Uniform confidence bands for kernel regression estimates with dependent data IN MEMORY OF F. MARMOL

Transcription

1 . Uniform confidence bands for kernel regression estimates with dependent data J. Hidalgo London School of Economics IN MEMORY OF F. MARMOL

2 AIMS AND ESTIMATION METHODOLOGY CONDITIONS RESULTS BOOTSTRAP

3 AIMS AND ESTIMATION: Given a set of data {y t } n t=1 f ( ) = A question of interest can be: regression f unction density f unction spectral density is f (x) smooth? What is its functional form?

4 Here we consider a regression function f ( ) =r ( ), thatis y t = r (x t )+u t ; t =1,...,n, (1.1) where the errors u t are assumed to follow a covariance stationary linear process, and where for simplicity we shall consider henceforth that x t = t/n. Our interests are

5 Test for a particular functional form for r (x), H 0 : r (x) =r 0 (x) x [0, 1]. Test for the continuity of r (x), H 0 : r + (x) =r (x) x [0, 1] where r + (x) =lim z x+ r (z) and r (x) = lim z x r (z).

6 We now describe the kernel estimator of r (x) at a point x q = q/n, 1+ň q n ň 1 is br a (q) :=br a (x q )= 1 ň nx t=1 y t K µ t q where K (x) is a symmetric in [ 1, 1], ň =[na] and a 0 as n. ň, (2.1)

7 To test for the continuity of r (x), we employ one-sided kernels of Rice (1984). These kernels were introduced to obtain consistent estimates at the boundary of their compact support. In our context, it appears useful as the implementation of the test requires the estimation of r + (x) and r (x). Let K + (x) and K (x) be one-sided kernels. Hence, our kernel estimators at the point x q are br a,± (q) = 1 ň nx t=1 µ t q y t K ±. (2.2) ň

8 METHODOLOGY: Tests are based on uniform confidence bands for br (x), thatis sup br (x) r (x). x [0,1] Main result is to obtain its limiting distribution.

9 For the kernel density estimator, say b f (x), off (x), Woodroofe(1967) showed that Pr ( b 1/2 n sup x Ξ n bf (x) f (x) c n <x ) d exp ³ 2e x, (1.2) where x R +, b n =( 2ň log a) and Ξ n is a grid of points in [0, 1]. This result was extended by Bickel and Rosenblatt (1973) when the supremum is over all x [0, 1].

10 In regression models, they were extended, among others, by Johnston (1982) for iid errors u t. However,we wish to relax the condition of independence of u t and in particular we wish to drop any mixing dependence condition on u t. For instance, we allow for strong dependence as in Csörgo and Mielniczuk (1995). Recall that when we allow the data not to be weakly dependent, mixing conditions are incompatible with this. See Ibragimov and Rozanov (1981).

11 However, contrary to Csörgo and Mielniczuk (1995), we do not take the supremum on Ξ n but, on [0, 1]. The possibility of the strong dependence of u t will imply that the normalization constants b n and c n in (1.2), depend on the degree of dependence of u t,measured by the long memory parameter, in contrast to when u t is strong mixing dependent.

12 CONDITIONS: C1 {u t } t Z is a covariance stationary linear process defined as u t = X j=0 ϑ j ε t j ; X j=0 ϑ 2 j <, withϑ 0 =1, where {ε t } t Z is a zero mean iid sequence with E ³ ε 2 t μ < for some >4. Also,forj 0, = σ 2 ε and E ³ ε t = ϑ j = jx k=0 ϑ k b k j, ϑ 0 = b 0 =1, and where, for k 1, ϑ k = (k) k d 1, d [0, 1/2) ; (k) (k +1) < 0 (k) k 1 with 0 (k) > 0 and P k=0 k 2 b k <.

13 Condition C1 implies that u t = X j=0 ϑ j ε 0 t j ; ε0 t = X j=0 One model satisfying (2.3) is the FARIMA(p, d, q) process (1 L) d Φ p (L) u t = Θ q (L) ε t, b j ε t j. (2.3) where (1 L) d = P k=0 ϑ k L k with ϑ k = Γ (k + d) / (Γ (d) Γ (k +1)) and Φ p (L) and Θ q (L) are the autoregressive and moving average polynomials with no common roots and outside the unit circle. The latter implies that Φ 1 p (L) Θ q (L) = P j=0 b j L j with b j = O ³ j a for any a>0.

14 Denoting B (λ) = X j=0 b j e ijλ ; λ ( π, π], the decomposition in (2.3) implies that the s.d.f of {u t } t Z is where g (λ) = f (λ) = σ2 ε g (λ) h (λ), 2π P j=0 ϑ j e ijλ 2 and h (λ) = B (λ) 2. The condition P k=0 k 2 b k < implies that h (λ) is twice continuously differentiable for all λ [0,π]. An example of the function (k) is (k) =log ξ k for any ξ>0.

15 Moreover, because C1, ϑ k is of bounded variation, that is P k=0 ϑk ϑ k+1 <, and ϑ k is also a quasi-monotonic sequence, see Yong (1974, p.4), wehavethat g (λ) Dλ 2d as λ 0+, 0 <D<, seeyong s(1974) Theorems III-11 and 12. Finally, we have that γ j = Z π π g (λ) cos (jλ) dλ ζ (d) j 2d 1 as j, d µ 0, 1, (2.4) 2 =2π j0, d =0, where ab is the Kronecker delta and ζ (d) =2Γ (1 2d)cos ³ π ³ 1 2 d.

16 We could allow for more than one singularity when examining the asymptotic properties of the supremum of br a (q). However, it eases the arguments for the validity of the bootstrap.

17 C2 For all x [0, 1], the regression function r (x) satisfies lim y x r (y) r (x) Q (x) x y τ = o (1) where 0 <τ 2 and Q (x) is a polynomial of degree [τ 1], with[z] denoting the integer part of z. C3 K :[ 1, 1] R is a symmetric continuous differentiable function in ( 1, 1) which integrates 1. C4 K + : [0, 1] R and K : [ 1, 0] R, where K + (x) = K ( x), K (1) + (x) = K(1) ( x), K + (0) = K + (1) = 0, R 1 0 K + (x) dx =1and R 10 xk + (x) dx =0.

18 C5 As n, (i) ň 1 0 and (ii) n 1 2 d a 1 2 d+τ D<, withτ as in Condition C2. If d =0and in C2 τ =2, then the optimal choice of a satisfies a = Dn 1/5 for some finite positive constant D, which corresponds to the choice of the bandwidth parameter by, say, cross-validation. Also, note that for a given degree of smoothness on r (x), thatisτ in C2, the bandwidth parameter a converges to zero slower as d increases.

19 We shall give a proposition before the main results. Proposition 1 Under Assumptions C1 and C3, sup 1 s n sx t=1 K t u t sx t=1 K t u t = o p ³ n d+1/4, (2.5) where u t = P j=0 ϑ j ε t j and {ε t } t Z is a zero mean iid sequence of standard normal random variables.

20 RESULTS We shall first examine the covariance of br a (q) at two different points x q1 x q2. Define b (q 1,q 2 )=(q 2 q 1 ) /ň and b := lim n b (q 1,q 2 ).

21 Proposition 2 Assuming C1 C3 and C5,foranyň < q 1 q 2 < n ň,asn, ň 1 2d Cov (br a (q 1 ),br a (q 2 )) ρ (b, d), (3.1) (a) if 0 < d < 1 2, ρ (b; d)=h θ (d) Z 1 1 Z 1 +b b 1 v w 2d 1 K (v) K (w b) dvdw, (3.2) (b) if d = 0, ρ (b; d)=h R 1 1 K (v) K (v b) dv.

22 The next proposition deals with the correlation structure of br a (q) as b 0 and when b as n. This will play an important role when studying the limit distribution of T r given in (3.3). Proposition 3 Under C1 C3 and C5,forsomeα (0, 2 ], asn, (a) ρ (b (q 1,q 2 ),d) ρ (b (q 1,q 1 );d) 1= D b (q 1,q 2 ) α + o ( b (q 1,q 2 ) α ) as b (q 1,q 2 ) 0 (b) ρ (b (q 1,q 2 ),d)log(b (q 1,q 2 )) = o (1) as b (q 1,q 2 ).

23 Proposition 4 Assuming C1 C3 and C5,foranyfinite collection q j, j = 1,..., p, such that ň < q j < n ň and q j1 q j2 nz > 0,asn, ň 1 2 d ρ 1 2 (0 ; d) ³ br a ³ qj r ³ qj j =1,...,p d N (0, diag (1,...,1 )).

24 The results are key to obtain the asymptotic distribution of T r = sup br a (q) r (q). (3.3) q=ň+1,...,n ň

25 Theorem 5 Let υ n =( 2 log a) 1 /2. Assuming that C1 C3 and C5 holds Prob ½υ n µň 1 2 d ¾ ρ 1 2 (0 ; d) T r ξ n x exp ³ 2e x, for x > 0, n where (a) If 0 < d < 1 /2,then ( µ1 ξ n = υ n + υn α log log a 1 +log à (2 π) α 2 α E for some 0 < E <, whereα is as given in Proposition 3.2, 0 <J α lim a 0 Z 0 e s Pr sup Y (t) > s 0 t [a] 1 ds <!) α 1 J α

26 and Y (t) is a stationary mean zero Gaussian process with covariance structure Cov (Y (t 1 ), Y (t 2 )) = t 1 α + t 2 α t 2 t 1 α. (b) If d = 0,then ξ n = υ n +υ 1 n log 1 2 π Ã Z 1 1 K (1 ) (x) 2 dx! 1 /2.

27 Atestfor H 0 : r (x) =r 0 (x), x (0, 1) at a κ significant level is based just on rejecting if υ n µ ň 1 2 d ρ 1 2 (0; d) T r ξ n is greater than x κ,the(1 κ) th quantile of the Gumbel limiting distribution.

28 We now describe the test for continuity or structural breaks, that is H 0 : r + (x) =r (x), x (0, 1). (3.5) Because under H 0, r + (x) = r (x) for all x [0, 1], we should expect that br a,+ (q) ' br a, (q), forallq = ň +1,...,n ň 1. This suggests that test for breaks can be based on T b r = sup br a,+ (q) br a, (q). (3.6) [ň]<q<n [ň]

29 Before we examine the properties of T b r,wefirst examine the covariance of br a,+ (q) br a, (q) at two different points x q1 x q2.

30 Proposition 6 Assuming C1, C2, C4 and C5,underH 0,foranyň < q 1 q 2 < n ň, as n, where ň 1 2d Cov br a,+ (q 1 ) br a, (q 1 ),br a,+ (q 2 ) br a, (q 2 ) ρ (b, d)=:ρ + (b, d)+ρ (b, d) ρ ± (b, d) ρ :(b, d), (a) if 0 < d < 1 2, ρ + (b; d)=h θ (d) Z 1 0 Z 1 +b b v w 2d 1 K + (v) K + (w b) dvdw,

31 Z 0 ρ (b; d) = H θ (d) ρ ± (b; d) = H θ (d) ρ (b; d) = H θ (d) Z b 1 Z 1 Z b 0 Z 0 1 b 1 v w 2d 1 K (v) K (w b) dvdw, b 1 v w 2d 1 K + (v) K (w b) dvdw, Z 1 +b b v w 2d 1 K (v) K + (w b) dvdw. (b) if d = 0, ρ + (b; d)=ρ (b; d) =4 1 H R 1 0 K + (v) K + (v b) dv and ρ ± (b; d)=ρ (b; d)=0.

32 The next proposition deals with the correlation structure of br a,+ (q) br a, (q) as b (q 1,q 2 ) 0. Proposition 7 Under C1, C2, C4 and C5,forsomeα (0, 2 ], asn, (a) ρ (b (q 1,q 2 ),d) ρ (b (q 1,q 1 );d) 1= D b (q 1,q 2 ) α + o ( b (q 1,q 2 ) α ) as b (q 1,q 2 ) 0 (b) ρ (b (q 1,q 2 ),d)log(b (q 1,q 2 )) = o (1) as b (q 1,q 2 ). Proposition 8 Assuming C1, C2, C.4 and C5,foranyfinite collection q j, j = 1,...,p, such that ň < q j < n ň and q j1 q j2 nz > 0,asn, ň 1 2 d ρ 1 2 (0 ; d) ³ br a,+ ³ qj br a, ³ qj j =1,...,p d N (0, diag (1,...,1 )).

33 Theorem 9 Let υ n =( 2 log a) 1 /2. Assuming that C1, C2, C4 and C5 holds µ Prob ½υ n ň 1 2 d ρ 1 2 (0 ; d) Tr b ¾ ξ n x exp ³ 2e x, for x > 0, n where (a) If 0 < d < 1 /2,then ( µ1 ξ n = υ n + υn α where E, J α and Y (t) as Theorem 5. log log a 1 +log à (2 π) α 2 α E!) α 1 J α (b) If d = 0,then ξ n = υ n +υ 1 n log 1 2 π à Z 1 0 K (1 ) + (x)2 dx! 1 /2.

34 Theorems 5 and 9 give asymptotic justification for T r or T b r given in (3.3) or (3.6), respectively, and so for the hypothesis testing in, say, (3.5) under H 0,weobserve that one of the normalization constants needed to achieve the asymptotic distribution depends on d. Also, and perhaps more importantly, not only ξ n depends on d but on J α which may be very difficult to compute except for the especial cases α =1or 2. In particular, we have that J 2 = υ n + υ 1 n log ³ π 1 (E/2) 1/2 υ 1 n and J 1 = υ n + log n (E/π) 1/ log log a 1o,whereE is a constant which depends on K and easy to obtain.

35 More specifically, although d can be estimated, we face one potential difficulty to obtain the uniform bounds of br a (q). FromtheproofofProposition3,α depends on K as well as on d, sothattoobtainj α does not seem an easy task. Under these circumstances, bootstrap algorithms appear to be a sensible procedure to go forward.

36 BOOTSTRAP - Major interest to make inferences when quantiles limiting distribution difficult or possible to obtain. - The bootstrap statistic, say T r, consistently estimates the distribution of T r under H 0 H a. That is T r d T r. - A second requirement, (to have good power), is that under H 1, T r also converges in bootstrap distribution.

37 Motivation a Rate of convergence to the Gumbel distribution is logarithmic. See for example Hall(1979, 1991) or Resnick (1987). Konakov and Piterbarg (1984) or Hall (1991) suggest the rate is the best possible one. b The normalization constants needed to achieve the asymptotic distribution of sup x [0,1] br (x) r (x) is model dependent and difficult to obtain.

38 Solution: Edgeworth expansions and bootstrap techniques.. Edgeworth expansions: Konakov and Piterbarg (1984) showedthatitiso ³ n δ for any arbitrarily small δ>0. Hall (1991) showed that bootstrap methods better job in that the accuracy is O ³ ň 1/2 log 2 n.

39 Bootstrap Time Series. - Time Domain. - Frequency Domain.

40 TIME DOMAIN 1. Moving block bootstrap (MBB) of Künsch (1989). 2. Subsampling of Politis and Romano (1994). 3. Sieve-bootstrap of Bühlmann (1997) following ideas in Kreiss (1988).

41 FREQUENCY DOMAIN Basically, these procedures are based on resampling the periodogram or (modulus) of DFT of {u t } n t=1. Specifically, the bootstrap approximates P j=0 ϑ j e ijλ 2. Examples: Franke and Härdle (1992) or Dahlhaus and Janas (1996), whereas among the latter, see Theiler et al. (1992) or Prichard and Theiler (1994) or more recently Hidalgo (2003). One problem with the latter bootstraps is that they may not be valid in some circumstances.

42 One example is when bootstrapping the sample mean, and the same happens to be in our context. See also Dahlhaus and Janas (1996) for some remarks and other examples for its not validity.

43 - Validity: common assumption is that the data satisfy some mixing dependence such as strong-mixing. - For data exhibit strong dependence, there is not justification for the validity of the sieve-bootstrap. - However, some results are given in Hall et al. (1998) for the subsampling algorithm. Requirement:v t = H (u t ) and u t to be Gaussian and H (u) =u or u 2 E ³ u 2. - For the MBB requires that H (u) =u see Lahiri (1993).

44 - When perform hypothesis testing, methods as subsampling would have less power than bootstraps schemes whose (asymptotic) distribution is the same under both the null and alternative hypotheses.

45 Bootstrap proposed: a Valid for data which do not satisfy the standard mixing conditions, such as strong mixing. b Similarities with the sieve-bootstrap. For example, it approximates the whole transfer function P j=0 ϑ j e ijλ instead of its modulus as that in, say, Theiler et al. (1992) or Hidalgo (1993). Ability to cope in a natural form with missing observations or unequal spaced data.

46 it is not a subset of the original data. bootstrap data, say {u t } t 1, is covariance stationary in the sense that Cov (u t,u s) is a function of t s.

47 We now describe the bootstrap algorithm. Suppose interest to bootstrap u = n 1 P n t=1 u t. Let for a generic sequence {a t } n t=1, w a ³ λj = 1 n 1/2 nx t=1 a t e itλ j. Suppose that in C1 ϑ k = I (k =0),sou t = P k=0 b k ε t k. Now, using the identity u t = 1 n 1/2 nx j=1 e itλ jw u ³ λj. (4.1)

48 Using Bartlett s approximation w u ³ λj B ³ e iλ j wε ³ λj, u t 1 n 1/2 nx j=1 e itλ jb ³ e iλ j wε ³ λj.

49 Now suppose that u t =(1 L) d X k=0 b k ε t k ; b 0 =1, The previous arguments may induce to employ the Bartlett s approximation and the approximation w u ³ λj ³ 1 e iλ j d B ³ e iλ j wε ³ λj, (4.2) u t 1 n 1/2 nx e itλ j j=1 ³ 1 e iλ j d B ³ e iλ j wε ³ λj. (4.3)

50 However, the lack of smoothness of ³ 1 e iλ j d around λj =0and results given in Robinson s (1995a) Theorem 1 at frequencies λ j for fixed j indicate that the approximation in (4.2) for those frequencies appears not to be appropriate. Observe thatthesefrequenciesarepreciselythosewhicharemorerelevantwhenexamining the asymptotic behaviour of u or, br a (x) or br a,+ (x) and br a, (x). Recall that the asymptotic variance of br a (x) depends only on f (λ) for λ in a neighbourhood of zero. For that reason, we replace ³ 1 e iλ j d by eg 1/2 ³ λ j ; d = n 1 X = n+1 γ ( ; d) e i λ j 1/2,

51 where So, eg ³ λ j ; d γ ( ; d) = ( 1) Γ (1 2d) Γ ( d +1)Γ (1 d) can be regarded as a truncated version of 1 e iλ j 2d.

52 Therefore we modify (4.3) as u t eu t =: 1 n 1/2 nx j=1 e itλ j eg 1/2 ³ λ j ; d B ³ e iλ j wε ³ λj, (4.4) whichistobeusedtoobtainthebootstrapsampleu 1,...,u n. The right side of (4.4) preserves (asymptotically) the covariance structure of u t. Thus, we replace d and B ³ e iλ j in the right of (4.4) by consistent estimators, say bd and b B ³ e iλ j, the problem of how to obtain a bootstrap sample u t, t =1,...,n, becomes that of how to perform a valid bootstrap algorithm for w ε ³ λj, j =1,...,n.

53 The previous arguments suggest the bootstrap algorithm to bootstrapping T r in (3.3), andwhichisdescribedinthefollowing7steps.

54 STEP 1 Obtain the centred residuals bu t = eu t n 1 P n t=1 eu t, t =1,...,n,where and br a (t) is given in (2.1). eu t = y t br a (t), t =1,..., n STEP 2 Let eu = ³ eu 1, eu 2,..., eu 0 n be a random sample with replacement from the empirical distribution of the standardized residuals ǔ t = eσ 1 u bu b t, eσ 2 bu = 1 n and obtain the discrete Fourier transform of eu, nx t=1 bu 2 t, η j := w ε ³ λj = 1 n 1/2 nx t=1 eu t e itλ j, j =1,..., n.

55 STEP 3 We estimate d by b d an estimator of d, say Robinson s (1995b) GSE, defined as where 0 < 1 < 1/2, and er (d) =log 1 m bd =arg mx j=1 min d [0, 1 ] λ 2d j I ³ bubu λj 2d er (d), (4.5) mx j=1 log λ j for integer m [1, ñ =[n/2]) and where I bubu (λ) = w bu (λ) 2 / (2π) is the periodogram of bu t,withm 1 + mn 1 0.

56 Let our estimator of σ 2 εh (λ) /2π be 1 bh (λ) = 2m +1 mx j= m 1 e i(λ+λ j) 2 b d Ibubu ³ λ + λj. Remark 1 In general g (λ) 6= by C1, seealso(2.4), h (λ) =g (λ) of h (λ). 1 e iλ 2d. However this is not a problem because 1 e iλ 2d h (λ) satisfiesthesameconditions

57 STEP 4 Consider M =[n/4m] and compute u t = 1 n 1/2 where nx e itλ n 1 X j j=1 = n+1 bγ ³, b d e i λ j 1/2 bb ³ e iλ j η j, (t =1,...,n), bb ³ e iλ j =1+ b b 1 e iλ j b b M e imλ j (4.6) and b b = 1 n ñ 1 X ba ³ e iλ j ei λ j, ( =1,...,M) j= ñ+1

58 with bc r =ñ 1 P ñ =1 log ³b h (λ ) cos (rλ ); for r =0,...,M and ba (λ) =exp MX r=1 bc r e irλ. The estimator b A (λ) of A (λ) comes from the canonical spectral decomposition of h (λ), see Brillinger (1981). Also, notice that bb ³ e iλ j 2 is an estimator of h ³ e iλ j.

59 STEP 5 Compute and br e (t) = br e,+ (t), t =1,...,ň br e (t), t = ň +1,..., n ň 1 br e, (t), t = n ň,..., n. y t = br e (t)+(2π) 1/2 e 1 2 bc 0 u t ; t =1,..., n, where br e,+ (t), andbr e,+ (t) and br e, (t) are defined as in (2.1) and (2.2) but with a bandwidth parameter e such that e 0 and a = o (e). Note that by the well-known Kolmogorov formula, 2πe bc 0 is an estimator of the variance of ε t, that is the one-step mean square prediction error. Moreover, the requirement that a = o (e) is typical when bootstrapping nonparametric regression

60 functions, see for instance Härdle and Marron (1991). Basically this is needed to make sure that the bias of the bootstrap nonparametric estimator converges to zero in probability. Recall that, in our setup, the bias of the nonparametric estimator of r + ( ) r ( ) converges to zero. STEP 6 Compute br a (q), q = ň +1,...,n ň 1, asin(2.1) but with y t replaced by yt and the same bandwidth parameter a employed in STEP 1 to compute br a (t). Our final step is: STEP 7 Compute the bootstrap version of T r as T r = sup br a (q) br a (q). (4.7) ň<q<n ň

61 Condition on m and the bandwidth parameters a and e. C6 As n, (i) (ne) 1 0, (ii) n 1 2 d e 5 2 d D<, wherewehavetaken τ =2in C3, (iii) D 1 n 1/3 <a<dn 1/4 and (iv) D 1 n 3/5 <m< Dn 3/4. Observe that C6 implies that a = o (e). However, the bandwidth e satisfies the same conditions as the generic bandwidth a in C5.

62 We modify Condition C1 to: C1 C1 holds, where the coefficients ϑ k are given by ϑ k = Γ (k + d) / (Γ (d) Γ (k +1)). The modification of C1 to C1 0 will imply that the function g (λ) in (2.4) is 1 e iλ 2d.

63 As was mentioned after STEP 5, the motivation to use a pilot bandwidth parameter e such that a = o (e) is to provide a correct adjustment of the bias, and as the next proposition indicates, this is the case in our context. To that end, remember that in the proof of Proposition 3.3, we obtain that ϕ a (q) =E (br a (q)) r (q) =o ³ ň d 1/2, whereas the bootstrap bias constructed from the resampled data is ϕ a (q) =E (br a (q) br a (q)).

64 Proposition 10 Assuming C1 0, C2, C3,withτ = 2 there, and C5, under H 0 H 1, as n, ϕ a (q) =o p ³ňd 1/2. Proposition 11 Assuming C1 0, C2, C3,withτ = 2 there, and C5,wehavethat for 0 q < n ň, 1 ň 2d ň+q X v=q+1 E ³ u t u t+v δ v = op (1 ). (4.8) Theorem 12 Under the same conditions of Proposition 4.2 and H 0 H 1,asn µ Prob ½υ n ň 1 2 b d ρ 1 ³ ¾ 2 0 ; d b Tr ξ n x P ³ exp 2e x, for x > 0, where υ n and ξ n were definedintheorem3.4. Y

65 We now comment on how to obtain the bootstrap analogue of T b r given in (3.6), that is when the interest is on testing for (3.5). STEP 1 Obtain the centred residuals bu t = eu t n 1 P n t=1 eu t, t =1,...,n,where eu t = y t br a,+ (t), t =1,...,ň y t 1 b 2 r a,+ (t)+br a, (t), t = ň +1,..., n ň 1 y t br a, (t), t = n ň,..., n, and br a,+ (t) and br a, (t) are given in (2.2). STEPS 2 TO 4 As Steps 2 to 4.

66 STEP 5 As in STEP 5 but br e (t) replace by 1 r be,+ (t)+br e, (t), t = ň +1,...,n ň 1 2 and y t = br e (t)+(2π) 1/2 e bc 0/2 u t ; t =1,...,n, where br e,+ (t) and br e, (t) are defined as in (2.2) but with a bandwidth parameter e such that e 0 and a = o (e). STEP 6 Compute br a,+ (q) and br a, (q), q = ň +1,...,n ň 1, asin(2.2) but with y t replaced by yt and the same bandwidth parameter a employed in STEP 1 to compute br a,+ (t) and br a, (t).

67 STEP 7 Compute the bootstrap version of Tr b as T b r = sup ň<q<n ň br a,+ (q) br a, (q).

68 Denote the bootstrap bias as ϕ a (q) =E ³ br a,+ (q) br a, (q). Proposition 13 Assuming C1 0, C2,withτ = 2 there, C4 and C6,underH 0 H 1, as n, ϕ a (q) =o p ³ňd 1/2. Theorem 14 Under the same conditions of Proposition 4.2 and H 0 H 1,asn µ Prob ½υ n ň 1 2 b d ρ 1 ³ ¾ 2 0 ; d b Tr b ξ n x P ³ exp 2e x, for x > 0, where υ n and ξ n were definedintheorem3.4. Y

69 Remark 2 To obtain a critical value x (β), forwhichexp ³ 2e x(β) =1 β, is the same as to find the value, say z n,suchthat lim Prob {T n r zn Y}=1 β. N Now, zn depends on ξ n and υ n and thus on the choice of the bandwidth parameter a. N In fact, since ξ n and υ n cannot be computed, in practice we would compute z n. N The latter implies that for the bootstrap to be valid, we need zn to satisfy that z n zn p 0, wherez n is such that lim n Prob {T r z n } =1 β, e.g. the corresponding value with the original data.

70 N But, for z n zn p 0 to hold true, it is obvious that we need the normalization constants ξ n and υ n tobethesameforbotht r and Tr, or at least their difference to converge to zero. N This is obviously possible only if the bandwidth parameter a isthesamewhen estimating the regression function with both the original y t and bootstrap y t data.