PUBLISHED VERSION. Copyright 2009 Cambridge University Press.

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1 PUBLISHED VERSION Gao, Jiti; King, Maxwell L.; Lu, Zudi; josteim, D.. Nonparametric specification testing for nonlinear time series wit nonstationarity, Econometric eory, 2009; 256 Suppl: Copyrigt 2009 Cambridge University Press PERMISSIONS ttp://journals.cambridge.org/action/stream?pageid=4088&level=2#4408 e rigt to post te definitive version of te contribution as publised at Cambridge Journals Online in PDF or HML form in te Institutional Repository of te institution in wic tey worked at te time te paper was first submitted, or for appropriate journals in PubMed Central or UK PubMed Central, no sooner tan one year after first publication of te paper in te journal, subject to file availability and provided te posting includes a prominent statement of te full bibliograpical details, a copyrigt notice in te name of te copyrigt older Cambridge University Press or te sponsoring Society, as appropriate, and a link to te online edition of te journal at Cambridge Journals Online. Inclusion of tis definitive version after one year in Institutional Repositories outside of te institution in wic te contributor worked at te time te paper was first submitted will be subject to te additional permission of Cambridge University Press not to be unreasonably witeld. 10 t December 2010 ttp://dl.andle.net/2440/56759

2 Econometric eory, 25, 2009, Printed in te United States of America. doi: /s NONPARAMERIC SPECIFICAION ESING FOR NONLINEAR IME SERIES WIH NONSAIONARIY JII GAO University of Adelaide MAXWELL KING Monas University ZUDI LU University of Adelaide DAG JØSHEIM University of Bergen is paper considers a nonparametric time series regression model wit a nonstationary regressor. We construct a nonparametric test for weter te regression is of a known parametric form indexed by a vector of unknown parameters. We establis te asymptotic distribution of te proposed test statistic. Bot te setting and te results differ from earlier work on nonparametric time series regression wit stationarity. In addition, we develop a bootstrap simulation sceme for te selection of suitable bandwidt parameters involved in te kernel test as well as te coice of simulated critical values. An example of implementation is given to sow tat te proposed test works in practice. 1. INRODUCION During te past two decades or so, tere as been muc interest in bot teoretical and empirical analysis of long-run economic and financial time series data. Models and metods used ave been based initially on parametric linear autoregressive moving average representations Granger and Newbold, 1977; Brockwell and Davis, 1990; and many oters and ten on parametric nonlinear time series models see, e.g., ong, 1990; Granger and eräsvirta, 1993; Fan and Yao, Suc parametric linear or nonlinear models, as already pointed out in existing studies, e autors would like to tank Robert aylor, one of te guest editors, and two referees for teir encouragement and constructive comments. e autors also acknowledge useful comments from te conference participants, Bruce Hansen and Peter Pillips in particular, of te Nottingam Conference in Honour of Professor Paul Newbold in September anks also go to Gowry Srianantakumar and Jiying Yin for teir excellent computing assistance and te Australian Researc Council for its continuing support of te Discovery Grants under grant numbers DP and DP Address correspondence to Jiti Gao, Scool of Economics, University of Adelaide, Adelaide SA 5005, Australia; jiti.gao@adelaide.edu.au. c 2009 Cambridge University Press /09 $

3 1870 JII GAO E AL. may be too restrictive in some cases. is leads to various nonparametric and semiparametric tecniques being used to model nonlinear time series data wit te focus of attention being on te case were te observed time series satisfies a type of stationarity. Bot estimation and specification testing as been systematically examined in tis situation Robinson, 1988, 1989; Masry and jøsteim, 1995, 1997; Li and Wang, 1998; Li, 1999; Fan and Linton, 2003; Fan and Yao, 2003; Gao, 2007; Li and Racine, 2007; and oters. e stationarity assumption is restrictive because many time series are nonstationary. ere is now a large literature on linear modeling of nonstationary time series see, for example, Dickey and Fuller, 1979; Pillips, 1987, 1997; Pillips and Perron, 1988; Lobato and Robinson, 1998; Pillips and Xiao, 1998; Kapetanios, Sin, and Snell, 2003; Robinson, 2003; and oters, but not muc as been done in te nonlinear situation. In parametric nonlinear and nonparametric nonlinear time series models wit nonstationarity, existing studies include Pillips and Park PP 1998, Karlsen and jøsteim K 1998, 2001, Park and Pillips PP 2001, Wang and Pillips WP 2009, Karlsen, Myklebust, and jøsteim KM 2007, Pillips 2007, and Cen, Gao, and Li CGL e paper by PP 1998 was among te first to discuss nonparametric kernel estimation in a nonparametric autoregression model wit integrated regressors. K 1998, 2001 independently discuss nonparametric kernel estimation of null recurrent time series. e paper by PP 2001 discusses estimation problems in various parametric nonlinear models wit integrated regressors. WP 2006 develop an alternative approac to nonparametric kernel estimation in bot autoregression and cointegration models wit integrated regressors. e KM 2007 paper provides a class of nonparametric versions of some of tose parametric models proposed in Engle and Granger Pillips 2007 discusses a nonparametric setting of parametric spurious time series models initially proposed in Granger and Newbold 1974 and ten Pillips More recently, CGL 2008 propose a semiparametric estimation in a partially linear model wit nonstationarity. In te field of model specification wit nonstationarity, tere are some existing studies see, for example, Hong and Pillips, 2005; Kasparis, 2007, 2008; and Vadim, All te cited papers consider specification testing in time series regression wit unit roots. e first two papers consider model specification testing in a cointegration setting, wile te tird paper discusses te applicability of te Bierens test in a class of nonlinear and nonstationary models and establises some corresponding results. e last paper develops a functional form test in dealing wit nonlinearity, nonstationarity, and spurious forecasts. In te original version of tis paper, Gao, King, Lu, and jøsteim 2007 also propose using a nonparametric kernel test for nonstationarity in an autoregressive model. In tis paper, we are interested in considering a nonlinear time series of te form Y t = mx t + σ 0 e t wit X t = X t 1 + u t, t = 1,2,...,, 1.1

4 MODEL ESING WIH NONSAIONARY DAA 1871 were m is an unknown function defined over R 1 =,, σ 0 > 0isan unknown parameter, {u t } is a sequence of independent and identically distributed i.i.d. errors, and {e t } is a sequence of martingale differences. We are ten interested in testing te following null ypotesis: H 0 : P mx t = m θ0 X t = 1 for all t 1, 1.2 were m θ0 x is a known parametric function of x indexed by a vector of unknown parameters, θ 0. Under H 0, model 1.1 becomes a nonlinear parametric model of te form Y t = m θ0 X t + σ 0 e t wit X t = X t 1 + u t, t = 1,2,...,. 1.3 PP 2001 extensively discuss some estimation problems for tis kind of parametric nonlinear time series model. o te best of our knowledge, te problem of testing 1.2 for te case were {X t } is nonstationary as not been discussed. is paper attempts to derive a simple kernel test for tis kind of parametric specification of te conditional mean function wen te regressors are integrated. In summary, te main contributions of tis paper are: i We propose a new test statistic for model 1.2. eoretical properties for te proposed test statistic are establised. ii In order to implement te proposed test in practice, we develop a new simulation procedure based on te assessment of bot te size and power of te proposed test. e rest of te paper is organized as follows: Section 2 establises a nonparametric kernel test procedure as well as its asymptotic distribution. A bootstrap simulation sceme is proposed in Section 3. Section 4 sows ow to implement te proposed test in practice. Section 5 concludes wit some remarks on extensions. Matematical details are given in te Appendix. Additional details are available from Appendixes B D of te original version by Gao et al NONPARAMERIC KERNEL ES Consider a test problem of te form H 0 : PmX t = m θ0 X t =1 versus H 1 : PmX t =m θ1 X t + X t,θ 1 =1 2.1 for all t 1 and some θ 1 a parameter space, were θ 0 denotes te true value of θ if H 0 is true, and,θ 1 is a sequence of unknown functions to ensure tat model 1.1 is a semiparametric time series model under H 1. Note tat eac x,θ 1 beaves like a kind of distance function between te null and alternative ypoteses. Suc structure allows te inclusion of eiter a global alternative or a sequence of local alternatives. Wen x,θ 1 x,θ 1,

5 1872 JII GAO E AL. we are interested in testing a parametric conditional mean function versus a global semiparametric conditional mean function. We are testing a parametric conditional mean function against a sequence of local alternatives wen { x,θ 1 } is a sequence of semiparametric functions. o construct a nonparametric kernel test, te main idea is to compare a parametric estimator of m under H 0 wit a nonparametric kernel estimator. In order to avoid introducing biases associated wit nonparametric kernel estimation Gao and King, 2004, we use a smooted version of te parametric estimator in te construction. Similarly to existing studies for te stationary time series case see, for example, Cap. 3 of Gao, 2007, we propose using a kernel-based test of te form M = M = ɛ s K X t X s ɛ t, 2.2 were K = K / wit K being a probability kernel function, is a bandwidt parameter, and ɛ t = Y t m θ X t, in wic θ is a consistent estimator of θ 0 under H 0. In tis paper, we consider θ as te nonlinear least squares estimator of θ 0 as defined in PP In order to establis te asymptotic distribution for M, we need to introduce te following assumption. Assumption 2.1. i e sequence {u t = X t X t 1 } is a sequence of i.i.d. random errors wit E[u t ] = 0, E[u 2 t ] = σ u 2, and μ 4 = E[u 4 t ] <. e marginal density function of {u t } is symmetric. e caracteristic function ψ of {u t } satisfies ψv dv <. ii e sequence {e t } is a sequence of martingale differences satisfying E[e t B t 1 ] = 0, E[e 2 t B t 1] = 1 a.s., E[e 3 t B t 1] = 0 a.s., and 0 <ν 4 = E[e 4 t B t 1 ] < a.s., were B t 1 = σ {e s :1 s t 1} is te σ -field generated by {e s :1 s t 1}. iii e sequences {u s : s 1} and {e t : t 1} are mutually independent. iv e function K is a symmetric and bounded probability density wit compact support CK. In addition, K x + y K x x y for all x CK and any given y, were x is a nonnegative bounded function for all x CK. Let K 3 denote te tree-time convolution of K wit itself. Let Q θ = 1/ t=1 Y t m θ X t 2. Define te nonlinear least squares estimator of θ 0 as te minimizer of Q θ over θ : θ = argmin θ Q θ. Assumption 2.2. i ere are unknown parameters θ 0 and σ 0 > 0 suc tat model 1.3 under H 0 is te true identifiable model.

6 MODEL ESING WIH NONSAIONARY DAA 1873 ii Here, lim = 0 and limsup 1/2 δ 0 = for some 0 <δ 0 < 1/5. iii e function m θ x is differentiable wit respect to θ for eac fixed x.in addition, under H i i = 0,1 te following equations old in probability: for 0 <δ 0 < 1/5τ denotes te transposed, lim R ij 3/2 2δ 0 θ θ i τ θ θ i j = 0, 2.3 j were, for i = 0,1, j = 1,2, and R ij = s=1 t s t s r ij s wit { mθi x τ } mθi x j x r ij s = φ dx, θ θ s in wic φ is te density function of te normal random variable N0,1. Remark 2.1. i Assumption 2.1i requires {u t } to be i.i.d. in order to ensure tat S t = t i=1 X i ave independent increments for all t 1. e last sentence of Assumption 2.1i imposes a mild condition on te caracteristic function, and it olds in many cases. e condition ψv dv < ensures certain convergence results. Let φ x be te density function of 1/ σ u t=1 u t. en Assumption 2.1i implies sup x φ x φx 0as, were φx = 1/ 2πe x2 /2 is te density function of te standard normal random variable N0,1. e proof is standard see, for example, Caps. 8 and 9 of Cow and eicer, Assumption 2.1ii is quite standard in tis kind of problem see, for example, Ass. 2.1 of PP, Obviously, Assumption 2.1ii covers te case were {e t } is a sequence of i.i.d. errors. Assumption 2.1iii imposes te independence between {e s } and {u t } for all s,t 1. Suc an independence assumption is somewat restrictive but may not be too unreasonable in tis kind of nonstationary problem. Assumption 2.1iv is also quite standard in tis kind of nonstationary situation. ii Assumption 2.2i ensures tat te true model 1.3 under H 0 is identifiable. Assumption 2.2ii imposes some minimum conditions on te bandwidt. Assumption 2.2iii imposes some tecnical conditions involving bot te form of m θ0 and te rate of convergence of θ to θ 0. For example, wen m θ0 x = α 0 + β 0 x and te rate of convergence of θ to θ 0 is of o P 3/8+δ 0/2 1/4 1, Assumption 2.2iii olds wit i = 0. In te case were m θ1 x = α 1 +β 1 x +γ 1 x1 exp λ 1 x 2 and te rate of convergence of θ to θ 1 is of o P 7/8+δ 0/2 1/4 1, Assumption 2.2iii olds wit i = 1. We state te main teorem of tis section; its proof is given in te Appendix. HEOREM 2.1. Consider model 1.1. Suppose tat Assumptions old wit i = 0 in Assumption 2.2iii. en, under H 0,

7 1874 JII GAO E AL. L = L = M σ D N0,1 as, were σ 2 = 2 t=1 s=1, t ɛ2 s K 2 X s X t ɛ 2 t. eorem 2.1 sows tat L converges in distribution to standard normality as. Existing studies for te stationary time series case already discuss te small sample performance of tis type of nonparametric kernel-based test. Wen using a normal distribution to approximate te exact finite-sample distribution of tis kind of test, te performance of bot te size and power functions is not good. In order to improve te finite sample performance of L, we propose using a bootstrap simulation metod. Suc a metod is known to work quite well in te stationary case. For eac given bandwidt satisfying certain teoretical conditions, instead of using an asymptotic value of l 0.05 = at te 5% level, for example, we use a simulated critical value for computing te size function and ten te power function. An optimal bandwidt is cosen suc tat te power function is maximized at te optimal bandwidt. Our finite-sample studies sow tat tere is little size distortion wen using suc a simulated critical value. Suc issues are discussed in detail in Section BOOSRAP SIMULAION SCHEME Section 3 discusses ow to simulate a critical value for te implementation of L in eac case. We ten examine its finite sample performance using one example in Section 4, below. Before we look at ow to implement L in practice, we propose a simulation sceme. Simulation Sceme 3.1 e exact α-level critical value, l α 0 <α<1, is te 1 α quantile of te exact finite-sample distribution of L. Because tere are unknown quantities, suc as parameters and functions, we cannot evaluate l α in practice. We terefore suggest coosing an approximate α-level critical value, lα by using te following simulation procedure: i For eac t = 1,2,...,, generate Yt = m θ X t + σ 0 et, were te original sample X 1,...,X acts in te resampling as a fixed design, {e t } is sampled independently eiter from a prespecified distribution or using a nonparametric bootstrap metod, σ 0 is an initial consistent estimator of σ 0, and θ is te nonlinear least squares estimator of θ 0 based on te original sample. ii Use te data set {Y t, X t : t = 1,2,..., } to reestimate θ 0,σ 0. Denote te resulting estimate by θ, σ. Compute te statistic L tat is te corresponding version of L by replacing θ, σ and {Y t, X t :1

8 MODEL ESING WIH NONSAIONARY DAA 1875 t } wit θ, σ and {Y t, X t :1 t } on te rigt-and side of L. iii Repeat te above steps M times and produce M versions of L, denoted by L m for m = 1,2,...,M. Use te M values of L m to construct teir empirical bootstrap distribution function. e bootstrap distribution of L given W ={X t,y t :1 t } is defined by P L x = P L x W. Let l α satisfy P L l α = α and ten estimate l α by l α. iv Define te size and power functions by α = P L lα H 0 and β = P L lα H 1. In practice, bot α and β may be approximated using Edgewort expansions similarly to 3.23 and 3.24 of Gao In order to study bot te size and power functions, we specify te form of a sequence of alternatives as H 1 : P mx t = m θ1 X t + X t,θ 1 = 1, 3.1 were { x,θ 1 } is a sequence of unknown functions satisfying certain conditions in Assumption 3.2 below. Under H 1, model 1.1 becomes Y t = mx t + ɛ t = m θ1 X t + X t,θ 1 + ɛ t, 3.2 were x,θ 1 can be estimated by x, θ 1, in wic θ 1 minimizes t=1 Yt m θ1 X t X t,θ 1 2, 3.3 and x,θ 1 = t=1 K b cv X t xy t m θ1 X t / t=1 K b cv X t x, wit b cv being cosen by a conventional cross-validation estimation metod. Similarly to te proof of Proposition 3.1 of Gao and Gijbels 2008 for te stationary case, it may be sown tat lim x, θ 1 / x,θ 1 = 1in probability for eac given x. Since bot te establisment and te proof of suc a consistency result require more detailed discussion, we wis to leave suc details for future researc. Let H ={ : α ε 0 <α<α+ε 0 } for some 0 <ε 0 <α. Coose an optimal bandwidt ĥ test suc tat ĥ test = arg max H β. 3.4 Since {e t } is stationary, existing results Sect. 3 of Gao and Gijbels, 2008 suggest using an approximate version of te form ĥ test = â 1/2 Ĉ 3/2, 3.5

9 1876 JII GAO E AL. were Ĉ 2 = t=1 2 X t, θ 1 px t μ 2 2 ν2 K 2 vdv 1 t=1 p2 X t 1 t=1 px t and â = 2K K 2 udu 3 ĉp wit ĉp = 3, in wic μ 2 = 1 t=1 Y t m θ X t 2, ν 2 = 1 t=1 p2 X t, px = 1 ĥcv t=1 K X t x wit ĥ cv being cosen by a conventional crossvalidation selection metod, and K 3 is te tree-time convolution of K ĥ cv wit itself. We ten use lα ĥ test in te computation of bot te size and power values of L ĥ test for eac case. Note tat, as sown in te Appendix, te leading term of L is given by L = t=1 s=1, t ɛ s K X t X s ɛ t σ 1 + t=1 s=1, t X s K X t X s X t, 3.6 σ 1 were σ 2 1 is proportional to 3/2 as explicitly given in Lemma A.1 of te Appendix. Equation 3.6 sows tat te first term contributes to te asymptotic normality under H 0 and te second term contributes to te asymptotic consistency of te test under H 1. us, in order to ensure tat te test statistic is asymptotically consistent, we need to impose Assumptions 3.1 and 3.2 below. Assumption 3.1. i ere are consistent estimators σ and σ suc tat, as, σ σ 0 P 0 and σ σ P 0. ii Let H 0 be true. en te following equation olds in probability: for 0 < δ 0 < 1/5, R j lim θ 3/2 2δ 0 θ τ θ θ j = 0, 3.7 j were, for j = 1,2, R j = t 1 s=1 1 s 1 t s r j s wit { m r j s = θ x τ m θ x } j x φ dx. θ θ s Assumption 3.2. Let H 1 be true. Suppose tat Assumption 2.2iii olds wit i = 1. In addition, te following equation olds for 0 <δ 0 < 1/5: D lim 3/4 δ =, were D = t 1 s=1 1 s 1 t s C s wit C s = 2 x,θ 1φ x s dx.

10 MODEL ESING WIH NONSAIONARY DAA 1877 Assumption 3.1i imposes only mild consistency conditions on σ and σ to ensure tat te bootstrap critical value lα is an asymptotically correct α-level critical value under any model in H 0. Similarly to Corollary 4.4 of PP 2001, one may impose conditions on te local integrability or te integrability of m θ to ensure tat Assumption 3.1i olds. Assumption 3.1ii corresponds to Assumption 2.2iii wit i = 0. Similarly to Remark 2.1iii, it can be verified tat Assumption 3.1ii olds wen m θ x belongs to a class of parametric functions. Assumption 3.2 requires tat te distance between H 0 and H 1 is large enoug to ensure tat te test is consistent under H 1. Similarly to Assumption 2.2iii, Assumption 3.2 involves bot te form of m θ1 under H 1 and te rate of convergence of θ to θ 1 wen te form of mx is cosen as mx = m θ1 x + x,θ 1. In bot teory and practice, various forms may be considered for m θ0 and m. For example, we consider te following forms: H 0 : m θ0 x = α 0 + β 0 x versus H 1 : mx = m θ1 x + x,θ 1 = α 1 + β 1 x + γ 1 x 2, 3.9 were θ 0 = α 0,β 0 is estimated by θ, and <α 1,β 1,γ 1 < are unknown parameters. In tis case, in order to verify Assumption 3.2, it suffices to sow tat, as, E [ t 1 s=1 X s 2 K X t X s Xt 2 ] 3/4 δ, wic follows from letting X st = X t X s t 1 = [ ] E Xs 2 K Xt X s Xt 2 t 1 [ ] E Xs 2 K Xt X s X s + X t X s 2 = t 1 xs 2 K xst x s + x st 2 f s x s f st x st dx s dx st letting y s = x s and y st = x st / = t 1 = 1 + o1 y 2 s K y st y s + y st 2 f s y s f st y st dy s dy st t x s y x 4 K yg s g st dxdy s t s t s

11 1878 JII GAO E AL. = φ0 1 + o1 = φ0 1 + o1 t 1 t x x 4 φ s t s s 1 1 s t s x x 4 φ dx s dx K ydy = C 7/2 1 + o1, 3.11 wic follows by using te normal distribution approximation metod as outlined in te proof of Lemma A.1 in te Appendix, were f s denotes te density function of X s f st denotes te density function of X t X s, g s denotes te density function of X s / s, and g st denotes te density function of X t X s / t s. is sows tat Assumption 3.2 olds. In general, we may consider testing various classes of parametric functions under H 0 against nonparametric and/or semiparametric alternatives under H 1. is is bot teoretically justifiable and practically implementable, because, as demonstrated by PP 2001, ms. 5.1 and 5.2, te rate of convergence for one class can be different from tat for anoter class. We state te following results of tis section. HEOREM 3.1. i Assume tat te conditions of eorem 2.1 old. In addition, if Assumption 3.1 olds, ten under H 0, we ave lim P L lα = α. ii Assume tat te conditions of eorem 2.1 old. In addition, if Assumptions 3.1 and 3.2 old, ten under H 1, we ave lim P L lα = 1. e proof of eorem 3.1 is given in te Appendix. eorem 3.1i implies tat eac lα is an asymptotically correct α-level critical value under any model in H 0, and eorem 3.1ii sows tat L is asymptotically consistent. In Section 4 we illustrate eorem 3.1 using a simulated example. 4. AN EXAMPLE OF IMPLEMENAION is section studies te finite-sample properties of te size and power functions of te proposed test. Example 4.1 Consider a nonlinear time series model of te form Y t = mx t,θ+ e t and X t = X t 1 + u t, t = 1,2,..., 4.1 were {e t } is a sequence of i.i.d. N0,1, {u t } is also a sequence of i.i.d. N0,1, X 0 = 0, and te forms of mx,θ are given as follows: H 0 : mx,θ 0 = θ 0 x versus H 1 : mx,θ 1 = θ 11 x + θ 12 x 2 and 4.2 H 0 : mx,θ 0 = θ 0 x versus H 1 : mx,θ 1 = θ 21 x + θ 22 x 1 e θ 23 x 2, 4.3

12 MODEL ESING WIH NONSAIONARY DAA 1879 were te θ s are cosen as follows: Case 1: θ 0 = θ 11 = θ 21 = 1 and θ 12 = θ 22 = θ 23 = 0.08; Case 2: θ 0 = θ 11 = θ 21 = 1 and θ 12 = θ 22 = Note tat Assumptions 2.2 and 3.2 bot old in tis case. e form of mx,θ 1 in 4.3 as been used in Kapetanios et al In tis section, we use an ordinary least squares OLS metod to estimate te unknown parameters involved in te models under H 0 and te proposed estimation metod in 3.3 for te unknown parameters and functions under H 1. In order to compare te performance of te proposed test based on different bandwidts, we evaluate te finite-sample performance of te proposed test associated wit bot te power-based optimal bandwidt ĥ test in 3.4 and an estimation-based optimal bandwidt of te form ĥ cv = argmin H wic m i X i ; = j=1, i K X j X i 1 i=1 Y i m i X i ; 2,in wit K x = Yj / l=1, i K X l X i x I [ 1,1] x and H = [ 1, 1 δ ] is cosen suc tat bot small and relatively large bandwidt values may be selected, were 0 <δ<1. Note tat ĥ test and ĥ cv eac as one version under H 0, but bot ave two versions for Cases 1 and 2 under H 1. o use some simple notation, we introduce itest = ĥ test and icv = ĥ cv for i = 0,1,2 to represent 0test and 0cv under H 0, and itest and icv under H 1 for Cases i wit i = 1,2. We ten define L itest = L itest and L icv = L icv for i = 0,1,2. For i = 0,1,2, let f itest denote te frequency of L itest > lα itest and f icv denote te frequency of L icv > lα icv. We consider cases were te number of replications in eac of te sample versions of te size and power functions was M = 1,000, wit B = 250 bootstrapping resamples {e t } involved in Simulation Sceme 3.1 from te standard normal distribution N0,1; te simulations were done for te cases of = 80, 200, 500, and 800. ables 4.1 and 4.2 sow tat bot te proposed test and te proposed simulation sceme are implementable and work well numerically for te cointegration case. First, te augmented test based on ĥ test is more powerful tan tat associated wit ĥ cv in eac case. Second, ables 4.1 and 4.2 sow tat te proposed test is applicable to bot linear and nonlinear alternatives. ird, able 4.2 sows tat te proposed test still as power even wen te distance between te null and an alternative is made deliberately close. For example, wen θ 12 and θ 22 are made ABLE 4.1. Simulated sizes at te 1%, 5%, and 10% levels 1% level 5% level 10% level f 0cv & f 0test f 0cv & f 0test f 0cv & f 0test

13 1880 JII GAO E AL. ABLE 4.2. Simulated power values at te 1%, 5%, and 10% levels Model 4.2 Model 4.3 f 1cv f 2cv f 1test f 2test f 1cv f 2cv f 1test f 2test 1% level % level % level as small as 5% and te sample is as medium-sized as = 80, te proposed test still as a power value greater tan te nominal level in eac case. Finally, ables 4.1 and 4.2 also sow tat te power increases wen te distance between te null ypotesis and an alternative increases. 5. CONCLUSION AND EXENSIONS We ave proposed a new nonparametric test for te conditional mean function wen te regressors are integrated. e asymptotic normal distribution of te proposed test statistic as been establised. In addition, we ave proposed a simulation sceme to implement te proposed test in practice. e finite-sample results sow tat bot te proposed test and te simulation sceme are practically applicable and implementable. As briefly mentioned in Section 1, we may also consider testing te conditional variance nonparametrically. Furtermore, bot te conditional mean and te conditional variance functions may be specified simultaneously. e main idea is tat to test H 01 : PmX t = m θ0 X t and σx t = σ θ0 X t = 1, 5.1 we may use a kernel-based test of te form L = [ Us K 1 X s X t U t + V s G 2 X s X t V t ], 5.2

14 MODEL ESING WIH NONSAIONARY DAA 1881 were = 1, 2 is a pair of bandwidt parameters, K and G are bot probability kernel functions, U t = Y t m θ X t, V t = Ut 2 σ 2 θ X t, and θ is an estimator of θ 0 under H 01. Analogously to eorem 2.1, we may establis a corresponding teorem for L. As te detail for tis case is extremely lengty and tecnical, we leave tis issue for future study. Anoter important extension would be to te case were X t = X t1,...,x td in 1.1 is a vector of d-dimensional nonstationary sequences. In tis case, we are interested in testing H 02 : P mx t = d i=1 m iθ0 X ti = 1 for all t 1, 5.3 were eac m iθ0 is a known function indexed by θ 0. Detailed construction of suc a test would involve some estimation procedures for additive models as used in Gao, Lu, and jøsteim 2006 in te stationary spatial case. Since suc an extension is not straigtforward, we leave it as a future topic. REFERENCES Brockwell, P. & R. Davis 1990 ime Series eory and Metods. Springer. Cen, J., J. Gao, & D. Li 2008 Semiparametric regression estimation in null recurrent time series. Working paper, University of Adelaide; available at ttp:// Cow, Y.S. & H. eicer 1988 Probability eory. Springer-Verlag. Dickey, D.A. & W.A. Fuller 1979 Distribution of estimators for autoregressive time series wit a unit root. Journal of te American Statistical Association 74, Engle, R.F. & C.W.J. Granger 1987 Co-integration and error correction: Representation, estimation and testing. Econometrica 55, Fan, J. & Q. Yao 2003 Nonlinear ime Series: Nonparametric and Parametric Metods. Springer. Fan, Y. & O. Linton 2003 Some iger-teory for a consistent nonparametric model specification test. Journal of Statistical Planning and Inference 109, Gao, J Nonlinear ime Series: Semiparametric and Nonparametric Metods. Capman & Hall/CRC. Gao, J. & I. Gijbels 2008 Bandwidt selection in nonparametric kernel testing. Journal of te American Statistical Association 484, Gao, J. & M.L. King 2004 Adaptive testing in continuous-time diffusion models. Econometric eory 20, Gao, J., M.L. King, Z. Lu, & D. jøsteim 2007 Specification esting in Nonlinear ime Series wit Nonstationarity. Working paper, University of Adelaide; available at ttp:// directory/jiti.gao. Gao, J., Z. Lu, & D. jøsteim 2006 Estimation in semiparametric spatial regression. Annals of Statistics 34, Granger, C.W.J. & P. Newbold 1974 Spurious regressions in econometrics. Journal of Econometrics 2, Granger, C.W.J. & P. Newbold 1977 Forecasting Economic ime Series. Academic Press. Granger, C.W.J. &. eräsvirta 1993 Modelling Nonlinear Dynamic Relationsips. Oxford University Press. Hall, P. & C. Heyde 1980 Martingale Limit eory and Its Applications. Academic Press.

15 1882 JII GAO E AL. Hong, S.H. & P.C.B. Pillips 2005 esting Linearity in Cointegrating Relations wit an Application to PPP. Cowles Foundation Discussion Paper 1541, Yale University. Kapetanios, G., Y. Sin, & A. Snell 2003 esting for a unit root in te nonlinear SAR framework. Journal of Econometrics 112, Karlsen, H.,. Myklebust, & D. jøsteim 2007 Nonparametric estimation in a nonlinear cointegration model. Annals of Statistics 35, Karlsen, H. & D. jøsteim 1998 Nonparametric Estimation in Null Recurrent ime Series. Working paper 50, Sonderforscungsbereic series 373, Humboldt University. Karlsen, H. & D. jøsteim 2001 Nonparametric estimation in null recurrent time series. Annals of Statistics 29, Kasparis, I e Bierens est for Certain Nonstationary Models. Discussion paper , University of Cyprus. Kasparis, I Detection of functional form misspecification in cointegrating relations. Econometric eory 24, Li, Q Consistent model specification tests for time series econometric models. Journal of Econometrics 92, Li, Q. & J. Racine 2007 Nonparametric Econometrics: eory and Practice. Princeton University Press. Li, Q. & S. Wang 1998 A simple consistent bootstrap tests for a parametric regression functional form. Journal of Econometrics 87, Lobato, I. & P.M. Robinson 1998 A nonparametric test for I0. Review of Economic Studies 65, Masry, E. & D. jøsteim 1995 Nonparametric estimation and identification of nonlinear ARCH time series. Econometric eory 11, Masry, E. & D. jøsteim 1997 Additive nonlinear ARX time series and projection estimates. Econometric eory 13, Park, J. & P.C.B. Pillips 2001 Nonlinear regressions wit integrated time series. Econometrica 69, Pillips, P.C.B Understanding spurious regressions in econometrics. Journal of Econometrics 33, Pillips, P.C.B ime series regression wit a unit root. Econometrica 55, Pillips, P.C.B Unit root tests. In S. Klotz ed., Encyclopedia of Statistical Sciences, vol. 1, Pillips, P.C.B Local limit teory and spurious regressions. Cowles Foundation Discussion Paper, Yale University. Pillips, P.C.B. & J. Park 1998 Nonstationary density estimation and kernel autoregression. Cowles Foundation Discussion Paper 1181, Yale University. Pillips, P.C.B. & P. Perron 1988 esting for a unit root in time series regression. Biometrika 75, Pillips, P.C.B. & Z. Xiao 1998 A primer on unit root testing. Journal of Economic Surveys 12, Robinson, P.M Root-N-consistent semiparametric regression. Econometrica 56, Robinson, P.M Hypotesis testing in semiparametric and nonparametric models for econometric time series. Review of Economic Studies 56, Robinson, P.M Efficient tests of nonstationary ypoteses. In Recent Developments in ime Series, vol. 1, pp Elgar Reference Collection. International Library of Critical Writings in Econometrics. ong, H Nonlinear ime Series: A Dynamical System Approac. Oxford University Press. Vadim, M Nonlinearity nonstationarity and spurious forecasts. Journal of Econometrics 142, Wang, Q. & P.C.B. Pillips 2009 Asymptotic teory for local time density estimation and nonparametric cointegrating regression. Econometric eory 25,

16 MODEL ESING WIH NONSAIONARY DAA 1883 APPENDIX is appendix provides matematical details for te proofs of te main teorems and teir associated lemmas. Additional derivations are available from Appendixes B D of te original version by Gao et al o avoid notational complication, we introduce te following notation: Let a st = K X t X s, ɛ t = σ 0 e t, and η t = 2 t 1 s=1 a stɛ s. Recall λ t θ 0 = m θ0 X t m θ X t. Observe tat, under H 0, M = ɛ s K X t X s ɛ t = ɛ s K X s X t ɛ t + λ s θ 0 K X t X s λ t θ ɛ s K X t X s λ t θ 0 M 1 + M 2 + M 3, A.1 σ 2 = 2 ɛ s 2 K 2 X t X s ɛ 2 t = 2 ɛs 2 K 2 X t X s ɛt λ 2 s θ 0 K 2 X t X s λ 2 t θ 0 + R, A.2 were R is te remainder term given by R = σ 2 2 ɛs 2 K 2 X t X s ɛ 2 t 2 λ 2 s θ 0 K 2 X t X s λ 2 t θ 0. In view of A.1 and A.2, to prove eorem 2.1 it suffices to sow tat, as, M 1 D N0,1, A.3 σ M i P 0 for i = 2,3, A.4 σ σ 2 σ 2 σ 2 P 0, A.5 were σ 2 = 2 t=1 s=1, t ɛ2 s a2 st ɛ2 t. We will return to te proof of A.4 and A.5 in te second alf of tis Appendix after aving proved Lemmas A.1 A.3. In order to prove A.3, we need to introduce a stocastic normalization procedure before we may apply Corollary 3.1 of Hall and Heyde 1980, p. 58 to our case. Let C 10 = 2σ0 4 K 2 u du and define a random variable of te form σ 2 10 = C 10 N, A.6

17 1884 JII GAO E AL. in wic N as te same definition as n in Karlsen and jøsteim It is te number of regenerations for te Markov cain {X t }. Note tat we use σ10 2 to express te explicit function of te random variable N for notational simplicity. More details about te definition of N are available from Appendix B of te Gao et al In addition, it follows from Appendix B tat te inequality 1/2 δ 0 N 1/2+δ 0 A.7 olds almost surely for large enoug and all 0 <δ 0 < 1/5. As sown in Lemma A.3 below, we ave, as, σ 2 σ10 2 P 1. A.8 In view of A.8, to prove A.3 it suffices to sow tat, as, M 1 σ 10 D N0,1. A.9 We now start to prove A.9. Before verifying te conditions of Corollary 3.1 of Hall and Heyde 1980, we introduce some notation. Let U t = η t ɛ t /σ 10 and,s = σ {U t :1 t s} be te σ field generated by {U t :1 t s}. Since N is independent of {e t :1 t } by construction, E[U t,t 1 ] = 0. By Corollary 3.1 of Hall and Heyde 1980, in order to prove A.9, it suffices to sow tat for all δ>0, ] E [Ut 2 I {[U t >δ]},t 1 P 0, A.10 [ ] E Ut 2,t 1 P 1. A.11 Given te definition of {U t }, in order to verify A.10 and A.11 it suffices to sow tat, as, 1 σ10 4 ηt 4 P 0, A.12 1 σ10 2 ηt 2 P 1. A.13 e proofs of A.12 and A.13 are given in Lemmas A.2 and A.3, respectively. A.1. Lemmas. Assumption 2.1i already assumes tat {u i } is a sequence of i.i.d. random variables and as a symmetric probability density function. Now we let f x and f st x be te density functions of u i and X st = X t X s, respectively, and g st x be te density function of V st = X st / t s. Clearly, f st x = g st x/ t s1/ t s, and by utilizing te usual normal approximation of V st D N0,1 as t s under te conventional central limit teorem conditions, it follows from Assumption 2.1i tat

18 MODEL ESING WIH NONSAIONARY DAA 1885 sup x R 1 g st x φx 0ast s. us, sup x R 1 g st x/ t s φx/ t s 0ast s, were φx = 1/ 2π exp{ x 2 /2}. Anoter key condition used in te following proofs is tat {e s } and {u t } are assumed to be mutually independent for all s,t 1. In order to complete te proof of eorem 2.1 we need to evaluate σ 2 1 = varm 1. Recall ɛ s = σ 0 e s, X st = X t X s = i=s+1 t u i, and define ξ st = K X st ɛ s ɛ t wit λ s θ 0 = m θ0 X s m θ X s. A.14 We assume witout loss of generality tat σu 2 = E[u2 t ] 1 and σ 0 2 = E[e2 1 ] 1 trougout tis Appendix. For i = 1,...,4, 1 s < t, and 1 s 2 < s 1 < t, we introduce te following notation: B i s,t = E [ K i X st ] = K i x f st xdx = t s K i x x g st dx, A.15 t s [ ] B i s 1,s 2,t = E K i X s 1 t K i X s 1 t + X s2 s 1 = 2 K i xk i x + y f s1 t x f s2,s 1 ydx dy = t s1 s1 s 2 K i xk i x + y x y g s1 t g s2,s t s1 1 dx dy, A.16 s1 s 2 C cdpq s,t = E [ e c s ed t e2 s 1p e 2 t 1q] 1 if c = d = 2, p = q = 0, = ν if c = d = 0, p = q = 2, 0 if c = d = p = q = 1, A.17 were ν 4 = E[e 4 t ]. Since {e t } and {u s } are assumed to be mutually independent for all s,t, we can obtain tat, for large enoug, σ 2 1 = var[ M 1 ] t 1 [ ] = 41 + o1 E ξst 2 = 4σ0 4 t o1 B 2 s,t C 2200 s,t. Lemma A.1 below derives te order of σ 2 1 and sows tat te rate of σ 2 1 diverging to is slower tan 2, wic is te corresponding rate for te stationary case.

19 1886 JII GAO E AL. LEMMA A.1. Assume tat te conditions of eorem 2.1 old. en, as, σ 2 1 = var[ M 1 ] = C 0 3/2 1 + o1, A.18 were C 0 = 16σ 4 0 K 2 udu/3 2π. Proof. Coose some positive integer 1 suc tat and. Observe tat t 1 s=1 E[ast 2 1 ] = s=1 t=s+1 E[a 2 st ] = A 1 + A 2, 0as A.19 were A 1 = 1 s=1 1 t s E[a 2 st ] = O = o 3/2, using te fact tat E[a 2 st ] k2 0 due to te boundedness of te kernel K by a constant k 0 > 0. Using A.15, we ave 1 A 2 = E[ast 2 ] s=1 +1 t s 1 1 = s=1 +1 t s 1 t s K 2 x x g st dx t s = d o1 K 2 1 xdx s=1 +1 t s 1 1 t s = 4 K 2 ydy d 0 3/2 1 + o1, 3 A.20 were d 0 = 1/ 2π. Equations A.19 and A.20 imply tat, for large enoug, t 1 s=1 E[ast 2 ] = 4 K 2 ydy 3 3/2 1 + o1. 2π erefore, it follows tat, for, t 1 4 [ ] E ξst 2 = 4σ0 4 t 1 s=1 B 2 s,t C 2200 s,t = C 0 3/2 1 + o1, A.21 A.22 were C 0 = 16σ 0 4 K 2 udu 3. us te proof of Lemma A.1 is completed. n 2π For 0 <δ 0 < 1 5, recall C 10 as defined in A.6 and let σ20 2 = C 10 1/2 δ 0. Wenow ave te following lemma. LEMMA A.2. Under te conditions of eorem 2.1, we ave, as, 1 σ10 4 ηt 4 P 0. A.23

20 MODEL ESING WIH NONSAIONARY DAA 1887 Proof. In view of A.7, we ave, for large enoug and any given δ>0, 1 P σ10 4 ηt 4 1 >δ P σ10 4 ηt 4 >δ, 1/2 δ 0 N 1/2+δ 0 + P N < 1/2 δ 0 or N > 1/2+δ 0 1 P σ20 4 us, in order to prove A.23, we only need to sow tat 1 [ ] σ20 4 E ηt 4 0. Observe tat ηt 4 >δ + o1 1 [ ] σ 20 4 δ E ηt 4 + o1. A.24 [ ] E ηt 4 t 1 t 1 t 1 t 1 = 16 E [ ] a s1 t a s2 t a s3 t a s4 t ɛ s1 ɛ s2 ɛ s3 ɛ s4. s 1 =1 s 2 =1 s 3 =1 s 4 =1 A.25 A.26 Since Assumption 2.1 imposes mutual independence on {u s } and {e t } for all s, t 1, in order to prove A.23, it suffices to sow tat, as, 1 t 1 t 1 [ ] σ20 4 E as 2 1 t a2 s 2 t 0, s 1 =1 s 2 =1, s 1 A.27 1 t 1 [ ] σ20 4 E ast 4 0. o prove A.27, using A.16 we ave t 1 s 1 =1 t 1 E s 2 =1, s 1 [ ] as 2 1 t a2 s 2 t A.28 = 4 2 t 1 s K 2 xk 2 x + y t=3 s 1 =2 s 2 =1 t s1 s1 s 2 x y g s1 t g s2,s t s1 1 dx dy s1 s 2 = o1j 2 02 d2 0 t=3 t 1 s 1 =2 s 1 1 s 2 =1 1 1 t s1 s1 s 2 = C 2 2 = o 3 2δ 0 2 = o σ 4 20, A.29 using te assumption tat lim 1/2 δ 0 = for 0 <δ 0 < 1/5, were C > 0 is some constant, J 02 = K 2 udu, and d 0 = 1/ 2π.

21 1888 JII GAO E AL. Similarly to A.20, using A.15 we ave t 1 E[a 4 t 1 st ] = K 4 x x g st dx t s t s = C o1 K 4 t 1 1 xdx t s = C 3/2 1 + o1 = o 3 2δ 0 2 = o σ20 4, A.30 using te assumption tat lim 1/2 δ 0 =, were C > 0 is some constant. Equations A.29 and A.30 complete te proofs of A.27 and A.28. is completes te proof of Lemma A.2. n LEMMA A.3. Let te conditions of eorem 2.1 old. en, as, 1 σ10 2 ηt 2 P 1. A.31 Proof. Observe tat ηt 2 t 1 2 t 1 = 2 a st ɛ s = 4 a 2 t 1 t 1 st ɛ2 s + 4 ɛ s1 a s1 t a s2 t ɛ s2. s 1 =1 s 2 =1, s 1 A.32 We first sow tat, as, 4 t 1 σ10 2 ast 2 ɛ2 s P 1. A.33 Similarly to te proofs of Lemmas A.1 and A.2, it can be sown tat t 1 ast ɛ 2 s 2 1 = o P σ10 2, A.34 using te assumption tat {ɛ t } is independent of {u s } for all s,t, and E[ɛ 2 1 ] = 1. In view of A.34, in order to prove A.33, it suffices to sow tat, as, 4 t 1 σ10 2 a 2 st = 2 σ 2 10 t=1 s=1 a 2 st P 1. A.35 Let Qu = K 2 u/ K 2 udu. en Q is a probability kernel. According to Lemma C.1 in Appendix C of Gao et al. 2007, we ave tat, as, 1 N Q s=1 Xs x P 1 A.36 uniformly in x R 1, were we ave used te result tat te invariant measure of te random walk {X t } can be taken to be a Lebesgue measure wit corresponding density

22 MODEL ESING WIH NONSAIONARY DAA 1889 px 1. e uniform convergence in A.36 strengtens te pointwise convergence of eorem 5.1 of Karlsen and jøsteim 2001 in te random walk case. For more details, refer to Gao et al and its Appendixes B D. us, te proof of A.35 follows from A.36 and 2 σ10 2 ast 2 = 1 t=1 s=1 t=1 = 1 t=1 1 N J 02 K 2 Xs X t s=1 1 Xs X t N Q s=1 P 1 A.37 as. In view of A.31 and A.32, in order to complete te proof of A.31 we need to sow tat 1 t 1 t 1 σ10 2 ɛ s1 a s1 t a s2 t ɛ s2 P 0 as. A.38 s 1 =1 s 2 =1, s 1 Similarly to A.24, te proof of A.38 follows from 1 ] t 1 t 1 2 E[ σ20 4 ɛ s1 a s1 t a s2 t ɛ s2 0, A.39 s 1 =1 s 2 =1, s 1 wic, using te same arguments as in A.25 A.30 and te fact tat {ɛ s } is a sequence of martingale differences and also independent of {u t }, follows from t 1 =2 t 2 =1 s 1 =1 s 2 =1 E [ ] a s1 t 1 a s2 t 1 a s1 t 2 a s2 t 2 = O 5/2 3 = o σ20 4. A.40 is terefore completes te proof of Lemma A.3. A.2. Proofs of eorems. Proof of eorem 2.1. In view of A.3, to complete te proof of eorem 2.1 it suffices to prove A.4 and A.5. We only give te proof of A.4, since te proof of A.5 is very similar. aylor expansions of m θ x wit respect to θ at θ 0 imply mθ0 x τ m θ x m θ0 x = θ θ 0 + o P θ θ 0 A.41 θ for eac given x. us, in order to prove A.4, using te same arguments as in A.24, it suffices to sow tat θ θ 0 τ mθ0 X s Xt X s mθ0 X t τ K θ θ 0 = o P σ 10. t=1 s=1 θ θ A.42 Note tat using te same arguments as in A.24, te proof of A.42 follows wen A.42 olds wit σ 10 replaced by σ 20. n

23 1890 JII GAO E AL. o do so, we first evaluate te following quantity. Straigtforward calculations imply tat, for large enoug letting Y 1 = X s and Y 12 = X t X s and ten x 1 = y 1 and x 2 = y 2 /, t 1 [ mθ0 X s τ ] Xt X s mθ0 X s + X t X s E K θ θ t 1 mθ0 y 1 τ y12 m θ0 y 1 + y 12 = K f s y 1 f st y 12 dy θ θ 1 dy 12 t 1 mθ0 x 1 τ mθ0 x 1 + x 2 = K x 2 f s x 1 f st x 2 dx 1 dx 2 θ θ t mθ0 x 1 τ mθ0 x 1 = 1 + o1 K x 2 s t s θ θ x1 x2 g s g st dx 1 dx 2 s t s t mθ0 x τ mθ0 x x = d o1 φ dx. s t s θ θ s A.43 is, along wit Assumption 2.2iii wit j = 1 and te Markov inequality, implies tat A.42 olds, wit σ 10 replaced by σ 20. is terefore proves A.4 for i = 2. Meanwile, it follows from A.3 tat 1 Xt X s σ 10 ɛ s K ɛ t = O P 1. A.44 t=1 s=1 us, te proof of A.4 for i = 3 follows from A.42 A.44 and ɛ s K Xt X s Xt X 2 s K λ t θ 0 t=1 s=1 Xt X s Xt X s ɛ s K ɛ t t=1 s=1 λ s θ 0 K λ t θ 0 t=1 s=1 = O P σ 10 o P σ 10 = o P σ10 2. A.45 Similarly to A.41 A.43, using Assumption 2.2iii wit j = 2, one may verify A.5. n Proof of eorem 3.1. Using ɛ t Y t m θ X t = m θ X t m θ X t + σ 0 e t,

24 MODEL ESING WIH NONSAIONARY DAA 1891 we ave M ɛ s K X s X t ɛ t = σ 0 es K X s X t σ 0 et + λ s K X s X t λ t + 2 σ 0 es K X s X t λ t, A.46 were λ t = m θ X t m θ X t. Using Assumptions 2.1, 2.2, and 3.1, in view of te notation of L introduced in Simulation Sceme 3.1 as well as te proof of eorem 2.1, we may sow tat, as, P L x x for all x, A.47 olds in probability wit respect to te distribution of te original sample {X t,y t :1 t }. In detail, in order to prove A.47, using te fact tat {e s } and {X t,y t } are independent for all s,t 1, we may sow tat te proofs of Lemmas A.2 and A.3 remain true by successive conditioning arguments. Let z α be te 1 α quantile of suc tat z α = 1 α. en it follows from A.47 tat, as, L P z α 1 z α = α in probability. A.48 is, togeter wit te construction tat P L l α = α, implies tat, as, l α z α P 0. A.49 Using te conclusion of eorem 2.1 and A.47 again, we ave, as, L P x P L x P 0 for all x,. A.50 is, along wit te construction tat P L α l = α again, sows tat, as, lim L P lα = α A.51 olds. erefore, te conclusion of eorem 3.1i is proved. o prove eorem 3.1ii, we need to decompose M as follows: M = ɛ s K X st ɛ t = ɛ s θ 1 K X st ɛ t θ 1 + λ s θ 1 K X st λ t θ λ s θ 1 K X st ɛ t θ 1, were ɛ t θ 1 = Y t mx t and λ t θ 1 = mx t m θ X t under H 1.

25 1892 JII GAO E AL. By te proof of eorem 2.1, in order to prove eorem 3.2ii it suffices to sow tat, under H 1, t=1 s=1, t λ sθ 1 K X t X s λ t θ 1 P. A.52 σ 10 Using aylor expansions to m θ wit respect to θ, weave mx t m θ X t = X t,θ 1 + m θ1 X t m θ X t τ m θ X t = X t,θ 1 + θ 1 θ θ=θ1. A.53 θ In view of A.52, using Assumption 2.2iii wit i = 1, in order to prove A.52 it suffices to sow tat t=1 s=1, t E[ X s,θ 1 K X t X s X t,θ 1 ]. A.54 σ 20 Note tat letting X st = X t X s t 1 s=1 [ E X s,θ 1 K Xt X s t 1 [ = X s,θ 1 K ] X t,θ 1 Xt X s ] X s + X t X s,θ 1 t 1 xst = x s,θ 1 K x s + x st,θ 1 f s x s f st x st dx s dx st letting y s = x s and y st = x st t 1 = y s,θ 1 K y st y s + y st f s y s f st y st dy s dy st t = 1 + o1 2 x s y s t s x,θ 1K yg s g st dx dy t s t = φ0 1 + o1 2 x s t s x,θ 1φ dx K ydy s t = φ0 1 + o1 C s. s t s A.55 e proof of eorem 3.1ii terefore follows from Assumption 3.2 and equations A.53 A.55. n