relevant regin Υ f (y t 1,..., y t p )-space. The crrespndance between the randm cefficient frmulatin f the QAR mdel (1) and the cnditinal quantile functin frmulatin (2) presuppses the mntnicity f the latter in τ. In the regin Υ where this mntnicity hlds (1) can be regarded as a valid mechanism fr simulating frm the QAR mdel (2). Of curse, mdel (1) can, even in the absence f this mntnicity, be taken as a valid data generating mechanism, hwever the link t the strictly linear cnditinal quantile mdel is n lnger valid. At pints where the mntnicity is vilated the cnditinal quantile functins crrespnding t the mdel described by (1) have linear kinks. Attempting t fit such piecewise linear mdels with linear specificatins can be hazardus. We will return t this issue in the discussin f Sectin 4. In the next sectin we briefly describe sme essential features f the QAR mdel. 2.2. Prperties f the QAR Prcess. The QAR(p) mdel (1) can be refrmulated in mre cnventinal randm cefficient ntatin as, (5) y t = µ 0 + α 1,t y t 1 + + α p,t y t p + u t where µ = Eθ 0 (U t ), u t = θ 0 (U t ) µ, and α j,t = θ j (U t ), fr j = 1,..., p. Thus, {u t } is an iid sequence f randm variables with distributin functin F ( ) = θ0 1 ( + µ), and the α j,t cefficients are functins f this u t innvatin randm variable. The QAR(p) prcess (5) can be expressed as an p-dimensinal vectr autregressin prcess f rder 1: Y t = Γ + A t Y t 1 + V t

with Γ = [ µ0 Rger Kenker and Zhijie Xia 5 0 p 1 ], A t = [ ] [ ] Ap 1,t α p,t ut, V I p 1 0 t =, p 1 0 p 1 where A p 1,t = [ α 1,t,..., α p 1,t ], Y t = [y t,, y t p+1 ], and 0 p 1 is the (p 1)- dimensinal vectr f zers. In the Appendix, we shw that under regularity cnditins given in the fllwing Therem, an F t -measurable slutin fr (5) can be fund. T frmalize the freging discussin and facilitate later asympttic analysis, we intrduce the fllwing cnditins. A.1: {u t } are iid randm variables with mean 0 and variance σ 2 <. The distributin functin f u t, F, has a cntinuus density f with f(u) > 0 n U = {u : 0 < F (u) < 1}. A.2: Let E(A t A t ) = Ω A, the eigenvalues f Ω A have mduli less than unity. A.3: Dente the cnditinal distributin functin Pr[y t < F t 1 ] as F t 1 ( ) and its derivative as f t 1 ( ), f t 1 is unifrmly integrable n U. Therem 2.1. Under assumptins A.1 and A.2, the time series y t given by (5) is cvariance statinary and satisfies a central limit therem 1 y t N ( µ y, ω 2 n y), where µ 0 µ y = 1 p j=1 µ ; ωy 2 = lim 1 p n E[ (y t µ y )] 2, and µ j = E(α j,t ), j = 1,..., p. T illustrate sme imprtant features f the QAR prcess, we cnsider the simplest case f QAR(1) prcess, (6) y t = α t y t 1 + u t, where α t = θ 1 (U t ) and u t = θ 0 (U t ) crrespnding t (4), whse prperties are summarized in the fllwing crllary. Crllary 2.1. If y t is determined by (6), and ωα 2 = E(α t) 2 < 1, under assumptin A.1, y t is cvariance statinary and satisfies a central limit therem 1 y t N ( 0, ω 2 n y), where with µ α = E(α t ) < 1. ω 2 y = 1 + µ α (1 µ α )(1 ω 2 α )σ2,

6 Quantile Autregressin In the example given in Sectin 2.1, α t = θ 1 (U t ) = min{γ 0 + γ 1 U t, 1} 1, and Pr ( α t < 1) > 0, the cnditin f Crllary 2 hlds and the prcess y t is glbally statinary but can still display lcal (and asymmetric) persistency in the presence f certain type f shcks (psitive shcks in the example). Crllary 2 als indicates that even with α t > 1 ver sme range f quantiles, as lng as ωα 2 = E(α t ) 2 < 1, y t can still be cvariance statinary in the lng run. Thus, a quantile autregressive prcess may allw fr sme (transient) frms f explsive behavir while maintaining statinarity in the lng run. Under the assumptins in Crllary 2, by recursively substituting in (6), we can see that j 1 (7) y t = β t,j u t j, where β t,0 = 1, and β t,j = α t i, fr j 1. j=0 is a statinary F t -measurable slutin t (6). In additin, if j=0 β t,jv t j cnverges in L p, then y t has a finite p-th rder mment. The F t -measurable slutin f (6) gives a dubly stchastic MA( ) representatin f y t. In particular, the impulse respnse f y t t a shck u t j is stchastic and is given by β t,j. On the ther hand, althugh the impulse respnse f the quantile autregressive prcess is stchastic, it des cnverge (t zer) in mean square (and thus in prbability) as j, crrbrating the statinarity f y t. If we dente the autcvariance functin f y t by γ y (h), it is easy t verify that γ y (h) = µ α h σ2 y, where σ2 y =. 1 ωα 2 Remark 2.1. Cmparing t the QAR(1) prcess y t, if we cnsider a cnventinal AR(1) prcess with autregressive cefficient µ α and dente the crrespnding prcess by y t, the lng-run variance f y t (given by ωy) 2 is (as expected) larger than that f y t. The additinal variance the QAR prcess y t cmes frm the variatin f α t. In fact, ωy 2 can be decmpsed int the summatin f the lng-run variance f y and an t additinal term that is determined by the variance f α t : ω 2 y = ω 2 y + i=0 σ2 σ 2 (1 µ α ) 2 (1 ω 2 α )Var(α t), where ω 2 y = σ2 /(1 µ α ) 2 is the lng-run variance f y t. We cnsider estimatin and related inference n the QAR mdel in the next tw sectins. 3. Estimatin Estimatin f the quantile autregressive mdel (3) invlves slving the prblem (8) min θ R p+1 ρ τ (y t x t θ),

Rger Kenker and Zhijie Xia 7 where ρ τ (u) = u(τ I(u < 0)) as in Kenker and Bassett (1978). Slutins, θ(τ), are called autregressin quantiles. Given θ(τ), the τ-th cnditinal quantile functin f y t, cnditinal n x t, culd be estimated by, ˆQ yt (τ x t ) = x t θ(τ), and the cnditinal density f y t can be estimated by the difference qutients, ˆf yt (τ x t 1 ) = (τ i τ i 1 )/( ˆQ yt (τ i x t 1 ) ˆQ yt (τ i 1 x t 1 )), fr sme apprpriately chsen sequence f τ s. If we dente E(y t ) as µ y, E(y t y t j ) as γ j, and let Ω 0 = E(x t x t ) = lim n 1 n x tx t, then [ ] 1 µy Ω 0 = µ y Ω y where γ 0 γ p 1 Ω y =...... γ p 1 γ 0 In the special case f QAR(1) mdel (6), Ω 0 = E(x t x t ) = diag[1, γ 0], γ 0 = E[yt 2]. Let Ω 1 = lim n 1 n f t 1[Ft 1 1 (τ)]x tx t, and define Σ = Ω 1 1 Ω 0Ω 1 1. The asympttic distributin f θ(τ) is summarized in the fllwing Therem. Therem 3.1. Under assumptins A.1 - A.3, Σ 1/2 n( θ(τ) θ(τ)) B k (τ), where B k (τ) represents a k-dimensinal standard Brwnian Bridge, k = p + 1. By definitin, fr any fixed τ, B k (τ) is N (0, τ(1 τ)i k ). In the imprtant special case with cnstant cefficients, Ω 1 = f[f 1 (τ)]ω 0, where f( ) and F ( ) are the density and distributin functins f u t, respectively. We state this result in the fllwing crllary. Crllary 3.1. Under assumptins A.1 - A.3, if the cefficients α jt are cnstants, then [ f[f 1 (τ)] 1 Ω 0 ] 1/2 n( θ(τ) θ(τ)) Bk (τ). An alternative frm f the mdel that is widely used in ecnmic applicatins is p 1 (9) y t = µ 0 + δ 0,t y t 1 + δ j,t y t j + u t, where, crrespnding t (5), p δ 0,t = α s,t, δ j,t = s=1 j=1 p α s,t, j = 1,, p 1. s=j+1

8 Quantile Autregressin In the abve transfrmed mdel, δ 0,t is the critical parameter crrespnding the largest autregressive rt. Let z t = (1, y t 1, y t 1,..., y t p+1 ), we may write the quantile regressin cunterpart f (9) as (10) Q yt (τ F t 1 ) = z t δ(τ). where δ(τ) = (α 0 (τ), δ 0 (τ), δ 1 (τ),, δ p 1 (τ)). The limiting distributins f the quantile regressin estimatrs δ(τ) can be btained frm ur previus analysis. If we define 1 0 0 0 0 1 1 1 J = 0 0 1 1., and = JΣJ.. 0 0 0 1 then we have, under assumptins A.1 - A.3, 1/2 n( δ(τ) δ(τ)) B k (τ). If we fcus ur attentin n the largest autregressive rt δ 0,t in the ADF type regressin (9) and cnsider the special case that δ j,t = cnstant fr j = 1,..., p 1, then, a result similar t Crllary 2 can be btained. Crllary 3.2. Under assumptins A.1-A.3, if δ j,t = cnstant fr j = 1,..., p 1, and δ 0,t 1 and δ 0,t < 1 with psitive prbability, then the time series y t given by (9) is cvariance statinary and satisfies a central limit therem. 4. Quantile Mntnicity As in ther linear quantile regressin applicatins, linear QAR mdels shuld be cautiusly interpreted as useful lcal apprximatins t mre cmplex nnlinear glbal mdels. If we take the linear frm f the mdel t literally then bviusly at sme pint, r pints, there will be crssings f the cnditinal quantile functins unless these functins are precisely parallel in which case we are back t the pure lcatin shift frm f the mdel. This crssing prblem appears mre acute in the autregressive case than in rdinary regressin applicatins since the supprt f the design space, i.e. the set f x t that ccur with psitive prbability, is determined within the mdel. Nevertheless, we may still regard the linear mdels specified abve as valid lcal apprximatins ver a regin f interest. It shuld be stressed that the estimated cnditinal quantile functins, ˆQ y (τ x) = x θ(τ),

Rger Kenker and Zhijie Xia 9 0 40 80 0 200 400 600 800 1000 Figure 1. QAR and Unit Rt Time-Series: The figure cntrasts tw time series generated by the same sequence f innvatins. The grey sample path is a randm walk with standard Gaussian innvatins; the black sample path illustrates a QAR series generated by the same innvatins with randm AR(1) cefficient.85 +.25Φ(u t ). The latter series althugh exhibiting explsive behavir in the upper tail is statinary as described in the text. are guaranteed t be mntne at the mean design pint, x = x, as shwn in Bassett and Kenker (1982), fr linear quantile regressin mdels. Crssing, when it ccurs, is generally cnfined t utlying regins f the design space. In ur randm cefficient view f the QAR mdel, y t = x t θ(u t) we express the bservable randm variable y t as a linear functin cnditining cvariates. But rather than assuming that the crdinates f the vectr θ are independent randm variables we adpt a diametrically ppsite viewpint that they are perfectly functinally dependent, all driven by a single randm unifrm variable. If the functins (θ 0,..., θ p ) are all mntnically increasing then the crdinates f the randm vectr α t are said t be cmntnic in the sense f Schmeidler (1986). 1 This is ften the case, but there are imprtant cases fr which this mntnicity fails. What then? What really matters is that we can find a linear reparameterizatin f the mdel that des exhibit cmntnicity ver sme relevant regin f cvariate space. Since 1 Randm variables X and Y n a prbability space (Ω, A, P ) are said t be cmntnic if there are mntne functins, g and h and a randm variable Z n (Ω, A, P ) such that X = g(z) and Y = h(z).

10 Quantile Autregressin (Intercept) 1.5 0.0 1.5 x 0.85 1.00 0.2 0.4 0.6 0.8 tau 0.2 0.4 0.6 0.8 tau Figure 2. Estimating the QAR mdel: The figure illustrates estimates f the QAR(1) mdel based n the black time series f the previus figure. The left panel represents the intercept estimate at 19 equally spaced quantiles, the right panel represents the AR(1) slpe estimate at the same quantiles. The shaded regin is a.90 cnfidence band. Nte that the slpe estimate quite accurate reprduces the linear frm f the QAR(1) cefficient used t generate the data. fr any nnsingular matrix A we can write, Q y (τ x) = x A 1 Aθ(τ), we can chse p + 1 linearly independent design pints {x s : s = 1,..., p + 1} where Q y (τ x s ) is mntne in τ, then chsing the matrix A s that Ax s is the sth unit basis vectr fr R p+1 we have Q y (τ x s ) = γ s (τ), where γ = Aθ. And nw inside the cnvex hull f f ur selected pints we have a cmntnic randm cefficient representatin f the mdel. In effect, we have simply reparameterized the design s that the p + 1 cefficients are the cnditinal quantile functins f y t at the selected pints. The fact that quantile functins f sums f nnnegative cmntnic randm variables are sums f their marginal quantile functins, see e.g. Denneberg(1994), allws us t interplate inside the cnvex hull. Of curse, linear extraplatin is als pssible but we must be cautius abut pssible vilatins f the mntnicity requirement in this regin. The interpretatin f linear cnditinal quantile functins as apprximatins t the lcal behavir in central range f the cvariate space shuld always be regarded as prvisinal; richer data surces can be expected t yield mre elabrate nnlinear specificatins that wuld have validity ver larger regins. Figure 1 illustrates a

Rger Kenker and Zhijie Xia 11 y 5 10 15 y 5 10 15 5 10 15 x 5 10 15 x Figure 3. QAR(1) Mdel f U.S. Shrt Term Interest Rate: The AR(1) scatterplt f the U.S. three mnth rate is superimpsed in the left panel with 49 equally spaced estimates f linear cnditinal quantile functins. In the right panel the mdel is augmented with a nnlinear (quadratic) cmpnent. The intrductin f the quadratic cmpnent alleviates sme nnmntnicity in the estimated quantiles at lw interest rates. realizatin f the simple QAR(1) mdel described in Sectin 2. The black sample path shws 1000 bservatins generated frm the mdel (4) with AR(1) cefficient θ 1 (u) =.85 +.25u and θ 0 (u) = Φ 1 (u). The grey sample path depicts the a randm walk generated frm the same innvatin sequence, i.e. the same θ 0 (U t ) s but with cnstant θ 1 equal t ne. It is easy t verify that the QAR(1) frm f the mdel satisfies the statinarity cnditins f Sectin 2.2, and despite the explsive character f its upper tail behavir we bserve that the series appears quite statinary, at least by cmparisn t the randm walk series. Estimating the QAR(1) mdel at 19 equally spaced quantiles yields the intercept and slpe estimates depicted in Figure 2. Figure 3 depicts estimated linear cnditinal quantile functins fr shrt term (three mnth) US interest rates using the QAR(1) mdel superimpsed n the AR(1) scatter plt. In this example the scatterplt shws clearly that there is mre dispersin at higher interest rates, with nearly degenerate behavir at very lw rates. The fitted linear quantile regressin lines in the left panel shw little evidence f crssing, but at rates belw.04 there are sme vilatins f the mntnicity requirement in the fitted quantile functins. Fitting the data using a smewhat mre

12 Quantile Autregressin (Intercept) 0.4 0.2 0.0 0.2 x 0.8 0.9 1.0 1.1 1.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 tau tau Figure 4. QAR(1) Mdel f U.S. Shrt Term Interest Rate: The QAR(1) estimates f the intercept and slpe parameters fr 19 equally spaced quantile functins are illustrated in the tw plts. Nte that the slpe parameter is, like the prir simulated example, explsive in the upper tail but mean reverting in the lwer tail. cmplex nnlinear (in variables) mdel by intrducing a anther additive cmpnent θ 2 (τ)(y t 1 δ) 2 I(y t 1 < δ) with δ = 8 in ur example we can eliminate the prblem f the crssing f the fitted quantile functins. In Figure 4 depicting the fitted cefficients f the QAR(1) mdel and their cnfidence regin, we see that the estimated slpe cefficient f the QAR(1) mdel has smewhat similar appearance t the simulated example. Even mre flexible mdels may be needed in ther settings. A B-spline expansin QAR(1) mdel fr Melburne daily temperature is described in Kenker(2000) illustrating this apprach. The statistical prperties f nnlinear QAR mdels and assciated estimatrs are much mre cmplicated than the linear QAR mdel that we study in the present paper. Despite the pssible crssing f quantile curves, we believe that the linear QAR mdel prvides a cnvenient and useful lcal apprximatin t nnlinear QAR mdels. Such simplied QAR mdels can still deliver imprtant insight abut dynamics, e.g. adjustment asymmetries, in ecnmic time series and thus prvides a useful tl in empirical diagnstic time series analysis.

Rger Kenker and Zhijie Xia 13 5. Inference On The QAR Prcess In this sectin, we turn ur attentin t inference in QAR mdels. Althugh ther inference prblems can be analyzed, we cnsider here the fllwing inference prblems that are f paramunt interest in many applicatins. The first hypthesis is the quantile regressin analg f the classical representatin f linear restrictins n θ: (1) H 01 : Rθ(τ) = r, with knwn R and r, where R dentes an q p-dimensinal matrix and r is an q-dimensinal vectr. In additin t the classical inference prblem, we are als interested in testing fr asymmetric dynamics under the QAR framewrk. Thus we cnsider the hypthesis f parameter cnstancy, which can be frmulated in the frm f: (2) H 02 : Rθ(τ) = r, with unknwn but estimable r. We cnsider bth the cases at specific quantiles τ (say, median, lwer quartile, upper quartile) and the case ver a range f quantiles τ T. 5.1. The Regressin Wald Prcess and Related Tests. Under the linear hypthesis H 01 : Rθ(τ) = r and the assumptins f Therem 3, we have (11) V n (τ) = n [ RΩ 1 1 Ω 0 Ω 1 1 R ] 1/2 (R θ(τ) r) Bq (τ), where B q (τ) represents a q-dimensinal standard Brwnian Bridge. Fr any fixed τ, B q (τ) is N (0, τ(1 τ)i q ). Therefre, the regressin Wald prcess can be cnstructed as W n (τ) = n(r θ(τ) r) [τ(1 τ)r Ω 1 1 Ω 0 Ω 1 1 R ] 1 (R θ(τ) r), where Ω 1 and Ω 0 are cnsistent estimatrs f Ω 1 and Ω 0. If we are interested in testing Rθ(τ) = r ver τ T, we may cnsider, say, the fllwing Klmgrv-Smirnv (KS) type sup-wald test: KSW n = sup W n (τ), τ T If we are interested in testing Rθ(τ) = r at a particular quantile τ = τ 0, a Chi-square test can be cnducted based n the statistic W n (τ 0 ). The limiting distributins are summarized in the fllwing Therem. Therem 5.1. Under the assumptins f Therem 3 and the linear restrictin H 01, W n (τ 0 ) χ 2 q, and KSW n = sup τ T W n (τ) sup Q 2 q (τ), τ T where Q q (τ) = B q (τ) / τ(1 τ) is a Bessel prcess f rder q, where represents the Euclidean nrm. Fr any fixed τ, Q 2 q(τ) χ 2 q is a centered Chi-square randm variable with q-degrees f freedm.

14 Quantile Autregressin 5.2. Testing Fr Asymmetric Dynamics. The hypthesis that θ j (τ), j = 1,, p, are cnstants ver τ (i.e. θ j (τ) = µ j ) can be represented in the frm f H 02 : Rθ(τ) = r by taking R = [0 p 1.I p ] and r = [µ 1,, µ p ], with unknwn parameters µ 1,, µ p. The Wald prcess and assciated limiting thery prvide a natural test fr the hypthesis Rθ(τ) = r when r is knwn. T test the hypthesis with unknwn r, apprpriate estimatr f r is needed. In many ecnmetrics applicatins, a n- cnsistent estimatr f r is available. If we lk at the prcess V n (τ) = [ n R Ω 1 Ω ] 1/2 1 Ω 1 0 1 R (R θ(τ) r), then under H 02, V n (τ) = [ n (12) R Ω 1 1 ] 1/2 [ Ω 0 Ω 1 1 R (R θ(τ) r) n R Ω 1 Ω ] 1/2 1 Ω 1 0 1 R ( r r) B q (τ) f(f 1 (τ)) [ RΩ 1 0 R ] 1/2 Z where Z = lim n( r r). The necessity f estimating r intrduces a drift cmpnent (f(f 1 (τ)) [ RΩ 1 0 R ] 1/2 Z) in additin t the simple Brwnian bridge prcess, invalidating the distributin-free character f the riginal Klmgrv-Smirnv (KS) test. T restre the asympttically distributin free nature f inference, we emply a martingale transfrmatin prpsed by Khmaladze (1981) ver the prcess V n (τ). Dente df(x)/dx as f, and define ġ(r) = (1, ( f/f)(f 1 (r))), and C(s) = 1 s ġ(r)ġ(r) dr, we cnstruct a martingale transfrmatin K n V n (τ) defined as: τ [ 1 ] Ṽ n (τ) = K V n (τ) = V n (τ) ġ n (s) Cn 1 (s) ġ n (r)d V n (r) ds, 0 where ġ n (s) and C n (s) are unifrmly cnsistent estimatrs f ġ(r) and C(s) ver τ T, and prpse the fllwing Klmgrv-Smirnv 2 type test based n the transfrmed prcess: (13) KH n = sup τ T Ṽn(τ). Under the null hypthesis, the transfrmed prcess Ṽn(τ) cnverges t a standard Brwnian mtin. Fr mre discussins f quantile regressin inference based n the martingale transfrmatin apprach, see, Kenker and Xia (2002) and references therein. s 2 A Cramer-vn-Mises type test based n the transfrmed prcess can als be cnstructed and analysed in a similar way.

Rger Kenker and Zhijie Xia 15 We assume the fllwing assumptins n the estimatrs. A.4: There exist estimatrs ġ n (τ), Ω 0 and Ω 1 satisfying: i.: sup τ ġ n (τ) ġ(τ) = p (1), ii.: Ω 0 Ω 0 = p (1), Ω 1 Ω 1 = p (1), n( r r) = O p (1). Therem 5.2. Under the assumptins A.1 - A.4 and the hypthesis H 02, Ṽ n (τ) W q (τ), KH n = sup Ṽn(τ) sup W q (τ), τ T where W q (r) is a q-dimensinal standard Brwnian mtin. The martingale transfrmatin is based n functin ġ(s) which needs t be estimated. There have been quite a few appraches in estimating the scre: f f (F 1 (s)). Prtny and Kenker (1989) studied adaptive estimatin and emplyed kernel-smthing methd in estimating the density and scre functins, unifrm cnsistency f the estimatrs is als discussed. Cx (1985) prpsed an elegant smthing spline apprach t the estimatin f f /f and Ng (1995) prvided an efficient algrithm fr cmputing this scre estimatr. Estimatin f Ω 0 is straightfrward: Ω 0 = n 1 t x tx t. Fr the estimatin f Ω 1, see, inter alia, Kenker and Bassett(1982), Kenker (1994), Pwell (1987), and Kenker and Machad (1999) fr related discussins. 6. Mnte Carl We cnducted a Mnte Carl experiment t examine the effectiveness f inference prcedures based n the QAR methd. T investigate the finite sample perfrmance f QAR based inference prcedures, we examine the empirical size and pwer f the prpsed tests and reprt the representative results in Tables 1-3. The data in ur experiments were generated frm mdel (6), where u t are i.i.d. randm variables. We are particularly interested in whether r nt the time series y t display asymmetric dynamics. Thus, we cnsider quantile autregressin (2) with p = 1 and test the hypthesis that α 1 (τ) = cnstant ver τ. Fr the tests, we cnsider the Klmgrv-Smirnv type test KH n given by (13) fr different sample sizes and different innvatin distributins. We chse T = [0.1, 0.9]. Bth the case where u t are standard nrmal variates and the case that u t are studentt distributed variables with 3 degrees f freedm are cnsidered. The number f repetitins is 1000, and tw sample sizes are examined: n = 100, and n = 300. When α t = cnstant, the empirical rejectin rates gives the size f test. we reprt the sizees f this test fr three chices f α t : (1) α t = 0.95; (2) α t = 0.9; (3) α t = 0.6. The first tw chices f α t (0.95 and 0.9) are large and clse t unity s that the crrespnding time series display cartain degree f (symmetric) persistence. τ T

16 Quantile Autregressin Under the alternatives, the prcesses display asymmetric dynamics. Fr the chice f alternatives, we cnsidered the fllwing fur chices f α t, { 1, ut 0, α t = ϕ 1 (u t ) = 0.8, u t < 0. { 0.95, ut 0, α t = ϕ 2 (u t ) = 0.8, u t < 0. α t = ϕ 3 (u t ) = min{0.5 + F u (u t ), 1} α t = ϕ 4 (u t ) = min{0.75 + F u (u t ), 1} These alternatives deliver prcesses with different types f asymmetric (r lcal) persistency. In particular, when α t = ϕ 1 (u t ), ϕ 3 (u t ), ϕ 4 (u t ), y t display unit rt behavir in the presence f psitive r large values f innvatins, but have a mean reversin tendency with negative shcks. The alternative α t = ϕ 2 (u t ) has lcal t (r weak) unit rt behavir in the presence f psitive f innvatins, and behave mre statinarily when there are negative shcks. The cnstructin f tests uses estimatrs f the density and scre. We estimate the density (r sparsity functin) using the apprach described in the text. Fr the scre functin ġ, we emply the adaptive kernel estimatr f Prtny and Kenker (1989). The density estimatin exerts imprtant influence n the finite sample perfrmance f ur test. Unsuitable bandwidth selectin can prduce pr estimates. Fr this reasn, we pay particular attentin t the bandwidth chice in density estimatin. In the experiments, we cnsider the bandwidth chices suggested by Hall and Sheather (1988) and Bfinger (1975) and rescaled versins f them. A bandwidth rule that Hall and Sheather (1988) suggested based n Edgewrth expansin fr studentized quantiles is h HS = n 1/3 zα 2/3 [1.5s(t)/s (t)] 1/3, where z α satisfies Φ(z α ) = 1 α/2 fr the cnstructin f 1 α cnfidence intervals, and s(t) = ϕ 0 (t) 1. In the absence f ther infrmatin abut the frm f s( ), we plug-in the Gaussian mdel t select bandwidth and btain h HS = n 1/3 z 2/3 α [1.5φ2 (Φ 1 (t))/(2(φ 1 (t)) 2 + 1)] 1/3. Anther bandwidth selectin has been prpsed by Bfinger (1975). The Bfinger bandwidth h B was derived based n minimizing the mean squared errr f the density estimatr and is f rder n 1/5 : h B = n 1/5 [4.5s 2 (t)/(s (t)) 2 ] 1/5. Again, we plug-in the Gaussian density and btain the fllwing bandwidth that has been widely used in practice h B = n 1/5 [4.5φ 4 (Φ 1 (t))/(2(φ 1 (t)) 2 + 1) 2 ] 1/5. The Mnte Carl results indicate that the Hall-Sheather bandwidth prvides a gd lwer bund and the Bfinger bandwidth prvides a reasnable upper bund

Rger Kenker and Zhijie Xia 17 fr bandwidth selectin in testing parameter cnstancy. Fr this reasn, we cnsider bandwidth values between h HS and h B. In particular, we cnsider rescaled versins f h B and h HS (θh B and δh HS where 0 < θ < 1 and δ > 1 are scalars) in ur Mnte Carl and representative results are reprted. Bandwidth values that are cnstant ver the whle range f quantiles are nt recmmended. The sampling perfrmance f tests using a cnstant bandwidth turned ut t be pr, and are inferir than bandwidth chices such as the Hall/Sheather r Bfinger bandwidth that varies ver the quantiles. Fr these reasn, we fcus n bandwidth h B, h HS, θh B, and δh HS. The scre functin was estimated by the methd f Prtny and Kenker (1989) and we chse the Silverman (1986) bandwidth in ur Mnte Carl. Our simulatin results shw that the test is mre affected by the estimatin f the density than that f the scre. Intuitively, the estimatr f the density plays the rle f a scalar and thus has the largest influence. The Mnte Carl results als indicates that the methd f Prtny and Kenker (1989) cupled with the Silverman bandwidth has reasnably gd perfrmance. Table 1 reprts the empirical size and pwer fr the case with Gaussian innvatins and sample size n = 100. Cnsidering the fact that many financial applicatins have ntriusly heavy-tailed behavir we cnsider prcesses with heavy-tailed distributins. Table 2 reprts results when u t are student-t innvatins. The sample size crrespnding t Table 2 is still n = 100. Results in Table 2 cnfirm that, using the quantile regressin based apprach, pwer gain can be btained in the presence f heavy-tailed disturbances. Experiments based n larger sample sizes are als cnductedand Table 3 reprts the size and pwer fr the case with Gaussian innvatins and sample size n = 300. Results in Table 3 is qualitatively similar t that f Table 1, but it als shws that, as the sample sizes increase, the tests d have imprved size and pwer prperties, crrbrating the asympttic thery. In summary, the Mnte Carl results indicate that, by chsing apprpriate bandwidth, the prpsed tests have reasnable size and pwer prperties. The test using a rescaled versin f Bfinger bandwidth (h = 0.6h B ) yields gd perfrmance in all three cases.

18 Quantile Autregressin Table 1: Testing Cnstancy f Cefficient α Bandwidth h = 3h HS h = h HS h = h B h = 0.6h B Empirical Size α t = 0.95 0.073 0.287 0.018 0.056 α t = 0.9 0.073 0.275 0.01 0.046 α t = 0.6 0.07 0.287 0.012 0.052 Empirical Pwer α t = ϕ 1 (u t ) 0.474 0.795 0.271 0.391 α t = ϕ 2 (u t ) 0.262 0.620 0.121 0.234 α t = ϕ 3 (u t ) 0.652 0.939 0.322 0.533 α t = ϕ 4 (u t ) 0.159 0.548 0.046 0.114 u t = N(0,1), n = 100, Number f replicatin =1000 Table 2: Testing Cnstancy f Cefficient α h = 3h HS h = h HS h = h B h = 0.6h B Empirical Size α t = 0.95 0.086 0.339 0.011 0.059 α t = 0.9 0.072 0.301 0.015 0.043 α t = 0.6 0.072 0.305 0.013 0.038 Empirical Pwer α t = ϕ 1 (u t ) 0.556 0.819 0.319 0.444 α t = ϕ 2 (u t ) 0.348 0.671 0.174 0.279 α t = ϕ 3 (u t ) 0.713 0.933 0.346 0.55 α t = ϕ 4 (u t ) 0.284 0.685 0.061 0.162 u t = t(3), n = 100, Number f replicatin =1000

Rger Kenker and Zhijie Xia 19 Table 3: Testing Cnstancy f Cefficient α h = 3h HS h = h HS h = h B h = 0.6h B Empirical Size α t = 0.95 0.081 0.191 0.028 0.049 α t = 0.9 0.098 0.189 0.03 0.056 α t = 0.6 0.097 0.16 0.02 0.045 Empirical Pwer α t = ϕ 1 (u t ) 0.974 0.992 0.921 0.937 α t = ϕ 2 (u t ) 0.831 0.923 0.685 0.763 α t = ϕ 3 (u t ) 0.998 1 0.971 0.989 α t = ϕ 4 (u t ) 0.557 0.897 0.235 0.392 u t = N(0,1), n = 300, Number f replicatin =1000 7. Empirical Applicatins There have been many claims and bservatins that sme ecnmic time series are asymmetric. Fr example, it has been bserved that increases in the unemplyment rate are sharper than declines; firms are mre apt t increase than t decrease in prices. It has als been argued that psitive shcks t the ecnmy may be mre persistent than negative shcks. Fr this reasn, studies have been cnducted n the existence f asymmetric behavir in these series. If an ecnmic time series displays asymmetric dynamics systematically, then apprpriate mdels are needed t incrprate such behavir. In this sectin, we apply the QAR mdel t tw ecnmic time series: unemplyment rates and retail gasline prices in the US. Our empirical analysis indicate that bth series display asymmetric dynamics. 7.1. Unemplyment Rate. Many studies n unemplyment suggest that the respnse f unemplyment t expansinary r cntractinary shcks may be asymmetric. An asymmetric respnse t different types f shcks has imprtant implicatins in plicy. In this sectin, we examine unemplyment dynamics using the prpsed prcedures. The data that we cnsider are quarterly and annual rates f unemplyment in the US. In particular, we lked at (seasnally adjusted) quarterly rates, starting frm the first quarter f 1948 and ending at the last quarter f 2003, with 224 bservatins. and the annual rates are frm 1890 t 1996. Many empirical studies in the unit rt literature have investigated unemplyment rate data. Nelsn and Plsser (1982) studied the unit rt prperty f annual US unemplyment rates in their seminal wrk n furteen macrecnmic time series. Evidence based n the unit rt tests suggests that the series is statinary. This series and ther type unemplyment rates have been ften re-examined in later analysis.

20 Quantile Autregressin We first apply regressin (10) n the unemplyment rates. We use the BIC criterin f Schwarz (1978) and Rissanen (1978) in selecting the apprpriate lag length f the autregressins. The selected lag length is p = 3 fr the annual data and p = 2 fr the quarterly data. The OLS estimatin f the largest autregressive rt is 0.718 fr the annual series and 0.941 fr the quarterly rates. Quantile autregressin was als perfrmed fr each deciles. The estimates f the largest autregressive rt at each quantile is reprted in Table 4. These estimated values are different ver different quantiles, displaying asymmetric dynamics ver the business cycle. In particular, we find that in the presence f negative shcks, the estimated autregressive rt is generally larger. Table 4: The Estimated Largest AR Rt at Each Decile f Unemplyment τ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Annual Data δ 0 (τ) 0.7406 0.7765 0.9293 0.8710 0.8588 0.7933 0.7270 0.6808 0.5991 Quarterly Data δ 0 τ) 0.912 0.908 0.931 0.919 0.951 0.959 0.967 0.962 0.953 We then test asymmetric dynamics using the martingale transfrmatin based Klmgrv-Smirnv prcedure (13) based n quantile autregressin (8). Accrding t the suggestin frm the Mnte Carl results, we chse the rescaled Hall and Sheather (1988) bandwidth 3h HS and the rescaled Bfinger (1975) bandwidth 0.6h B in estimating the density functin. The tests were cnstructed ver τ T = [0.05, 0.95] and results are reprted in Table 5. The empirical results indicate that asymmetric behavir exist in these series. Table 5: The Klmgrv Test sup τ T Ṽ n (τ) Bandwidth 0.6h B 3h HS 5% Critical Values Annual Rate 4.8962 5.1172 4.523 Quarterly Rate 4.4599 5.3637 3.393 7.2. Retail Gasline Price Dynamics. Our secnd applicatin investigates the asympttic price dynamics in the retail gasline market. It has been dcumented that many markets exist asymmetric price dynamics. In this sectin, we apply the QAR madel t the US retail gasline prices and investigate the existence f asymmetric price adjustment. We cnsider weekly data f US regular gasline retail price frm August 20, 1990 t Februry 16, 2004. The sample size is 699. Table 7 reprt the OLS based augmented Dickey-Fuller regressin estimatin results and the ADF tests fr the null hypthesis f a unit rt (again we use the BIC criterin t select the lag length (p = 4) f the autregressins.) The evidence we btain is marginal; the unit rt null is rejected by the cefficient based test ADF α, but can nt be rejected by the t rati based test ADF t.

Rger Kenker and Zhijie Xia 21 We next cnsider quantile regressin based n the ADF mdel (9) t examine the persistency behavir f the gasline price series at varius quantiles. In particular, Table 6 reprts the estimates f the largest autregressive rts δ 0 (τ) at each decile. The evidence based n these pint estimates f the largest autregressive rt at each quantile suggests that the gasline price series has asymmetric dynamics. Frm the table we can see that there exists asymmetry in persistency. The largest autregressive cefficient estimate δ 0 (τ) has different values ver different quantiles, displaying asymmetric dynamics ver the business cycle. In particular, δ 0 (τ) mntnically increases when we mve frm lwer quantiles t higher quantiles. The autregressive cefficient values at the lwer quantiles are relatively small, indicating that the lcal behavir f the gasline price wuld be statinary. Hwever, at higher quantiles, the largest autregressive rt is clse t r even slightly abve unity, cnsequently the time series display unit rt r lcally explsive behavir at upper quantiles. Table 6: The Estimated Largest AR Rt at Each Decile f Gasline Price τ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 δ 0 (τ) 0.948 0.958 0.971 0.980 0.996 1.005 1.016 1.024 1.047 Table 7: Unit Rt Tests Test Statistic 5% Critical Values OLS Based ADF α test 17.1385 14.1 OLS Based ADF t test 2.6731 2.86 sup τ T Ṽ n (τ) 8.34774 5.560 QKS α 35.7941 13.2181 QCM α 320.407 19.7209 We next perfrm frmal tests fr the null hypthesis that the gasline price series has cnstant autregressive cefficents. We apply the martingale transfrmatin based Klmgrv-Smirnv prcedure (13) based n quantile autregressin (2), cnstancy f cefficients is rejected. The calculated Klmgrv-Smirnv statistic (using the rescaled Bfinger (1975) bandwidth 0.6h B is 8.347735 (lag length p = 4), which is larger than the 5% level critical value (5.56). Hwever, taking int accunt f the unit rt behavir under the null, we cnsider the fllwing (cefficient-based) empirical quantile prcess U n (τ) = n( δ 0 (τ) 1), and the Klmgrv-Smirnv (KS) r Cramer-vn-Mises (CvM) type tests: (14) QKS α = sup U n (τ), QCM α = τ T τ T U n (τ) 2 dτ.

22 Quantile Autregressin Using the results f unit rt quantile regressin asympttics prvided by Kenker and Xia (2003), we have, under the unit rt hypthesis, [ 1 1 ] 1 1 (15) U n (τ) U(τ) = B 2 f(f 1 y B (τ)) y dbψ τ. where B w (r) and B τ ψ (r) are limiting prcesses f n 1/2 [nr] y t and n 1/2 [nr] ψ τ (u tτ )). We adpt the apprach f Kenker and Xia (2003) and apprximate the distributins f the limiting variates by resampling methd and cnstruct btstrap tests fr the unit rt hypthesis based n (14). Table 7 reprts the QKS α and QCM α tests fr the null hypthesis f a cnstant unit rt. The 5% level critical values calculated based n the resampling prcedure are als reprted in the table. The cnstant unit rt hypthesis is rejected at 5% level by bth tests. These results, tgether with the pint estimates reprted in Table 6, indicate that the gasline price series has asymmetric adjustment dynamics and thus is nt well characterized as a cnstant cefficient unit rt prcess. 0 0

Rger Kenker and Zhijie Xia 23 8. Appendix: Prfs 8.1. Prf f Therem 1. Giving a p-th rder autregressin prcess (5), we dente E(α j,t ) = µ j, and assume that 1 µ j 0. Let µ = µ 0 /(1 p j=1 µ j), and dente we have y t = y t µ (16) y t = α 1,t y t 1 + + α p,t y t p + v t, where v t = u t + µ p (α l,t µ l ). l=1 It s easy t see that Ev t = 0 and Ev t v s = 0 fr any t s since Eα l,t = µ l and u t are independent. In rder t derive statinarity cnditins fr the prcess y t, we first find an F t -measurable slutin fr (16). We define the p 1 randm vectrs and the p p randm matrix Y t = [y t,, y t p+1 ], V t = [v t, 0,, 0] A t = [ ] Ap 1,t α p,t, I p 1 0 p 1 where A p 1,t = [ α 1,t,..., α p 1,t ] and 0 p 1 is the (p 1)-dimensinal vectr f zers, then [ ] E(V t V t ) = σv 2 0 1 (p 1) = Σ 0 (p 1) 1 0 (p 1) (p 1) and the riginal prcess can be written as By substitutin, we have Y t = A t Y t 1 + V t Y t = V t + A t V t 1 + A t A t 1 V t 2 + [A t A t m+1 ]V t m + [A t A t m ]Y t m 1 where = Y t,m + R t,m Y t,m = m B j V t j, R t,m = B m+1 Y t m 1, and B j = j=0 { j 1 l=0 A t l, j 1. I, j = 0. The statinarity f an F t -measurable slutin fr y t invlves the cnvergence f { m j=0 B jv t j } and {R t,m } as m increases, fr fixed t. Fllwing a similar analysis as Nichlls and Quinn (1982, Chapter 2), We need t verify that vece [ Y t,m Y t,m] cnverges as m. Ntice that B j is independent with V t j and {u t, t = 0, ±1, ±2, }.

24 Quantile Autregressin are independent randm variables, thus, {B j V t j } j=0 is an rthgnal sequence in the sense that E[B j V t j B k V t k ] = 0 fr any j k. Thus vece [ [ ] [ ] m m m ] Y t,m Y t,m = vece ( B j V t j )( B j V t j ) = vece B j V t j V t j B j j=0 j=0 Ntice that vec(abc) = (C A)vec(B), and B k ), we have [ m ] vece B j V t j V t j B j j=0 If we dente = E = E = [ m j=0 ( j 1 [ m j=0 j 1 m j=0 l=0 A = E[A t ] = ( j l=0 A l j=0 ) ( j ) k=0 B k (B j B j )vec(v t j V t j ) ] ) A t l l=0 ( j 1 l=0 = j k=0 (A k ) ] A t l vec(v t j V t j ) E(A t l A t l )vece(v t j V t j ) [ ] µp 1 α p, I p 1 0 p 1 where µ p 1 = [ α 1,..., α p 1 ], then A t = A + Ξ t, where E(Ξ t ) = 0, and then E(A t l A t l ) = E [(A + Ξ t ) (A + Ξ t )] = A A + E(Ξ t Ξ t ) = Ω A [ ] m m vece ( B j V t j )( B j V t j ) = j=0 j=0 m Ω j A vec(σ). The critical cnditin fr the statinarity f the prcess y t is that m j=0 Ωj A cnverges as m. The matrix Ω A may be represented in Jrdan cannical frm as Ω A = P ΛP 1, where Λ has the eigenvalues f Ω A alng its main diagnal. If the eigenvalues f Ω A have mduli less than unity, Λ j cnverges t zer at a gemetric rate. Ntice that Ω j A = P Λj P 1, fllwing a similar analysis as Nichlls and Quinn (1982, Chapter 2), Y t (and thus y t ) is statinary and can be represented as Y t = B j V t j. j=0 The central limit therem then fllws frm Billingsley (1961) (als see Nichlls and Quinn (1982, Therem A.1.4)). j=0

Rger Kenker and Zhijie Xia 25 8.2. Prf f Therem 3. If we dente v = n( θ(τ) θ(τ)), then ρ τ (y t θ(τ) x t ) = ρ τ (u tτ (n 1/2 v) x t ), where u tτ = y t x t θ(τ). Minimizatin f (8) is equivalent t minimizing: (17) Z n (v) = [ ρτ (u tτ (n 1/2 v) x t ) ρ τ (u tτ ) ]. If v is a minimizer f Z n (v), we have v = n( θ(τ) θ(τ)). The bjective functin Z n (v) is a cnvex randm functin. Knight (1989) (als see Pllard (1991) and Knight (1998)) shws that if the finite-dimensinal distributins f Z n ( ) cnverge weakly t thse f Z( ) and Z( ) has a unique minimum, the cnvexity f Z n ( ) implies that v cnverges in distributin t the minimizer f Z( ). We use the fllwing identity: if we dente ψ τ (u) = τ I(u < 0), fr u 0, (18) ρ τ (u v) ρ τ (u) = vψ τ (u) + (u v){i(0 > u > v) I(0 < u < v)} = vψ τ (u) + v 0 {I(u s) I(u < 0)}ds. Thus the bjective functin f minimizatin prblem can be written as [ ρτ (u tτ (n 1/2 v) x t ) ρ τ (u tτ ) ] = (n 1/2 v) x t ψ τ (u tτ ) + 0 (n 1/2 v) x t {I(u tτ s) I(u tτ < 0)}ds We first cnsider the limiting behavir f W n (v) = 0 (n 1/2 v) x t {I(u tτ s) I(u tτ < 0)}ds. Fr cnvenience f asympttic analysis, we dente W n (v) = ξ t (v), ξ t (v) = 0 (n 1/2 v) x t {I(u tτ s) I(u tτ < 0)}ds. We further define ξ t (v) = E{ξ t (v) F t 1 }, and W n (v) = n ξ t(v), then {ξ t (v) ξ t (v)} is a martingale difference sequence. Ntice that u τt = y t x t α(τ) = y t F 1 t 1 (τ)

26 Quantile Autregressin W n (v) = = = (n 1/2 v) x t E{ 0 (n 1/2 v) x t 0 (n 1/2 v) x t Under assumptin A.3, 0 W n (v) = = 1 2n [I(u tτ s) I(u tτ < 0)] F t 1 } [ s+f 1 t 1 (τ) F 1 t 1 (τ) f t 1 (r)dr ] ds [ Ft 1 (s + F 1 t 1(τ)) F t 1 (F 1 n 1/2 v x t 0 s t 1(τ)) f t 1 (F 1 t 1 (τ))sds + p(1) f t 1 (F 1 t 1 (τ))v x t x t v + p(1) By ur assumptins and statinarity f y t, we have W n (v) 1 2 v Ω 1 v ] sds Using the same argument as Herce(1996), the limiting distributin f t ξ t(v) is the same as that f t ξ t(v). Fr the behavir f the first term, n 1/2 n x tψ τ (u tτ ), in the bjective functin, ntice that x t F t 1 and E[ψ τ (u tτ ) F t 1 ] = 0, x t ψ τ (u tτ ) is a martingale difference sequence and thus n 1/2 n x tψ τ (u tτ ) satisfies a central limit therem. Fllwing the arguments f Prtny (1984) and Gutenbrunner and Jurevckva (1992), the autregressin quantile prcess is tight and thus the limiting variate viewed as a randm functin f τ, is a Brwnian bridge ver τ T, n 1/2 x t ψ τ (u tτ ) Ω 1/2 0 B k (τ). Fr each fixed τ, n 1/2 n x tψ τ (u tτ ) cnverges t a q-dimensinal vectr nrmal variate with cvariance matrix τ(1 τ)ω 0. Thus, = Z n (v) [ ρτ (u tτ (n 1/2 v) x t ) ρ τ (u tτ ) ] = (n 1/2 v) x t ψ τ (u tτ ) + v Ω 1/2 0 B k (τ) + 1 2 v Ω 1 v = Z(v) (n 1/2 v) x t 0 {I(u tτ s) I(u tτ < 0)}ds.

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