Bootstrap prediction intervals for Markov processes

Transcription

1 arxiv: arxiv: Bootstrap prediction intervals for Markov processes Li Pan and Dimitris N. Politis Li Pan Department of Matematics University of California San Diego La Jolla, CA , USA Dimitris N. Politis Department of Matematics University of California San Diego La Jolla, CA , USA Abstract: Given time series data X 1,..., X n, te problem of optimal prediction of X n+1 as been well-studied. Te same is not true, owever, as regards te problem of constructing a prediction interval wit prespecified coverage probability for X n+1, i.e., turning te point predictor into an interval predictor. In te past, prediction intervals ave mainly been constructed for time series tat obey an autoregressive model tat is linear, nonlinear or nonparametric. In te paper at and, te scope is expanded by assuming only tat {X t} is a Markov process of order p 1 witout insisting tat any specific autoregressive equation is satisfied. Several different approaces and metods are considered, namely bot Forward and Backward approaces to prediction intervals as combined wit tree resampling metods: te bootstrap based on estimated transition densities, te Local Bootstrap for Markov processes, and te novel Model-Free bootstrap. In simulations, prediction intervals obtained from different metods are compared in terms of teir coverage level and lengt of interval. Keywords and prases: Confidence intervals, Local Bootstrap, Model-Free Prediction. Corresponding autor: Dimitris N. Politis, Department of Matematics, University of California San Diego, La Jolla, CA , USA; dpolitis@ucsd.edu; tel: ( ; fax: ( Te website ttp:// politis/dpsoftware.tml contains relevant software for te implementation of metods developed in tis paper. 1

2 L. Pan and D. Politis/Bootstrap for Markov processes 2 1. Introduction Prediction is a key objective in time series analysis. Te teory of optimal linear and nonlinear point predictors as been well developed. Te same is not true, owever, as regards te problem of constructing a prediction interval wit prespecified coverage probability, i.e., turning te point predictor into an interval predictor. Even in te related problem of regression, te available literature on prediction intervals is not large; see e.g. Geisser (1993, Carroll and Ruppert (1991, Olive (2007, Olive (2015, Patel (1989, Scmoyer (1992, and Stine (1985. Recently, Politis (2013 as re-cast te prediction problem including prediction intervals in a Model-Free setting. An autoregressive (AR time series model, be it linear, nonlinear, or nonparametric, bears a formal resemblance to te analogous regression model. Indeed, AR models can typically be successfully fitted by te same metods used to estimate a regression, e.g., ordinary Least Square (LS regression metods for parametric models, and scatterplot smooting for nonparametric ones. Tere are several papers for prediction intervals for AR models (typically linear tat represent a broad spectrum of metods; see e.g. Alonso et al. (2002, Box and Jenkins (1976, Breidt et al. (1995, Masarotto (1990, Pascual et al. (2004, Tombs and Scucany (1990, and Wolf and Wunderli (2015. Recently, Pan and Politis (2015 presented a unified approac towards prediction intervals wen a time series {X t } obeys an autoregressive model tat is eiter linear, nonlinear or nonparametric. We expand te scope by assuming only tat {X t } is a Markov process of order p 1 witout insisting tat any specific autoregressive equation is satisfied. Recall tat Pan and Politis (2015 identified two different general approaces towards building bootstrap prediction intervals wit conditional validity, namely te Forward and Backward recursive scemes. We will address bot Forward and Backward approaces in te setting of Markovian data; see Section 2 for details. In terms of te actual resampling mecanism, we will consider te following tree options: 1. Te bootstrap metod based on kernel estimates of te transition density of te Markov processes as proposed by Rajarsi (1990; see Section Te Local Bootstrap for Markov processes as proposed by Paparoditis and Politis (1998 and Paparoditis and Politis (2002; see Section Te Model-Free Bootstrap for Markov Processes; tis is a novel resampling sceme tat stems from te Model-Free Prediction Principle of Politis (2013. To elaborate, te key idea is to transform a given complex dataset into one tat is i.i.d. (independent, identically distributed; aving done tat, te prediction problem is greatly simplified, and tat includes te construction of prediction intervals. In te case of a Markov Process, tis simplification can be accomplised using te Rosenblatt (1952 transformation; see Section 6. In te case of time series tat satisfy an autoregressive equation tat is nonlinear and/or nonparametric, Pan and Politis (2015 noted tat te Backward approac was not generally feasible. Recall tat, under causality, AR models are special cases of Markov processes. Hence, in Section 5 we propose a ybrid approac for nonparametric autoregressions in wic te forward step uses te autoregressive equation explicitly wile te backward step uses one of te tree aforementioned Markov bootstrap procedures. In te following, Section 2 will describe te setting of te prediction problem under consideration, and te construction of bootstrap prediction intervals. All prediction intervals studied in te paper at and are asymptotically valid under appropriate conditions. We will assess and compare te finitesample performance of all te metods proposed via Monte Carlo simulations presented in Section 7. Appendix A is devoted to sowing tat a Markov process remains Markov after a time-reversal; tis is needed to justify te use of all Backward bootstrap approaces. Finally, Appendix B discusses te problem of prediction intervals in r step aead prediction for r 1.

3 L. Pan and D. Politis/Bootstrap for Markov processes 3 2. Prediction and Bootstrap for Markov Processes 2.1. Notation and Definitions Here, and trougout te rest of te paper, we assume tat X = {X t, t = 1, 2, } is a real-valued, strictly stationary process tat is Markov of order p. Letting Y t = (X t, X t 1,, X t p+1, we define F (y = P [Y p y], F (x, y = P [X p+1 x, Y p y], F (x y = P [X p+1 x Y p = y], (2.1 for x R, y R p ; in te above, we ave used te sort-and {Y p y} to denote te event {te it coordinate of Y p is less or equal to te it coordinate of y for all i = 1,..., p}. Let f(y, f(x, y, f(x y be te corresponding densities of te distributions in eq. (2.1. We will assume trougout te paper tat tese densities are wit respect to Lebesgue measure. However, our results in Sections 3 and 4, i.e., bootstrap based on estimated transition densities and Local Bootstrap, could be easily generalized to te case of densities taken wit respect to counting measure; i.e., te case of discrete random variables. Remark 6.4 sows a modification tat also renders te Model-Free bootstrap of Section 6 valid for discrete data. Let X 1 = x 1, X 2 = x 2,, X n = x n denote te observed sample pat from te Markov cain X, and let y n = (x n,, x n p+1. Denote by ˆX n+1 te cosen point predictor of X n+1 based on te data at and. Because of te Markov structure, tis predictor will be a functional of ˆf n ( y n wic is our data-based estimator of te conditional density f( y n. For example, te L 2 optimal predictor would be given by te mean of ˆf n ( y n ; similarly, te L 1 optimal predictor would be given by te median of ˆf n ( y n. To fix ideas in wat follows will will focus on te L 2 optimal predictor, usually approximated by ˆX n+1 = x ˆf n (x y n dx, wit te understanding tat oter functionals of ˆf n ( y n can be accommodated equally well. Remark 2.1. An integral suc as x ˆf n (x y n dx can be calculated by numerical integration, e.g. using te adaptive quadrature metod. However, te L 2 optimal predictor can be approximated in several different ways tat are asymptotically equivalent. Te most straigtforward alternative is a kernel smooted estimator of te autoregression scatterplot, i.e., estimator (5.3 defined in te sequel. Remark 6.2 lists some furter alternative options. Beyond te point predictor ˆX n+1, we want to construct a prediction interval tat will contain X n+1 wit probability 1 α asymptotically; te following definition is elpful. Definition 2.1. Asymptotic validity of prediction intervals. Let L n, U n be functions of te data X 1,, X n. Te interval [L n, U n ] will be called a (1 α100% asymptotically valid prediction interval for X n+1 given X 1,, X n if P (L n X n+1 U n 1 α as n (2.2 for all (X 1,, X n in a set tat as (unconditional probability equal to one. Te probability P in (2.2 sould be interpreted as conditional probability given X 1,, X n altoug it is not explicitly denoted; ence, Definition 2.1 indicates conditional validity of te prediction interval [L n, U n ]. Remark 2.2. Asymptotic validity is a fundamental property but it does not tell te wole story; see Pan and Politis (2015 for a full discussion. To elaborate, one could simply let L n and U n be te

4 L. Pan and D. Politis/Bootstrap for Markov processes 4 α/2 and 1 α/2 of te conditional density estimator ˆf n ( y n respectively. If ˆf n ( y n is consistent for f n ( y n, ten tis interval would be asymptotically valid; neverteless, it would be caracterized by pronounced under-coverage in finite samples since te nontrivial variability in te estimate ˆf n ( y n is ignored. In order to capture te finite-sample variability involved in model estimation some kind of bootstrap algoritm is necessary. Tus, consider a bootstrap pseudo series X1,, Xn constructed according to one of te metods mentioned in te Introduction. Let ˆf n( y n be te corresponding estimator of f( y n as obtained from te bootstrap data X1,, Xn. To acieve conditional validity, we will ensure tat te last p values in te bootstrap world coincide wit te last p values in te real world, i.e., tat (Xn,, Xn p+1 = y n. Finally, we construct te predictor ˆX n+1 using te same functional, i.e., mean, median, etc., as used in te construction of ˆXn+1 in te real world but, of course, tis time te functional is applied to ˆf n( y n. For example, te L 2 optimal predictor in te bootstrap world will be given by ˆX n+1 = x ˆf n(x y n dx. Bootstrap probabilities and expectations are usually denoted by P and E, and tey are understood to be conditional on te original data X 1 = x 1,, X n = x n. Since Definition 2.1 involves conditional validity, we will understand tat P and E are also conditional on Xn p+1 = x n p+1,, Xn = x n wen tey are applied to future events in te bootstrap world, i.e., events determined by {Xs for s > n}; tis is not restrictive since we will ensure tat our bootstrap algoritms satisfy tis requirement. Definition 2.2. Te predictive root is te error in prediction, i.e., X n+1 ˆX n+1. Similarly, te bootstrap predictive root is te error in prediction in te bootstrap world, i.e., X n+1 ˆX n+1. Remark 2.3. Construction of prediction intervals in tis paper will be carried out via approximating te quantiles of te predictive root wit tose of te bootstrap predictive root. To see wy, suppose te (conditional probability P (X n+1 ˆX n+1 a is a continuous function of a in te limit as n. If one can sow tat sup P (X n+1 ˆX n+1 a P (Xn+1 ˆX P n+1 a 0, a ten standard results imply tat te quantiles of P (Xn+1 ˆX n+1 a can be used to consistently estimate te quantiles of P (X n+1 ˆX n+1 a, tus leading to asymptotically valid prediction intervals. Indeed, all prediction intervals tat will be studied in tis paper are asymptotically valid under appropriate conditions. However, as mentioned earlier, it is difficult to quantify asymptotically te extent to wic a prediction interval is able to capture bot sources of variation, i.e., te variance associated wit te new observation X n+1 and te variability in estimating ˆX n+1 ; ence, te prediction intervals in tis paper will be compared via finite-sample simulations. Finally, note tat Pan and Politis (2015 also defined prediction intervals based on studentized predictive roots. For concreteness, in tis paper we will focus on te simple notion of Definition 2.2 but generalization to studentized predictive roots is straigtforward Forward vs. Backward Bootstrap for Prediction Intervals Consider te bootstrap sample X 1,, X n. As mentioned in Section 2.1, in order to ensure conditional validity it would be elpful if te last p values in te bootstrap world coincided wit te last p values in te real world, i.e., tat (X n,, X n p+1 = y n (x n,, x n p+1. For te

5 L. Pan and D. Politis/Bootstrap for Markov processes 5 application to prediction intervals, note tat te bootstrap also allows us to generate X n+1 so tat te statistical accuracy of te predictor ˆX n+1 can be gauged. However, under a usual Monte Carlo simulation, none of te simulated bootstrap series will ave teir last p values exactly equal to te original vector y n as needed for prediction purposes. Herein lies te problem, since te beavior of te predictor ˆX n+1 needs to be captured conditionally on te original vector y n. To avoid tis difficulty in te set-up of a linear AR(p model, Tombs and Scucany (1990 proposed to generate te bootstrap data X 1,, X n going backwards from te last p values tat are fixed at (X n,, X n p+1 = y n ; tis is te backward bootstrap metod tat was revisited by Breidt et al. (1995 wo gave te correct algoritm of finding te backward errors. Note tat te generation of X n+1 must still be done in a forward fasion using te fitted AR model conditionally on te value X n. Going beyond te linear AR(p model, a backward bootstrap for Markov processes was proposed by Paparoditis and Politis (1998 via teir notion of Local Bootstrap. We will elaborate on te backward Local Bootstrap and oter backward bootstrap metods for Markov processes in te sequel. A key result ere it tat a Markov process remains Markov after a time-reversal; see our Appendix A. Neverteless, te natural way Markov processes evolve is forward in time, i.e., one generates X t given X t 1, X t 2,..., X t p. Tus, it is intuitive to construct bootstrap procedures tat run forward in time, i.e., to generate X t given X t 1, X t 2,..., X t p. Indeed, most (if not all of te literature on bootstrap confidence intervals for linear AR models uses te natural time order to generate bootstrap series. However, recall tat predictive inference is to be conducted conditionally on te last p values given by y n in order to be able to place prediction bounds around te point predictor ˆX n+1. In order to maintain te natural time order, i.e., generate bootstrap series forward in time, but also ensure tat X n+1 is constructed correctly, i.e., conditionally on te original y n, Pan and Politis (2015 introduced te forward bootstrap metod for prediction intervals, tat comprises of te following two steps. In describing it, we will use te notion of fitting a Markov model by estimating te transition density f(x y as will be discussed in detail in Section 3; different notions of Markov bootstrap, e.g., te Local Bootstrap, work analogously. A. Coose a starting vector (X1 p, X2 p,..., X0 appropriately, e.g., coose it at random as one of te stretces (subseries of lengt p found in te original data X 1,, X n. Ten, use te fitted Markov model, i.e., use te estimated transition density ˆf n (x y, in order to generate bootstrap data Xt recursively for t = 1,..., n. Now re-fit te Markov model using te bootstrap data X1,, Xn, i.e., obtain ˆf n(x y as an estimate of te transition density. B. Re-define te last p values in te bootstrap world, i.e., let (Xn,, Xn p+1 = y n, and generate te future bootstrap observation Xn+1 by a random draw from density ˆf n( y n. Also compute te one-step aead bootstrap point predictor ˆX n+1 = x ˆf n(x y n dx. Note tat te forward bootstrap idea as been previously used for prediction intervals in linear AR models by Masarotto (1990 and Pascual et al. (2004 but wit some important differences; for example, Masarotto (1990 omits te important step B above. Pan and Politis (2015 found tat te forward bootstrap is te metod tat can be immediately generalized to apply for nonlinear and nonparametric autoregressions as well, tus forming a unifying principle for treating all AR models; indeed, for nonlinear and/or nonparametric autoregressions te backward bootstrap seems infeasible. Neverteless, as will be sown in te next two sections, te backward bootstrap becomes feasible again under te more general setup of Markov process data. In Section 5 we will return briefly to te setup of a nonlinear and/or nonparametric autoregression and propose a ybrid approac in wic te forward step uses te autoregressive equation explicitly wile te backward step uses one of te tree Markov bootstrap procedures mentioned in te Introduction.

6 L. Pan and D. Politis/Bootstrap for Markov processes 6 3. Bootstrap Based on Estimates of Transition Density Rajarsi (1990 introduced a bootstrap metod tat creates pseudo-sample pats of a Markov process based on an estimated transition density; tis metod can form te basis for a forward bootstrap procedure for prediction intervals. Since te time-reverse of a Markov cain is also a Markov cain see Appendix A, it is possible to also define a backward bootstrap based on an estimated backward transition density Forward Bootstrap Based on Transition Density Recall tat x 1, x 2,, x n is te observed sample pat from te Markov cain X, and y t = (x t,, x t p+1. In wat follows, te prase generate z f( will be used as sort-and for generate z by a random draw from probability density f(. Algoritm 3.1. Forward Bootstrap (1 Coose a probability density K on R 2 and positive bandwidts 1, 2 to construct te following kernel estimators: ˆf n (x, y = ˆf n (y = 1 n K( x x i, y y i 1 (n p 1 2 i=p (3.1 ˆf n (x, ydx (3.2 ˆf n (x y = ˆf n (x, y ˆf n (y (3.3 for all x R, y R p, and were is a norm in R p. (2 Calculate te point predictor ˆx n+1 = x ˆf n (x y n dx. (3 (a Generate y p = (x p,, x 1 wit probability density function ˆf n ( given by (3.2. Alternatively, let y p be one of te stretces of p observations tat are present as a subset of te original series x 1,..., x n ; tere are n p + 1 suc stretces coose one of tem at random. (b Generate x p+1 ˆf n ( y p given by (3.3. (c Repeat (b to generate x t+1 ˆf n ( y t for t = p,, n 1, were y t = (x t,, x t p+1. (d Construct ˆf n(x y in a similar way as in (3.3 wit te same kernel and bandwidts but based on te pseudo-data x 1, x 2,, x n instead of te original data. (e Calculate te bootstrap point predictor ˆx n+1 = x ˆf n(x y n dx. (f Generate te bootstrap future value x n+1 ˆf n ( y n. (g Calculate te bootstrap root replicate as x n+1 ˆx n+1. (4 Repeat (3 B times; te B bootstrap root replicates are collected in te form of an empirical distribution wose α-quantile is denoted q(α. (5 Te (1 α100% equal-tailed, bootstrap prediction interval for X n+1 is given by [ˆx n+1 + q(α/2, ˆx n+1 + q(1 α/2]. (3.4

7 L. Pan and D. Politis/Bootstrap for Markov processes Backward Bootstrap Based on Transition Density Letting Y t = (X t, X t 1,, X t p+1 and x R, y R p as before, we can define te backwards transition distribution as F b (x y = P [X 0 x Y p = y] wit corresponding density f b (x y. Similarly, we define te backwards joint distribution as F b (x, y = P [X 0 x, Y p y] wit corresponding density f b (x, y. Having observed a sample pat x 1, x 2,, x n of a Markov cain, Appendix A sows tat te time-reversed sample-pat x n, x n 1,, x 1 can be considered as a sample pat of anoter Markov cain wit transition distribution and density given by F b (x y = P [X 0 x Y p = y] and f b (x y respectively. Note tat te densities f b (x, y and f b (x y admit kernel estimators as follows: ˆf bn (x, y = 1 (n p 1 2 ˆf bn (x y = ˆf bn (x, y ˆf bn (y n i=p+1 K( x x i p, y y i ( Te above equation can be used to form an alternative estimator of te unconditional density f(y, namely ˆf bn (y = ˆf bn (x, ydx. Te algoritm for backward bootstrap based on transition density is very similar to tat of te corresponding forward bootstrap. Te only difference is in Step (3 were we generate te pseudo series (x 1,, x n in a time-reversed fasion. Te backward bootstrap algoritm is described below were te notation y t = (x t,, x t p+1 is again used. Algoritm 3.2. Backward Bootstrap (1-(2 Same as te steps in Algoritm 3.1. (3 (a Let y n = y n. (b Generate x n p ˆf bn ( y n = y n (3.6 (c Repeat (b going backwards in time to generate x t ˆf bn ( y t+p for t = n p, n p 1,, 1. (d Generate bootstrap future value x n+1 ˆf n ( y n. [Note: tis is again going forward in time, using te forward transition density exactly as in te Forward Bootstrap Algoritm 3.1.] (e Construct ˆf n(x y in a similar way as in (3.3 wit te same kernel and bandwidts but based on te pseudo-data x 1, x 2,, x n instead of te original data. (f Calculate te bootstrap root replicate as x n+1 ˆx n+1. (4-(5 Same as te steps in Algoritm Asymptotic Properties For simplicity, in tis section we focus on a Markov sequence X of order one, i.e., p = 1. Te following tecnical assumptions are needed to ensure asymptotic validity of te prediction intervals of Sections 3.1 and 3.2. (α 1 X = {X 1, X 2, } forms an aperiodic, strictly stationary and geometrically ergodic and φ mixing Markov cain on (R, B were B is Borel-σ algebra over R.

8 L. Pan and D. Politis/Bootstrap for Markov processes 8 (α 2 F (y, F (x, y and F (x y defined in eq. (2.1 wit p = 1 are absolutely continuous, and ave uniformly continuous and bounded densities f(y, f(x, y and f(x y respectively. (α 3 Assume a compact subset S R exists suc tat f(y δ > 0 for eac y S. Also assume X t S for all t 1. Remark 2.1 of Rajarsi (1990 provides a discussion on te (nonrestrictiveness of assumption α 3 wic sould ave little effect in practice. Let K(x, y be an appropriately cosen probability density on R 2 ; also let 1 = 2 = for simplicity. Te required conditions on te kernel K and te bandwidt are specified in assumptions (β 1 -β 3. (β 1 K(x, y is uniformly continuous in (x, y, and K(x, y 0 as (x, y. (β 2 K(x, y is of bounded variation on S S. (β 3 As n, we ave = (n 0, n and m=1 mk+1 (m 4(k+1 < for some k 3. Under assumptions (α 1 -(α 3 and (β 1 -(β 3, te following results are proved by Rajarsi (1990. sup ˆf n (x, y f(x, y 0 a.s. (3.7 x,y sup ˆf n (x f(x 0 a.s. (3.8 x sup ˆf n (x y f(x y 0 a.s. (3.9 x,y Focusing on te forward bootstrap, te above tree equations are enoug to sow tat sup P (X n+1 x P (Xn+1 x 0 a.s. (3.10 x To argue in favor of asymptotic validity by appealing to Remark 2.3, we ave to center te distributions appearing in eq. (3.10. Recall tat te predictor of future value is ˆX n+1 = x ˆf n (x y n dx, and te bootstrap predictor is ˆX n+1 = x ˆf n(x y n dx. Now it is not ard to sow tat ˆX n+1 xf(x yn dx a.s. and ˆX n+1 xf(x y n dx as well; details can be found in Pan (2013. Terefore, it follows tat ˆX n+1 ˆX n+1 0 a.s., and appealing to Remark 2.3 we ave te following. Corollary 3.1. Under assumptions (α 1 -(α 3 and (β 1 -(β 3, te prediction interval constructed from te forward bootstrap of Algoritm 3.1 is asymptotically valid. As Appendix A sows, te time-reverse of a Markov process is also a Markov process. Hence, similar arguments leading to Corollary 3.1 can be used to prove te following. Corollary 3.2. Under assumptions (α 1 -(α 3 and (β 1 -(β 3, te prediction interval constructed from te backward bootstrap of Algoritm 3.2 is asymptotically valid. Remark 3.1. [On bandwidt coice] Bandwidt coice is as difficult as it is important in practice. Rajarsi (1990 used te bandwidt coice = 0.9An 1/6 were A = min(ˆσ, IQR 1.34, ˆσ is te estimated standard deviation of te data, and IQR is te interquartile range. However, our simulations indicated tat suc a bandwidt coice typically gives prediction intervals tat exibit significant over-coverage. Note tat te last requirement of assumption (β 3 implies tat n 4 0; tis convergence can be very slow provided k is large but in any case te order of sould be at most O(n 1/4. Terefore, te practical bandwidt coice was modified to = 0.9An 1/4. Te cross-validation metod for bandwidt selection is not recommended ere as it results into an of order n 1/5.

9 4. Te Local Boostrap for Markov processes L. Pan and D. Politis/Bootstrap for Markov processes 9 Te Local Bootstrap for Markov processes was proposed by Paparoditis and Politis (2002. Altoug it still assumed tat te random variables X 1, X 2, are continuous, possessing probability densities, te Local Boostrap generates bootstrap sample pats based on an estimated transition distribution function tat is a step function as opposed to generating bootstrap sample pats based on an estimated transition density as in Section 3; in tat sense, Rajarsi s (1990 metod is to te Local Boostrap wat te smooted bootstrap for i.i.d. data is to Efron s (1979 original bootstrap tat resamples from te empirical distribution function Forward Local Bootstrap As before, let x 1,, x n be te observed sample pat or a Markov cain of order p, and let Y t = (X t, X t 1,, X t p+1 and y t = (x t, x t 1,, x t p+1. Following Paparoditis and Politis (2002, te estimator of te one-step aead transition distribution function will be given by te weigted empirical distribution n 1 j=p F n (x y = 1 (,x](x j+1 W b (y y j n 1 m=p W (4.1 b(y y m were W b ( = 1/bW ( /b wit W ( being a bounded, Lipscitz continuous and symmetric probability density kernel in R p, and b > 0 is a bandwidt parameter tending to zero. Te Local Boostrap generation of pseudo-data will ten be based on te estimated conditional distribution ˆF n (x y. However, since te latter is a step-function, i.e., it is te distribution of a discrete random variable, in wat follows it is easier to work wit te probability mass function associated wit tis discrete random variable. Algoritm 4.1. Forward Local Bootstrap (1 Coose a resampling kernel W and bandwidt b; ere b can be selected by cross validation. Ten calculate te predictor ˆx n+1 as n 1 j=p W b(y n y j x j+1 n 1 m=p W. b(y n y m (2 (a Set starting value y p to a subseries consisting of any consecutive p values from {x 1,, x n }. (b Suppose x 1,, x t for t p ave already been generated. Let J be a discrete random variable taking its values in te set {p,, n 1} wit probability mass function given by Ten let x t+1 = x J+1 for t = p. P (J = s = W b (yt y s n 1 m=p W b(yt y m. (c Repeat (b for t = p + 1, p + 2,... to generate x p+1,, x n. (d Calculate te bootstrap predictor ˆx n+1 as n 1 j=p W b(y n yj x j+1 n 1 m=p W, b(y n ym were y t = (x t,, x t p+1.

10 L. Pan and D. Politis/Bootstrap for Markov processes 10 (e Re-define yn = y n, and ten generate x n+1 = x J+1 as in step (b, were J is a discrete random variable taking its values in te set {p,, n 1} wit probability mass function given by W b (y n y s P (J = s = n 1 m=p W b(y n y m. (f Calculate te bootstrap prediction root replicate as x n+1 ˆx n+1 (3 Repeat step (2 B times; te B bootstrap root replicates are collected in te form of an empirical distribution wose α-quantile is denoted q(α. (4 Te (1 α100% equal-tailed, forward Local Bootstrap prediction interval for X n+1 is given by [ˆx n+1 + q(α/2, ˆx n+1 + q(1 α/2] Backward Local Bootstrap As sown in Appendix A, te time-reverse of a Markov process is also a Markov process. So in tis section we will employ te Backward Local Bootstrap tat was introduced by Paparoditis and Politis (1998 for te purpose of constructing prediction intervals; teir example was a first order autoregressive process wit conditionally eteroscedastic errors, i.e., te model X t = φx t 1 + ɛ t α 0 + α 1 X 2 t 1 were {ɛ t} are i.i.d wit mean 0 and variance 1. We will now generalize tis idea to te Markov(p case; in wat follows, te Backward Local Bootstrap employs an estimate of te backward conditional distribution given by n p j=1 F bn (x y = 1 (,x](x j W b (y y j+p n p m=1 W b (y y (4.2 m+p were b is te backward bandwidt wic can be different from te forward bandwidt b. Algoritm 4.2. Backward Local Bootstrap (1 Same as te Forward Local Bootstrap in Algoritm 4.1. (2 (a Set starting value y n = y n. (b Suppose yt+p as already been generated were 1 t n p. Let J be a discrete random variable taking its values in te set {1, 2,, n p} wit probability mass function given by W b (yt+p y s+p P (J = s = n p m=1 W b (y t+p y m+p. Ten let x t = x J. (c Repeat (b to generate x n p,, x 2, x 1 backwards in time, i.e., first generate x n p, ten generate x n p 1, etc. (d Let J be a discrete random variable taking its values in te set {p, p + 1,, n 1} wit probability mass function given by P (J = s = W b (y n y s n 1 m=p W b(y n y m. Ten let x n+1 = x J+1. [Note: tis is again going forward in time exactly as in te Forward Local Bootstrap Algoritm 4.1.]

11 L. Pan and D. Politis/Bootstrap for Markov processes 11 (e Calculate te bootstrap predictor ˆx n+1 by n 1 j=p W b(y n y j x j+1 n 1 m=p W b(y n y m (f Calculate te bootstrap prediction root replicate as x n+1 ˆx n+1. (3-(4 Same as te steps of te Forward Local Bootstrap Algoritm Asymptotic Properties Paparoditis and Politis (2002 under teir assumptions (A1-(A3 and (B1-(B2 proved tat sup F n (x y F (x y 0 a.s. (4.3 x,y were te transition distribution estimator F (x y was defined in (4.1; tis is sufficient to sow eq. (3.10 for te Forward Local Bootstrap. Te argument regarding ˆX n+1 ˆX n+1 0 is similar as in Section 3.3, and ence asymptotic validity follows. Corollary 4.1. Under assumptions (A1-(A3 and (B1-(B2 of Paparoditis and Politis (2002, te prediction interval constructed from te Forward Local Bootstrap of Algoritm 4.1 is asymptotically valid. Due to Appendix A, it is easy to see tat te same consistency can be sown for te backwards transition distribution estimator F bn (y x defined in (4.2, i.e., tat sup F bn (x y F b (x y 0 a.s. x,y from wic asymptotic validity of te Backward Local Bootstrap follows. Corollary 4.2. Under assumptions (A1-(A3 and (B1-(B2 of Paparoditis and Politis (2002, te prediction interval constructed from te Backward Local Bootstrap of Algoritm 4.2 is asymptotically valid. 5. Hybrid Backward Markov Bootstrap for Nonparametric Autoregression In tis section only, we will consider te special case were our Markov(p process is generated via a nonparametric autoregression model, i.e., one of te two models below: AR wit omoscedastic errors: AR wit eteroscedastic errors: X t = m(x t 1,..., X t p + ɛ t wit ɛ t i.i.d.(0, σ 2 (5.1 X t = m(x t 1,..., X t p + σ(x t 1,..., X t p ɛ t wit ɛ t i.i.d.(0, 1. (5.2 As before, we assume tat {X t } is strictly stationary; we furter need to assume causality, i.e., tat ɛ t is independent of {X t 1, X t 2,...} for all t. As usual, te recursions (5.1 and (5.2 are meant to

12 L. Pan and D. Politis/Bootstrap for Markov processes 12 run forward in time, i.e., X p+1 is generated given an initial assignment for X 1,..., X p ; ten, X p+2 is generated given its own p-past, etc. Recently, Pan and Politis (2015 presented a unified approac for prediction intervals based on forward, model-based resampling under one of te above two models. It was noted tat a backward model-based bootstrap is not feasible except in te special case were te conditional expectation function m(x t 1,..., X t p is affine in its arguments, i.e., a linear AR model; see e.g. Breidt et al. (1995 and Tombs and Scucany (1990. Using te ideas presented in Sections 3 and 4, we can now propose a ybrid Backward Markov Bootstrap for Nonparametric Autoregression in wic forward resampling is done using te model, i.e., eq. (5.1 or (5.2, wereas te backward resampling is performed using te Markov property only Hybrid Backward Markov Bootstrap Algoritms Consider Markov Processes generated from eiter omoscedastic model (5.1 or eteroscedastic model (5.2. Given a sample {x 1, x 2,, x n }, te algoritms of ybrid backward Markov bootstrap based on transition density for nonparametric model wit i.i.d errors and eteroscedastic errors are described in Algoritm 5.1 and 5.2 respectively; te corresponding algoritms based on Local Bootstrap are described in Algoritm 5.3 and 5.4. Algoritm 5.1. Hybrid backward Backward Markov bootstrap based on transition density for nonparametric model wit omoscedastic errors (1 Select a bandwidt and construct te kernel estimates ˆm(y i for i = p,, n 1, were ˆm(y i = n 1 yi yt t=p K( x t+1 n 1 yi yt t=p K( ; (5.3 as before, y t = (x t, x t 1,, x t p+1, and is a norm in R p. (2 Compute te residuals: ˆɛ i = x i ˆm(y i 1, for i = p + 1,, n. (3 Center te residuals: ˆr i = ˆɛ i (n p 1 n t=p+1 ˆɛ t, for i = p + 1,, n; let te empirical distribution of ˆr t denoted by ˆF ɛ. (a Construct te backward transition density estimate ˆf bn as eq. (3.6. (b Let y n = y n. (c Generate x n p ˆf bn ( y n = y n. Repeat it to generate x t ˆf bn ( y t+p for t = n p,, 1 backwards in time, i.e., first for t = n p, ten for t = n p 1, etc. (d Compute te future bootstrap observation X n+1 by te AR formula X n+1 = ˆm(y n + ɛ n+1 = ˆm(y n + ɛ n+1, were ɛ n+1 is generated from ˆF ɛ. Ten re-estimate m( based on te pseudo data, i.e., ˆm (y = and let ˆX n+1 = ˆm (y n = ˆm (y n n 1 i=p K( y y i n 1 i=p K( y y i x i+1,

13 L. Pan and D. Politis/Bootstrap for Markov processes 13 (e Calculate te bootstrap root replicate as X n+1 ˆX n+1. (4 Steps (a-(e in te above are repeated B times, and te B bootstrap root replicates are collected in te form of an empirical distribution wose α-quantile is denoted q(α. (5 Ten, a (1 α100% equal-tailed predictive interval for X n+1 is given by [ ˆm(x n + q(α/2, ˆm(x n + q(1 α/2] Algoritm 5.2. Hybrid backward Markov bootstrap based on transition density for nonparametric model wit eteroscedastic errors (1 Select te bandwidt and construct te estimates { ˆm(y i, ˆσ(y i } for i = p,, n 1, were (2 Compute te residuals: ˆσ 2 (y i = ˆm(y i = n 1 t=1 n 1 yi yt t=p K( x t+1 n 1 yi yt t=p K( yi yt K( n 1 t=1 (x t+1 ˆm(y t 2. yi yt K( ˆɛ i = x i ˆm(y i 1, ˆσ(y i 1 for i = p + 1,, n. (3 Tis step is similar as step (3 in Algoritm 5.1; te only difference is in step (d wen te future bootstrap observation X n+1 is computed as follows: X n+1 = ˆm g (y n + ˆσ g (y nɛ n+1 = ˆm g (y n + ˆσ g (y n ɛ n+1. Here ˆm g and ˆσ g are over-smooted estimates of m and g computed in te same way as ˆm and ˆσ but using bandwidt g tat is bigger tan ; see Pan and Politis (2015 for a discussion. (4-(5 Same as tose in Algoritm 5.1. Algoritm 5.3. Hybrid backward local bootstrap for nonparametric model wit omoscedastic errors Te algoritm is identical to Algoritm 5.1 wit te exception of steps 3 (a to (c tat ave to be canged as follows. (3 (a Select a resampling bandwidt b and kernel W. (b Let y n = y n Suppose y t+p ave already been generated for 1 t n p. Let J be a discrete random variable taking its values in te set {1, 2,, n p} wit probability mass function given by, (c Ten x t = x J. P (J = s = W b (yt+p y s+p n p m=1 W b (y t+p y m+p, Repeat (b to generate x n p,, x 2, x 1 backwards in time. Algoritm 5.4. Hybrid backward local bootstrap for nonparametric model wit eteroscedastic errors

14 L. Pan and D. Politis/Bootstrap for Markov processes 14 (1 (2 Same as te corresponding steps in Algoritm 5.2 (3 (a (c Same as steps (a (c in Algoritm 5.3 (d (e Same as te corresponding steps in Algoritm 5.2 (4 (5 Same as te corresponding steps in Algoritm 5.2 Remark 5.1. Te ybrid algoritms use a model-based resampling based on fitted residuals. As discussed by Pan and Politis (2015, usage of predictive residuals may be preferable. According to te two models (5.1 or (5.2, te predictive residuals are respectively defined as ˆɛ (t t = x t ˆm (t t (y t 1 or ˆɛ (t t = xt ˆm(t t (y t 1 ˆσ (t t (y t 1 were ˆm (t and ˆσ (t are smooting estimators calculated from te original dataset aving te t-t point deleted. Finally, to define ybrid backward bootstrap intervals based on predictive residuals we just need to replace te fitted residuals {ˆɛ i } in step (2 of Algoritms by te predictive residuals {ˆɛ (t t }. 6. Bootstrap Prediction Intervals for Markov Processes Based on te Model-Free Prediction Principle We now return to te setup of data from a general Markov process tat does not necessarily satisfy a model equation suc as (5.1 or (5.2. In wat follows, we will introduce te Model-Free Bootstrap for Markov Processes; tis is a novel approac tat stems from te Model-Free Prediction Principle of Politis (2013. Te key idea is to transform a given complex dataset into one tat is i.i.d., and terefore easier to andle; aving done tat, te prediction problem is greatly simplified, and tis includes te construction of prediction intervals. In te case of a Markov Process, tis simplification can be practically accomplised using te Rosenblatt (1952 transformation. Instead of generating one-step aead pseudo data by some estimated conditional distribution, e.g. te transition density given in (3.3 or te transition distribution function given in (4.1, te Model-Free Bootstrap resamples te transformed i.i.d. data, and ten transforms tem back to obtain te desired one-step aead prediction. Note tat te bootstrap based on kernel estimates of te transition density of Section 3, and te Local Bootstrap of Section 4 can also be considered model-free metods as tey apply in te absence of a model equation suc as (5.1 or (5.2. Te term Model-Free Bootstrap specifically refers to te transformation approac stemming from te Model-Free Prediction Principle of Politis ( Teoretical Transformation Let X = {X 1, X 2, } be a stationary time series from a Markov process of order p, and Y t 1 = (X t 1,, X t p. Given Y t 1 = y R p, we denote te conditional distribution of X t as D y (x = P (X t x Y t 1 = y. (6.1 Tis is te same distribution discussed in eq. (2.1; canging te notation will elp us differentiate between te different metods. For some positive integer i p, we also define te distributions wit partial conditioning as follows D y,i (x = P (X t x Y (i t 1 = y (6.2 were Y (i t 1 = (X t 1,, X t i and y R i. In tis notation, we can denote te unconditional distribution as D y,0 (x = P (X t x wic does not depend on y. Trougout tis section, we assume tat, for any y and i, te function D y,i ( is continuous and invertible over its support.

15 L. Pan and D. Politis/Bootstrap for Markov processes 15 A transformation from our Markov(p dataset X 1,, X n to an i.i.d. dataset η 1,, η n can now be constructed as follows. Let η 1 = D y,0 (X 1 ; η 2 = D Y (1 1,1 (X 2; η 3 = D Y (2 2,2 (X 3; ; η p = D Y (p 1 p 1,p 1 (X p (6.3 and η t = D Yt 1 (X t for t = p + 1, p + 2,, n. (6.4 Note tat te transformation from te vector (X 1,, X m to te vector (η 1,, η m is one-to-one and invertible for any natural number m by construction. Hence, te event {X 1 = x 1,, X t = x m } is identical to te event {η 1 = ζ 1,, η t = ζ m } wen te construction of ζ t follows (6.3 and (6.4, i.e., ζ 1 = D y,0 (x 1 ; ζ 2 = D y (1 1,1(x 2; ζ 3 = D y (2 2,2(x 3; ; ζ p = D Y (p 1 p 1,p 1 (x p (6.5 and ζ t = D yt 1 (x t for t = p + 1, p + 2,, n (6.6 were y t 1 = (x t 1,, x t p and y (i t 1 = (x t 1,, x t i. It is not difficult to see tat te random variables η 1,, η n are i.i.d. Uniform(0,1; in fact, tis is an application of te Rosenblatt (1952 transformation in te case of Markov(p sequences. For example, te fact tat η 1 is Uniform(0,1 is simply due to te probability integral transform. Now for t > p, we ave P (η t z η t 1 = ζ t 1,..., η 1 = ζ 1 = P (η t z X t 1 = x t 1,..., X 1 = x 1 = P (η t z Y t 1 = y t 1, X t p 1 = x t p 1,..., X 1 = x 1 by te discussion preceding eq. (6.5. Letting y be a sort-and for y t 1, we ave: P (η t z η t 1 = ζ t 1,..., η 1 = ζ 1 =P (D y (X t z Y t 1 = y, X t p 1 = x t p 1,..., X 1 = x 1 =P (X t D 1 y (z Y t 1 = y (by te Markov property =D y (Dy 1 (z =z (wic is uniform and does not depend on y. Hence, P (η t z η t 1 = ζ t 1,..., η 1 = ζ 1 = z, i.e., for t > p, η t is a random variable tat is independent of te its own past and as a Uniform (0,1 distribution. Te same is true for η t wit 1 < t < p; te argument is similar to te above but using te D y,t ( distribution instead of D y (. All in all, it sould be clear tat te random variables η 1,, η n are i.i.d. Uniform(0, Estimating te Transformation from Data To estimate te teoretical transformation from data, we would need to estimate te distributions D y,i ( for i = 0, 1,..., p 1 and D y (. Note, owever, tat D y,i ( for i < p can in principle be computed from D y ( since te latter uniquely specifies te wole distribution of te stationary Markov process. Hence, it sould be sufficient to just estimate D y ( from our data. Anoter way of seeing tis is to note tat te p variables in eq. (6.3 can be considered as edge effects or initial conditions ; te crucial part of te transformation is given by eq. (6.4, i.e., te one based on D y (. Given observations x 1,, x n, we can estimate D y (x by local averaging metods suc as te kernel estimator n i=p+1 ˆD y (x = 1 {x i x}k( y yi 1 n k=p+1 K(. (6.7 y y k 1

16 L. Pan and D. Politis/Bootstrap for Markov processes 16 Note tat ˆD y in (6.7 is a step function; we can use linear interpolation on tis step function to produce an estimate D y tat is piecewise linear and strictly increasing (and terefore invertible; see Politis (2010, Section 4.1 for details on tis linear interpolation. Consequently, we define u t = D yt 1 (x t, for t = p + 1,, n. (6.8 Estimator ˆD y is consistent for D y under regularity conditions; see eq. (4.3 and te associated discussion. Furtermore, te consistency of D y follows from its close proximity to ˆD y ; see te related discussion in Politis (2010. It ten follows tat u t η t were η t was defined in Section 6.1, and tus {u t for t = p+1,, n} are approximately i.i.d. Uniform(0,1. Hence, te goal of transforming our data x 1,, x n to a sequence of (approximately i.i.d. random variables u t as been acieved; note tat te initial conditions u 1,..., u p were not explicitly generated in te above as tey are not needed in te Model-free bootstrap algoritms Basic Algoritm for Model-free Prediction Interval Given observations x 1,, x n, te basic model-free (MF algoritm for constructing te predictive interval can be described as follows. As before, y t 1 = (y t 1,, y t p and y t 1 = (x t 1,, x t p. Algoritm 6.1. Model-Free (MF Metod (1 Use eq. (6.8 to obtain te transformed data u p+1,, u n. (2 Calculate ˆx n+1, te predictor of x n+1, by ˆx n+1 = 1 n p n t=p+1 D 1 y n (u t. (6.9 (3 (a Resample randomly(wit replacement te transformed data u p+1,, u n to create te pseudo data u M, u M+1,, u 0, u 1,, u n 1, u n and u n+1 for some large positive integer M. (b Draw x M,, x M+p 1 randomly from any consecutive p values of te dataset (x 1,, x n ; let y M+p 1 = (x M+p 1,, x M. (c Generate x t = D 1 y t 1(u t for t = M + p,, n. (d Calculate te bootstrap future value x n+1 = 1 D y n (u n+1. n n p t=p+1 (e Calculate te bootstrap predictor ˆx n+1 = 1 interpolated version of te step function given by n ˆD y(x i=p+1 = 1 {x i x} K( y y i 1. n k=p+1 K( y y k 1 1 D y n (u t were D y is a linearly (f Calculate te bootstrap root x n+1 ˆx n+1. (4 Repeat step (3 B times; te B bootstrap root replicates are collected in te form of an empirical distribution wose α-quantile is denoted q(α. (5 Te (1 α100% equal-tailed predictive interval for X n+1 is given by [ˆx n+1 + q(α/2, ˆx n+1 + q(1 α/2].

17 Some remarks are in order. L. Pan and D. Politis/Bootstrap for Markov processes 17 Remark 6.1. Algoritm 6.1 is in effect a Forward bootstrap algoritm for prediction intervals according to te discussion of Section 2. Constructing a backward bootstrap analog of Algoritm 6.1 is straigtforward based on te Markov property of te reversed Markov process sown in Appendix A. One would ten need a reverse construction of te teoretical transformation of Section 6.1. To elaborate on te latter, we would instead let η t = G Yt+p (X t for t = n p, n p 1,..., 1 were G y (x = P (X t x Y t+p = y is te backwards analog of D y (x; te η t for t = n,..., n p+1 can be generated using te backwards analogs of D y,i (x. Te details are straigtforward and are omitted especially since te finite-sample performance of te two approaces is practically indistinguisable. Remark 6.2. As mentioned in Section 2, tere exist different approximations to te conditional expectation wic serves as te L 2 optimal predictor. Te usual one is te kernel smooted estimator (5.3 but eq. (6.9 gives an alternative approximation; we ave used it in Algoritm 6.1 because it follows directly from te Model-Free Prediction Principle of Politis (2013. However, te two approximations are asymptotically equivalent, and tus can be used intercangeably. To see wy, note tat 1 n p n t=p+1 1 D y 1 n (u t D y 1 n (udu 0 x ˆf n (x y n dx ˆm(y n = 1 0 ˆD 1 y n (udu 1 n p n t=p+1 ˆD 1 y n (u t were ˆD 1 y ( indicates te quantile inverse of te step-function ˆD y (. Remark 6.3. Recall tat ˆD y (x is a local average estimator, i.e., averaging te indicator 1 {xi x} over data vectors Y t tat are close to y. If y is outside te range of te data vectors Y t, ten obviously estimator ˆD y (x can not be constructed, and te same is true for D y (x. Similarly, if y is at te edges of te range of Y t, e.g., witin of being outside te range, ten ˆD y (x and D y (x will not be very accurate. Step 1 of Algoritm 6.1 can ten be modified to drop te u i s tat are obtained from an x i wose y i 1 is witin of te boundary; see Politis (2013 for a related discussion. Remark 6.4. [Discrete-valued Markov processes] Since te transformed data u p+1,..., u n are approximately i.i.d. Uniform(0,1, te resampling in step 3(a of Algoritm 6.1 could alternatively be done using te Uniform distribution, i.e., generate u M, u M+1,, u 0, u 1,, u n 1, u n and u n+1 as i.i.d. Uniform(0,1. Algoritm 6.1 still works fine wit tis coice but can not obviously be extended to include te use of predictive residuals as Section 6.4 proposes; see also te discussion in te Rejoinder of Politis (2013. Interestingly, generating u i as i.i.d. Uniform(0,1, and replacing 1 1 all occurences of D y ( by te quantile inverse ˆD y ( in Algoritm 6.1, makes te algoritm valid for te situation were te time series X t is discrete-valued, i.e., wen te true D y (x is indeed a step function in x Better Model-free Prediction Intervals: te Predictive Model-free Metod In Pan and Politis (2015, model-based predictive residuals were used instead of te fitted residuals to improve te performance of te predictive interval. From eq. (6.7, we see tat te conditional distribution of interest is D yn (x = P (X t x Y t 1 = y n wic is estimated by n i=p+1 ˆD yn (x = 1 {x i x}k( yn yi 1 n. yn yi 1 k=p+1 K(

18 L. Pan and D. Politis/Bootstrap for Markov processes 18 Since x n+1 is not observed, te above estimated conditional distribution for x n+1 treats te pair (y n, x n+1 as an out-of-sample pair. To mimic tis situation in te model-free set-up, we can use te trick of Pan and Politis (2015, i.e., to calculate an estimate of D yn (x based on a dataset tat excludes te pair (y t 1, x t for t = p + 1,, n. In oter words, define te delete-one estimator ˆD (t y t 1 (x t = n i=p+1,i t 1 {x i x t}k( yt 1 yi 1 k=p+1,k t K(, for t = p + 1,, n. yt 1 y k 1 Linear interpolation on (t ˆD y (x gives (t D y (x, and we can ten define u (t t = D (t y t 1 (x t ; (6.10 ere, te u (t t serve as te analogs of te predictive residuals studied in Pan and Politis (2015 in a nonparametric regression setup. Algoritm 6.2. Predictive Model-Free (PMF Metod Te algoritm is identical to Algoritm 6.1 after substituting u (p+1 p+1,, u(n n in place of u p+1,, u n. Remark 6.5. If y t 1 is far from oter y i s (for i = p + 1,, n and i t, ten te denominator of ˆD y (t t 1 (x t can be zero wic leads to an undefined value of u (t t. We omit all tese undefined u (t t s in te practical application of te above algoritm Smooted Model-Free Metod In Section 6.2, we estimated te transition distribution D y (x = P (X t x Y t 1 = y by ˆD y (x as defined in (6.7. Noting tat D y (x is, by assumption, continuous in x wile ˆD y (x is not, te linearly interpolated, strictly increasing estimator D y (x was used instead. However, D y (x is piecewice linear, and terefore not smoot in te argument x. In tis section, we employ an alternative estimator of te conditional transition density tat is smoot in x. Note tat te function 1 {xi x} is te cumulative distribution of a point mass distribution. To smoot tis point mass distribution, we substitute 1 {xi x} in eq. (6.7 wit Λ( x xi 0 were Λ( is an absolutely continuous, strictly increasing cumulative distribution function, and 0 is a positive bandwidt parameter. Te new estimator D y (x is defined by D y (x = n x xi i=p+1 Λ( 0 K( y yi 1 n k=p+1 K( y y k 1 and te transformed data {v t for t = p + 1,, n} can be calculated by, (6.11 v t = D yt 1 (x t. (6.12 Substituting D y (x for D y (x and {v t } for {u t } in Algoritm 6.1, we ave te smooted model free metod as follows: Algoritm 6.3. Smooted Model-Free Metod (SMF (1 Use eq. (6.12 to obtain te transformed data v p+1,, v n.

19 (2 Calculate ˆx n+1, te predictor of x n+1, by L. Pan and D. Politis/Bootstrap for Markov processes 19 ˆx n+1 = 1 n p n t=p+1 D 1 x n (v t (3 (a Resample randomly (wit replacement te transformed data v p+1,, v n to create te pseudo data v M, v M+1,, v 0, v 1,, v n 1, v n and v n+1 for some large positive integer M. (b Draw x M,, x M+p 1 from any consecutive p values of te dataset (x 1,, x n ; let y M+p 1 = (x M+p 1,..., x M. (c Generate x t = D 1 y t 1(v t for t = M + p,, n. (d Calculate te bootstrap future value x n+1 = (e Calculate ˆx n+1 = 1 n p n t=p+1 1 D y n (vt were D 1 y n (v n+1. n D y(x i=p+1 = Λ( x x i 0 K( y y i 1. n k=p+1 K( y y k 1 (f Calculate te bootstrap root x n+1 ˆx n+1. (4 Repeat step (3 B times; te B bootstrap root replicates are collected in te form of an empirical distribution wose α-quantile is denoted q(α. (5 Te (1 α100% equal-tailed predictive interval for X n+1 is given by [ˆx n+1 + q(α/2, ˆx n+1 + q(1 α/2]. Remark 6.6. As in Remark 6.3, Step 1 (a of Algoritm 6.3 can be modified to drop te v i s tat are obtained from an x i s wose y i 1 is witin of te boundary. Remark 6.7. [On Bandwidt Coice] As suggested by Li and Racine (2007 Capter 6.2, te optimal smooting of Dy (x wit respect to Mean Squared Error (MSE requires tat 0 = O p (n 2/5 and = O p (n 1/5 ; ence, in te algoritm s implementation, we cose troug cross validation, and ten let 0 = 2. As in Section 6.4, we can also use te delete-x t estimator D (t y (x t = n xt xi i=p+1,i t Λ( 0 in order to construct te transformed data: K( yt 1 yi 1 k=p+1,k t K( yt 1 y k 1 for t = p + 1,, n v (t t = D (t y t 1 (x t for t = p + 1,, n. Tis leads to te Predictive Smooted Model-Free Algoritm. Algoritm 6.4. Predictive Smooted Model-Free (PSMF Metod Te algoritm is identical to Algoritm 6.3 after substituting v (p+1 p+1,, v(n n in place of v p+1,, v n.