Time Series Nonparametric Regression Using Asymmetric Kernels with an Application to Estimation of Scalar Diffusion Processes

Transcription

1 ime Series Nonarametric Regression Using Asymmetric Kernels with an Alication to Estimation of Scalar Diffusion Processes Nikolay Gosodinov y Concordia University and CIREQ Masayuki Hirukawa z Northern Illinois University, Concordia University and CIREQ December 007 Abstract his aer considers a nonstandard kernel regression for strongly mixing rocesses when the regressor is nonnegative. he nonarametric regression is imlemented using asymmetric kernels [Gamma (Chen, 000b), Inverse Gaussian and Recirocal Inverse Gaussian (Scaillet, 004) kernels] that ossess some aealling roerties such as lack of boundary bias and adatability in the amount of smoothing. he aer investigates the asymtotic and nite-samle roerties of the asymmetric kernel Nadaraya-Watson, local linear, and re-weighted Nadaraya-Watson estimators. Pointwise weak consistency, rates of convergence and asymtotic normality are established for each of these estimators. As an imortant economic alication of asymmetric kernel regression estimators, we reexamine the roblem of estimating scalar di usion rocesses. Keywords: Nonarametric regression; strong mixing rocesses; Gamma kernel; Inverse Gaussian kernel; Recirocal Inverse Gaussian kernel; di usion estimation. JEL classi cation numbers: C3; C4; C; E43; G3. We would like to thank Evan Anderson, Bruce Hansen, Arthur Lewbel, Peter Phillis, Ximing Wu, and articiants at the 007 MEG Conference for helful comments. he rst author gratefully acknowledges nancial suort from FQRSC, IFM and SSHRC. y Deartment of Economics, Concordia University, 455 de Maisonneuve Blvd. West, Montreal, Quebec H3G M8, Canada; hone: (54) (ext. 3935); fax: (54) ; gosodin@alcor.concordia.ca; web: htt://alcor.concordia.ca/~gosodin/. z Deartment of Economics, Northern Illinois University, Zulauf Hall 55, DeKalb, IL 605, USA; hone: (85) ; fax: (85) 75-09; mhirukawa@niu.edu; web: htt://

2 Introduction he goal of this aer is to roose a nonstandard kernel-tye estimator for nonarametric regression using time series data when the suort of the regressor has a boundary. Suose that for a stationary, strongly mixing rocess f(x t ; Y t )g t R, we are interested in estimating the regression function m (x) E f (Y t ) X t xg ; () where () is a known measurable function. Examles of () include conditional distribution function and r th -order conditional moment estimation of Y t given X t x when (Y ) fy yg and (Y ) Y r ; r > 0, resectively, and X t may denote a lagged value of Y t : An interesting situation, that often arises in economics and nance, is when the regressor X t in () is nonnegative. In this case, the local constant or Nadaraya-Watson (NW) estimator (Nadaraya, 964; Watson, 964) based on a standard, symmetric kernel su ers from bias near the origin that does not vanish even asymtotically. his is due to the fact that the symmetric kernels assign strictly ositive weights outside the suort of X t. Accordingly, several boundary correction techniques have been roosed in the context of nonarametric regression such as boundary kernels (Gasser and Müller, 979) and Richardson extraolation (Rice, 984). he local linear (LL) estimator by Fan and Gibels (99) is also known to automatically adat the boundary bias. On the other hand, there is a growing literature on emloying asymmetric kernels as an alternative device for boundary bias correction. In density estimation for ositive observations, Chen (000b) introduces the Gamma kernel, and Scaillet (004) rooses the Inverse Gaussian and Recirocal Inverse Gaussian kernels. hese asymmetric kernels have several attractive roerties. First, they are free of boundary bias because the suort of the kernels match that of the density. Second, the shae of the asymmetric kernel varies according to the ositions of design oints, and, as a result, the amount of smoothing changes in an adative manner. hird, the asymmetric kernels achieve the otimal (in integrated mean squared error sense) rate of convergence within the class of nonnegative kernel estimators. hroughout this aer, we refer to asymmetric kernels as kernel functions with suort on the nonnegative real line. Bouezmarni and Rolin (003), Brown and Chen (999), Chen (999, 000a), and Jones and Henderson (007) consider estimation of density and regression functions de ned over the unit interval using di erent versions of asymmetric kernels.

3 Finally, their variances decrease as the osition at which smoothing is made moves away from the boundary. his roerty is articularly advantageous when the suort of the density has sarse regions. Subsequently, Bouezmarni and Scaillet (005) demonstrate weak convergence of the integrated absolute error for asymmetric kernel density estimators, whereas Hagmann and Scaillet (007) investigate the local multilicative bias correction for asymmetric kernel density estimators that is analogous to the one by Hort and Jones (996) in the symmetric kernel case. Besides density estimation, Chen (00) alies asymmetric kernels to the LL estimator, and Fernandes and Monteiro (005) establish the central limit theorem for a class of asymmetric kernel functionals. Furthermore, while all studies cited above are based on iid samling, Bouezmarni and Rombouts (006a,b) extend asymmetric kernel density and hazard estimation to ositive time series data. In line with these recent develoments, this aer rooses a nonarametric regression estimator for deendent data using asymmetric kernels. We consider the NW, LL and re-weighted Nadaraya- Watson (RNW; Hall and Presnell, 999) estimators and study their asymtotic and nite-samle behavior. While the NW estimator includes a design bias term that deends on the density function of the regressor, the LL estimator is free of this bias term. On the other hand, unlike the LL estimator, the NW estimator always yields estimated values within the range of observations f (Y t )g t and can reserve monotonicity and nonnegativity in conditional distribution estimation or nonnegativity in conditional variance estimation, for examle. he RNW estimator is known to incororate the strengths of the NW and LL estimators and has been used for nonarametric regression estimation (Cai, 00), quantile estimation (Hall, Wol and Yao, 999; Cai, 00), and conditional density estimation (De Gooier and Zerom, 003). We adat the three estimators to asymmetric kernels and strongly mixing data, and establish ointwise weak consistency and asymtotic normality. We believe that our asymtotic results constitute an imortant theoretical comlement to the results for time series nonarametric regression with symmetric kernels such as Lu and Linton (007) and Masry and Fan (997). Although we focus on the single regressor case throughout, the basic idea of our methodology is exected to hold in the multile regressor context.

4 As an imortant economic alication of the asymmetric kernel regression estimators, we consider the roblem of estimating time-homogeneous drift and di usion functions in scalar di usion rocesses. Using the in nitesimal generator and aylor series exansions, Stanton (997) derives higher-order aroximation formulae of the drift and di usion functions that are estimated nonarametrically by the NW estimator. An interesting emirical nding that emerges from this work is that the drift function for the US short-term interest rate aears to exhibit substantial nonlinearity. In contrast, Chaman and Pearson (000) argue that the documented nonlinearity in the short rate drift could be surious due to the oor nite-samle roerties of the Stanton s (997) estimator at high values of interest rates where the data are sarse. Fan and Zhang (003) estimate the rst-order aroximations of the drift and di usion functions by the LL estimator, and conclude that there is little evidence against linearity in the short rate drift. Bandi (00), Durham (003) and Jones (003) also do not nd emirical suort for nonlinear mean reversion in short-term rates. We exect that the use of the asymmetric kernel estimators can shed additional light on the nonarametric estimation of sot rate di usion models. he remainder of the aer is organized as follows. Section develos asymtotic roerties of the asymmetric kernel regression estimators and discusses their ractical imlementation. Section 3 conducts a Monte Carlo simulation exeriment that examines the nite samle erformance of these estimators in the context of scalar di usion rocesses for sot interest rates. Section 4 summarizes the main results of the aer. All roofs are given in the aendix. his aer adots the following notational conventions: () R 0 y ex ( y) dy; > 0 is the Gamma function; G (; ), IG (; ) and RIG (; ) symbolize the Gamma, Inverse Gaussian, and Recirocal Inverse Gaussian distributions with arameters (; ), resectively; fg is the indicator function; N denotes the set of ositive integers f; ; :::g, bxc signi es integer art of x; and c (> 0) denotes a generic constant, the quantity of which varies from statement to statement. he exression X d Y reads A random variable X obeys the distribution Y. For integers n and k such that 0 k n, n k n! k!(n k)! denotes the number of combinations of size k taken from n obects. Finally, the exression X Y is used whenever X Y! as!. 3

5 Nonarametric Regression Using Asymmetric Kernels for ime Series Data. Nonarametric Regression Estimators Consider the roblem of estimating nonarametric regression () using a samle f(x t ; Y t )g t, where X t 0 is assumed throughout. For a given design oint x > 0, the NW, LL and RNW asymmetric kernel estimators are de ned as ^m nw (x) ^m ll (x) ^m rnw (x) P t (Y t) K x;b (X t ) P t K ; x;b (X t ) w t (x) (Y t ) ; t P t (Y t) t (x) K x;b (X t ) P t ; t (x) K x;b (X t ) where K x;b (u) is an asymmetric kernel function with a smoothing arameter b. he LL estimator satis es ^m ll (x) ^ 0 (x), where ^ (x) h^0 (x) ; ^ i (x) solves the otimization roblem ^ (x) arg min (x) f (Y t ) 0 (x) (x) (X t x)g K x;b (X t ) : t he weight functions for the LL estimator fw t (x)g t are given by w t (x) fs (x) S (x) (X t x)g K x;b (X t ) S 0 (x) S (x) S (x) ; S (x) (X t x) K x;b (X t ) ; 0; ; : t On the other hand, the weight functions for the RNW estimator f t (x)g t satisfy t (x) 0; t (x) ; t (X t x) t (x) K x;b (X t ) 0: () t Since f t (x)g t that satisfy () are not uniquely determined, they are seci ed as arameters that maximize the emirical log-likelihood P t log f t (x)g subect to these constraints. hen, as shown in Cai (00, 00), f t (x)g t are de ned as t (x) f + (X t x) K x;b (X t )g ; (3) 4

6 where is the Lagrange multilier associated with P t (X t x) t (x) K x;b (X t ) 0 that can be determined by maximizing the ro le emirical log-likelihood L ; fx t g t ; x log f + (X t x) K x;b (X t )g : t We consider several candidates for asymmetric kernels: Gamma density K G with arameters (xb + ; b) roosed by Chen (000b), Inverse Gaussian (IG) density K IG with arameters (x; b) and Recirocal Inverse Gaussian (RIG) density K RIG with arameters ( (x b) ; b) roosed by Scaillet (004). hese densities are given by K G(xb+;b) (u) uxb ex ( ub) fu > 0g ; b xb+ (xb + ) K IG(x;b) (u) ex + x bu 3 u fu > 0g ; K RIG((x b);b) (u) bu ex u bx x u x b x b b + x b u fu > 0g : As is the case with symmetric kernels, the asymmetric kernel RNW estimator shares some attractive roerties of both NW and LL estimators. By construction, min t f (Y t )g ^m rnw (x) max t f (Y t )g for any x, and the RNW estimator always generates nonnegative estimates in nite samles whenever () is nonnegative, as the NW estimator does. Moreover, the weight functions for the LL estimator fw t (x)g t satisfy the moment conditions similar to () w t (x) ; t (X t x) w t (x) 0: t Hence, the bias roerties of the RNW estimator are exected to be as good as that of the corresonding LL estimator, and better than that of the NW estimator for interior design oints.. Asymtotic Proerties of Estimators In this section we establish ointwise weak consistency with rates and asymtotic normality of the NW, LL and RNW estimators for strongly mixing rocesses. Before stating regularity conditions Chen (000b) also rooses another version of the Gamma kernel function K G (u; b (x) ; b) u b (x) u ex b b b (x) fu > 0g ; ( b (x)) where xb if x b b (x) (xb) : 4 + if x [0; b) However, this version is not considered here, because asymtotic roerties of the LL and RNW estimators using K G (u; b (x) ; b) for interior x (satisfying xb! ) are rst-order equivalent to those when K G (u; xb + ; b) is emloyed. 5

7 for our main results, we rovide the de nition of an -mixing rocess for reference. Let F b a denote the -algebra generated by the stationary sequence f(x t ; Y t )g b ta and (k) su AF 0 ;BF k Pr (A \ B) Pr (A) Pr (B) ; k : hen, the stationary rocess f(x t ; Y t )g t is said to be strongly mixing or -mixing if (k)! 0 as k! (Rosenblatt, 956). Also, let f (x) be the marginal density of the regressor X t, and de ne (x) V ar f (Y t ) X t xg. o obtain our main results, the following regularity conditions are required: (A) For a given design oint x > 0, m 00 (x), f 00 (x) and (x) are bounded and continuous. (A) su x0 f (x) M <, 0 < m inf x0 f (x), and su u0;v0 f t;s (u; v) M <. (A3) n E (Y ) o X u 0 + u l and E [max f (Y t ) ; (Y s ) ; (Y t ) (Y s )g X t u; X s v] 0 + u m + v n ; 8u; v 0, for some >, for some 0 ; ; 0 ; ; 0, and for some l; m; n N. (A4) he strong mixing coe cient (k) satis es P k ka f (k)g < for some a >. (A5) he smoothing arameter b b satis es b! 0 and b! b! 0 and b! for the Gamma and RIG kernels for the IG kernel as!. (A6) n here exists a sequence s N such that s!, s o b o, and b (s )! 0 as!. (A7) he smoothing arameter b b additionally satis es b 5! [0; ) as!. Similar conditions to (A)-(A7) are commonly used in the literature of LL (Lu and Linton, 007; Masry and Fan, 997) and RNW estimation (Cai, 00, 00; De Gooier and Zerom, 003) with deendent data. he condition (A3) is insired by Hansen (006), who derives uniform convergence 6

8 rates of nonarametric density and regression estimators using deendent data even when unbounded suort kernels are emloyed. Both Hansen (006) and this aer allow the two conditional moments to diverge. An imortant di erence is that while his condition controls the divergence rates of the conditional moments in comarison with the rate of decay in tails of the marginal density of regressors, (A3) assumes the existence of olynomial dominating functions, taking into account that all three asymmetric kernels have moments of any nonnegative integer order, as indicated in the roof of Lemma B in the aendix. he conditions (A5) and (A7) for the smoothing arameter b are required to establish the asymtotic normality of the estimators and ensure that the bias and the variance converge to zero, and the remainder term in the bias exression is asymtotically negligible. (A4) imlies that the strong mixing coe cient has the size ( ) ( ). o establish heorem (oint asymtotic normality of regression and rst-order derivative estimators), we need to relace (A4) and (A5) by the stronger conditions (A4 ) and (A5 ) stated below. Note that (A4 ) and (A5) are required to aroximate the variance of the rst-order derivative estimator, and to ensure that the variance converges to zero, resectively. In contrast, the original conditions (A4) and (A5) su ce to demonstrate the asymtotic results for the LL estimator only. (A4 ) he strong mixing coe cient satis es P k ka f (k)g < for some a > 3 ( ). (A5 ) he smoothing arameter b b satis es b! 0 and b 3! b! 0 and b 6! for the Gamma and RIG kernels for the IG kernel as!. Now we resent kernel-seci c results on weak consistency and asymtotic normality of the three estimators. Since the results deend on the kernel emloyed, we denote the NW estimator using the Gamma kernel as ^m nw G (x), for examle. A similar notational convention is alied to the LL and RNW estimators. We also mean by interior x and boundary x that the design oint x satis es xb! and xb! > 0 as!, resectively. 7

9 heorems, and 3 establish the ointwise weak consistency and asymtotic normality of the asymmetric kernel NW, LL and RNW estimators for interior x. heorem. If conditions (A)-(A7) hold, then for interior x, b ^m nw G (x) m (x) b b m 0 (x) + xf 0 (x) f (x) ^m nw IG (x) m (x) x 3 m 0 (x) f 0 (x) ^m nw RIG (x) m (x) x + x m00 (x) b d! N (0; V G ) ; f (x) + m00 (x) b! N (0; V IG ) ; d b! N (0; V RIG ) ; m 0 (x) f 0 (x) f (x) + m00 (x) where V G x (x) f(x), V IG x 3 (x) f(x) and V RIG V G : d Proof. See Aendix A. heorem. If conditions (A)-(A3), (A4 ), (A5 ), (A6)-(A7) hold, then for interior x, b; ^ G (x) (x) xm00 (x) b d 0 0! N ; V G ; x b; ^ IG (x) (x) x3 m 00 (x) b d 0 0! N ; V IG ; x 3 b; ^ RIG (x) (x) xm00 (x) b d 0 0! N ; V RIG ; x where (x) [m (x) ; m 0 (x)] and b; b 0 0 b. Proof. See Aendix A. Corollary. If conditions (A)-(A7) hold, then for interior x, b ^m ll G (x) b b ^m ll IG (x) ^m ll RIG (x) m (x) m (x) m (x) xm00 d (x) b! N (0; V G ) ; x3 m 00 d (x) b! N (0; V IG ) ; xm00 (x) b d! N (0; V RIG ) : heorem and Corollary can be further extended to the th -order local olynomial estimation, rovided that m() has a bounded continuous th -order derivative and the mixing condition is roerly strengthened. 8

10 heorem 3. If conditions (A)-(A7) hold, then for interior x, b ^m rnw G (x) m (x) b b Proof. See Aendix A. ^m rnw IG (x) m (x) ^m rnw RIG (x) m (x) xm00 d (x) b! N (0; V G ) ; x3 m 00 d (x) b! N (0; V IG ) ; xm00 d (x) b! N (0; V RIG ) : he next two theorems derive the ointwise weak consistency and asymtotic normality of NW and LL estimators for boundary x. Before roceeding, we modify the conditions (A6) and (A7). Note that two alternative relacements of (A7), namely, (A7 ) and (A7 ), are required for asymtotic normality of NW and LL estimators, resectively. (A6 ) here exists a sequence s N such that 8 n < s!, s o (b ) o, and (b) (s )! 0 for the Gamma and RIG kernels n : s!, s o b o, and b (s )! 0 for the IG kernel as!. (A7 ) as!. he smoothing arameter b b additionally satis es 8 < b 3! [0; ) for the Gamma kernel b 0! [0; ) for the IG kernel : b 5! [0; ) for the RIG kernel (A7 ) he smoothing arameter b b additionally satis es b 5! [0; ) b 0! [0; ) for the Gamma and RIG kernels for the IG kernel as!. heorem 4. If conditions (A)-(A5), (A6 ), (A7 ) hold, then for boundary x, b f ^m nw G (x) m (x) m 0 (x) bg! d N 0; VG B ; b ^m nw IG (x) m (x) 3 m 0 (x) f 0 (x) b ^m nw RIG (x) m (x) ( + ) f (x) + m00 (x) b 4 d! N 0; V B m 0 (x) f 0 (x) f (x) + m00 (x) b 9 IG ; d! N 0; V B RIG ;

11 where VG B (+) (x) + (+) f(x), V IG B (x) 3 f(x) and VRIG B +(76) 3 +(33) 5 (x) f(x). Proof. See Aendix A. heorem 5. If conditions (A)-(A3), (A4 ), (A5 ), (A6 ), (A7 ) hold, then for boundary x, m b; ^ 00 (x) ( ) b d G (x) (x)! N ; 4b 0 b;3 ^ IG (x) (x) 3 m 00 (x) b 4 d 0 0! N ; V IG B ; " #) 3 m b; (^ 00 (x) ( + ) b d 0 RIG (x) (x)! N ; b 0 RIG VRIG B ; VG B ( + ) where RIG 4 b; b n 3 4(+) 0 0 b Proof. See Aendix A. n 3 4(+) o (348) 3 +(53) 5 +(76) 3 +(33) 5 and b;3 b 0 0 b. n (+) o (348) 3 +(53) 5 +(76) 3 +(33) 5 + (58) 3 +(58) 5 +(3364) 7 +(76) 3 +(33) 5 o 3 5 ; Corollary. If conditions (A)-(A5), (A6 ), (A7 ) hold, then for boundary x, b ^m ll G (x) m (x) b ^m ll IG (x) m (x) b ^m ll RIG (x) m (x).. Discussion of results ( ) m00 (x) b 3 m00 (x) b 4 ( + ) m00 (x) b d! N 0; d! N 0; V B IG ; + 5 ( + ) V G B ; d! N 0; V B RIG : Choice of estimator and kernel function. In case of an interior design oint, the results in heorem, Corollary and heorem 3 reveal that the LL and RNW estimators eliminate the design bias term of the NW estimator without any e ect on the variance. An immediate consequence of heorem 3 and Corollary is that each RNW estimator is rst-order equivalent to the corresonding LL estimator, as is the case with symmetric kernels. Furthermore, we can see from heorems -3 that for each of NW, LL and RNW estimators, variances decrease with x, i.e. as the osition in which smoothing is made moves away from the boundary. his roerty is articularly advantageous when the suort of the regressor has sarse regions. 0

12 Now turning our attention to the roerties of the di erent kernel functions, we note that the estimators based on the Gamma and RIG kernels are rst-order equivalent for interior x. he asymtotic bias of the IG-based estimators is larger than that of the Gamma and RIG estimators when x > ; however, the larger bias is comensated by a much smaller variance. For examle, in the secial case of a linear function m(x), the estimators ^m ll IG (x) and ^mrnw IG (x) dominate their Gamma and RIG counterarts for x > but not for x < which is the situation in our interest rate alication. Some interesting ndings emerge from the boundary design oint case. First, comaring heorems and 4 or Corollaries and, we see that for each of the NW and LL estimators, imrovement in order of magnitude in the bias term is achieved at the exense of in ating the variance. Indeed, if the smoothing arameter b is chosen to satisfy (A5) and (A7), then the bias of the NW and LL estimators becomes asymtotically negligible over the boundary region, and thus only the variance matters. Second, for the IG and RIG kernels, the LL estimator eliminates the design bias term of the NW estimator even over the boundary region, whereas the Gamma NW and LL estimators do not have common bias terms. More imortantly, heorem 4 and Corollary show that ^m nw G (x) and ^mll G (x) do not share the same asymtotic variance for boundary x, which is tyically the case for interior x, IG, RIG and symmetric kernels. 3 For examle, for 0:5, the variance of ^m ll G (x) is twice as big as the variance of ^m nw G (x) and it may well be the case that the Gamma-based NW estimator is referred over the Gamma LL estimator even though the latter may have a smaller bias. Figure lots the di erences in the asymtotic variances of ^m nw G (x), ^mnw RIG (x) and ^mll G (x) as a function of [0:; ],4 and shows the substantial e ciency advantages of the Gamma NW estimator at the extreme design oints. 3 he mean of G (xb + ; b) is not the design oint x but x + b, whereas both IG (x; b) and RIG ( (x b) ; b) have mean x. Hence, as xb!, S (x) O (b) for the Gamma kernel, whereas S (x) O b for the IG and RIG kernels. hen, the term involving m 0 (x) dominates the bias of ^m nw G (x) for boundary x, and as a result, ^mnw G (x) and ^m ll G (x) do not have common bias terms. Likewise, the reason why ^mnw G (x) and ^mll G (x) do not share the same asymtotic variance for boundary x is that when the Gamma kernel is emloyed, the scale-adusted outer roduct matrix S y S 0 (x) b S (x) b S (x) b has non-negligible o -diagonal elements in the limit as xb! ; for details, S (x) see Lemma A7 in Aendix A. 4 he estimator ^m nw IG (x) is not lotted in the diagram, because it has a slower rate of convergence than ^mnw G (x), ^m nw RIG (x) and ^mll G (x) for boundary x.

13 Also, unlike the case for interior x, asymtotic indeendence between LL regression and rstorder derivative estimators does not necessarily hold for boundary x; in fact, asymtotic variancecovariance matrices of Gamma and RIG-based LL estimators in heorem 5 have non-zero o -diagonal elements when xb! is assumed. Moreover, we do not rovide a theorem for the RNW estimator in the boundary case. he di culty for establishing the asymtotic roerties of the RNW arises from the fact that when x is located in a articularly small boundary region (of order O (b)), there are not enough observations less than x for the constraint () to hold, and, as a result, the RNW estimator is not well de ned. Even though the asymtotic behavior of the RNW estimator for boundary x does not a ect its global roerties, these observations indicate that the numerical erformance of the RNW estimator near the boundaries could be rather oor which is con rmed by our simulation results resented below. Mean squared error. It follows directly from heorem 3 and Corollary that the mean squared errors (MSE) of the three LL (and thus RNW) estimators for interior x are aroximated by MSE ^m ll G (x) 4 x fm 00 (x)g b + (x) b x f (x) ; MSE ^m ll IG (x) 4 x6 fm 00 (x)g b + (x) b x 3 f (x) ; MSE ^m ll RIG (x) 4 x fm 00 (x)g b + (x) b x f (x) : In contrast, heorem suggests that the MSEs of their corresonding NW estimator for interior x are aroximated by MSE f ^m nw G (x)g MSE f ^m nw IG (x)g MSE f ^m nw RIG (x)g m 0 (x) + xf 0 (x) f (x) m 0 (x) x3 f 0 (x) f (x) + x m00 (x) b + + x3 m00 (x) m 0 (x) xf 0 (x) f (x) + x m00 (x) b + b b + b b (x) x f (x) ; (x) x 3 f (x) ; (x) x f (x) ; and the NW estimators contain an additional design bias term that deends on the density of the regressor f (x) while the variance terms remain unchanged. hese results agree with the case of standard symmetric kernels.

14 Otimal smoothing arameter. From the MSE exressions, it can be easily inferred that the otimal smoothing arameters of the LL (and thus RNW) estimators for interior x are b G " # 5 (x) fm 00 (x)g x f (x) 5 ; b IG " # 5 (x) fm 00 (x)g x 3 f (x) 5 ; b RIG " # 5 (x) fm 00 (x)g x f (x) 5 : Note that the otimal smoothing arameters are b O 5 O a, where a is the MSEotimal bandwidth for the LL estimator using second-order symmetric kernels. Also, at the otimum, ( ) 45 m 00 (x) (x) 45 ; MSE ^m ll G (x) 5 4 f (x) MSE ^m ll IG (x) 5 4 ( ) 45 m 00 (x) (x) 45 ; f (x) ( ) 45 m 00 (x) (x) 45 ; MSE ^m ll RIG (x) 5 4 f (x) and each otimal MSE is identical and does not deend on x (the deendence of each otimal MSE on x comes only through f (x) and (x)). In addition, the otimal MSE is the same as that of the LL estimator using the Gaussian kernel. herefore, as argued by Chen (000b) and Scaillet (004), we can see that, for interior x; the three asymmetric kernels de ned over [0; ) have the same ointwise e ciency as the Gaussian kernel over ( ; )..3 Imlementation and Selection of Smoothing Parameter he ractical imlementation of the roosed nonarametric estimators requires a choice of smoothing arameter. While the revious section rovides some guidance in this direction, the exressions for the otimal smoothing arameters deend on unknown functions of the data and a uniform lug-in rule is di cult to obtain. Note also that the otimal smoothing arameters for the asymmetric kernels deend exlicitly on the design oint and, in rincile, they should take di erent values at each x. Hagmann and Scaillet (007), however, argue for a uniform smoothing arameter since the deendence on the design oint x may deteriorate the adatability of asymmetric kernels. 3

15 In this aer, we adot a cross-validation (CV) aroach to choosing a uniform smoothing arameter for nonarametric curve estimation based on asymmetric kernels. Since the data are deendent, the leave-one-out CV is not aroriate. Instead, we work with the h-block CV version of Györ et al. (989) and Burman et al. (994) where h data oints on both sides of observation t are removed from the samle and the function m(x) is estimated from the remaining (h + ) observations. he idea behind this method is that, due to the strong mixing roerty of the data, the blocks of length h are asymtotically indeendent although the block size may need to shrink (at certain rate) relative to the total samle size in order to ensure the consistency of the rocedure. Let ^m (t h):(t+h) (X t ) denote the estimate from observations ; ; :::; t h ; t + h + ; :::;. hen, the smoothing arameter can be selected by minimizing the least squares cross-validation function CV (b) arg min bb h (Yt ) ^m (t h):(t+h) (X t ) (X t ) ; (4) th+ where () is a weighting function that has comact suort and is bounded by. Minimizing CV (b) is asymtotically equivalent to minimizing the true exected rediction error rovided that h goes to zero at some rate as h! and! (Chu, 989; Györ et al., 989). Alternatively, if one assumes that h is a nontrivial fraction of the samle size so that h is a xed constant as h! and!, CV (b) has to be corrected as in Burman et al. (994). 5 While the corrected CV (b) of Burman et al. (994) may rovide a better nite-samle aroximation to the true exected rediction error, this rocedure is comutationally more involved and in our numerical exeriments the smoothing arameter is chosen by minimizing (4) with (X t ) : 3 Monte Carlo Exeriment: Di usion Models of Sot Rate he nonarametric estimation of continuous-time di usion rocesses, that are used to describe the underlying dynamics of sot interest rates, has been an active area of recent research (Bandi and Phillis, 003; Florens-Zmirou, 993; Jiang and Knight, 997; Nicolau, 003; among others). In this section, we assess the nite-samle roerties of our roosed asymmetric kernel estimators in the 5 he asymtotic otimality of the h-block cross validation bandwidths for mixing data in Chu (989), Györ et al. (989) and Burman et al. (994) is derived for symmetric kernels. While it is useful to extend these results to asymmetric kernels, it is beyond the scoe of this aer. 4

16 context of a di usion rocess of sot rate and evaluate the economic imortance of the results in terms of comuted bond and otion ricing errors. he data for the rst simulation exeriment is generated from the CIR model (Cox et al., 985) dr t ( r t ) dt + r t dw t ; (5) where W t is a standard Brownian motion. his model is convenient because the transition and marginal densities are known and the bond and call otion rices are available in closed form (Cox et al., 985). 5; 000 samle aths for the sot interest rate of length 600 observations are simulated using the rocedure described in Chaman and Pearson (000). After drawing an initial value from the marginal Gamma density, the interest rate rocess is constructed recursively by drawing random numbers from the transition non-central chi-square density and using the values for, and and a time ste between two consecutive observation equal to 5 corresonding to weekly data. We consider two arameter con gurations that are used in Chaman and Pearson (000) - (; ; ) (0:459; 0:0857; 0:0783) and (0:85837; 0:0857; 0:566) ; that roduce ersistent interest rate rocess with monthly autocorrelations of 0.98 and 0.93, resectively. he two seci cations are calibrated to generate data with the same unconditional mean variance. he strong mixing roerty of the rocess generated by (5), is demonstrated by Carrasco et al. (007). he exressions for the rice of a zero-couon discount bond and a call otion on a zero-couon discount bond have an analytical form and are given in Cox et al. (985). We follow Jiang (998) and Phillis and Yu (005) and comute the rices of a three-year zero-couon discount bond and a oneyear Euroean call otion on a three-year discount bond with a face value of $00 and an exercise rice of $87 with an initial interest rate of 5% by simulating sot rate data from the estimated di usion rocess. he simulated bond and derivative rices are then comared to the analytical rices based on the true values of the arameters. More seci cally, the rice of a zero-couon bond with face value P 0 and maturity ( t) is comuted as Pt P 0 E t ex 5 Z t rudu ;

17 where r t r t ; dr t [b(r t ) b (r t )]dt + b(r t )dw t ; and b(r t ); b(r t ) and b (r t ) denote the nonarametric estimates of the drift, di usion and market rice of risk functions, resectively. For simlicity, the market rice of risk is assumed to be equal to zero since its comutation requires another interest rate rocess of di erent maturity. he exectation is evaluated by Monte Carlo simulation using a discretized version of the dynamics of the sot rate. he rice of a call otion with maturity (n t) on a zero-couon bond with maturity ( t); face value P 0 and exercise rice K is comuted as Ct n E t ex E t ex Z n t Z n t rudu max (Pn K; 0) rudu max P 0 E n ex Z N rvdv K; 0 ; where n < and samle aths for rt are simulated from the nonarametrically estimated discretized model of sot rate. In order to evaluate if the roosed estimators cature well the shae of the true function, data are also generated from the nonlinear di usion model of Ahn and Gao (999) dr t ( r t ) r t dt + r :5 t dw t ; (6) where the drift is a quadratic function of the interest rate. he strong mixing roerties of the rocess generated by (6) can be inferred by verifying the conditions in Chen et al. (999). As argued by Ahn and Gao (999), s t r t follows a square-root rocess with non-central chi-square transitional density which facilitates the simulation of interest rate data. he articular arameterization that we emloy in simulating the data from (6) is (; ; ) (3; 0:; ) which is similar to the values estimated by Ahn and Gao (999) from actual data. We consider the NW estimators with Gaussian and Gamma kernels and the LL and RNW estimators with Gamma kernel. he LL estimator with Gaussian kernel roduces substantially larger biases than these estimators and is not reorted. First, Figures to 5 resent the nite-samle roerties of the asymmetric NW estimators of the drift function from the CIR model. Figures and 4 lot the median drift estimates of the Gamma, IG and RIG NW estimators for both arameterizations and a xed smoothing arameter. 6

18 In agreement with the theoretical results in Section., the Gamma and RIG estimators exhibit very similar behavior and rovide a very good aroximation to the true drift function. In contrast, the IG drift function estimator is much more biased (the bias of the IG estimator is still substantial for larger smoothing arameters) and we do not consider this estimator further in the aer. Figures 3 and 5 lot the 90% Monte Carlo con dence bands of the Gamma and RIG estimators and reveal that the Gamma estimator is less variable than the RIG estimator esecially for the more ersistent seci cation. In the rest of the aer, we only reort the results from the Gamma NW estimator noting that the RIG NW estimator delivers very similar results. In order to comare the roerties of the Gamma NW with the Gaussian NW, Gamma RNW and Gamma LL estimators, we choose a common algorithm for selecting the smoothing arameter based on h-block cross validation with h 30 (our exeriments with di erent values of h delivered very similar results.) It is interesting to note that Gamma NW and RNW select signi cantly smaller smoothing arameters than the Gaussian NW and Gamma LL estimators. he median Monte Carlo estimates lotted in Figures 6 and 8 show that the Gamma NW and Gaussian NW are almost unbiased whereas the bias of the Gamma LL is rather large for both interior and boundary design oints. It aears that the Gamma LL estimator is more sensitive to the high ersistence in the data and its behavior imroves for less ersistent seci cations. While the Gamma NW is only slightly less biased than the Gaussian NW, the asymmetric kernel estimator exhibits smaller variability (Figure 7) near the boundaries. he behavior of the asymmetric RNW estimator is similar to the Gamma NW estimator but it tends to be much more noisy. Finally, Figures 9 and 0 lot the drift function estimates from the nonlinear di usion seci cation of Ahn and Gao (999). As in the case of linear drift, the Gamma kernel estimator rovides a very good aroximation of the true drift function. he symmetric (Gaussian) NW estimator exhibits larger bias and variability for interest rates above 9% whereas the local linear estimator again tends to erform rather oorly comared to the asymmetric kernel estimator. In summary, the Gamma NW aears to be the best erforming nonarametric estimator of the drift function of highly ersistent di usion rocesses considered in the simulation exeriments. 7

19 he economic signi cance of the imroved estimation of di usion models of sot rate is evaluated by comaring bond and otion ricing errors based on di erent nonarametric estimators for the CIR model with (; ; ) (0:459; 0:0857; 0:0783). For reference, we include also the bond and otion rices comuted analytically from the OLS estimates of ; and obtained from the discretized version of the model. he results are resented in able. Desite the fact that the OLS estimator uses knowledge of the true shaes of the drift and di usion functions, the bond and esecially the call otion rices are substantially underestimated due mainly to the severe downward bias of the OLS estimator in autoregressive models (Phillis and Yu, 005). In contrast, the bond and derivative rices based on both symmetric and asymmetric kernel estimators are much less biased and actually roduce slightly ositive ricing errors. he bias of the Gamma estimator is smaller than its Gaussian counterart but more imortantly, the Gamma-based bond and otion rices enoy much smaller variability and tighter con dence intervals than the symmetric kernel-based rices. 4 Conclusion his aer rooses several asymmetric kernel estimators of conditional moment functions based on deendent data and nonnegative conditioning variables. he consistency, rate of convergence and asymtotic normality of these estimators are established for both interior and boundary design oints. We show that the asymmetric kernel estimators ossess some aealing roerties such as lack of boundary bias and/or adatability in the amount of smoothing. he aer adots a block cross-validation method for deendent data in choosing the smoothing arameter. he nite-samle erformance of the estimators is evaluated in the context of a scalar di usion rocess of sot interest rate. Several interesting directions for future research include construction of bootstra con dence bands and bootstra-based seci cation testing, establishing uniform rates of convergence and rate imrovement via multilicative bias correction. 8

20 A Aendix A: Proofs of heorems In this aendix, we resent the roofs only for the Gamma kernel because the roofs for the IG and RIG kernels are similar. Note that aroximations to the moments of the IG and RIG kernels can be obtained by following Scaillet (004) and alying Lemmata B and B. A. Proofs of heorems and he roofs of heorems and require the following three lemmata. Before roceeding, de ne t (Y t ) m (X t ). Lemma A. Let S S 0 (x) b S (x) b S (x) b S (x) : If the conditions (A)-(A5) hold, then for interior x, S G;! SG S IG;! SIG S RIG;! SRIG 0 f (x) ; 0 x 0 0 x 3 f (x) ; 0 f (x) : 0 x Proof of Lemma A. Using Lemma B, n o n o E fs G; (x)g E (X x) K G(xb+;b) (X ) E ( ;x x) f ( ;x ) ; where ;x d G (xb + ; b). aking a second-order aylor exansion of f (;x ) around ;x x yields ( ;x x) f ( ;x ) ( ;x x) f (x)+( ;x x) + f 0 (x)+ ( ;x x) + f 00 (x)+o n( ;x x) +3o : herefore, by Lemma B, for interior x, E fs G;0 (x)g f (x) + O (b) ; E fs G; (x)g ff (x) + xf 0 (x)g b + O b ; E fs G; (x)g xf (x) b + O b : 9

21 Since strong mixing imlies ergodicity, we can aly Birkho s ergodic theorem to establish the results. Lemma A. Let t 0 (x) b (x) " b P P t tk x;b (X t ) t (X t x) t K x;b (X t ) # : Also for an arbitrary vector c R, de ne Q c t. If the conditions (A)-(A3), (A4 ) and (A5) hold, then for interior x, V ar b Q G; V ar b Q IG; V ar b Q RIG; (" # )! c V G c c 0 x x (x) f (x) c; 0 4 (" # )! c V IG c c 0 x 3 x (x) f (x) c; (" # )! c V RIG c c 0 x x (x) f (x) c: 0 4 Proof of Lemma A. It su ces to demonstrate that n o V ar b G;0 (x) n o V ar b b G; (x) n o Cov b G;0 (x) ; b b G; (x) x (x) f (x) + o () ; (7) x 4 (x) f (x) + o () ; (8) o () : (9) (i) Proof of (7). It follows from E ( t X t ) 0 that ( n o V ar b G;0 X (x) V ar t G;0 (0) + b 4 t K G(xb+;b) (X t ) ) G;0 () ; (0) where G;0 () b E + K G(xb+;b) (X ) K G(xb+;b) (X + ) is the th -order autocovariance of the stationary rocess b 4 t K G(xb+;b) (X t ). For the rst term on the right-hand side of (0), by the law of iterated exectations and Lemma B, n o G;0 (0) b E (X ) KG(xb+;b) (X ) b A b; (x) E ( ;x ) f ( ;x ) b b x + o b E ( ;x ) f ( ;x ) ; 0

22 where ;x d G (xb + ; b). aking a aylor exansion of ( ;x ) f ( ;x ) around ;x x and using Lemma B, we have E ( ;x ) f ( ;x ) (x) f (x) + O (b) so that G;0 (0) x (x) f (x) + o () : On the other hand, for a constant a satisfying (A4 ), ick a sequence d 0 b ( )(a). hen, the second term on the right-hand side of (0) is bounded by X d 0 G;0 () X G;0 () + G;0 () U + U : d 0 For U, using t (Y t ) E f (Y t ) X t g and the law of iterated exectations gives G;0 () b E E ( (Y ) (Y + ) X ; X + ) K G(xb+;b) (X ) K G(xb+;b) (X + ) +E E ( (Y ) X ; X + ) E ( (Y + ) X + ) K G(xb+;b) (X ) K G(xb+;b) (X + ) +E E ( (Y ) X ) E ( (Y + ) X ; X + ) K G(xb+;b) (X ) K G(xb+;b) (X + ) +E E ( (Y ) X ) E ( (Y + ) X + ) K G(xb+;b) (X ) K G(xb+;b) (X + ) b (U + U + U 3 + U 4 ) : As indicated in the roof of Lemma B, G (xb + ; b) has moments of any nonnegative integer order, and all these moments are O (). hen, by (A) and (A3), U Z Z 0 0 E f (Y ) (Y + ) X u; X + vg K G(xb+;b) (u) K G(xb+;b) (v) f ;+ (u; v) dudv c Z Z 0 0 O () : ( 0 + u m + v n ) K G(xb+;b) (u) K G(xb+;b) (v) dudv In addition, using (a conditional moment version of) Hölder s inequality, n E f (Y t ) X t ug E f (Y t ) X t ug E (Y t ) o Xt u : h n Without loss of generality, assume 0 so that 0 + u l max ; E (Y t ) oi Xt u. hen, (A3) imlies that E f (Y t ) X t ug 0 + u l 0 + u l : ()

23 Using (), (A) and (A3), we have U Z Z 0 0 E f (Y ) X u; X + vg E f (Y + ) X + vg K G(xb+;b) (u) K G(xb+;b) (v) f ;+ (u; v) dudv c Z Z 0 0 O () : ( 0 + u m + v n ) 0 + v l K G(xb+;b) (u) K G(xb+;b) (v) dudv Similarly, U 3 O () can be shown. Furthermore, by () and (A), U 4 Z Z 0 0 E f (Y ) X ug E f (Y + ) X + vg K G(xb+;b) (u) K G(xb+;b) (v) f ;+ (u; v) dudv Z c 0 + u l K G(xb+;b) (u) du 0 O () : Hence, G;0 () O b, which establishes that U O d 0 b O b fa ( )g(a)! 0: For U, we can aly Davydov s lemma (Corollary A. in Hall and Heyde, 980) to obtain G;0 () 8 f ()g E b 4 K G(xb+;b) (X ) : o nd the bound for E b 4 K G(xb+;b) (X ), note that since g (z) z (z 0) is increasing and convex, x y (x + y) ( x) + ( x) + x + y : ( y) ( y) Substituting x (Y ) and y E f (Y ) X g yields h (Y ) + E f (Y ) X g i h i (Y ) + E f (Y ) X g : ()

24 hen, we have Z E b 4 K G(xb+;b) (X ) n cb 4 E (Y ) o X u KG(xb+;b) (u) f (u) du 0 Z + E f (Y ) X ug KG(xb+;b) (u) f (u) du 0 cb 4 (U + U ) : Again, as argued in the roof of Lemma B, G (xb + ; b) has moments of any nonnegative integer order and all these moments are O (). hen, it follows from Lemma B, (A), (A3), and () that each of U and U is bounded by Z ca b; (x) u l K G(xb+;b) (u) du O fa b; (x)g O b ( ) : herefore, E b 4 K G(xb+;b) (X ) O b 4, and thus U O b f ()g d 0 because O b ( the roof of this art. O b ( ) d a 0 X d 0 a f ()g! 0; ) d a 0 O (), d 0!, and P a f ()g <. his comletes Remark. We can demonstrate (7) even after relacing (A4 ) by a weaker condition (A4). Observe that given (A4) and d 0 b ( )(a), each of U and U still becomes o (). (ii) Proof of (8). We have n o V ar b b G; (x) ( X V ar t G; (0) + b 4 (X t x) t K G(xb+;b) (X t ) ) G; () ; (3) where G; () b E (X x) (X + x) + K G(xb+;b) (X ) K G(xb+;b) (X + ) is the th - order autocovariance of the stationary rocess b 4 (X t x) t K G(xb+;b) (X t ). By the law of iterated exectations and Lemma B, the rst term on the right-hand side of (3) reduces to n o G; (0) b E (X x) (X ) KG(xb+;b) (X ) n o b A b; (x) E ( ;x x) ( ;x ) f ( ;x ) b b x + o b n o E ( ;x x) ( ;x ) f ( ;x ) ; 3

25 where ;x d G (xb + ; b). so that By a aylor exansion and Lemma B, we have n o E ( ;x x) ( ;x ) f ( ;x ) xb + b G; (0) x 4 (x) f (x) + o () : (x) f (x) + O b On the other hand, the second term on the right-hand side of (3) is bounded by X d G; () X G; () + G; () V + V ; where the sequence d is de ned as d b 3( )(a) for a constant a satisfying (A4 ). For V, the same logic as in art (i) yields d G; () b E E f (Y ) (Y + ) X ; X + g X x K G(xb+;b) (X ) X + x K G(xb+;b) (X + ) +b E E f (Y ) X ; X + g X x K G(xb+;b) (X ) E f (Y + ) X + g X + x K G(xb+;b) (X + ) +b E E f (Y ) X g X x K G(xb+;b) (X ) E f (Y + ) X ; X + g X + x K G(xb+;b) (X + ) +b E E f (Y ) X g X x K G(xb+;b) (X ) E f (Y + ) X + g X + x K G(xb+;b) (X + ) V + V + V 3 + V 4 : Observe that by (A), we have f (u) m so that Using (A3), f ;+ (u; v) f (u) f (v) M m ) f ;+ (u; v) cf (u) f (v) : (4) Z Z V cb ( 0 + u m ) u x K G(xb+;b) (u) f (u) du v x K G(xb+;b) (v) f (v) dv 0 0 Z Z + v n v x K G(xb+;b) (v) f (v) dv u x K G(xb+;b) (u) f (u) du 0 0 cb (V V + V 3 V 4 ) : 4

26 he Cauchy-Schwarz inequality imlies that V Z Z 0 (u x) K G(xb+;b) (u) f (u) du + u m (u 0 Z ( 0 + u m ) K G(xb+;b) (u) f (u) du 0 ( 0 V + V ) V 3 : 0 x) K G(xb+;b) (u) f (u) du By a aylor exansion and Lemma B, we have V O (b) and V O (b). In addition, V 3 O (), and thus V O b. Similarly, each of V, V 3 and V 4 is at most O b. Hence, V O b. Alying the same rocedure, we can also demonstrate that each of V, V 3 and V 4 is bounded by O b. Hence, we can conclude that G; () O b, which establishes that V O d b O b fa 3( )g(a)! 0: For V, we can aly again Davydov s lemma to obtain G; () 8 f ()g E b 4 (X x) K G(xb+;b) (X ) : It follows from (), (), (4), and Lemma B that E b 4 (X x) K G(xb+;b) (X ) Z n cb 4 u x E (Y ) o X u KG(xb+;b) (u) f (u) du 0 Z + u x E f (Y ) X ug KG(xb+;b) (u) f (u) du 0 cb 4 A b; (x) Z cb 4 A b; (x) V : 0 u x 0 + u l K G(xb+;b) (u) f (u) du By the Cauchy-Schwarz inequality, V Z Z 0 u x K G(xb+;b) (u) f (u) du + u l u 0 Z u l K G(xb+;b) (u) f (u) du ( 0 V + V ) V 3 : 0 x K G(xb+;b) (u) f (u) du Recall that > 4. Hence, by Lemma B, each of V and V is at most O b. Clearly, 5

27 V 3 O (), and thus we have V O (b) so that E b 4 (X x) K G(xb+;b) (X ) b 4 O b ( ) O (b) O b 3 34 : herefore, V O b 3 3 f ()g d O b 3( ) d a X d a f ()g! 0; because O b 3( ) d a O (), d!, and P a f ()g <. his comletes the roof of this art. (iii) Proof of (9). We have n Cov ( X Cov t G;3 (0) + b G;0 (x) ; o b b G; (x) b 4 (X t x) t K G(xb+;b) (X t ) t G;3 () ; (5) b 4 t K G(xb+;b) (X t ) ; where G;3 () E (X + x) + K G(xb+;b) (X ) K G(xb+;b) (X + ) is the th -order crosscovariance of the stationary rocesses b 4 t K G(xb+;b) (X t ) and b 4 (X t x) t K G(xb+;b) (X t ). By the law of iterated exectations and Lemma B, the rst term on the right-hand side of (5) reduces to n o G;3 (0) E (X x) (X ) KG(xb+;b) (X ) A b; (x) E ( ;x x) ( ;x ) f ( ;x ) b x + o b E ( ;x x) ( ;x ) f ( ;x ) ; X ) where ;x d G (xb + ; b). By a aylor exansion and Lemma B, we can see that E ( ;x x) ( ;x ) f ( ;x ) O (b) ; and thus G;3 (0) o b o (). On the other hand, alying the same rocedures as in arts (i) and (ii), we can also establish that the second term on the right-hand side of (5) is o (). his comletes the roof. 6

28 Lemma A3. If the conditions (A)-(A3), (A4 ), (A5)-(A6) hold, then for interior x, b Q G; d! N (0; c V G c) ; b Q IG; d! N (0; c V IG c) ; b Q RIG; d! N (0; c V RIG c) : Proof of Lemma A3. We emloy the small-block and large-block argument. Partition the set f; : : : ; g into q + subsets with large block of size r and small block of s. Put Also let & G; c Z G;, where for 0; : : : ;. hen, Z G; q r + s : b 4 + K G(xb+;b) (X + ) b 4 (X + x) + K G(xb+;b) (X + ) b Q G; 0 & G;: Furthermore, de ne the random variables, for 0 q, It follows that G; (r +s )+r X i(r +s ) 0 b Q G; X We will show that 0 & G;i; G; qx G; + E fex (itq G;;)g 0 (+)(r +s ) X i(r +s )+r & G;i; G;q G; + G;q q Q 0 iq (r +s ) & G;i: A (Q G;; + Q G;; + Q G;;3 ) : E Q G;; E Q G;;3! 0; (6)! 0; (7) E ex it G;! 0; (8) q X E G; 0 q X h n E G; G; (c V G c) oi 0! c V G c; (9)! 0 (0) for every > 0. (6) and (7) imly that Q G;: and Q G;;3 are asymtotically negligible, (8) imlies that the summands G; in Q G;; are asymtotically mutually indeendent, and (9) 7

29 and (0) are the standard Lindeberg-Feller conditions for asymtotic normality of Q G;; under indeendence. Hence, the lemma follows if we can show (6)-(0). We rst choose the block sizes. (A6) imlies that a sequence N such that!, s b! 0, and and the small-block size by s. b (s )! 0 as!. De ne the large-block size r by $ b % r It follows that s r! 0; r! 0; r b! 0; (s )! 0 () r as!. he roofs of (6)-(0) are given subsequently. (i) Proof of (6). Observe that E Q G;; q X 0 q V ar X G; + i0 qx 0; 6i For F, it follows from stationarity and Lemma A that 0 F q V Xs & G; On the other hand, F can be further rewritten as F qx qx Cov G;i; G; F + F : A q s fc V G c + o ()g O (q s ) : sx sx i0 0; 6i l 0 l 0 Cov G;mi+l ; G;m+l ; where m (r + s ) + r. Since i 6, we have (m i + l ) (m + l ) r. hen, by stationarity, Xr F l 0 l l +r X Cov G;l ; G;l Cov G;0 ; G; : Note that the arguments used in the roof of Lemma A imly that P r Cov G;0 ; G; o (). herefore, F o ( ), and thus, by (), r E q s s Q G;; O + o () O r + s + o ()! 0: 8

30 (ii) Proof of (7). Using a similar argument to the one used in the roof of (6), we have, by (), E Q G;;3 f q (r + s )g V ar & X G;0 + Cov G;0 ; G; o () fc V G c + o ()g + o ()! 0: 0 (iii) Proof of (8). Observe that G;a is F a i a -measurable with i a a (r + s ) + and a a (r + s ) + r. Alying Lemma B3 with V ex it G; and () yields E fex (itq G;;)g q Q 0 E ex it G; 6q (s + ) 6 r + s (s + )! 0: (iv) Proof of (9). By stationarity and Lemma A, we have E G; V ar G; r fc V G c + o ()g : herefore, by (), q X E q r r G; fc V G c + o ()g r + s 0 c V G c! c V G c: (v) Proof of (0). We emloy a truncation argument because is not necessarily bounded. Let L f Lg for some xed truncation oint L > 0. Also let & L G; c Z L G;, where Z L G; b 4 L + K G(xb+;b) (X + ) b 4 (X + x) L + K G(xb+;b) (X + ) : Furthermore, de ne Q L G; b 4 0 (r +s )+r X & L G;; L G; i(r +s ) & L G;i: In addition, let ~& L G; c ~Z L G;, where ~Z L G; and ~ L f > Lg. Finally, de ne so that Q G; QL G; + ~ Q L G;. b 4 ~ L +K G(xb+;b) (X + ) b 4 (X + x)~ L +K G(xb+;b) (X + ) ~Q L G; b 4 0 ~& L G; 9

31 Since both K G(xb+;b) (u) and uk G(xb+;b) (u) are bounded above, we have (X + x) K G(xb+;b) (X + ) < () for 0; : : : ; so that & L clb 4. G; hen, L G; clr b 4 ; (3) and thus, by (), L G; r c b! 0: It follows that at all for su ciently large. n L Pr G; (c V G c) o 0 (4) hen, alying (3) and (4), we have q X h n L E G; L G; (c V G c) oi 0 r c b q X 0 n L Pr G; (c V G c) o! 0: In other words, (0) holds for the truncated variables. Consequently, we have the following asymtotic normality result where V L G V ar ZL G; b Q L G; X x. 0 & L G; d! N 0; c V L Gc ; (5) he remaining task for establishing (0) is to show that as rst! and then L!, b V ar ~Q L G;! 0: (6) Indeed, nex E nex E + t ex c VGc L o it b Q G; o it b Q L G; t ex c V G c t ex c VGc L + E t ex c V G c n ex it b Q ~ o L G; E + E + E 3 : 30