Local Whittle Likelihood Estimators and Tests for non-gaussian Linear Processes

Transcription

1 Local Whittle Likelihood Estimators and Tests for non-gaussian Linear Processes By Tomohito NAITO, Kohei ASAI and Masanobu TANIGUCHI Department of Mathematical Sciences, School of Science and Engineering, Waseda University Abstract In this paper, we propose a local Whittle likelihood estimator for spectral densities of non-gaussian linear processes and a local Whittle likelihood ratio test statistic for the problem of testing whether the spectral density of a non-gaussian stationary linear process belongs to a parametric family or not. Introducing a local Whittle likelihood of a spectral density f θ (λ) around λ, we propose a local estimator ˆθ = ˆθ(λ) of θ which minimizes the local Whittle likelihood around λ, and use fˆθ(λ) (λ) as an estimator of the true spectral density. For the testing problem, we use a local Whittle likelihood ratio test statistic based on the local Whittle likelihood estimator. The asymptotics of these statistics are elucidated. It is shown that their asymototic distributions do not depend on non-gaussianity of the processes. Because our models include nonlinear stationary time series models, we can apply the results to stationary GARCH processes. Advantages of the proposed estimator and test are demonstrated by a few simulated numerical examples. AMS 2000 subject classifications: Primary 62G07; 62M15. Secondary 62G10; 62G20. Keywords and phrases: Non-Gaussian linear process, local Whittle likelihood estimator, spectral density, local likelihood ratio test. Corresponding author. 1

2 1 Introduction The study of spectral density functions of stationary processes provides an effective approach to estimate their underlying models and various methods for testing problems. There are two ways to estimate the spectral density directly. The one is the parametric method which is studied in the literature (e.g., Brockwell and Davis (1991)). Also Hosoya and Taniguchi (1982) constructed a very general framework of the Whittle estimator for a class of non- Gaussian linear processes. Further, Giraitis and Robinson (2001) proposed to use the Whittle estimation procedure for the squared ARCH processes. The other is the nonparametric method based on smoothed periodogram (e.g., Hannan (1970)). For i.i.d. observations, Hjort and Jones (1996) proposed a new probability density estimator fˆθ(x) (x) which minimizes a local likelihood for f θ around x. They showed that fˆθ(x) (x) has the same asymptotic variance as the ordinary nonparametric kernel estimator but potentially a smaller bias. For a Gaussian stationary process, Fan and Kreutzberger (1998) proposed a local polynomial estimator based on the Whittle likelihood. Then it was shown that it has advantages over the least-squares based on log-periodogram. In this paper, for a class of non-gaussian linear processes, we introduce a local Whittle likelihood of the spectral density f θ (λ) and propose the local Whittle estimator fˆθ(λ) (λ) around each frequency λ [ π, π]. Then we elucidate the asymptotics of ˆθ(λ) and fˆθ(λ) (λ). Next we consider the problem of testing whether the spectral density of a class of stationary processes belongs to a parametric family or not. For this testing problem, Fan and Yao (2003, Section 9.3.2) and Fan and Zhang (2004) proposed a generalized likelihood ratio tests based on the Whittle likelihood and a local Whittle estimator. Then they elucidated the asymptotics of the generalized likelihood ratio tests under the null hypothesis. Because the results above rely on Gaussianity of the process concerned, in this paper, we drop this assumption, and discuss the problem of testing whether the spectral density of a class of non-gaussian linear processes belongs to a parametric family or not. A local Whittle likelihood ratio test is proposed. Then it is shown that the asymptotic distribution of the test converges in distribution to a normal distribution. An interesting feature is that the asymptotics of the estimator and test statistic do not depend on non-gaussianity of the process. Because we do not assume Gaussianity of the process concerned, we can apply the results to stationary nonlinear processes which include GARCH processes. Some examples for a class of GARCH models are given. Numerical studies for the local Whittle likelihood ratio test are also provided. This paper is organized as follows. Section 2 describes the model, assumptions and estimator ˆθ(λ). In Section 3, we derive the asymptotics of ˆθ(λ) and fˆθ(λ) (λ). Section 4 introduces test statistic, then the asymptotic distribution of it is derived under the null hypothesis. Numerical examples for a class of GARCH (1, 1) processes are provided in Section 5. They illuminate some interesting features of our approach. The proofs of the main theorems are relegated to Section 6. Throughout this paper, we denote the set of all integers by J, and denote Kronecker s delta by δ(m, n). 2 Setting In this section we describe the model, assumptions and estimators. Throughout this paper we consider a class of non-gaussian linear processes, which include not only ARMA but also squared GARCH processes etc. Assume that {z(n) : n J} is a general linear process defined by z(n) = G(j)e(n j), n J, (2.1) j=0 2

3 where {e(n)} is a white noise process satisfying E[e(n)] = 0 Furtheremore, we assume that E[e(n)e(m)] = δ(m, n)σ 2, E[e(n) 4 ] <. G(j) 2 <. (2.2) Then {z(n)} is a second-order stationary process with spectral density j=0 2 f(w) = σ2 2π G(j)e iwj. (2.3) j=0 Henceforth we denote by P the set of all spectral density funcitons of the form (2.3). Write the autocovariance function of z(n) as γ( ). Then we assume, Assumpton 1. n 2 γ(n) <. (2.4) Let z(1), z(2),..., z(n) be an observed stretch of {z(n)}, and denote the periodogram of {z(n)} by I N (λ) = 1 N 2 z(n)e inλ. 2πN n=1 Assumption 2. Let K( ) be a kernel function which satisfies: (1) K is a real bounded nonnegative even function with a bounded support. (2) K(t)dt = 1, t2 K(t)dt = κ 2 <, tj K 2 (s)ds <, j = 0, 1, 2 (3) For k(x) = K(t)eitx dt, there exists an even integrable monotone decreasing function k(x) such that k(x) k(x) on [0, ). Assumption 3. For a bandwidth h decreasing as N, we assume that 1 N 1 2 h + N 1 5 h 0. (2.5) Let {f θ (λ) : θ Θ R q } be a parametric family of spectral density functions of P where Θ is a compact set. Here we impose the following assumption. Assumption 4. f θ (w) is three times continuously differentiable with respect to w and θ. We define a local distance function D λ (, ) around a given local point λ for spectral densities {f t } by π { D λ (f t, f) = K h (λ w) log f t (w) + f(w) } dw, f t (w) π where K h (x) = 1 h K(x/h). If we replace f by I N, we call D λ (f θ, I N ) the locall Whittle likelifood function under f θ. 3

4 Define T λ (f) Θ by D λ (f Tλ (f), f) = min t θ D λ(f t, f), Henceforth we sometimes write T λ (f) as θ 0 (λ) which is called a pseudo-true value of θ. As an estimator of θ 0 (λ), we use ˆθ(λ) defined by ˆθ(λ) = T λ (I N ) = arg min t Θ D λ(f t, I N ). We can use fˆθ(λ) (λ) as a local estimator of f and call this the local Whittle likelihood estimator. For simplicity we sometimes write f θ(λ) (λ) as f θ (λ). 3 Estimating Theory In this section, we investigate the asymptotics of ˆθ(λ) and fˆθ(λ). Fan and Kreutsberger (1998) showed the asymptotics of a local polynomial estimator of the spectral density based on the Whittle likelihood for Gaussian linear processes. Here we set down the following assumption. Assumption 5. j Q 1,j 2,j 3= e(j 1, j 2, j 3 ) < where Q e (j 1, j 2, j 3 ) is the joint fourth order cumulant of e(n), e(n + j 1 ), e(n + j 2 ), e(n + j 3 ). Next we show the asymptotic distribition of ˆθ(λ) as N. Theorem 3.1 Suppose that the {z(n)} given in (2.1) and K( ) satisfy Assumptions 1-5, T λ (g) exists uniquely and lies in Int Θ, and that M(λ) = 2 θ θ is a nonsingular matrix. Then if N, 1 (f τ θ 0 (λ)f(λ) + log f θ0 (λ)) Nh {ˆθ(λ) θ 0 (λ) 1 2 h2 σkm(λ) Ṽ = where σ 2 K = t 2 K(t)dt. N(0, M(λ) 1 Ṽ (M(λ) 1 ) τ ) { 2πf 2 (λ) K2 (x)dx θ f 1 θ 0 4πf 2 (λ) K2 (x)dx θ f 1 θ 0 λ 2 {f(λ) θ f 1 θ 0 } (λ)} (λ) θ f 1 τ θ 0 (λ) λ 0 (λ) θ f 1 τ θ 0 (λ) λ = 0 The following theorem establishes the asymptotic normality of local Whittle estimator fˆθ(λ) (λ). Theorem 3.2 Under the same conditions as Theorem 3.1 ( ) ( Nh fˆθ(λ) (λ) f(λ) N 0, ( θ τ f θ 0 (λ))m(λ) 1 Ṽ M(λ) 1 ( ) θ f θ 0 (λ)) The proofs of the theorems are relegated to Section 5. Remark 3.1 In Theorem 3.2, we can see that the asymptotic variance and bias of the local Whittle estimator depend on only f(λ), f θ0 (λ) and K. Thus the asymptotic distribution of the local Whittle estimator does not depend on non-gaussianity of the process. 4

5 4 Testing Theory When we estimate the spectral density of an observed time series, it is a significant problem whether the spectral density is parametric or not. For this, Fan and Zhang (2004) applies local linear polynomial teqniques to the log-periodogram of Gaussian process. Consider the problem of testing whether f(λ) belongs to a specific parametric family {f θ ( ) : θ Θ} or not, i.e., H 0 : f( ) = f θ ( ) v.s. H 1 : f( ) f θ ( ). (4.1) Although we do not assume Gaussianity of {X t }, the Whittle likelihood function under H 0 is expressed as l(θ) = N k=1 { log f θ (λ k ) + I } N(λ k ), λ k = π + 2πk/N (k = 1,, N). f θ (λ k ) We call ˆθ WH = arg max θ Θ l(θ) the Whittle likelihood estimator of θ. In this paper, for H 1, we use the following local Whittle likelihood function around λ [ π, π]: l loc (θ) = N k=1 { log f θ (λ k ) + I } N (λ k ) K h (λ λ k ), (4.2) f θ (λ k ) where K h ( ) is an appropriate kernel function. Let ˆθ LW (λ) = arg max θ Θ l loc (θ), and we regard fˆθlw (λ) (λ) as a sort of nonparametric estimator of the spectral density f(λ). For the testing problem (4.1), we use the following likelihood ratio test statistic T LW = l(ˆθ WH ) l(ˆθ LW (λ)) { } { } N = log (λ fˆθwh k ) + I N(λ k ) N + log fˆθlw (λ (λ fˆθwh k ) k ) (λ k) + I N(λ k ) fˆθlw (λ k ) (λ k) = k=1 k=1 k=1 N { } log fˆθlw (λ k ) (λ k) log (λ fˆθwh k ) + I N (λ k )(fˆθlw (λ k ) (λ k) 1 (λ fˆθwh k ) 1 ). Actually if T LW > z α, a selected level, we reject H 0, otherwise, accept it. We derive the asymptotics of T LW under H 0 of (4.1). Write T LW as T LW = {l(θ) l(ˆθ LW (λ))} {l(θ) l(ˆθ WH )} = T LW,1 T LW,2. It is known that T LW,2 = O P (1) under appropriate regularity conditions (e.g., Taniguchi and Kakizawa (2000, Section 3.1)). In what follows, it is seen that T LW,1 is asymptotically of order in probability tending to. Hence, in order to derive the asymptotic distribution of T LW we have only to derive that of T LW,1. For this, furthermore, we impose the following assumption. Assumption 6. (i) {z(t)} is kth-order stationary with all of whose moments exist. (ii) The joint kth-order cumulant Q z (j 1,, j k 1 ) of z(t), z(t + j 1 ),, z(t + j k 1 ) satisfies j 1,,j k 1 = for l = 1,, k 1 and any k, k = 2, 3,. (1 + j l ) Q z (j 1,, j k 1 ) < 5

6 Then we get the following theorem whose proof is relegated to Section 5 Theorem 4.1. Suppose that Assumptions 1-4 and 6 hold. Then, under H 0, where and µ N = 1 h σ 2 N = 2 h [ π K(0) π π π π π σ 1 N (T LW µ N ) d N(0, 1), θ f θ(λ)(λ) 1 M(λ) 1 θ f θ(λ)(λ) 1 f θ(λ) (λ) 2 dλ { θ f θ(λ)(λ) 1 M(λ) 1 θ f θ(λ)(λ) 1 { θ f θ(λ)(λ) 1 M(λ) 1 θ f θ(λ)(λ) 1 M(λ) = 2 θ θ log f θ(λ)(λ). } 2 f θ(λ)(λ) 4 dλ } 2 f θ(λ)(λ) 4 dλ K(ω) 2 dω K(ω) 2 dω Remark 4.1. It should be noted that the asymptotics of T LW also do not depend on non- Gaussianity of the process. This seems interesting. 5 Numerical Examples We study performance of the local Whittle likelihood estimators for simulated data and fearture of the asymptotic distribution of T LW. Consider the following GARCH(1, 1) model: ] x(t) = σ(t)ε(t) σ 2 (t) = c + bx 2 (t 1) + aσ 2 (t 1) (5.1) where c > 0, a 0, b 0 and a + b < 1, and ε(t) i.i.d.(0, 1). Let z(t) = x 2 (t), then we can write z(t) = c + (b + a)z(t 1) + e(t) ae(t 1), where e(t) = z 2 (t) σ 2 (t) = {ε 2 (t) 1}{c + bx 2 (t 1) + aσ 2 (t 1)}. Here {e(t)} becomes a white noise process with σ 2 e = Var[e(t)]. Therefore, {z(t)} becomes an ARMA(1, 1) model, and the spectral density f z (λ) of {z(t)} is given by f z (λ) = σ2 e 1 ae iλ 2 2π 1 (b + a)e iλ. To estimate f z, we consider the local Whittle likelihood estimator by fitting a family of spectral densities {f θ, θ Θ} given by f θ (λ) = σ2 2π 1 αeiλ 2 = σ2 2π (1 2α cos λ + α2 ) 1 where θ = (α, σ 2 ) τ. The integral of local Whittle likelihood is approximated by the sum of Fourier frequencies w n = 2πn/N. For I N (λ) = 1 2πN N n=1 z(n)einλ 2, we calculate D λ (f θ, I N ) over grid points on [0.05, 0.95] [1, 25] for θ to derive ˆθ(λ) for each λ. We compare performance of the local Whittle estimator with that of the traditional estimator, ˆf(λ) = π π W M (λ t)i N (t)dt 6

7 where W M ( ) is the Daniell window (see Hannan (1970)). For spectral density estimator f(λ), we define the residual sum of squares (RSS) by where λ j = jπ/p. RSS( f) p 1 = (f(λ j ) f(λ j ) 2 j=0 Figure 1 is about here. Figure 2 is about here. The estimated fˆθ(λ) (λ) and periodogram I N (λ) are plotted in Figure 1. Figure 2 provides the graphs of fˆθ (solid line), the smoothed periodogram ˆf (dashed line) and the true spectral density f (dotted line) with c 0 = 1, a = 0.1, b = 0.2, h = 0.5 and M = 4. Then we observe that RSS(fˆθ) = and RSS( ˆf) = , which implies that the local Whittle estimator is better than the traditional one. Further, we repeated this experiment ten times, and calculated the sample mean of RSS, say RSS. Then we get RSS(fˆθ) = and RSS( ˆf) = Hence the local Whittle estimator fˆθ is recommendable. Recall that ˆθ LW (λ) is the minimizer of (4.2), and that T LW,1 = l(θ) l(ˆθ LW (λ)). Let c = 1.0, a = 0.1, b = 0.3 in (5.1), then generate x(1),, x(500). Based on them we can calculate T LW,1. Here we consider the local Whittle likelihood estimator by fitting a family of MA(1) spectral densities to estimate f z (λ), i.e.,{f θ, θ Θ} given by f θ (λ) = σ2 2π 1 + αeiλ 2 where θ = (α, σ 2 ) τ. We repeated this procedure 1000 times. Then, Figures 3 and 4 gives the empirical distributions of T LW,1 for the cases of (i) ε(t) i.i.d. N(0, 1) and (ii) ε(t) = ɛ(t)/ 5/4 i.i.d.(0, 1) with ɛ(t) i.i.d. t(10), respectively. Figure 3 is about here. Figure 4 is about here. These figures are similar, hence, confirm the result of Theorem 4.1. References [1] Brillinger, D. R. (1981). Time Series: Data Analysis and Theory, expanded ed. Holden- Day, San Francisco. [2] Brockwell, P. J. and Davis, R.A. (1991). Time series: theory and methods, 2nd edn. Springer-Verlag, New York. [3] Fan, J. and Kreutsberger, E. (1998). Automatic local smoothing for spectral density estimation. Scand. J. Statist. 25, [4] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Parametric and Nonparametric Methods. Springer-Verlag, New York. [5] Fan, J. and Zhang, W. (2004). Generalized likelihood ratio tests for spectral density. Biometrika

8 [6] Fuller, W. A. (1996). Introduction to Statistical Time Series, second ed. Wiley, New York. [7] Giraitis, L. and Robinson, P. M. (2001). Whittle estimation of ARCH models. Econometric Theory, 17, [8] Hannan, E. J. (1970). Multiple Time Series. John Wiley, New York. [9] Hjort, N. L. and Jones, M. C. (1996). Locally parametric nonparametric density estimation. Ann. Statist. 24, [10] Hosoya, Y. and Taniguchi, M. (1982). A central limit theorem for stationary processes and the parameter estimation of linear processes. Ann. Statist. 10, [11] Taniguchi, M. (1987). Minimum contrast estimation for spectral densities of stationary processes. J. Roy. Statist. Soc., Series B 49, [12] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer, New York. TOMOHITO, NAITO. DEPARTMENT OF MATHEMATICAL SCIENCES SCHOOL OF SCIENCE AND ENGINEERING WASEDA UNIVERSITY OKUBO SHINJUKU-KU TOKYO JAPAN. tomohito-n@toki.waseda.jp KOHEI, ASAI. DEPARTMENT OF MATHEMATICAL SCIENCES SCHOOL OF SCIENCE AND ENGINEERING WASEDA UNIVERSITY OKUBO SHINJUKU-KU TOKYO JAPAN. asai@toki.waseda.jp MASANOBU, TANIGUCHI. DEPARTMENT OF MATHEMATICAL SCIENCES SCHOOL OF SCIENCE AND ENGINEERING WASEDA UNIVERSITY OKUBO SHINJUKU-KU TOKYO JAPAN. taniguchi@waseda.jp 8

9 Figure 1: local Whittle estimator(solid) and periodogram(dotted) 9

10 Figure 2: local Whittle estimator (solid), smoothed periodogram (dashed) and true spectral density (dotted) 10

11 10 8 Percent of Total tlw1 : gaussian Figure 3: empirical distribution of T LW,1 when the innovation process is Gaussian 11

12 10 8 Percent of Total tlw1 : t-distribution Figure 4: empirical distribution of T LW,1 when the innovation process is t-distributed 12