Quantile Autoregression

Transcription

1 Quantile Autregressin Rger Kenker University f Illinis, Urbana-Champaign University f Minh June 2017 Centercept Lag(y) Rger Kenker (UIUC) QAR Braga / 32

2 Intrductin In classical autregressin mdels y t = αy t 1 + u t, y t = h(y t 1, α) + u t, cnditining cvariates influence nly the lcatin f the cnditinal distributin f the respnse: Respnse = Signal + IID Nise. But why shuld nise always be s nice and well-behaved? Rger Kenker (UIUC) QAR Braga / 32

3 A Mtivating Example tday's max temperature yesterday's max temperature Daily Temperature in Melburne: An AR(1) Scatterplt Rger Kenker (UIUC) QAR Braga / 32

4 Estimated Cnditinal Quantiles f Daily Temperature tday's max temperature yesterday's max temperature Daily Temperature in Melburne: A Nnlinear QAR(1) Mdel Rger Kenker (UIUC) QAR Braga / 32

5 Cnditinal Densities f Melburne Daily Temperature Yesterday's Temp 11 Yesterday's Temp 16 Yesterday's Temp 21 density density density tday's max temperature tday's max temperature tday's max temperature Yesterday's Temp 25 Yesterday's Temp 30 Yesterday's Temp 35 density density density tday's max temperature tday's max temperature tday's max temperature Lcatin, scale and shape all change with y t 1. When tday is ht, tmrrw s temperature is bimdal! Rger Kenker (UIUC) QAR Braga / 32

6 Linear AR(1) and QAR(1) Mdels The classical linear AR(1) mdel y t = α 0 + α 1 y t 1 + u t, with iid errrs, u t : t = 1,, T, implies E(y t F t 1 ) = α 0 + α 1 y t 1 and cnditinal quantile functins are all parallel: Q yt (τ F t 1 ) = α 0 (τ) + α 1 y t 1 with α 0 (τ) = F 1 u (τ) just the quantile functin f the u t s. But isn t this rather bring? What if we let α 1 depend n τ t? Rger Kenker (UIUC) QAR Braga / 32

7 A Randm Cefficient Interpretatin If the cnditinal quantiles f the respnse satisfy: Q yt (τ F t 1 ) = α 0 (τ) + α 1 (τ)y t 1 then we can generate respnses frm the mdel by replacing τ by unifrm randm variables: y t = α 0 (u t ) + α 1 (u t )y t 1 u t iid U[0, 1]. This is a very special frm f randm cefficient autregressive (RCAR) mdel with cmntnic cefficients. Rger Kenker (UIUC) QAR Braga / 32

8 On Cmntnicity Definitin: Tw randm variables X, Y : Ω R are cmntnic if there exists a third randm variable Z : Ω R and increasing functins f and g such that X = f(z) and Y = g(z). If X and Y are cmntnic they have rank crrelatin ne. Frm ur pint f view the crucial prperty f cmntnic randm variables is the behavir f quantile functins f their sums, X, Y cmntnic implies: X+Y(τ) = F 1 X (τ) + F 1 Y (τ) F 1 X and Y are driven by the same randm (unifrm) variable. Rger Kenker (UIUC) QAR Braga / 32

9 The QAR(p) Mdel Cnsider a p-th rder QAR prcess, Q yt (τ F t 1 ) = α 0 (τ) + α 1 (τ)y t α p (τ)y t p Equivalently, we have randm cefficient mdel, y t = α 0 (u t ) + α 1 (u t )y t α p (u t )y t p x t α(u t ). Nw, all p + 1 randm cefficients are cmntnic, functinally dependent n the same unifrm randm variable. Rger Kenker (UIUC) QAR Braga / 32

10 Vectr QAR(1) representatin f the QAR(p) Mdel Y t = µ + A t Y t 1 + V t where µ = [ µ0 0 p 1 ] [ at α, A t = p (u t ) I p 1 0 p 1 a t = [α 1 (u t ),..., α p 1 (u t )], Y t = [y t,, y t p+1 ], v t = α 0 (u t ) µ 0. ] [ vt, V t = 0 p 1 ] It all lks rather cmplex and multivariate, but it is really still nicely univariate and very tractable. Rger Kenker (UIUC) QAR Braga / 32

11 Sluching Tward Asymptpia We maintain the fllwing regularity cnditins: A.1 {v t } are iid with mean 0 and variance σ 2 <. The CDF f v t, F, has a cntinuus density f with f(v) > 0 n V = {v : 0 < F(v) < 1}. A.2 Eigenvalues f Ω A = E(A t A t ) have mduli less than unity. A.3 Dente the cnditinal CDF Pr[y t < y F t 1 ] as F t 1 (y) and its derivative as f t 1 (y), f t 1 is unifrmly integrable n V. Rger Kenker (UIUC) QAR Braga / 32

12 Statinarity Therem 1: Under assumptins A.1 and A.2, the QAR(p) prcess y t is cvariance statinary and satisfies a central limit therem with 1 n (y t µ y ) N ( 0, ω 2 n y), t=1 µ 0 µ y = 1 p j=1 µ, p µ j = E(α j (u t )), j = 0,..., p, ω 2 y = lim 1 n n E[ (y t µ y )] 2. t=1 Rger Kenker (UIUC) QAR Braga / 32

13 Example: The QAR(1) Mdel Fr the QAR(1) mdel, r with u t iid U[0, 1]. Q yt (τ y t 1 ) = α 0 (τ) + α 1 (τ)y t 1, y t = α 0 (u t ) + α 1 (u t )y t 1, if ω 2 = E(α 2 1 (u t)) < 1, then y t is cvariance statinary and 1 n (y t µ y ) N ( 0, ω 2 n y), t=1 where µ 0 = Eα 0 (u t ), µ 1 = E(α 1 (u t ), σ 2 = V(α 0 (u t )), and µ y = µ 0 (1 µ 1 ), ω2 y = (1 + µ 1 )σ 2 (1 µ 1 )(1 ω 2 ), Rger Kenker (UIUC) QAR Braga / 32

14 Qualitative Behavir f QAR(p) Prcesses The mdel can exhibit unit-rt-like tendencies, even temprarily explsive behavir, but episdes f mean reversin are sufficient t insure statinarity. Under certain cnditins,the QAR(p) prcess is a semi-strng ARCH(p) prcess in the sense f Drst and Nijman (1993). The impulse respnse f y t+s t a shck u t is stchastic but cnverges (t zer) in mean square as s. Rger Kenker (UIUC) QAR Braga / 32

15 Estimated QAR(1) v. AR(1) Mdels f U.S. Interest Rates Centercept Lag(y) Data: Seasnally adjusted mnthly: April, 1971 t June, D 3-mnth T-bills really have a unit rt? Rger Kenker (UIUC) QAR Braga / 32

16 Estimatin f the QAR mdel Estimatin f the QAR mdels invlves slving, ˆα(τ) = argmin α n t=1 ρ τ (y t x t α), where ρ τ (u) = u(τ I(u < 0)), the -functin. Fitted cnditinal quantile functins f y t, are given by, ˆQ t (τ x t ) = x t ˆα(τ), and cnditinal densities by the difference qutients, ˆf t (τ x t 1 ) = 2h ˆQ t (τ + h x t 1 ) ˆQ t (τ h x t 1 ), Rger Kenker (UIUC) QAR Braga / 32

17 The QAR Prcess Therem 2: Under ur regularity cnditins, nω 1/2 (ˆα(τ) α(τ)) B p+1 (τ), a (p + 1)-dimensinal standard Brwnian Bridge, with Ω = Ω 1 1 Ω 0Ω 1 1. n Ω 0 = E(x t x t ) = lim n 1 x t x t, Ω 1 = lim n 1 n t=1 t=1 f t 1 (F 1 t 1 (τ))x tx t. Rger Kenker (UIUC) QAR Braga / 32

18 Inference fr QAR mdels Fr fixed τ = τ 0 we can test the hypthesis: H 0 : Rα(τ) = r using the Wald statistic, W n (τ) = n(rˆα(τ) r) [R ˆΩ 1 1 ˆΩ 0 ˆΩ 1 1 R ] 1 (Rˆα(τ) r) τ(1 τ) This apprach can be extended t testing n general index sets τ T with the crrespnding Wald prcess. Rger Kenker (UIUC) QAR Braga / 32

19 Asympttic Inference Therem: Under H 0, W n (τ) Q 2 m(τ), where Q m (τ) is a Bessel prcess f rder m = rank(r). Fr fixed τ, Q 2 m(τ) χ 2 m. Klmgrv-Smirv r Cramer-vn-Mises statistics based n W n (τ) can be used t implement the tests. Fr knwn R and r this leads t a very nice thery estimated R and/r r testing raises new questins. The situatin is quite analgus t gdness-f-fit testing with estimated parameters. Rger Kenker (UIUC) QAR Braga / 32

20 Example: Unit Rt Testing Cnsider the augmented Dickey-Fuller mdel y t = δ 0 + δ 1 y t 1 + p δ j y t j + u t. We wuld like t test this cnstant cefficients versin f the mdel against the mre general QAR(p) versin: j=2 Q yt (τ x t ) = δ 0 (τ) + δ 1 (τ)y t 1 + p δ j (τ) y t j The hypthesis: H 0 : δ 1 (τ) = δ 1 = 1, fr τ T = [τ 0, 1 τ 0 ], is cnsidered in K and Xia (JASA, 2004). j=2 Rger Kenker (UIUC) QAR Braga / 32

21 Example: Tw Tests When δ 1 < 1 is knwn we have the candidate prcess, V n (τ) = n(ˆδ 1 (τ) δ 1 )/ ˆω 11. where ˆω 2 11 is the apprpriate element frm ˆΩ 1 1 ˆΩ 0 ˆΩ 1 1. Fluctuatins in V n (τ) can be evaluated with the Klmgrv-Smirnv statistic, sup V n (τ) sup B(τ). τ T τ T When δ 1 is unknwn we may replace it with an estimate, but this disrupts the cnvenient asympttic behavir. Nw, ˆV n (τ) = n((ˆδ 1 (τ) δ 1 ) (ˆδ 1 δ 1 ))/ ˆω 11 Rger Kenker (UIUC) QAR Braga / 32

22 Martingale Transfrmatin f ˆV n (τ) Khmaladze (1981) suggested a general apprach t the transfrmatin f parametric empirical prcesses like ˆV n (τ) : Ṽ n (τ) = ˆV n (τ) τ 0 [ ġ n (s) C 1 n (s) 1 s ] ġ n (r)d ˆV n (r) ds where ġ n (s) and C n (s) are estimatrs f ġ(r) = (1, (ḟ/f)(f 1 (r))) ; C(s) = 1 s ġ(r)ġ(r) dr. This is a generalizatin f the classical Db-Meyer decmpsitin. Rger Kenker (UIUC) QAR Braga / 32

23 Restratin f the ADF prperty Therem Under H 0, Ṽ n (τ) W(τ) and therefre sup Ṽ n (τ) sup W(τ), τ T τ T with W(r) a standard Brwnian mtin. The martingale transfrmatin f Khmaladze annihilates the cntributin f the estimated parameters t the asympttic behavir f the ˆV n (τ) prcess, thereby restring the asympttically distributin free (ADF) character f the test. Rger Kenker (UIUC) QAR Braga / 32

24 Three Mnth T-Bills Again Centercept Lag(y) A test f the lcatin-shift hypthesis yields a test statistic f 2.76 which has a p-value f rughly 0.01, cntradicting the cnclusin f the cnventinal Dickey-Fuller test. Rger Kenker (UIUC) QAR Braga / 32

25 QAR Mdels fr Lngitudinal Data In estimating grwth curves it is ften valuable t cnditin nt nly n age, but als n prir grwth and pssibly n ther cvariates. Autregressive mdels are natural, but cmplicated due t the irregular spacing f typical lngitudinal measurements. Finnish Height Data: {Y i (t i,j ) : j = 1,..., J i, i = 1,..., n.} Partially Linear Mdel [Pere, Wei, K, and He (2006)]: Q Yi (t i,j )(τ t i,j, Y i (t i,j 1 ), x i ) = g τ (t i,j ) + [α(τ) + β(τ)(t i,j t i,j 1 )]Y i (t i,j 1 ) + x i γ(τ). Rger Kenker (UIUC) QAR Braga / 32

26 Parametric Cmpnents f the Cnditinal Grwth Mdel τ Bys Girls ˆα(τ) ˆβ(τ) ˆγ(τ) ˆα(τ) ˆβ(τ) ˆγ(τ) (0.020) (0.011) (0.011) (0.024) (0.011) (0.020) (0.019) (0.025) (0.029) (0.024) (0.024) (0.007) (0.006) (0.009) (0.009) (0.016) (0.016) (0.007) (0.009) (0.013) (0.017) (0.017) (0.018) (0.022) (0.021) (0.027) (0.027) (0.030) (0.038) (0.007) (0.006) (0.008) (0.012) (0.015) (0.024) (0.010) (0.009) (0.008) (0.009) (0.015) (0.018) (0.027) Estimates f the QAR(1) parameters, α(τ) and β(τ) and the mid-parental height effect, γ(τ), fr Finnish children ages 0 t 2 years. Rger Kenker (UIUC) QAR Braga / 32

27 Frecasting with QAR Mdels Given an estimated QAR mdel, ˆQ yt (τ F t 1 ) = x t ˆα(τ) based n data: y t : t = 1, 2,, T, we can frecast ŷ T +s = x T +sˆα(u s), s = 1,, S, where x T +s = [1, ỹ T +s 1,, ỹ T +s p ], U s U[0, 1], and ỹ t = { yt if t T, ŷ t if t > T. Cnditinal density frecasts can be made based n an ensemble f such frecast paths. Rger Kenker (UIUC) QAR Braga / 32

28 Linear QAR Mdels May Pse Statistical Health Risks Lines with distinct slpes eventually intersect. [Euclid: P5] Quantile functins, Q Y (τ x) shuld be mntne in τ fr all x, intersectins imply pint masses r even wrse. What is t be dne? Cnstrained QAR: Quantiles can be estimated simultaneusly subject t linear inequality restrictins. Nnlinear QAR: Abandn linearity in the lagged yt s, as in the Melburne temperature example, bth parametric and nnparametric ptins are available. Rger Kenker (UIUC) QAR Braga / 32

29 Nnlinear QAR Mdels via Cpulas An interesting class f statinary, Markvian mdels can be expressed in terms f their cpula functins: G(y t, y t 1,, y y p ) = C(F(y t ), F(y t 1 ),, F(y y p )) where G is the jint df and F the cmmn marginal df. Differentiating, C(u, v), with respect t u, gives the cnditinal df, H(y t y t 1 ) = u C(u, v) (u=f(y t ),v=f(y t 1 )) Inverting we have the cnditinal quantile functins, Q yt (τ y t 1 ) = h(y t 1, θ(τ)) Rger Kenker (UIUC) QAR Braga / 32

30 Example 1 (Fan and Fan) x y τ= 0.9 τ= 0.8 τ= 0.7 τ= 0.6 τ= 0.5 τ= 0.4 τ= 0.3 τ= 0.2 τ= 0.1 Mdel: Q yt (τ y t 1 ) = ( τ)y t 1 + Φ 1 (τ). Rger Kenker (UIUC) QAR Braga / 32

31 Example 2 (Near Unit Rt) β(τ) τ Mdel: Q yt (τ y t 1 ) = 2 + min{ τ, 1}y t 1 + 3Φ 1 (τ). Rger Kenker (UIUC) QAR Braga / 32

32 Cnclusins QAR mdels are an attempt t expand the scpe f classical linear time-series mdels permitting lagged cvariates t influence scale and shape as well as lcatin f cnditinal densities. Efficient estimatin via familiar linear prgramming methds. Randm cefficient interpretatin nests many cnventinal mdels including ARCH. Wald-type inference is feasible fr a large class f hyptheses; rank based inference is als an attractive ptin. Frecasting cnditinal densities is ptentially valuable. Many new and challenging pen prblems.... Rger Kenker (UIUC) QAR Braga / 32