An empirical likelihood goodness-of-fit test for time series

Transcription

1 J. R. tatist. oc. B (2003) 65, Part 3, pp An empirical likeliood goodness-of-fit test for time series ong Xi Cen, National University of ingapore, ingapore Wolfgang Härdle Humboldt-Universität zu Berlin, Germany and Ming Li National University of ingapore, ingapore [Received January Final revision January 2003] ummary. tandard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. Wen a parametric regression model is compared wit a nonparametric model, goodness-of-fit testing can be naturally approaced by evaluating te likeliood of te parametric model witin a nonparametric framework. We employ te empirical likeliood for an α-mixing process to formulate a test statistic tat measures te goodness of fit of a parametric regression model. Te tecnique is based on a comparison wit kernel smooting estimators. Te empirical likeliood formulation of te test as two attractive features. One is its automatic consideration of te variation tat is associated wit te nonparametric fit due to empirical likeliood s ability to tudentize internally. Te oter is tat te asymptotic distribution of te test statistic is free of unknown parameters, avoiding plug-in estimation. We apply te test to a discretized diffusion model wic as recently been considered in financial market analysis. Keywords: α-mixing; Empirical likeliood; Goodness-of-fit test; Nadaraya Watson estimator; Parametric models; Power of test; quare-root processes; Weak dependence 1. Introduction Te analysis and prediction of time series is standard in statistics. Te tecniques tat are employed usually rely on te actual model assumed to represent and generate te time series dynamics. Mismodelling migt result in biased prediction and an incorrect parameter specification. Te aim of tis paper is to sow ow empirical likeliood (EL) (Owen, 1990) may be used to construct simple test procedures for te goodness of fit of standard time series models. uppose tat {.X i, Y i /} n is a strictly stationary time series wit Y i R and X i R d. Let m.x/ = E.Y X = x/ be te conditional mean function, f be te density of te design X and σ 2.x/ = var.y X = x/ be te conditional variance of Y given X = x, a set to be specified later. uppose tat {m θ θ Θ} is a parametric model for te mean function m and tat ˆθ is an estimator of θ under tis parametric model. Te interest is to test te null ypotesis Address for correspondence: ong Xi Cen, Department of tatistics and Applied Probability, National University of ingapore, Kent Ridge, ingapore. stacsx@nus.edu.sg 2003 Royal tatistical ociety /03/65663

2 664. X. Cen, W. Härdle and M. Li H 0 : m.x/ = m θ.x/ for all x against a series of nonparametric alternatives H 1 : m.x/ = m θ.x/+ c n n.x/, were c n is a non-random sequence tending to 0 as n and n.x/ is a sequence of bounded functions. Te problem of testing a parametric mean regression against a nonparametric alternative is not new for an independent and identically distributed setting. Härdle and Mammen (1993) proposed a test statistic based on an L 2 -distance between a nonparametric kernel estimator of te conditional mean and te ypotesized parametric function. Tis kind of goodness-of-fit test, comparing nonparametric wit parametric fits, was also treated in Eubank and piegelman (1990), Hart (1997) and te references terein. Kreiss et al. (1998) ave extended te Härdle Mammen test to a time series context by implementing te wild bootstrap. Linearity tests based on local polynomial estimators were considered in Hjellvik et al. (1998). Koul and tute (1999) proposed a test for time series based on te empirical processes constructed on te residuals of parametric fits under ypotesis H 0. Te attractions of teir test are (a) its independence of a smooting parameter and (b) its capacity to test H 1 wit c n of order n 1=2, wic is smaller tan n 1=2 d=4 for te Härdle Mammen test. As tey pointed out, owever, te test is not applicable for d>1 and it requires te conditional variance function σ 2. / to be constant. Recently Horowitz and pokoiny (2001) proposed a test for independent data tat applies te Härdle Mammen test simultaneously on a set of bandwidt values. Tey sowed tat te test is consistent for H 1 wit c n = O.n 1=2 [log{log.n/}]/. An adaption of a similar sceme for te EL test is possible, wic would mean an improved resolution of te test tat is proposed in te current paper. Te device tat we use to formulate goodness-of-fit tests is EL (Owen, 1990, 2001), wic is a computer-intensive nonparametric alternative to te bootstrap. It as been sown to sare some key properties wit parametric likeliood, e.g. Wilks s teorem and Bartlett correctability; see Hall and La cala (1990), Qin and Lawless (1994) and Cen (1996). Kitamura (1997) considered blocked EL for parameters associated wit weakly dependent processes. Monti (1997) constructed confidence regions for a parameter of a stationary time series via Wittle s estimation metod. For independent data, EL as been employed for testing conditional moment restrictions (Tripati and Kitamura, 2000) and for nonparametric functions based on a local approximation (Fan and Zang, 2000). Te EL goodness-of-fit statistic tat is proposed ere is based on an EL ratio of te parametric model over a set witin te domain of te design density f. Te EL test is carried out by simulating a Gaussian random field wit known mean and covariance function. Te known covariance function is due to EL s ability to tudentize internally, leading to asymptotic pivotalness. Te test proposed as te advantages of (a) putting te goodness-of-fit measure witin te context of its variation and (b) avoiding secondary plug-in estimation, as te asymptotic distribution of te test statistic is free of unknown parameters. Te paper is structured as follows. In ection 2, we construct an EL ratio for m.x/, te basic building-block of te test proposed. In ection 3, we formulate te test statistic by integrating te EL ratio over te set and establis te asymptotic equivalence of te test statistic wit an integral of a squared Gaussian random field. Te test procedure is proposed in ection 4 and is applied to test a parametric diffusion model in ection 5. ection 6 reports simulation results. All te tecnical details including assumptions are given in Appendix A.

3 2. Kernel estimator and empirical likeliood Empirical Likeliood Goodness-of-fit Test 665 We first introduce a nonparametric kernel estimator for m. Let = {x R d f.x/ β} for some β > 0 be a compact set. Witout loss of generality we assume tat = [0, 1] d. Let Λ be a univariate rt-order kernel wic is compactly supported on [ 1, 1] suc tat Λ.t/ dt = 1, t l Λ.t/ dt = 0 if 1 l r 1, t r Λ.t/ dt = κ r 0 for an integer r 2, and let K be a d-dimensional product kernel of Λ, i.e. K.t 1,...,t d / = d Λ.t i /: Let be a positive smooting bandwidt wic is used to smoot in every component of X, implying tat te scale in eac component is rougly te same. Wen te scales of te components are different, tey can be standardized by using teir standard deviations. Let K.u/ = d K. 1 u/. Te nonparametric estimator of m.x/ considered is te Nadaraya Watson estimator ˆm.x/ = n / n Y i K.x X i / K.x X i /:.2:1/ Let mˆθ.x/ = n / n K.x X i /mˆθ.x i/ K.x X i / be te smooted parametric model. To avoid te issue of te bias tat is associated wit te nonparametric fit, te test statistic tat we sall consider is based on te difference between mˆθ and ˆm, rater tan between ˆm and mˆθ. Te local linear estimator can be used to replace te Nadaraya Watson estimator in estimating m because of its attractive bias properties. However, as we compare ˆm wit mˆθ, te bias issue is unimportant. We now introduce EL for a testing problem. At an arbitrary x, let p i.x/ be non-negative weigts allocated to.x i, Y i /. Te EL for mˆθ.x/ is { n } L{ mˆθ.x/} = max p i.x/,.2:2/ subject to n p i.x/ = 1 and ( ) n x Xi p i.x/ K {Y i mˆθ.x/} = 0: By introducing Lagrange multipliers, te optimal weigts are given as ( ) ] x 1 p i.x/ = n [1 1 Xi + λ.x/ K {Y i mˆθ.x/}.2:3/

4 666. X. Cen, W. Härdle and M. Li were λ.x/ is te root of n K{.x X i /=}{Y i mˆθ.x/} = 0:.2:4/ 1 + λ.x/k{.x X i /=}{Y i mˆθ.x/} Te maximum EL is acieved at p i.x/ = n 1 corresponding to te Nadaraya Watson estimator ˆm.x/. Te log-el ratio is l{ mˆθ.x/} = 2log[L{ mˆθ.x/}nn ] wic will be te basic buildingblock for te proposed EL goodness-of-fit statistic. We first evaluate λ.x/ in te following lemma. Lemma 1. Under assumptions (a) (g) given in Appendix A, sup λ.x/ =o p {.n d / 1=2 log.n/}: Te proof of lemma 1 is detailed in Cen et al. (2002). Let γ.x/ be a random process wit x. Trougout tis paper te notation γ.x/ = Õ p.δ n / means sup γ.x/ =O p.δ n / for a sequence {δ n }, and similarly for γ.x/ =õ p.δ n /. Let ( ) x Xi w i.x/ = K {Y i mˆθ.x/}, Ū j.x/ =.n d / 1 for integers j. Expansion of expression (2.4) yields n w i.x/ j n Ū 1.x/ λ.x/ Ū 2.x/ + λ 2.x/.n d / 1 w i.x/ λ.x/ w i.x/ +õ p{.n d / 3=2 log 3.n/} = 0:.2:5/ It may be sown by te tecnique tat is used in te proof of lemma 1 tat.n d / 1 n w i.x/ λ.x/ w i.x/ = Õ p.1/: Inverting equation (2.5), we ave λ.x/ = Ū2 1.x/ Ū 1.x/ +õ p {.n d / 1 log 2.n/}: Tis, togeter wit equation (2.3) and lemma 1, means tat l{ mˆθ.x/} = 2 n [ ( ) ] x Xi log 1 + λ.x/ K {Y j mˆθ.x/} = 2.n d / λ.x/ Ū 1.x/.n d / λ 2.x/ Ū 2.x/ +õ p {.n d / 1=2 log 3.n/} =.n d / Ū2 1.x/ Ū2 1.x/ +õ p{.n d / 1=2 log 3.n/}:.2:6/ Let v.x; / = d K 2.x y/ f.y/ σ2.y/ dy, b.x; / = d K.x y/ f.y/ dy and V.x; / = v.x; /=b 2.x; /. Note tat V.x; /=n d is te asymptotic variance of ˆm.x/ provided tat n d. It may be sown tat sup ˆm.x/ m θ.x/ =o p {.n d / 1=2 log.n/} and sup ˆf.x/ b.x/ =o p {.n d / 1=2 log.n/}. From te proof of lemma 1, we ave Ū 1.x/ = b.x; /{ ˆm.x/ m θ.x/}+õ p {.n d / 1 log 2.n/} and sup Ū 2.x/ v.x; / =o p {.n d / 1=2 log.n/+ 2 }. Tese and equation (2.6) mean tat l{ mˆθ.x/} = { ˆm.x/ m θ.x/} 2 nd + Õ{.n d / 1=2 log 3.n/ + 2 log 2.n/},.2:7/ V.x; / implying tat l{ mˆθ.x/} is asymptotically equivalent to a tudentized L 2-distance between mˆθ.x/ and ˆm.x/.

5 3. Goodness-of-fit test statistic Empirical Likeliood Goodness-of-fit Test 667 On te basis of te property of l{ mˆθ.x/} revealed in equation (2.7), te EL-based goodness-of-fit statistic proposed is l n. mˆθ / = l{ mˆθ.x/} dx: From equation (2.7), { ˆm.x/ l n. mˆθ / = mθ.x/} 2 nd dx + O p {.n d / 1=2 log 3.n/ + 2 log 2.n/}:.3:1/ V.x; / Härdle and Mammen (1993) proposed T n = n d=2 { ˆm.x/ mˆθ.x/}2 π.x/ dx as a measure of goodness of fit were π.x/ is a given weigt function. Equation (3.1) indicates tat l n. mˆθ / is asymptotically equivalent to d=2 T n wit π.x/ = V 1.x; / wic is proportional to f.x/=σ 2.x/. Te differences between te two test statistics are tat (a) te EL automatically tudentizes so tere is no need to estimate V.x; / and (b) te EL can capture features of data suc as te skewness and kurtosis. Te test statistic proposed by Kreiss et al. (1998) is equivalent to te Härdle Mammen statistic wit π.x/ = f 2.x/, wic is designed to downweigt low density areas. However, tis may lead to a loss of power. Te test statistic proposed can be readily extended to testing a parametric specification, σθ 2, of te volatility function σ 2.x/. We need only to replace te second constraint in equation (2.2) wit ( ) n x Xi p i.x/ K [{Y i ˆm.X i /} 2 σ.x/] = 0, 2ˆθ were σ like 2ˆθ mˆθ is a kernel smoot of. Te advantage of te internal tudentizing tat σ2ˆθ is offered by te EL becomes more apparent in tis case as explicit variance estimation of te kernel estimator of σ 2 involves fourt-order moments and so is more difficult. Te smooting bandwidt can be cosen by any bandwidt selector tat produces wic minimizes te mean integrated square error of te curve estimation, e.g. tose from te crossvalidation or te plug-in metods. Tis is an area tat as been intensively studied in nonparametric curve estimation. Teorem 1. Under assumptions (a) (g) in Appendix A, l n. mˆθ / and N 2.s/ ds ave te same asymptotic distribution as n, were N is a normal process on = [0, 1] d suc tat E{N.s/} = d=4 n.s/ V 1=2.s; / and { Ω.s, t/ =: cov{n.s/, N.t/} = f.s/ σ 2.s/ f.t/ σ 2.t/ } K.2/.0/ 1 K.2/ ( s t were K.2/ is te convolution of K. Te proof of te teorem is given in Appendix A. As K is a compact kernel on [ 1, 1] d, Ω.s, t/ = 0if s t > 2, wic means tat N.s/ and N.t/ are independent if s t > 2. As f.s/ σ 2.s/ = f.s/ σ 2.t/ + O./ wen s t 2, ( ) s t Ω.s, t/ = K.2/.0/ 2 K.2/ + O./:.3:2/ ),

6 668. X. Cen, W. Härdle and M. Li Hence, te leading order term of te covariance function is completely known and will facilitate a test based on te simulation of te normal process. Te proof of teorem 1 contains te following corollary. Corollary 1. Under assumptions (a) (f) in Appendix A, d=2 {l n. mˆθ / µ 0} d N.0, σ0 2/ as n were σ0 2 = 2 K.4/.0/K.2/.0/ 2 and µ 0 = 1 + d=2 V 1.s/ 2 n.s/ ds. Te corollary indicates tat l n. mˆθ / is O p. d=2 /, vanising as n. However, for a given, l n. mˆθ / is a monotone function of d=2 {l n. mˆθ / 1} wic is O p.1/. Hence, a test based on l n. mˆθ / is equivalent to a test based on te standardized statistic d=2 {l n. mˆθ / 1}. 4. Goodness-of-fit test An α-level test based on te asymptotic normality of l n. mˆθ / rejects ypotesis H 0 if d=2 {l n. mˆθ / 1} >z α {2K.4/.0/}=K.2/.0/ were z α is te.1 α/-quantile of N.0, 1/. Te corollary implies tat te asymptotic power of te test under H 1 is [ 1 Φ z α K.2/.0/ ] V 1.s/ 2 n.s/ ds {2K.4/.0/} wic is sensitive to alternatives in all directions and converges to 1 if c n is of a larger order tan n 1=2 d=4, ence implying tat te test is consistent. Te asymptotic normal test may not work well for a finite sample owing to te many approximations tat are used in establising te asymptotic normality, wic subsequently make it too far from te finite sample distribution of l n. mˆθ /. Indeed, obtaining a better approximation to te distribution of te test statistic motivated te use of te wild bootstrap (Härdle and Mammen, 1993; Kreiss et al., 1998; Hjellvik et al., 1998). Rater tan resort to te wild bootstrap, we propose to use te distribution of N 2.s/ ds to approximate tat of l n. mˆθ /. Tis proposal, as well as te result contained in teorem 1, is motivated by te observation tat te distribution of N.x/ mimics tat of.n d / 1=2 V 1.x; / { ˆm.x/ mˆθ.x/} at eac x. In particular, let ( ) s t Ω 0.s, t/ = K.2/.0/ 2 K.2/ wic is te leading term of Ω.s, t/, te covariance of N, and let N 0 be a Gaussian random field wit zero mean function and Ω 0 as its covariance function. ince te law of N 0 is completely known on given an, te distribution of N 0 2.s/ ds can be obtained by simulating {N 0.s/ s } over a lattice in. We use an algoritm proposed by Wood and Can (1994). Our numerical experience indicates tat te best effect is acieved wen te lattice used in discretizing l{ mˆθ.x/} dx is also used in simulating {N 0.s/ s }. Let w α be te simulated upper α-level quantile of N 0 2.s/ ds. Ten, te EL test proposed is to reject ypotesis H 0 if l n. mˆθ / w α. Fig. 1 contains te simulated null densities of l n. mˆθ / and N 0 2.s/ ds as well as two normal densities: te full scale asymptotic normal N.1, d σ0 2/ and N.1, σ2 1 / densities, were

7 Empirical Likeliood Goodness-of-fit Test 669 density T (a) density T (b) density T (c) density T (d) Fig. 1. Approximations to te density of l n. ˆmˆθ / ( ) by tose of N 2 0.s/ ds (... ), N.1, d σ 2 0 / ( ) and N.1, σ 2 1 / ( ) under models (6.1) and (6.2) wit n D 500: (a) D 0:05, k n D 300; (b) D 0:05, k n D 600; (c) D 0:06, k n D 300; (d) D 0:06, k n D 600 ( ) s t 2 σ1 2 = K.2/.0/ 2 K.2/ ds dt is te quantity tat d σ0 2 approximates. Let k n be te number of lattice points used in discretizing bot l{ mˆθ.x/} dx and N 0 2.s/ ds. Altoug Fig. 1 is for only two values of and k n as part of a compreensive simulation study tat is reported in ection 6, it sows typical results. Te simulation of N 0 2.s/ ds provides quite a satisfactory approximation to te distribution of l n. mˆθ /. Of te two normal approximations, N.1, σ2 1 / is better tan N.1, d σ0 2 / owing to its better approximation to te variance. Te simulated distribution nicely reflects te skewness of l n. mˆθ /, wic te two normal proxies cannot pick up. It is also wort noting tat te test based on te normal densities will typically ave a muc larger size tan te nominal level α. Wesee tat te distributions of l n. mˆθ / and N 0 2.s/ ds are insensitive to te number of lattice points used. 5. Testing te Cox Ingersoll Ross model for tandard and Poors 500 data In matematical finance, interest rates, stocks and oter financial products are modelled by diffusion processes wit specific parametric assumptions on te drift and diffusion functions. A well-known model for modelling te dynamic of interest rates is te Cox Ingersoll Ross model (Cox et al., 1985) given in equation (5.2) below. pecification tests for diffusion models ave been considered by Aït-aalia (1996), Hong and Li (2001) and oters. In tis section we apply te empirical likeliood test for a financial market model proposed by Platen (1999) on te tandard and Poors 500 sare index data wic contain te daily closing values of te index for 5479 trading days from December 31st, 1976, to December 31st, In Fig. 2(a), te index series sows an exponential trend wic is estimated by using te metod of Härdle et al. (2000). Fig. 2(a) also displays a residual process X.t/ at te bottom after removing te exponential trend. In matematical finance, we assume a specific dynamic form for tis X.t/ process. More precisely, Platen (1999) assumed te following model for an index process:

8 670. X. Cen, W. Härdle and M. Li Process Time (a) Difference of X(t) X(t) process (b) CV score Bandwidt (c) P-value Bandwidt (d) Fig. 2. (a) Raw tandard and Poors 500 data and teir exponential trend (top) and te X.t/ process (bottom), (b) X i D X.i1/ versus Y i D X ic1 X i (C) and te parametric fit for te conditional mean ( ), (c) cross-validation scores CV and (d) P -values of te EL test { t }.t/ =.0/X.t/ exp η.s/ ds,.5:1/ 0 wit a diffusion component X.t/ solving a stocastic differential equation dx.t/ = α{1 X.t/} dt + σ X 1=2.t/ dw.t/,.5:2/ were W.t/ is a Brownian motion and α and σ are parameters. Discretizing tis series wit sampling interval leads to observations.x i, Y i / sown in Fig. 2(b) wit Y i = X.i+1/ X i and X i = X i, wic is α mixing, and fulfil all te oter conditions assumed on te basis of te results of Genon-Catalot et al. (2000). We applied te EL test to verify te parametric mean function m.x/ = a.1 x/ specified by te Cox Ingersoll Ross model. Te process X.t/ was restored from te observed residuals by te approac tat was introduced in Härdle et al. (2000). Te estimate for te drift parameter a was â = 0:00968 by using metods based on te marginal distribution and te autocorrelation structure of X. Cross-validation was used to find a suitable value of. Te cross-validation score function CV attained a minimum at = 0:107 as sown in Fig. 2(c). As CV is known for its slow convergence to te optimal bandwidt, te prescribed served as a reference only. Furter investigation sowed tat an -value larger tan 0.06 oversmooted te drift function wereas an -value smaller tan 0.03 undersmooted. Terefore, te EL test was carried out for a set of -values ranging from 0.02 to Te P-values plotted in Fig. 2(d) are all quite large. o, te data conform wit te conditional mean structure specified by te Cox Ingersoll Ross model. 6. imulation In tis section we report results from a simulation study tat was designed to evaluate te performance of te EL test proposed. Te test based on te wild bootstrap given in Kreiss et al. (1998) serves as a comparison. Te simulation considers testing of time series models for d = 1, 2.

9 For d = 1, te model was Empirical Likeliood Goodness-of-fit Test 671 Y i = θ 1 X i + θ 2 Xi 2 + c n cos.8x i / + σ 0 X i " i,.6:1/ X i = γx i 1 + ρη i, i = 1,2,...,n,.6:2/ were θ =.θ 1, θ 2 / =.0:3, 0:1/ and σ 0 = 0:5; {" i } n and {η i} n are mutually independent and identically distributed innovations, wic are all independent of X i.wefixed" i N.0, 1/ and η i Unif[ 0:5, 0:5] wit ρ = 1 and γ = 0. Te coice of ρ was to make te density of X i bounded away from 0 witin = [ 0:5, 0:5] wic contains about 90% of observed X i. Tree c n -values were used: c n = 0, corresponding to ypotesis H 0, c n = 0:002 and c n = 0:004. For d = 2, we considered an autoregressive conditional eteroscedastic ARCH (2) model Y i = θ 1 Y i 1 + θ 2 Y i 2 + c n n.y i 1, Y i 2 / + σ 0 σ.y i 1, Y i 2 /" i.6:3/ were θ =.θ 1, θ 2 / =.0:3, 0:3/, " i IID N.0, 1/, σ 0 = 0:7 and σ.x, y/ = n.x, y/ =.0:3x 2 + 0:2y 2 + 0:2/. Tree c n -values, 0, 0.04 and 0.06, were used, and = [ 0:5, 0:5] 2. In bot models, te parameter θ was estimated by maximizing te conditional quasi-likeliood given te X i s. We fixed te nominal size of te tests at α = 5%, took n = 500 and n = 1000 and cose k n, te number of lattice points used in approximating te integral l{ mˆθ.x/} dx, to be 300 for d = 1 and 900 for d = 2. Te increase in k n reflected te increase in te dimensionality of X i. In te simulation of N 0 2.s/ ds, te same lattice was used to ensure a better approximation. Te simulated power of te EL and te wild bootstrap tests for a set of -values is summarized in Fig. 3 for d = 1 and in Fig. 4 for d = 2. Bot Fig. 3 and Fig. 4 sow tat te power of te EL test was consistently iger tan tat of te bootstrap test, wereas te size of te EL test was in general te same as tat of te bootstrap test. Te performance of te bootstrap test for te univariate model was disappointing especially wen n = 500. Te power was improved for d = 2 wit n = 1000, altoug te EL test was still noticeably better. Te better power of te EL test was probably because it could obtain contributions of lack of fit from areas of low density were te wild bootstrap downweigted. For a given c n > 0, te iger power tat is observed Power Bandwidt (a) Power Bandwidt (c) Power Bandwidt (b) Power Bandwidt (d) Fig. 3. Power of te EL test for (a) n D 500 and (b) n D 1000 and of te wild bootstrap test for (c) n D 500 and (d) n D 1000 under models (6.1) and (6.2):..., c n D 0:0; , c n D 0:02;, c n D 0:04;, nominal 5%

10 672. X. Cen, W. Härdle and M. Li Power Bandwidt (a) Power Bandwidt (c) Power Bandwidt (b) Power Bandwidt (d) Fig. 4. Power of te EL test for (a) n D 500 and (b) n D 1000 and of te wild bootstrap test for (c) n D 500 and (d) n D 1000 under models (6.3):..., c n D 0:0; , c n D 0:04;, c n D 0:06;, nominal 5% from n = 500 to n = 1000 was due to an increased distance between H 1 and H 0 altoug c n was te same. One overall feature of te simulation was tat te powers of bot tests were not very sensitive to te value of. Te simulation was also conducted wit real parameter values, giving te same pattern of results except tat te sizes of bot tests were closer to 5% and te power was sligtly smaller. Acknowledgements Te autors tank T. Kleinow, J. Gao, two referees, te Associate Editor and te Joint Editor for comments and suggestions wic improved te presentation of te paper. Te autors acknowledge support by te National University of ingapore and te Deutsc Forscungsgemeinscaft via onderforscungsbereic 373 Quantifikation und imulation ökonomiscer Prozesse at Humboldt-Universität zu Berlin. Appendix A: Tecnical details A.1. Assumptions Let Fk l be te σ-algebra of events generated by {.X i, Y i /, k i l} for l k. Te measure for dependence between a α-mixing time series is α.k/ = sup P.AB/ P.A/ P.B/ : A F i 1, B F i+k Te assumptions tat are required to establis te results given in te paper are te following: (a) Λ is a univariate rt-order kernel wic is compactly supported in [ 1, 1] and is Lipscitz continuous, d<4; te smooting bandwidt = O.n 1=.d+2r/ /. (b) f, m and σ 2 ave continuous derivatives up to te second order in and bot f and σ 2 are bounded below in. (c) ˆθ is a parametric estimator of θ witin te family of te parametric model, and sup mˆθ.x/ m θ.x/ =O p.n 1=2 /:

11 Empirical Likeliood Goodness-of-fit Test 673 (d) n.x/, te local sift in ypotesis H 1, is uniformly bounded wit respect to x and n, and c n = n 1=2 d=4, wic is te order of te difference between H 0 and H 1. (e) E[exp{a 0 Y 1 m.x 1 / }] < for some a 0 > 0; E. Y i k X i /< for some k>1; for all i, E{Y i m.x i / Ω i 1 } = 0 were Ω i 1 is te σ-field generated by {.X j+1, Y j } i 1 j=1. (f) Te conditional density of X given Y, f X Y A 1 <, te conditional joint density of.x 1, X l / given.y 1, Y l / is bounded for all l>1and te joint density of.x 1, Y 1, X s, Y s, X t, Y t / for t>s>1is continuous and bounded by a constant tat is free of s and t. (g) Te process {.X i, Y i /} is strictly stationary and α mixing, and α.k/ aρ k for some a>0and ρ.0, 1/. Assumptions (a) and (b) on te kernel and bandwidt are standard in nonparametric curve estimation. Te assumption of d<4is to make te bias in kernel variance estimation a smaller order of d=2. Te kernel metod will encounter te curse of dimension wen d 4 anyway. Te bandwidt selected by eiter cross-validation or te plug-in metod satisfies te order specified in assumption (a). Assumptions (c) and (d) are common in nonparametric goodness-of-fit tests, wereas assumptions (e), (f) and (g) are standard assumptions for dependent processes. In particular, assumption (g) means tat te data are geometric α mixing. It can be seen from te proof tat te geometric α-mixing condition can be weakened to α.k/ Ck s.d/ were s.d/ > 2 and is a monotone function of d. It is convenient tecnically to assume geometric α-mixing. For a univariate linear causal process Y t = Σ s=0 g t sξ s wit independent and identically distributed innovation {ξ s } s=0, Gorodeskii (1977) sowed tat te linear process is α mixing under certain conditions and establised te rate for te α-mixing coefficient. Pam and Tran (1985) sowed tat, if eac coefficient g t of te process is O.γ t /,0< γ < 1, ten te process is geometric α mixing. For a Markov process Y i = m.x i / + σ.x i /" i were X i =.Y i 1,...,Y i p / are lagged values and te " i are independent and identically distributed random variables, Masry and Tjøsteim (1995) provided conditions for geometric ergodicity and geometric α-mixing of te process. Trougout te proof we sall use C to denote positive constants wic may take different values at different places. A.2. Proof of teorem 1 Te proof of teorem 1 is carried out by proving tat bot l n. mˆθ / and N 2.s/ ds ave te same limiting normal distribution. We first prove tat l n. mˆθ / is asymptotically normally distributed. Let V.x/ = R.K/ σ 2.x/=f.x/ suc tat V.x; / = V.x/ + O. 2 /. From equation (3.1) l n. mˆθ / = n + o. d=2 / were n = n d V 1.x/{ ˆm.x/ m θ.x/} 2 dx: Let H n1.x/ = n 1 Σ K.x X i /" i and H n2.x/ = c n n 1 Σ K.x X i / n.x/. Ten, n = n d V 1.x/ f 2.x/{H 2 n1.x/ + H 2 n2.x/} dx + 2A n were A n = n d V 1.x/ f 2.x/ H n1.x/ H n2.x/ dx = n 1=2 3d=4 " i V 1.x/ f 1.x/ n.x/ K.x X i / dx{1 + o p. d=2 /} since H n2.x/ = K.x y/ n.y/ f.y/ dy + o p {.n d / 1=2 log.n/} = n.x/ f.x/ + o p. d=2 /. Let s 0.X i / = K.x X i /V 1.x/ f 1.x/ n.x/ dx and W n0 = n 1 Σ " i s 0.X i /. Clearly, A n = n 1=2 3d=4 W n0 {1 + o p. d=2 /}. ince E.W n0 / = 0 and var.w n0 / = n 1 E{" 2 i s2 0.X i/} Cn 1, A n = O p. 3d=4 /. Terefore, l n. mˆθ / = n1 + n2 + n3 + o p. d=2 /.A:1/

12 674. X. Cen, W. Härdle and M. Li were n1 = n 1 d " i " j V 1.x/ f 2.x/ K.x X i /K.x X j / dx, i j n2 = n 1 d " 2 i V 1.x/ f 2.x/ K 2.x X i/ dx, n3 = d=2 V 1.x/ f 2.x/{n 1 K.x X i / n.x i /} 2 dx: As n 1 Σ K.x X i / n.x i / = n.x/ f.x/ +õ p {n 1=2 d=2 log.n/ + r }, n3 = d=2 V 1.x/ 2 n.x/ dx + o p. d=2 /:.A:2/ Note tat E. n2 / = 1 + O. 2 / and var. n2 / = O.n 1 d /. Hence n2 = 1 + o p. d=2 /:.A:3/ Let φ ij = " i " j V 1.x/ f 2.x/ K.x X i /K.x X j / dx and n1 0 = Σ 1 i<j nφ ij so tat n1 = 2n 1 d n1 0. Note tat n1 0 is a degenerate U-statistic. A central limit teorem for degenerate U-statistics for absolutely regular processes as been establised in Hjellvik et al. (1996). By reading teir proof closely, we find tat te assumption of te absolute regularity is only to use an inequality of Yosiara (1976) wic can be replaced by te Davydov inequality (Bosq (1998), page 19) for α-mixing processes; see Gao and King (2001) for an updated proof. Tis means tat we can use te teorem in Hjellvik et al. (1996) for α-mixing processes. Let σij 2 = var.φ ij/ and σn 2 = Σ 1 i<j nσij 2. Let E i be expectations wit respect to ξ i =:.X i, Y i /.From remark B to teorem A of Hjellvik et al. (1996), σn 2 n2 σn0 2 =2asn were σ 2 n0 = E ie j {" 2 i "2 j V 1.x/ f 2.x/ V 1.y/ f 2.y/ } K.x X i /K.x X j /K.y X i /K.y X j / dx dy = d K.4/.0/K.2/.0/ 2 {1 + o.1/}: Tis means tat var. n1 / = 2 d K.4/.0/K.2/.0/ 2 {1 + o.1/}:.a:4/ Let P.ξ i /, P.ξ i, ξ j /, P.ξ i, ξ j, ξ k / and P.ξ i, ξ j, ξ k, ξ l / be te probability measures of ξ i,.ξ i, ξ j /,.ξ i, ξ j, ξ k / and.ξ i, ξ j, ξ k, ξ l / for different i, j, k, l {1,...,n} respectively. Define, for some constant δ > 0, { } M n1 = max E φ 1j φ ij 1+δ, φ 1j φ ij 1+δ dp.ξ 1 / dp.ξ i, ξ j /, 1<i<j n max { M n2 = max max E φ 1j φ ij 2.1+δ/, φ 1j φ ij 2.1+δ/ dp.ξ 1 / dp.ξ i, ξ j /, 1<i<j n } φ 1j φ ij 2.1+δ/ dp.ξ 1, ξ i / dp.ξ j /, φ 1j φ ij 2.1+δ/ dp.ξ 1 / dp.ξ i / dp.ξ j /, M n5 = max 1<i<j M n4 = M n3 = max.e φ 1jφ ij 2 /, 1<i<j n max 1<i, j, k n max E φ 1i φ 1j dp.ξ 1 / { ( max P 2.1+δ/, )} φ 1j φ jk 2.1+δ/ dp, φ 1i φ 1j dp.ξ 1 / 2.1+δ/ dp.ξ i / dp.ξ j /,

13 Empirical Likeliood Goodness-of-fit Test M n6 = max 1<i<j E φ 1i φ 1j dp.ξ 1 /, were, in M n4, i, j and k are mutually different and te maximization over P is taken over P.ξ 1, ξ i, ξ j, ξ k /, P.ξ 1 /P.ξ i, ξ j, ξ k /, P.ξ 1 /P.ξ i /P.ξ j, ξ k / and P.ξ 1 /P.ξ i /P.ξ j /P.ξ k /. According to teorem A of Hjellvik et al. (1996), to sow tat σn 10 n1 is asymptotically standard normally distributed, it is sufficient to ceck tat for some δ > 0 and as n max[σ 2 n {n2.m 1=.1+δ/ n1 + M 1=2.1+δ/ n5 + M 1=2 n6 /, n3=2.m 1=2.1+δ/ n2 + M 1=2 n3 + M1=2.1+δ/ n4 /}] 0:.A:5/ Rater tan evaluating all te M nl -terms, we present ere only te order of magnitude of M n1 and M n6 as te oter terms can be evaluated similarly. Let p, q>1suc tat p 1 + q 1 = 1. From condition (b) ( ) ( ) A 1 =: E φ 1j φ ij 1+δ C 2d.1+δ/ E " 1" i " 2 X1 X j j K.2/ K.2/ Xi X j 1+δ { ( ) ( )} C 2d.1+δ/.E " 1 " i " 2 j.1+δ/p / [E 1=p K.2/ X1 X j K.2/ Xi X.1+δ/q ] 1=q j : Condition (e) implies tat E " 1 " i " 2 j.1+δ/p <C. Let f 1,i,j be te joint density of.x 1, X i, X j /. Condition (f ) means tat { ( ) ( )} E K.2/ X1 X j K.2/ Xi X.1+δ/q j = 2d { K.2/.u/ K.2/.v/ }.1+δ/q f 1,i,j.z u, z v, z/ du dv dz C 2d :.A:6/ Terefore, A 1=.1+δ/ 1 C 2d{1 1=q.1+δ/} = o. d / if we coose q suc tat 1 <q<2=.1 + δ/. To evaluate te oter term in M n1, let E i,j be expectations wit respect to.ξ i, ξ j / and B 1 =: E 1 E i,j φ 1j φ ij 1+δ ( [ { ( ) C 2d.1+δ/ E 1 " 1 1+δ E i, j K.2/ X1 X j C 2d.1+δ/+2d=q : ( )} K.2/ Xi X.1+δ/q ] 1=q ) j Hence B 1=.1+δ/ 1 C 2d{1 1=q.1+δ/} = o. d / if we coose q suc tat 1 <q<2=.1 + δ/. Combining te results on A 1 and B 1,weaveσn 2n2 M 1=.1+δ/ n1 0asn. Now let us consider M n6. Let A 6 = E i,j.e 1 φ 1i φ 1j / 2. imilar to inequality (A.6), { ( ) ( )} E 1 φ 1i φ 1j C " i " j 2d E 1 " 2 X1 X j 1 K.2/ K.2/ Xi X j { ( C " i " j 2d+d=q K.2/ w + X )} i X q j f.w/ dw:.a:7/ Define anoter pair of p 1, q 1 > 1 suc tat p q 1 1 = 1. From inequality (A.7), [ ( A 6 C 4d+2d=q E i,j (" 2 i "2 j { K.2/ w + X )} i X q 2=q ) j f.w/ dw] C 4d+2d=q+d=q 1 :.A:8/ Hence M 1=2 n6 C d.2 1=q 1=2q1/. By coosing q 1 (after coosing p and q as above) suc tat 0 < 2 1=q 1=2q 1 < 1, M 1=2 n6 = o. d /,soσn 2n2 M 1=2 n6 = o.1/. Hence we establis expression (A.5) and te asymptotic normality of n1 0.

14 676. X. Cen, W. Härdle and M. Li In summary of equations (A.1) (A.4) and te asymptotic normality of n1 0,weave { } d=2 l n. mˆθ / 1 d=2 V 1.x/ 2 d.x/ dx N.0, σ 2 0 / as n, were σ0 2 = 2 K.4/.0/ K.2/.0/ 2. In wat follows we sall prove te asymptotic normality of T =: N 2.s/ ds. Let N 1.s/ = N.s/ d=4 n.s/ f.s/= V.s/: Ten N 1.s/ is a Gaussian process wit zero mean and covariance Ω. plit T as T 1 + T 2 + T 3 were T 1 = N 2 1.s/ ds, T 2 = 2 d=4 V 1=2.s/ n.s/ N 1.s/ ds, T 3 = d=2 V 1.s/ 2 n.s/ ds: As Ω is bounded, Ω.t, t/ dt <. From results on stocastic integrals and equation (3.2), E.T 1 / = Ω.s, s/ ds = 1, var.t 1 / = 2 Ω 2.s, t/ ds dt = 2 K.2/.0/ 2 K.2/ {.s t/=} 2 ds dt{1 + o.1/}.a:9/.a:10/ = d K.4/.0/K.2/.0/ 2 + o. d /:.A:11/ Terefore, var.t 1 / = 2 d K.4/.0/K.2/.0/ 2 + o. 2d /. It is obvious tat E.T 2 / = 0 and var.t 2 / = 4 d=2 V 1=2.s/ n.s/ Ω.s, t/ V 1=2.t/ n.t/ ds dt: As n and V 1 are bounded in, tere are constants C 1 and C 2 suc tat var.t 2 / C 1 d=2 Ω.s, t/ ds dt C 2 3d=2 : Hence d=2 T 2 p 0asn.AsT 3 is non-random, we ave E.T / = 1 + d=2 V 1.s/ 2 n.s/ ds + o.d=2 /,.A:12/ var.t / = 2 d K.4/.0/ K.2/.0/ 2 + o. d /:.A:13/ It remains to prove te asymptotic normality of d=2 {N2 0.s/ 1} ds. Let us first consider te case of d = 1. Let δ 1 and δ 2 be sequences tending to 0 as n, δ = δ 1 + δ 2, and r = [1=δ]. In particular, we assume tat δ 2 = o.δ 1 /, = o.δ 2 / and δ 2 > 2. Fori = 1,...,r, let V i =.i 1/δ+δ1.i 1/δ {N 2 1.s/ 1} ds, iδ V i = {N 2 1.s/ 1} ds:.i 1/δ+δ 1 ince δ 2 > 2, te covariance function and te strict stationarity of N 1.s/ mean tat {V i } r are independent and identically distributed. and {V i }r

15 Empirical Likeliood Goodness-of-fit Test 677 A standard derivation sows tat var.v i / = σ 2 0 δ 1{1 + o.1/} and var.v i / = σ2 0 δ 2{1 + o.1/} were σ 2 0 = 2 K.4/.0/K.2/.0/ 2. According to te central limit teorem, as n, Note tat r V i =.rδ 1 σ 2 0 / d N.0, 1/, r V i =.rδ 2 σ 2 0 / d N.0, 1/: 1.σ 2 0 / 1=2 {N 2 0.s/ 1} ds = r.σ2 0 / 1=2 V i +.σ 2 0 / 1=2 0 +.σ 2 0 / 1=2 1 ince rδ 1 1 and rδ 2 0, results (A.14) mean tat r.σ 2 0 / 1=2 d V i N.0, 1/,.σ 2 0 / 1=2 r V i p 0: {N 2 0 rδ r V i.a:14/.s/ 1} ds:.a:15/ It can be easily sown tat.σ 0 2/ 1=2 1 {N 2 rδ 0.s/ 1} ds p 0asn. Tus, we establis te asymptotic normality for d = 1. For te case of d>1, let d 1 = [0, 1] d 1 and T.s/ = d 1 {N1 2.s, t/ 1} dt were s [0, 1]. For i = 1,...,r, define V i = V i =.i 1/δ+δ1.i 1/δ iδ T.s/ ds, T.s/ ds:.i 1/δ+δ 1 were δ 1, δ 2 and r are te same quantities defined earlier for d = 1. Because of te use of a product kernel and strict stationarity, {V i } r and {V i }r are independent and identically distributed random variables. Te asymptotic normality can be proved in a similar fasion. References Aït-aalia, Y. (1996) Testing continuous-time models of te spot interest rate. Rev. Finan. tud., 9, Bosq, D. (1998) Nonparametric statistics for stocastic processes. Lect. Notes tatist., 110. Cen,. X. (1996) Empirical likeliood for nonparametric density estimation. Biometrika, 83, Cen,. X., Härdle, W. and Li, M. (2002) An empirical likeliood goodness-of-fit test for time series. Researc Report 3/2001. Department of tatistics and Applied Probability, National University of ingapore. (Available from ttp:// Cox, J. C., Ingersoll, J. E. and Ross,. A. (1985) A teory of term structure of interest rates. Econometrica, 53, Eubank, R. L. and piegelman, C. H. (1990) Testing te goodness of fit of a linear model via nonparametric regression tecniques. J. Am. tatist. Ass., 85, Fan, J. and Zang, J. (2000) ieve empirical likeliood ratio tests for nonparametric functions. Researc Report. Department of tatistics, Cinese University of Hong Kong. Gao, J. T. and King, M. (2001) Estimation and model specification testing in nonparametric and semiparametric time series econometric models. Tecnical Report. cool of Matematics and tatistics, University of Western Australia, Pert. Genon-Catalot, V., Jeanteau, T. and Laredo, C. (2000) tocastic Volatility models as idden markov models and statistical applications. Bernoulli, 6, Gorodeskii, V. V. (1977) On te strong mixing property for linear sequences. Teory Probab. Applic., 22, Hall, P. and La cala, B. (1990) Metodology and algoritms of empirical likeliood. Int. tatist. Rev., 58,

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