A note on the asymptotic normality of the kernel deconvolution density estimator with logarithmic chi-square noise
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Zu, Yang Article A note on the asymptotic normality of the kernel deconvolution density estimator with logarithmic chisquare noise Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Zu, Yang (2015) : A note on the asymptotic normality of the kernel deconvolution density estimator with logarithmic chi-square noise, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 3, Iss. 3, pp. 561-576, https://doi.org/10.3390/econometrics3030561 This Version is available at: https://hdl.handle.net/10419/171839 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Econometrics 2015,3, 561-576; doi:10.3390/econometrics3030561 OPEN ACCESS econometrics ISSN 2225-1146 www.mdpi.com/journal/econometrics Short Note A Note on the Asymptotic Normality of the Kernel Deconvolution Density Estimator with Logarithmic Chi-Square Noise Yang Zu Department of Economics, City University London, Northampton Square, EC1V 0HB London, UK; E-Mail: [email protected]; Tel.: +44-0-20-7040-8619; Fax: +44-0-20-7040-8580. Academic Editor: Kerry Patterson Received: 29 April 2015 / Accepted: 17 July 2015 / Published: 21 July 2015 Abstract: This paper studies the asymptotic normality for the kernel deconvolution estimator when the noise distribution is logarithmic chi-square; both identical and independently distributed observations and strong mixing observations are considered. The dependent case of the result is applied to obtain the pointwise asymptotic distribution of the deconvolution volatility density estimator in discrete-time stochastic volatility models. Keywords: kernel deconvolution estimator; asymptotic normality; volatility density estimation JEL classifications: C13, C22, C46, C58 1. Introduction Consider the measurement error model: Y=X+ε, where Xis the signal, while εis the noise. Assume Xis independent of ε;Xhas density fX, and εhas density k, so the density of Y, denoted as fY, is the convolution of fXand k: fY=fX∗k, where the ∗denotes convolution.
Econometrics 2015,3562 Assume we observe the realizations Y1, . . . , Ynof Yand that the function kis fully known; one possible estimator for fXfrom the noisy observations Y1, . . . , Ynis the kernel deconvolution estimator: ˆ fX(x) = 1 2πZ+∞ −∞ e−itx φK(th)d φfY(t) φk(t)dt, (1) where: d φfY(t) = 1 n n X j=1 eitYj, is the empirical characteristic function of density fY,K(x)is a kernel function, φKand φkare the Fourier transform of Kand k, respectively1. The kernel deconvolution estimator was first proposed for the measurement error model by Carroll and Hall [1] and Stefanski and Carroll [2]. Define the kernel deconvolution function as follows: νh(x) := 1 2πZ+∞ −∞ φK(t) φk(t/h)e−itxdt; the kernel deconvolution estimator can be written compactly as: ˆ fX(x) = 1 nh n X j=1 νhx−Yj h.(2) In this paper, I show the asymptotic normality for the estimator ˆ fX(x)when the distribution of εis logarithmic chi-square. The asymptotic distribution of the kernel deconvolution estimator has been considered in Fan [3], Fan and Liu [4], Van Es and Uh [5] and Van Es and Uh [6] for identically independently distributed (i.i.d.) observations. Masry [7] and Kulik [8] consider various cases for the weakly-dependent observations. However, none of the above research allows the error distribution to be the logarithmic chi-square distribution. I consider both the identical and independently distributed (i.i.d.) observations and strong mixing observations in this paper, which complements the above-mentioned literature. The results obtained in this paper can be applied to obtain the asymptotic distribution of the deconvolution volatility density estimator. The problem of estimating volatility density has been gaining increasing interest in econometrics in recent years; see, e.g., Van Es, Spreij, and Van Zanten [9] and Van Es, Spreij, and Van Zanten [10] for the kernel deconvolution estimator, Comte and Genon-Catalot [11] for the penalized projection estimator and Todorov and Tauchen [12] for the study in the context of high-frequency data. Kernel deconvolution with logarithmic chi-square noise arises naturally when estimating the volatility density in stochastic volatility (SV) models. Existing research (e.g., Van Es, Spreij, and Van Zanten [9] and Van Es, Spreij, and Van Zanten [10]) focuses on the convergence rates of the estimators, and the asymptotic distribution of the estimators is not available. 1The characteristic function of a random variable with density fis defined as φf=R R eitxf(x)dx.
Econometrics 2015,3563 In Section 2, I review the probabilistic properties of the logarithmic chi-square distribution; Section 3 presents the asymptotic normality of the estimator, for both i.i.d. observations and dependent observations; Section 4discusses the application of the results to volatility density estimation in SV models; Section 5concludes the paper. 2. Logarithmic Chi-Square Distribution The logarithmic chi-square distribution is obtained by taking the logarithm of a chi-square distribution with degrees of freedom of one. The density function of logarithmic chi-square distribution is: k(x) = 1 √2πe1 2xe−1 2ex. The density function of the logarithmic chi-square distribution is asymmetric and is plotted in Figure 1. Figure 1. Density function of the logarithmic chi-square distribution. The characteristic function of the logarithmic chi-square distribution is: φk(t) = 1 √π2itΓ1 2+ it, where Γ(.)is the gamma function. Fan [3] studies the quadratic mean convergence rate of the kernel deconvolution estimator; it turns out that the convergence rate of the estimator depends heavily on the type of error distribution. In particular, it is determined by the tail behaviour of the modulus of the characteristic function of the error distribution: the faster the modulus function goes to zero in the tail, the slower the converge rate. The following lemma, which is from Van Es, Spreij, and Van Zanten [10], gives the tail behaviour of |φk(t)|. Lemma 1. (Lemma 5.1 of Van Es, Spreij, and Van Zanten [10]) For |t|→∞, we have: |φk|=√2e−1 2π|t|1 + O1 |t|,(3)
Econometrics 2015,3564 and: Re φk(t) = |φk|cos htlog √1+4t2−ti+O1 |t|,(4) Im φk(t) = |φk|sin htlog √1+4t2−ti+O1 |t|.(5) From (3), it is known that the modulus of φk(t)decays exponentially fast as |t|→∞. It thus belongs to the super-smooth density according to the classification in Fan [13]. According to Fan [13], the optimal convergence rate of the estimator is (log n)−2, when h= (log n)−1. Figure 2plots the modulus function |φk|and its approximation √2e−1 2π|t|; we notice that the two functions almost coincide at both tails. Figure 2. Modulus function of the characteristic function of logarithmic chi-square distribution and its approximation: the higher peak curve is the approximating function √2e−1 2π|t|. From (4) and (5), it is known that in both tails, neither the real part nor imaginary part of the characteristic function can dominate the other; this violates the assumptions in the previous works by, e.g., Fan [3] and Masry [7], on studying the asymptotic normality; for super-smooth error distributions, these papers assume either the real part or the imaginary part to be dominant. 3. Asymptotic Normality In this paper, I consider one particular kernel function, namely the sinc kernel function: (C1) The sinc kernel function is defined as: K(x) = sin(x) πx, with Fourier transform2: φK(t) = I{|t|⩽1}. 2In this paper, I follow the convention to define the Fourier transform of a function fto be φf=R+∞ −∞ eitxf(x)dx.
Econometrics 2015,3565 The sinc kernel function is favoured in theoretical literature because of the simplicity of its Fourier transform and is thus used here.3 3.1. i.i.d. Observations In this section, I prove the asymptotic normality of the estimator when the observations are i.i.d. Theorem 1. When the observations are i.i.d. and εis distributed as logarithmic chi-square, if Assumption (C1) holds, when exp (1/h)/n →0as n→ ∞and h→0, it holds that, ˆ fX(x)−Kh∗fX(x) q1 2π2nexp (π/h)fY(x)→dN(0,1), where Kh(x) := (1/h)K(x/h). Proof. Denote: Zj=1 hνhx−Yj h, then: ˆ f(x) = 1 n n X j=1 Zj. First: Eˆ f(x) = EZ1 =E"1 2πZ+∞ −∞ e−itx φK(th)d φfY(t) φk(t)dt# =1 2πZ+∞ −∞ e−itx φK(th)Ehd φfY(t)i φk(t)dt =1 2πZ+∞ −∞ e−itx φK(th)φfY(t) φk(t)dt =1 2πZ+∞ −∞ e−itxφK(th)φfX(t)dt =Kh∗fX(x), 3Usually, for practical implementations, the following kernels: K1(x) = 48 cos x πx41−15 x2−144 sin x πx52−5 x2, with Fourier transform: φK1(t) = I{|t|⩽1}1−t23, are used because they have better numerical properties; see Delaigle and Gijbels [14] for the discussions.
Econometrics 2015,3566 Second, I evaluate Var Z1, Var Z1= Var 1 hνhx−Y1 h =1 h2 Eνhx−Y1 h2 −Eνhx−Y1 h2! =1 h2 Zνhx−y h2 fY(y)dy−(Kh∗fX(x))2! =1 h2hZνh(y)2dyfY(x)−(Kh∗fX(x))2 =1 2π2exp π hfY(x) (1 + o(1)) ,(6) where the last equality is obtained because Kh∗fX(x)→fX(x)as h→0, and R|νh(x)|2dx= h 2π2exp π h(1 + o(1)). The latter result is shown as follows, Z|νh(x)|2dx=1 2πZ|φνh(u)|2du =1 2πZ φK(u) φk(u/h) 2 du =h πZ1/h 0 1 φk(u) 2 du =h π ZM 0 1 φk(u) 2 du+Z1/h M 1 φk(u) 2 du!, where Mis a very big number. The first term in the brackets is a constant depending on M; the order of the second term can be evaluated as follows, Z1/h M 1 φk(u) 2 du=1 2πexp π h−exp (πM) =1 2πexp π h(1 + o(1)), where I use the fact that when Mis big, |φk(u)|can be replaced by its asymptotic approximation. The second term clearly dominates the first term, which is a constant, such that: Z|νh(x)|2dx=h 2π2exp π h(1 + o(1)).(7) Here, I use the argument of Butucea [15] to split the integral and show that the tail part of the integral dominates. A sufficient condition for asymptotic normality is the Lyapunov condition, which reduces to:
Econometrics 2015,3567 E|Z1−EZ1|2+δ nδ/2[Var(Z1)]1+δ/2→0,(8) for i.i.d. data. For an upper bound for the numerator, E|Z1−EZ1|2+δ⩽E|Z1|2+δ+|EZ1|2+δ ⩽2E|Z1|2+δ =2 h2+δZ+∞ −∞ νhx−y h 2+δ fY(y)dy ⩽C h2+δZ+∞ −∞ νhx−y h 2+δ dy(9) Now, notice the result from Van Es, Spreij, and Van Zanten [10] and Masry [16] that, for p > 24, kνhkp⩽kνhk1−2/p ∞kνhk2/p 2. An upper bound for kνhk∞is easy to get, as5: kνhk∞= sup x 1 2πZφK(t) φk(t/h)e−itxdt ⩽1 2πZ φK(t) φk(t/h)dt ⩽√2 π2hexp π 2h, while kνhk2 2is known from (7), such that: Z|νh(z)|pdz⩽kνhkp−2 ∞kνhk2 2 ⩽C×hp−2exp π(p−2) 2h×hexp π h =C×hp−1exp πp 2h, for p > 2. Therefore, take p= 2 + δand use the result in (9); it then holds that: E|Z1−EZ1|2+δ⩽C×exp π(2 + δ) 2h,(10) this, together with (6), implies that Lyapunov’s condition (8) holds, which completes the proof. 4This is easy to see by noticing R|νh(x)|pdx⩽R|νh(x)|2|supxνh(x)|p−2dxfor p > 2. 5Here, again, the splitting integral argument as in proving (7) is used; I omit the details for the ease of exposition.
Econometrics 2015,3568 3.2. Strong Mixing Observations In this section, I consider the model: Y=X+ε,(11) where X’s realizations of X1,··· , Xnare strictly stationary and strong mixing, while the noise realizations ε1,··· ,εnare i.i.d. logarithmic chi-square variables, independent of X, such that the observations Y1,··· , Ynare also strictly stationary and strong mixing. There are various concepts of dependence; here, I consider the case of αmixing, also called strong mixing, which is the weakest among all of the dependence concepts. Definition 1. Let {Xt},t=··· ,−1,0,1,··· be an infinite sequence of strictly stationary random variables and Fj ibe the σ-algebra generated by {Xt, i ⩽t⩽j}; then, the α-mixing coefficient is defined as: α(k) = sup A∈F0 −∞,B∈F+∞ k|P(A)P(B)−P(AB)|. The sequence {Xt},t=··· ,−1,0,1,···, is called α-mixing if α(k)→0as k→ ∞. For the dependent case, a bounded assumption on the joint density of observations is also needed. (C2) The probability density function of any joint distribution (Yi, Yj),1⩽i < j ⩽n, exists and is bounded by a constant. Now, I give the asymptotic normality theorem. Notice that the mixing assumption here is a litter weaker than that in Masry [7]. Theorem 2. In model (11), let X1, X2,··· , Xnbe strictly stationary, α-mixing with: ∞ X k=1 α(k)1−2/δ<∞,(12) for some δ>2; the noises ε1,··· ,εnare i.i.d. logarithmic chi-square variables, independent of X; if (C1) and (C2) hold, when exp (1/h)/n →0as n→ ∞and h→0, then: ˆ fX(x)−Kh∗fX(x) q1 2π2nexp (π/h)fY(x)→dN(0,1). Proof. First, by strict stationarity and using the ergodic theorem for strong mixing sequences, similarly as in the proof of Theorem 1, Eˆ f(x) = Kh∗fX(x). Next, the variance of the estimator is evaluated; first: Var ˆ f(x)=1 nVar (Z1) + 2 n2 n−1 X j=1 (n−j) Cov (Z1, Zj+1).
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