- Email: [email protected]
A new kernel for long-run variance estimates in seasonal time series models
Economics Letters 76 (2002) 165–171 www.elsevier.com / locate / econbase
A new kernel for long-run variance estimates in seasonal time series models Dong Wan Shin*, Man-Suk Oh Department of Statistics, Ewha University, Seoul 120 -750, South Korea Received 24 July 2001; accepted 12 December 2001
Abstract A new kernel for estimating long-run variances of stationary seasonal time series is proposed. The proposed kernel has an oscillating pattern which is in harmony with that of the autocovariance functions of seasonal time series. A Monte-Carlo experiment shows that the estimator based on the proposed kernel outperforms estimators based on existing kernels such as the Bartlett kernel, Parzen kernel, and Tukey–Hanning kernel for two typical monthly time series processes with moderate autocorrelations. 2002 Elsevier Science B.V. All rights reserved. Keywords: Autocovariance function; Efficiency; Seasonality JEL classification: C13; C14; C22
1. Introduction Heteroskedasticity and autocorrelation consistent (HAC) estimators of the long-run variances of stationary processes are required in many areas of econometrics such as linear regressions with heteroskedastic and serially correlated errors, semiparametric unit root tests, and generalized method of moments estimations of nonlinear simultaneous equation models. Several methods for HAC estimation were developed by Newey and West (1987), Gallant and White (1988, pp. 97–103), Andrews (1991), Phillips (1995), and many others; see the references in Andrews (1991). However, neither of these methods is designed for seasonal data, though in practice many data sets in economics and business are observed on quarterly or monthly bases. Therefore, it would be useful to develop an estimator which works well for typical seasonal time series data. In this article we propose a new kernel and an estimator based on the new kernel which take account of * Corresponding author. Tel.: 182-2-3277-2614; fax: 182-2-3277-2307. E-mail addresses: [email protected] (D.W. Shin), [email protected] (M.-S. Oh). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00048-4
2002 Elsevier Science B.V. All rights reserved.
166
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
seasonal behaviors. The proposed estimator is nonnegative and consistent. In addition, a Monte-Carlo simulation shows that the estimator based on the proposed kernel performs better than estimators based on the existing kernels, for typical seasonal models with moderate autocorrelations. This article is organized as follows. Section 2 proposes the new kernel and establishes nonnegativeness and consistency of the long-run variance estimator based on the proposed kernel. Section 3 presents a Monte Carlo study for two monthly multiplicative autoregressive (AR) and moving average (MA) models.
2. A new kernel and its properties It is very common that sample autocovariance functions (ACFs) of many empirical d-seasonal time series z t have spikes around seasonal lags h 5 0, 6d, 62d, . . . , where d is a given integer denoting period of seasonality such as 4 or 12. This fact is illustrated in Fig. 1 which shows plots of the population ACFs for multiplicative monthly AR and MA models of (1 2 0.4B)(1 2 0.4B 12 )z t 5 a t and 12 z t 5 (1 1 0.4B)(1 1 0.4B )a t , respectively, where B is the back-shift operator such that Bz t 5 z t 21
Fig. 1. Autocovariance functions for the multiplicative AR model, (1 2 0.4B)(1 2 0.4B 12 )z t 5 a t , and the multiplicative MA model, z t 5 (1 1 0.4B)(1 1 0.4B 12 )a t .
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
167
and a t is a white noise. This indicates that sample autocovariances for seasonal lags tend to be more significant than those for other lags. This point would be more conspicuous for the monthly case (d 5 12) than quarterly case (d 5 4). We discuss kernels through estimation of the long-run variance s 2 5 o `h 52` gh of a stationary process z t , where gh 5 cov(z 0 , z h ). Consider a regression model y t 5 x t9 b 1 z t in which z t is a stationary zero mean process, hy t , x t , t 5 1, . . . , nj is the set of observations, and b is an unknown regression parameter. A widely-used estimator of s 2 is
O ,
2
s¯ 5
h 52 ,
w h , gˆ h ,
w h , 5 1 2 uhu /( , 1 1),
(1)
where gˆ h 5 n 21 o nt 5uh u11 zˆ t zˆ t 2uh u , zˆ t 5 y t 2 x t9 bˆ , bˆ is a consistent estimator of b, and , is an integer called bandwidth. The kernel w h , in (1) is the Bartlett kernel. Newey and West (1987) showed 2 nonnegativeness and consistency of the estimator s¯ . Other widely-used kernels are the Parzen kernel and the Tukey–Hanning kernel. See Andrews (1991) for details about these kernels. Fig. 2 displays these kernels with , 5 3d 1 r 5 42 for the monthly case of d 5 12 where r 5 [d / 2], the integer part of d / 2. From Fig. 2, one can see an undesirable fact that, for all the three existing kernels, the weights of the terms gˆ h for h around d, 2d, . . . are smaller than those for h around d 2 r, 2d 2 r, . . . , respectively. For example, the weight w, ,12 of gˆ 12 is smaller than the weight w, 6 of gˆ 6 which tends to be less significant than gˆ 12 . Figs. 1 and 2 lead us to consider a new kernel which distributes more weight on gˆ h for h around the seasonal lags d, 2d, . . . than for h around d 2 r, 2d 2 r, . . . , respectively. We consider a new kernel w i w h , defined in the following estimator
OwO ,
r
sˆ 2 5
i 52r 1
i
h52 ,
w h , gˆ hd 1i , w i 5 1 2 uiu /(r c, ),
(2)
where r 1 5 [(d 2 1) / 2] and c, is a sequence of real numbers to be defined later. Fig. 2 shows a plot of the kernel for , 5 3 and c, 5 , 0.1 . The plot depicts values of the new kernel w i w h , in which h and i vary such that hd 1 i 5 0, 1, . . . , 3d 1 r. Noting that the shape of this kernel is in more harmony with the seasonal patterns of the ACFs than the other kernels, we expect better performance of the proposed estimator in typical seasonal models. The following two theorems show nonnegativeness and consistency of the estimator based on the proposed kernel. 2 Theorem 1. If c, 5 1, then sˆ $ 0. p
2 21 / 2 ) as n → `. Then sˆ 2 → s 2 , Theorem 2. Suppose , → `, , /n → 0, c, → `, and ( bˆ 2 b ) 5 Op (n p where → denotes convergence in probability.
The above two theorems suggest that we choose c, which is close to 1 in a small sample to achieve nonnegativeness and tends to ` as n → ` to achieve consistency. By choosing c, 5 , t for t close to 0, say 0.1, we can achieve nonnegativeness in nearly all finite samples as well as consistency in large samples. If n is small then c, is close to one and hence nonnegativeness is guaranteed for nearly all 2 samples, and if n is not small then consistency would render sˆ to be nonnegative.
168
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
Fig. 2. The proposed and the three popular kernels for l 5 3d 1 r with d 5 12.
3. A Monte-Carlo study We compare finite sample performance of the proposed kernel and other existing kernels for estimating s 2 . Performance of the proposed kernel relative to Bartlett kernel is first investigated 2 2 through comparing the two estimators, s¯ and sˆ . To match the bandwidths for the two estimators, 2 , d 1r 2 2 we consider s¯ 5 o h 52 , d 2r w h, , d 1r gˆ h instead of (1). Note that the bandwidths for this s¯ and sˆ are , d 1 r and ,, respectively. The two values correspond to the same bandwidth in that the two 2 2 estimators s¯ and sˆ use the same set of sample autocovariances h gˆ h , h 5 0, 1, . . . , , d 1 rj. Thus, comparison of the two estimators is meaningful in that we are comparing the proposed kernel and Bartlett kernel using the same bandwidth. We consider y t 5 z t with typical multiplicative seasonal AR and MA processes given by (1 1 f B)(1 1 F B 12 )z t 5 a t and z t 5 (1 1 u B)(1 1 Q B 12 )a t , respectively, which are frequently encountered in modeling monthly time series data. The parameter combinations are set so that f, F, u, and Q takes values in h0, 60.2, 60.4, 60.6, 60.8j; n 5 120, 480; , 5 1, 3, 5. For each parameter combination, 10,000 sets of hz t , t 5 1, . . . , nj are generated using standard normal errors a t generated
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
169
2 from RNNOA, a FORTRAN subroutine in IMSL (1989). In constructing our estimator sˆ , c, is 0.1 2 chosen to be , , which yields positive sˆ for all the samples from all the parameter combinations considered here. Finite sample mean square error (MSE) of the two estimators are compared by looking into the 2 2 2 2 2 2 2 relative efficiency of sˆ over s¯ given by Eff( sˆ , s¯ ) 5 MSE( s¯ ) / MSE( sˆ ), where MSE( s¯ ) and 2 2 2 MSE( sˆ ) are finite sample MSEs of s¯ and sˆ , respectively, computed from 10,000 random generations. Tables 1 and 2 present the square roots of finite sample efficiency values for the multiplicative AR and MA models, respectively. In Table 1 for the AR model, one can see that the efficiency values are
Table 1 2 2 12 Square roots of finite sample relative efficiencies of sˆ over s¯ for model y t 5 z t , (1 1 f B)(1 1 F B )z t 5 a t , where a t is a sequence of i.i.d. N(0, 1) errors and variance estimators are constructed using Bartlett kernel. The number of replications is 10,000
f
n 5 120
F
20.8 20.6 20.4 20.2 0
0.2
0.4
0.6
0.8
20.8 20.6 20.4 20.2 0
0.2
0.4
0.6
0.8
, 51 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.96 0.92 0.88 0.84 0.82 0.92 0.89 1.48 1.83
0.98 0.97 0.95 0.95 1.02 1.28 1.78 2.42 3.22
1.00 1.01 1.02 1.05 1.20 1.53 1.87 2.22 3.04
1.02 1.03 1.07 1.15 1.32 1.56 1.74 1.99 2.80
1.03 1.06 1.12 1.23 1.40 1.53 1.58 1.82 2.56
1.04 1.09 1.18 1.33 1.45 1.42 1.47 1.69 2.40
1.06 1.13 1.26 1.45 1.41 1.26 1.32 1.53 2.22
1.08 1.18 1.39 1.58 1.22 1.07 1.17 1.40 2.04
1.08 1.22 1.60 1.49 0.98 0.88 0.99 1.30 2.03
0.90 0.85 0.77 0.67 0.66 0.59 0.74 0.61 0.52
0.94 0.91 0.84 0.79 0.81 1.09 1.56 1.68 0.48
0.98 0.97 0.95 0.95 1.12 1.63 2.08 2.32 0.53
1.01 1.01 1.03 1.10 1.37 1.66 1.83 2.31 0.58
1.03 1.06 1.12 1.27 1.50 1.46 1.58 2.13 0.61
1.06 1.11 1.22 1.48 1.46 1.25 1.37 1.89 0.63
1.09 1.17 1.38 1.74 1.20 1.03 1.14 1.59 0.63
1.14 1.28 1.66 1.80 0.88 0.84 0.97 1.28 0.55
1.15 1.29 1.55 1.29 0.90 0.94 1.06 1.28 0.64
, 53 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.93 0.89 1.06 0.96 1.19 1.08 1.05 1.28 1.38
0.98 0.99 1.05 1.16 1.24 1.38 1.54 1.71 1.79
1.01 1.06 1.15 1.27 1.39 1.52 1.62 1.65 1.74
1.04 1.10 1.23 1.35 1.47 1.55 1.58 1.53 1.63
1.06 1.14 1.27 1.41 1.50 1.53 1.48 1.42 1.54
1.09 1.18 1.34 1.47 1.51 1.48 1.40 1.33 1.46
1.11 1.23 1.40 1.50 1.47 1.41 1.27 1.21 1.35
1.14 1.28 1.44 1.44 1.35 1.23 1.12 1.10 1.25
0.98 0.85 0.89 1.04 0.96 1.04 1.21 1.06 1.12
0.80 0.77 0.73 0.79 0.90 0.85 0.85 1.24 1.52
0.92 0.90 0.92 1.00 1.08 1.25 1.49 1.78 1.16
1.01 1.01 1.09 1.21 1.36 1.51 1.67 1.56 1.00
1.07 1.11 1.23 1.36 1.50 1.57 1.56 1.34 0.94
1.12 1.19 1.33 1.47 1.52 1.52 1.42 1.19 0.90
1.17 1.28 1.45 1.53 1.50 1.41 1.26 1.08 0.87
1.25 1.40 1.58 1.52 1.38 1.22 1.08 0.96 0.84
1.29 1.47 1.52 1.35 1.16 1.04 0.95 0.89 0.83
0.69 0.79 0.68 0.66 0.72 0.81 1.09 1.17 1.14
, 55 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.95 0.92 1.13 1.08 1.06 1.04 1.07 1.25 1.54
1.00 1.07 1.16 1.27 1.34 1.45 1.52 1.63 1.61
1.04 1.14 1.27 1.36 1.46 1.53 1.60 1.61 1.56
1.08 1.19 1.32 1.43 1.51 1.55 1.56 1.53 1.51
1.10 1.22 1.36 1.46 1.51 1.53 1.52 1.49 1.43
1.12 1.27 1.40 1.49 1.51 1.50 1.44 1.35 1.36
1.16 1.34 1.45 1.49 1.48 1.43 1.37 1.27 1.27
1.16 1.29 1.36 1.35 1.34 1.27 1.19 1.13 1.15
0.71 0.74 0.72 0.71 0.59 0.82 0.83 0.88 1.03
0.95 0.92 0.83 1.01 1.00 1.04 1.02 1.18 1.27
1.00 1.00 1.09 1.17 1.25 1.33 1.45 1.58 1.40
1.10 1.14 1.25 1.36 1.42 1.51 1.59 1.64 1.52
1.19 1.24 1.35 1.44 1.50 1.54 1.56 1.54 1.52
1.25 1.33 1.43 1.48 1.54 1.52 1.49 1.42 1.47
1.33 1.40 1.49 1.51 1.50 1.46 1.38 1.30 1.36
1.39 1.47 1.50 1.46 1.39 1.32 1.23 1.13 1.26
1.33 1.40 1.34 1.28 1.20 1.13 1.06 1.02 1.18
0.54 0.46 0.57 0.60 0.69 0.49 0.68 0.72 0.73
n 5 480
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
170
Table 2 2 2 12 Square roots of finite sample relative efficiencies of sˆ over s¯ for model y t 5 zt , z t 5 (1 1 u B)(1 1 Q B )a t , where a t is a sequence of i.i.d. N(0, 1) errors and variance estimators are constructed using Bartlett kernel. The number of replications is 10,000
u
n 5 120
Q
20.8
20.6
20.4
20.2
0
0.2
0.4
0.6
0.8
20.8
20.6
20.4
20.2
0
0.2
0.4
0.6
0.8
, 51 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.70 0.65 0.56 0.43 0.37 0.28 0.22 0.21 0.20
0.87 0.83 0.73 0.60 0.48 0.47 0.55 0.63 0.66
1.07 1.04 0.99 0.94 1.07 1.40 1.59 1.62 1.67
1.19 1.19 1.21 1.28 1.42 1.43 1.39 1.38 1.37
1.26 1.28 1.33 1.46 1.42 1.29 1.26 1.24 1.24
1.29 1.32 1.40 1.55 1.34 1.23 1.20 1.19 1.18
1.31 1.35 1.43 1.57 1.32 1.20 1.17 1.16 1.16
1.32 1.36 1.45 1.59 1.30 1.19 1.16 1.15 1.15
1.30 1.31 1.41 1.71 1.25 1.20 1.18 1.14 1.15
0.67 0.59 0.48 0.36 0.26 0.19 0.16 0.13 0.14
0.86 0.79 0.66 0.48 0.33 0.27 0.29 0.32 0.35
1.03 0.99 0.89 0.74 0.71 1.06 1.48 1.73 1.87
1.14 1.12 1.09 1.09 1.36 1.65 1.63 1.60 1.59
1.20 1.20 1.22 1.34 1.52 1.38 1.32 1.29 1.28
1.23 1.24 1.30 1.50 1.43 1.24 1.19 1.18 1.18
1.24 1.27 1.34 1.56 1.37 1.18 1.14 1.14 1.13
1.25 1.28 1.37 1.62 1.35 1.15 1.13 1.12 1.11
1.26 1.28 1.34 1.70 1.33 1.15 1.12 1.12 1.08
, 53 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.58 0.52 0.40 0.28 0.21 0.17 0.19 0.15 0.15
0.80 0.73 0.62 0.56 0.58 0.62 0.66 0.68 0.70
0.97 0.95 0.94 1.07 1.27 1.36 1.43 1.45 1.47
1.07 1.09 1.21 1.42 1.51 1.49 1.48 1.47 1.46
1.13 1.19 1.34 1.54 1.50 1.44 1.42 1.40 1.39
1.16 1.23 1.43 1.56 1.49 1.42 1.37 1.36 1.35
1.18 1.25 1.44 1.56 1.46 1.39 1.36 1.34 1.33
1.19 1.26 1.47 1.58 1.46 1.38 1.34 1.33 1.32
1.21 1.29 1.47 1.61 1.55 1.51 1.21 1.38 1.40
0.61 0.47 0.31 0.21 0.13 0.12 0.09 0.08 0.08
0.79 0.66 0.49 0.39 0.37 0.39 0.41 0.42 0.42
0.94 0.87 0.78 0.82 0.96 1.10 1.18 1.22 1.23
1.03 1.02 1.06 1.27 1.45 1.52 1.54 1.56 1.55
1.08 1.11 1.25 1.48 1.52 1.50 1.46 1.45 1.45
1.12 1.16 1.34 1.56 1.50 1.45 1.40 1.39 1.39
1.13 1.19 1.40 1.58 1.49 1.40 1.37 1.35 1.33
1.13 1.20 1.43 1.58 1.47 1.39 1.35 1.32 1.32
1.12 1.23 1.49 1.50 1.38 1.30 1.37 1.34 1.31
, 55 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.57 0.49 0.39 0.24 0.20 0.13 0.14 0.15 0.15
0.76 0.69 0.62 0.59 0.63 0.68 0.72 0.72 0.75
0.95 0.94 1.01 1.14 1.28 1.37 1.38 1.39 1.41
1.05 1.11 1.28 1.46 1.51 1.49 1.50 1.48 1.49
1.12 1.20 1.42 1.55 1.51 1.49 1.48 1.46 1.45
1.15 1.25 1.47 1.54 1.52 1.47 1.46 1.44 1.44
1.16 1.27 1.49 1.56 1.50 1.45 1.43 1.41 1.42
1.17 1.28 1.52 1.56 1.48 1.46 1.41 1.42 1.42
1.20 1.31 1.49 1.58 1.63 1.49 1.27 1.47 1.43
0.56 0.44 0.27 0.18 0.11 0.10 0.11 0.11 0.11
0.76 0.61 0.47 0.42 0.44 0.46 0.48 0.49 0.49
0.91 0.84 0.84 0.96 1.08 1.15 1.19 1.22 1.22
1.01 1.02 1.17 1.38 1.45 1.50 1.51 1.51 1.51
1.06 1.13 1.35 1.52 1.53 1.52 1.51 1.49 1.49
1.09 1.19 1.45 1.56 1.52 1.49 1.47 1.47 1.45
1.10 1.22 1.49 1.56 1.51 1.47 1.45 1.44 1.44
1.11 1.23 1.50 1.55 1.50 1.45 1.44 1.44 1.42
1.12 1.31 1.58 1.54 1.46 1.36 1.36 1.31 1.55
n 5 480
greater than 1 in most cases. Specifically, when n 5 120 the efficiency values are greater than 1 except for a few cases of f 5 68. When n 5 480 the efficiency values are greater than 1 except for some cases of extreme values of f or F. In Table 2 for the MA model, the efficiency values are greater than 1 in most cases of u between 2 0.4 and 0.8. However, when u 5 2 0.8 the square roots of efficiency values are quite low, being between 0.08 and 0.70. We can now say that, for seasonal time series models, the proposed kernel provides better estimates of s 2 if the serial correlation is not large. We have conducted the same experiment using the kernels of Parzen and Tukey–Hanning and observed that the relative performances of the proposed kernel over these two kernels are similar to those in Tables 1 and 2. Details can be obtained upon request.
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
171
4. Proofs Proof of Theorem 1. Let n* 5 , d. Let Z* 5 (z 0 , . . . , z n*21 )9. Let G * 5 var(Z*) and Gˆ be n* 3 n* matrices whose (i, j) elements are gui 2j u and gˆ ui2j u , respectively. Let 1, 5 (1, . . . , 1)9 be the , 3 1 vector of ones. Let e, k 5 (0, . . . , 0, 1, 0, . . . , 0)9 be the , 3 1 column vector in which the kth element is one and all other elements are zero, k 5 1, . . . , d. Let e k 5 (e 9, k , e 9, k , . . . , e ,9 k )9 5 e, k ^ 1, , where ^ denotes the Kronecker product. We have, if uk 2 , u 5 i, e 9k G *e, 5 e 19 G *e i 11 and e 9k Gˆ e, 5 e 91 Gˆ e i 11 . 21 21 Observe that e 19 G *e i 11 5 , [gi 1 o ,h51 (1 2 uhu /, )(ghd2i 1 ghd 1i )] 5 , o ,h 52( , 21 ) (1 2 uhu /, )ghd 1i , i 5 , 21 , 21 21 0, . . . , d 2 1, because o h 51 (1 2 uhu /, )ghd2i 5 o h51 (1 2 uhu /, )g2hd 1i 5 o h52( , 21 ) (1 2 uhu /, )ghd 1i . 21 ˆ hd 1i . Let gˆ *i 5 e 19 Gˆ e i 11 , i 5 0, . . . , r 2 1. Then gˆ *i 5 , Similarly, e 91 Gˆ e i 5 o ,h52( , 21) (1 2 uhu /, ) g , 21 o h 52( , 21 ) (1 2 uhu /, ) gˆ hd 1i . Let Gˆ * be the (r 2 1) 3 (r 2 1) matrix whose (i, j)-element is gˆ *ui2j u . Note 21 r21 r21 r ˆ *i 5 , r o i52r that 1 9r Gˆ *1 r 5 o i51 o j51 gˆ *ui 2j u 5 r o ir21 (1 2 uiu /r) o h, 52( 52(r 21) (1 2 uiu /r) g , 21 ) (1 2 uhu / 1 2 2 ˆ , ) gˆ hd 1i 5 , rsˆ . Now nonnegativeness of sˆ follows from the fact that G * 5 (e 1 u . . . ue r )9Gˆ (e 1 u . . . ue r ) is nonnegative definite because Gˆ is nonnegative definite and (e 1 u . . . ue r ) is of full rank. h Proof of Theorem 2. We give a proof for the situation in which n 5 md is an integer multiple of d because proofs for other situations are obvious given the result for n 5 md. For each i [ h 2 r 1 , 21 m 21 2 r 1 1 1, . . . , rj and k [ h1, . . . , dj, note that m o s 50 zˆ sd 1k zˆ sd 1k 1hd 1i is the sample cross covariance function of hzˆ sd 1k and zˆ sd 1k 1hd 1i , s 5 0, . . . , m 2 1j, where zˆ t 5 0 for t , 1 or t . n. Since ( bˆ O 2 b ) 5 Op (n 21 / 2 ) and , → `, , 2 / n → 0, Theorem 1 of Andrews (1991) is applicable to yield p , 21 m21 ` o h 52 , w h , m o s50 zˆ sd 1k zˆ sd 1k 1hd 1i →o h 52` ghd 1i , for each i [ h 2 r 1 , 2 r 1 1 1, . . . , rj and k [ 21 n 21 d 21 h1, . . . , dj. Averaging for k 5 1, . . . , d and noting that gˆ hd 1i 5 n o t51 zˆ t zˆ t1hd 1i 5 d o k51 m p m 21 , ` o s50 zˆ sd 1k zˆ sd 1k1hd 1i , we get o h52 , w h , gˆ hd 1i →o h 52` ghd 1i , i 5 2 r 1 , . . . , r. Since c, → `, it p 2 r , r ` ` 2 follows that sˆ 5 o i52r 1 (1 2 uiu /(rc, )) o h52 , w h , gˆ hd 1i →o i52r 1 o h52` ghd 1i 5 o h52` gh 5 s . h Acknowledgements This work was supported by the MOST through national R&D program for women’s university (grant [ 00-B-WB-06-A-03).
References Andrews, D.W.K., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858. Gallant, A.R., White, H., 1988. A Unified Theory of Estimation and Inference For Nonlinear Dynamic Models. Basil Blackwell, New York. IMSL, 1989. User’s Manual. IMSL, Houston, TX. Newey, W.K., West, K.D., 1987. Simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708. Phillips, P.C.B., 1995. Fully modified least squares and vector autoregression. Econometrica 63, 1023–1079.
A new kernel for long-run variance estimates in seasonal time series models Dong Wan Shin*, Man-Suk Oh Department of Statistics, Ewha University, Seoul 120 -750, South Korea Received 24 July 2001; accepted 12 December 2001
Abstract A new kernel for estimating long-run variances of stationary seasonal time series is proposed. The proposed kernel has an oscillating pattern which is in harmony with that of the autocovariance functions of seasonal time series. A Monte-Carlo experiment shows that the estimator based on the proposed kernel outperforms estimators based on existing kernels such as the Bartlett kernel, Parzen kernel, and Tukey–Hanning kernel for two typical monthly time series processes with moderate autocorrelations. 2002 Elsevier Science B.V. All rights reserved. Keywords: Autocovariance function; Efficiency; Seasonality JEL classification: C13; C14; C22
1. Introduction Heteroskedasticity and autocorrelation consistent (HAC) estimators of the long-run variances of stationary processes are required in many areas of econometrics such as linear regressions with heteroskedastic and serially correlated errors, semiparametric unit root tests, and generalized method of moments estimations of nonlinear simultaneous equation models. Several methods for HAC estimation were developed by Newey and West (1987), Gallant and White (1988, pp. 97–103), Andrews (1991), Phillips (1995), and many others; see the references in Andrews (1991). However, neither of these methods is designed for seasonal data, though in practice many data sets in economics and business are observed on quarterly or monthly bases. Therefore, it would be useful to develop an estimator which works well for typical seasonal time series data. In this article we propose a new kernel and an estimator based on the new kernel which take account of * Corresponding author. Tel.: 182-2-3277-2614; fax: 182-2-3277-2307. E-mail addresses: [email protected] (D.W. Shin), [email protected] (M.-S. Oh). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00048-4
2002 Elsevier Science B.V. All rights reserved.
166
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
seasonal behaviors. The proposed estimator is nonnegative and consistent. In addition, a Monte-Carlo simulation shows that the estimator based on the proposed kernel performs better than estimators based on the existing kernels, for typical seasonal models with moderate autocorrelations. This article is organized as follows. Section 2 proposes the new kernel and establishes nonnegativeness and consistency of the long-run variance estimator based on the proposed kernel. Section 3 presents a Monte Carlo study for two monthly multiplicative autoregressive (AR) and moving average (MA) models.
2. A new kernel and its properties It is very common that sample autocovariance functions (ACFs) of many empirical d-seasonal time series z t have spikes around seasonal lags h 5 0, 6d, 62d, . . . , where d is a given integer denoting period of seasonality such as 4 or 12. This fact is illustrated in Fig. 1 which shows plots of the population ACFs for multiplicative monthly AR and MA models of (1 2 0.4B)(1 2 0.4B 12 )z t 5 a t and 12 z t 5 (1 1 0.4B)(1 1 0.4B )a t , respectively, where B is the back-shift operator such that Bz t 5 z t 21
Fig. 1. Autocovariance functions for the multiplicative AR model, (1 2 0.4B)(1 2 0.4B 12 )z t 5 a t , and the multiplicative MA model, z t 5 (1 1 0.4B)(1 1 0.4B 12 )a t .
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
167
and a t is a white noise. This indicates that sample autocovariances for seasonal lags tend to be more significant than those for other lags. This point would be more conspicuous for the monthly case (d 5 12) than quarterly case (d 5 4). We discuss kernels through estimation of the long-run variance s 2 5 o `h 52` gh of a stationary process z t , where gh 5 cov(z 0 , z h ). Consider a regression model y t 5 x t9 b 1 z t in which z t is a stationary zero mean process, hy t , x t , t 5 1, . . . , nj is the set of observations, and b is an unknown regression parameter. A widely-used estimator of s 2 is
O ,
2
s¯ 5
h 52 ,
w h , gˆ h ,
w h , 5 1 2 uhu /( , 1 1),
(1)
where gˆ h 5 n 21 o nt 5uh u11 zˆ t zˆ t 2uh u , zˆ t 5 y t 2 x t9 bˆ , bˆ is a consistent estimator of b, and , is an integer called bandwidth. The kernel w h , in (1) is the Bartlett kernel. Newey and West (1987) showed 2 nonnegativeness and consistency of the estimator s¯ . Other widely-used kernels are the Parzen kernel and the Tukey–Hanning kernel. See Andrews (1991) for details about these kernels. Fig. 2 displays these kernels with , 5 3d 1 r 5 42 for the monthly case of d 5 12 where r 5 [d / 2], the integer part of d / 2. From Fig. 2, one can see an undesirable fact that, for all the three existing kernels, the weights of the terms gˆ h for h around d, 2d, . . . are smaller than those for h around d 2 r, 2d 2 r, . . . , respectively. For example, the weight w, ,12 of gˆ 12 is smaller than the weight w, 6 of gˆ 6 which tends to be less significant than gˆ 12 . Figs. 1 and 2 lead us to consider a new kernel which distributes more weight on gˆ h for h around the seasonal lags d, 2d, . . . than for h around d 2 r, 2d 2 r, . . . , respectively. We consider a new kernel w i w h , defined in the following estimator
OwO ,
r
sˆ 2 5
i 52r 1
i
h52 ,
w h , gˆ hd 1i , w i 5 1 2 uiu /(r c, ),
(2)
where r 1 5 [(d 2 1) / 2] and c, is a sequence of real numbers to be defined later. Fig. 2 shows a plot of the kernel for , 5 3 and c, 5 , 0.1 . The plot depicts values of the new kernel w i w h , in which h and i vary such that hd 1 i 5 0, 1, . . . , 3d 1 r. Noting that the shape of this kernel is in more harmony with the seasonal patterns of the ACFs than the other kernels, we expect better performance of the proposed estimator in typical seasonal models. The following two theorems show nonnegativeness and consistency of the estimator based on the proposed kernel. 2 Theorem 1. If c, 5 1, then sˆ $ 0. p
2 21 / 2 ) as n → `. Then sˆ 2 → s 2 , Theorem 2. Suppose , → `, , /n → 0, c, → `, and ( bˆ 2 b ) 5 Op (n p where → denotes convergence in probability.
The above two theorems suggest that we choose c, which is close to 1 in a small sample to achieve nonnegativeness and tends to ` as n → ` to achieve consistency. By choosing c, 5 , t for t close to 0, say 0.1, we can achieve nonnegativeness in nearly all finite samples as well as consistency in large samples. If n is small then c, is close to one and hence nonnegativeness is guaranteed for nearly all 2 samples, and if n is not small then consistency would render sˆ to be nonnegative.
168
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
Fig. 2. The proposed and the three popular kernels for l 5 3d 1 r with d 5 12.
3. A Monte-Carlo study We compare finite sample performance of the proposed kernel and other existing kernels for estimating s 2 . Performance of the proposed kernel relative to Bartlett kernel is first investigated 2 2 through comparing the two estimators, s¯ and sˆ . To match the bandwidths for the two estimators, 2 , d 1r 2 2 we consider s¯ 5 o h 52 , d 2r w h, , d 1r gˆ h instead of (1). Note that the bandwidths for this s¯ and sˆ are , d 1 r and ,, respectively. The two values correspond to the same bandwidth in that the two 2 2 estimators s¯ and sˆ use the same set of sample autocovariances h gˆ h , h 5 0, 1, . . . , , d 1 rj. Thus, comparison of the two estimators is meaningful in that we are comparing the proposed kernel and Bartlett kernel using the same bandwidth. We consider y t 5 z t with typical multiplicative seasonal AR and MA processes given by (1 1 f B)(1 1 F B 12 )z t 5 a t and z t 5 (1 1 u B)(1 1 Q B 12 )a t , respectively, which are frequently encountered in modeling monthly time series data. The parameter combinations are set so that f, F, u, and Q takes values in h0, 60.2, 60.4, 60.6, 60.8j; n 5 120, 480; , 5 1, 3, 5. For each parameter combination, 10,000 sets of hz t , t 5 1, . . . , nj are generated using standard normal errors a t generated
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
169
2 from RNNOA, a FORTRAN subroutine in IMSL (1989). In constructing our estimator sˆ , c, is 0.1 2 chosen to be , , which yields positive sˆ for all the samples from all the parameter combinations considered here. Finite sample mean square error (MSE) of the two estimators are compared by looking into the 2 2 2 2 2 2 2 relative efficiency of sˆ over s¯ given by Eff( sˆ , s¯ ) 5 MSE( s¯ ) / MSE( sˆ ), where MSE( s¯ ) and 2 2 2 MSE( sˆ ) are finite sample MSEs of s¯ and sˆ , respectively, computed from 10,000 random generations. Tables 1 and 2 present the square roots of finite sample efficiency values for the multiplicative AR and MA models, respectively. In Table 1 for the AR model, one can see that the efficiency values are
Table 1 2 2 12 Square roots of finite sample relative efficiencies of sˆ over s¯ for model y t 5 z t , (1 1 f B)(1 1 F B )z t 5 a t , where a t is a sequence of i.i.d. N(0, 1) errors and variance estimators are constructed using Bartlett kernel. The number of replications is 10,000
f
n 5 120
F
20.8 20.6 20.4 20.2 0
0.2
0.4
0.6
0.8
20.8 20.6 20.4 20.2 0
0.2
0.4
0.6
0.8
, 51 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.96 0.92 0.88 0.84 0.82 0.92 0.89 1.48 1.83
0.98 0.97 0.95 0.95 1.02 1.28 1.78 2.42 3.22
1.00 1.01 1.02 1.05 1.20 1.53 1.87 2.22 3.04
1.02 1.03 1.07 1.15 1.32 1.56 1.74 1.99 2.80
1.03 1.06 1.12 1.23 1.40 1.53 1.58 1.82 2.56
1.04 1.09 1.18 1.33 1.45 1.42 1.47 1.69 2.40
1.06 1.13 1.26 1.45 1.41 1.26 1.32 1.53 2.22
1.08 1.18 1.39 1.58 1.22 1.07 1.17 1.40 2.04
1.08 1.22 1.60 1.49 0.98 0.88 0.99 1.30 2.03
0.90 0.85 0.77 0.67 0.66 0.59 0.74 0.61 0.52
0.94 0.91 0.84 0.79 0.81 1.09 1.56 1.68 0.48
0.98 0.97 0.95 0.95 1.12 1.63 2.08 2.32 0.53
1.01 1.01 1.03 1.10 1.37 1.66 1.83 2.31 0.58
1.03 1.06 1.12 1.27 1.50 1.46 1.58 2.13 0.61
1.06 1.11 1.22 1.48 1.46 1.25 1.37 1.89 0.63
1.09 1.17 1.38 1.74 1.20 1.03 1.14 1.59 0.63
1.14 1.28 1.66 1.80 0.88 0.84 0.97 1.28 0.55
1.15 1.29 1.55 1.29 0.90 0.94 1.06 1.28 0.64
, 53 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.93 0.89 1.06 0.96 1.19 1.08 1.05 1.28 1.38
0.98 0.99 1.05 1.16 1.24 1.38 1.54 1.71 1.79
1.01 1.06 1.15 1.27 1.39 1.52 1.62 1.65 1.74
1.04 1.10 1.23 1.35 1.47 1.55 1.58 1.53 1.63
1.06 1.14 1.27 1.41 1.50 1.53 1.48 1.42 1.54
1.09 1.18 1.34 1.47 1.51 1.48 1.40 1.33 1.46
1.11 1.23 1.40 1.50 1.47 1.41 1.27 1.21 1.35
1.14 1.28 1.44 1.44 1.35 1.23 1.12 1.10 1.25
0.98 0.85 0.89 1.04 0.96 1.04 1.21 1.06 1.12
0.80 0.77 0.73 0.79 0.90 0.85 0.85 1.24 1.52
0.92 0.90 0.92 1.00 1.08 1.25 1.49 1.78 1.16
1.01 1.01 1.09 1.21 1.36 1.51 1.67 1.56 1.00
1.07 1.11 1.23 1.36 1.50 1.57 1.56 1.34 0.94
1.12 1.19 1.33 1.47 1.52 1.52 1.42 1.19 0.90
1.17 1.28 1.45 1.53 1.50 1.41 1.26 1.08 0.87
1.25 1.40 1.58 1.52 1.38 1.22 1.08 0.96 0.84
1.29 1.47 1.52 1.35 1.16 1.04 0.95 0.89 0.83
0.69 0.79 0.68 0.66 0.72 0.81 1.09 1.17 1.14
, 55 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.95 0.92 1.13 1.08 1.06 1.04 1.07 1.25 1.54
1.00 1.07 1.16 1.27 1.34 1.45 1.52 1.63 1.61
1.04 1.14 1.27 1.36 1.46 1.53 1.60 1.61 1.56
1.08 1.19 1.32 1.43 1.51 1.55 1.56 1.53 1.51
1.10 1.22 1.36 1.46 1.51 1.53 1.52 1.49 1.43
1.12 1.27 1.40 1.49 1.51 1.50 1.44 1.35 1.36
1.16 1.34 1.45 1.49 1.48 1.43 1.37 1.27 1.27
1.16 1.29 1.36 1.35 1.34 1.27 1.19 1.13 1.15
0.71 0.74 0.72 0.71 0.59 0.82 0.83 0.88 1.03
0.95 0.92 0.83 1.01 1.00 1.04 1.02 1.18 1.27
1.00 1.00 1.09 1.17 1.25 1.33 1.45 1.58 1.40
1.10 1.14 1.25 1.36 1.42 1.51 1.59 1.64 1.52
1.19 1.24 1.35 1.44 1.50 1.54 1.56 1.54 1.52
1.25 1.33 1.43 1.48 1.54 1.52 1.49 1.42 1.47
1.33 1.40 1.49 1.51 1.50 1.46 1.38 1.30 1.36
1.39 1.47 1.50 1.46 1.39 1.32 1.23 1.13 1.26
1.33 1.40 1.34 1.28 1.20 1.13 1.06 1.02 1.18
0.54 0.46 0.57 0.60 0.69 0.49 0.68 0.72 0.73
n 5 480
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
170
Table 2 2 2 12 Square roots of finite sample relative efficiencies of sˆ over s¯ for model y t 5 zt , z t 5 (1 1 u B)(1 1 Q B )a t , where a t is a sequence of i.i.d. N(0, 1) errors and variance estimators are constructed using Bartlett kernel. The number of replications is 10,000
u
n 5 120
Q
20.8
20.6
20.4
20.2
0
0.2
0.4
0.6
0.8
20.8
20.6
20.4
20.2
0
0.2
0.4
0.6
0.8
, 51 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.70 0.65 0.56 0.43 0.37 0.28 0.22 0.21 0.20
0.87 0.83 0.73 0.60 0.48 0.47 0.55 0.63 0.66
1.07 1.04 0.99 0.94 1.07 1.40 1.59 1.62 1.67
1.19 1.19 1.21 1.28 1.42 1.43 1.39 1.38 1.37
1.26 1.28 1.33 1.46 1.42 1.29 1.26 1.24 1.24
1.29 1.32 1.40 1.55 1.34 1.23 1.20 1.19 1.18
1.31 1.35 1.43 1.57 1.32 1.20 1.17 1.16 1.16
1.32 1.36 1.45 1.59 1.30 1.19 1.16 1.15 1.15
1.30 1.31 1.41 1.71 1.25 1.20 1.18 1.14 1.15
0.67 0.59 0.48 0.36 0.26 0.19 0.16 0.13 0.14
0.86 0.79 0.66 0.48 0.33 0.27 0.29 0.32 0.35
1.03 0.99 0.89 0.74 0.71 1.06 1.48 1.73 1.87
1.14 1.12 1.09 1.09 1.36 1.65 1.63 1.60 1.59
1.20 1.20 1.22 1.34 1.52 1.38 1.32 1.29 1.28
1.23 1.24 1.30 1.50 1.43 1.24 1.19 1.18 1.18
1.24 1.27 1.34 1.56 1.37 1.18 1.14 1.14 1.13
1.25 1.28 1.37 1.62 1.35 1.15 1.13 1.12 1.11
1.26 1.28 1.34 1.70 1.33 1.15 1.12 1.12 1.08
, 53 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.58 0.52 0.40 0.28 0.21 0.17 0.19 0.15 0.15
0.80 0.73 0.62 0.56 0.58 0.62 0.66 0.68 0.70
0.97 0.95 0.94 1.07 1.27 1.36 1.43 1.45 1.47
1.07 1.09 1.21 1.42 1.51 1.49 1.48 1.47 1.46
1.13 1.19 1.34 1.54 1.50 1.44 1.42 1.40 1.39
1.16 1.23 1.43 1.56 1.49 1.42 1.37 1.36 1.35
1.18 1.25 1.44 1.56 1.46 1.39 1.36 1.34 1.33
1.19 1.26 1.47 1.58 1.46 1.38 1.34 1.33 1.32
1.21 1.29 1.47 1.61 1.55 1.51 1.21 1.38 1.40
0.61 0.47 0.31 0.21 0.13 0.12 0.09 0.08 0.08
0.79 0.66 0.49 0.39 0.37 0.39 0.41 0.42 0.42
0.94 0.87 0.78 0.82 0.96 1.10 1.18 1.22 1.23
1.03 1.02 1.06 1.27 1.45 1.52 1.54 1.56 1.55
1.08 1.11 1.25 1.48 1.52 1.50 1.46 1.45 1.45
1.12 1.16 1.34 1.56 1.50 1.45 1.40 1.39 1.39
1.13 1.19 1.40 1.58 1.49 1.40 1.37 1.35 1.33
1.13 1.20 1.43 1.58 1.47 1.39 1.35 1.32 1.32
1.12 1.23 1.49 1.50 1.38 1.30 1.37 1.34 1.31
, 55 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8
0.57 0.49 0.39 0.24 0.20 0.13 0.14 0.15 0.15
0.76 0.69 0.62 0.59 0.63 0.68 0.72 0.72 0.75
0.95 0.94 1.01 1.14 1.28 1.37 1.38 1.39 1.41
1.05 1.11 1.28 1.46 1.51 1.49 1.50 1.48 1.49
1.12 1.20 1.42 1.55 1.51 1.49 1.48 1.46 1.45
1.15 1.25 1.47 1.54 1.52 1.47 1.46 1.44 1.44
1.16 1.27 1.49 1.56 1.50 1.45 1.43 1.41 1.42
1.17 1.28 1.52 1.56 1.48 1.46 1.41 1.42 1.42
1.20 1.31 1.49 1.58 1.63 1.49 1.27 1.47 1.43
0.56 0.44 0.27 0.18 0.11 0.10 0.11 0.11 0.11
0.76 0.61 0.47 0.42 0.44 0.46 0.48 0.49 0.49
0.91 0.84 0.84 0.96 1.08 1.15 1.19 1.22 1.22
1.01 1.02 1.17 1.38 1.45 1.50 1.51 1.51 1.51
1.06 1.13 1.35 1.52 1.53 1.52 1.51 1.49 1.49
1.09 1.19 1.45 1.56 1.52 1.49 1.47 1.47 1.45
1.10 1.22 1.49 1.56 1.51 1.47 1.45 1.44 1.44
1.11 1.23 1.50 1.55 1.50 1.45 1.44 1.44 1.42
1.12 1.31 1.58 1.54 1.46 1.36 1.36 1.31 1.55
n 5 480
greater than 1 in most cases. Specifically, when n 5 120 the efficiency values are greater than 1 except for a few cases of f 5 68. When n 5 480 the efficiency values are greater than 1 except for some cases of extreme values of f or F. In Table 2 for the MA model, the efficiency values are greater than 1 in most cases of u between 2 0.4 and 0.8. However, when u 5 2 0.8 the square roots of efficiency values are quite low, being between 0.08 and 0.70. We can now say that, for seasonal time series models, the proposed kernel provides better estimates of s 2 if the serial correlation is not large. We have conducted the same experiment using the kernels of Parzen and Tukey–Hanning and observed that the relative performances of the proposed kernel over these two kernels are similar to those in Tables 1 and 2. Details can be obtained upon request.
D.W. Shin, M.-S. Oh / Economics Letters 76 (2002) 165 – 171
171
4. Proofs Proof of Theorem 1. Let n* 5 , d. Let Z* 5 (z 0 , . . . , z n*21 )9. Let G * 5 var(Z*) and Gˆ be n* 3 n* matrices whose (i, j) elements are gui 2j u and gˆ ui2j u , respectively. Let 1, 5 (1, . . . , 1)9 be the , 3 1 vector of ones. Let e, k 5 (0, . . . , 0, 1, 0, . . . , 0)9 be the , 3 1 column vector in which the kth element is one and all other elements are zero, k 5 1, . . . , d. Let e k 5 (e 9, k , e 9, k , . . . , e ,9 k )9 5 e, k ^ 1, , where ^ denotes the Kronecker product. We have, if uk 2 , u 5 i, e 9k G *e, 5 e 19 G *e i 11 and e 9k Gˆ e, 5 e 91 Gˆ e i 11 . 21 21 Observe that e 19 G *e i 11 5 , [gi 1 o ,h51 (1 2 uhu /, )(ghd2i 1 ghd 1i )] 5 , o ,h 52( , 21 ) (1 2 uhu /, )ghd 1i , i 5 , 21 , 21 21 0, . . . , d 2 1, because o h 51 (1 2 uhu /, )ghd2i 5 o h51 (1 2 uhu /, )g2hd 1i 5 o h52( , 21 ) (1 2 uhu /, )ghd 1i . 21 ˆ hd 1i . Let gˆ *i 5 e 19 Gˆ e i 11 , i 5 0, . . . , r 2 1. Then gˆ *i 5 , Similarly, e 91 Gˆ e i 5 o ,h52( , 21) (1 2 uhu /, ) g , 21 o h 52( , 21 ) (1 2 uhu /, ) gˆ hd 1i . Let Gˆ * be the (r 2 1) 3 (r 2 1) matrix whose (i, j)-element is gˆ *ui2j u . Note 21 r21 r21 r ˆ *i 5 , r o i52r that 1 9r Gˆ *1 r 5 o i51 o j51 gˆ *ui 2j u 5 r o ir21 (1 2 uiu /r) o h, 52( 52(r 21) (1 2 uiu /r) g , 21 ) (1 2 uhu / 1 2 2 ˆ , ) gˆ hd 1i 5 , rsˆ . Now nonnegativeness of sˆ follows from the fact that G * 5 (e 1 u . . . ue r )9Gˆ (e 1 u . . . ue r ) is nonnegative definite because Gˆ is nonnegative definite and (e 1 u . . . ue r ) is of full rank. h Proof of Theorem 2. We give a proof for the situation in which n 5 md is an integer multiple of d because proofs for other situations are obvious given the result for n 5 md. For each i [ h 2 r 1 , 21 m 21 2 r 1 1 1, . . . , rj and k [ h1, . . . , dj, note that m o s 50 zˆ sd 1k zˆ sd 1k 1hd 1i is the sample cross covariance function of hzˆ sd 1k and zˆ sd 1k 1hd 1i , s 5 0, . . . , m 2 1j, where zˆ t 5 0 for t , 1 or t . n. Since ( bˆ O 2 b ) 5 Op (n 21 / 2 ) and , → `, , 2 / n → 0, Theorem 1 of Andrews (1991) is applicable to yield p , 21 m21 ` o h 52 , w h , m o s50 zˆ sd 1k zˆ sd 1k 1hd 1i →o h 52` ghd 1i , for each i [ h 2 r 1 , 2 r 1 1 1, . . . , rj and k [ 21 n 21 d 21 h1, . . . , dj. Averaging for k 5 1, . . . , d and noting that gˆ hd 1i 5 n o t51 zˆ t zˆ t1hd 1i 5 d o k51 m p m 21 , ` o s50 zˆ sd 1k zˆ sd 1k1hd 1i , we get o h52 , w h , gˆ hd 1i →o h 52` ghd 1i , i 5 2 r 1 , . . . , r. Since c, → `, it p 2 r , r ` ` 2 follows that sˆ 5 o i52r 1 (1 2 uiu /(rc, )) o h52 , w h , gˆ hd 1i →o i52r 1 o h52` ghd 1i 5 o h52` gh 5 s . h Acknowledgements This work was supported by the MOST through national R&D program for women’s university (grant [ 00-B-WB-06-A-03).
References Andrews, D.W.K., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858. Gallant, A.R., White, H., 1988. A Unified Theory of Estimation and Inference For Nonlinear Dynamic Models. Basil Blackwell, New York. IMSL, 1989. User’s Manual. IMSL, Houston, TX. Newey, W.K., West, K.D., 1987. Simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708. Phillips, P.C.B., 1995. Fully modified least squares and vector autoregression. Econometrica 63, 1023–1079.