07350015%2E2014%2E907059

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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:46

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Comment

Shiqing Ling & Ke Zhu

To cite this article: Shiqing Ling & Ke Zhu (2014) Comment, Journal of Business & Economic Statistics, 32:2, 202-203, DOI: 10.1080/07350015.2014.907059

To link to this article: http://dx.doi.org/10.1080/07350015.2014.907059

Published online: 16 May 2014.

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202 Journal of Business & Economic Statistics, April 2014

Comment

Shiqing L

ING

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China ([email protected])

Ke Z

HU

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, HaiDian District, Beijing, China ([email protected])

1. DISCUSSIONS

Congratulation to authors because of their interesting es-timation approach. In this discussion, we compare the fi-nite performance of the non-Gaussian quasi maximum likeli-hood estimator (NGQMLE) with that of the Laplacian QMLE (LQMLE). Suppose that the data sample {yt}nt=1 is

gener-distribution, the log-likelihood function (ignoring some con-stants) can be written as

Ln( ˜θ)=

where ˜vt( ˜θ) satisfies the following iteration:

˜

Then, the LQMLE is defined as

˜

θn=arg min

˜

θ

Ln( ˜θ),

see, for example, Berkes and Horv´ath (2004) and Zhu and Ling (2011). Unlike the NGQMLE, the LQMLE requires that E_|ε˜t| =1 for its identification, and only needs the finite

sec-ond moment of ˜εt for its asymptotically normal distribution.

In view of the relationship between θ and ˜θ, the LQMLE of

θisθn=(σn, a1n, . . . , apn, b1n, . . . , bqn), whereσn=√w˜n/r,

ain=α˜in/ω˜n, andbj n=β˜j n.

We generate 1000 replications of sample size n₌500 and 1000 from models (1.1)–(1.2) with the true parameter (σ, a1, b1)=(0.5,0.6,0.3), where the innovationsεtare chosen

as Student’stand generalized Gaussian distributions such that Eεt =0 and var(εt)=1.Table 1reports the sample bias and

root mean square error (RMSE) of each estimator. To make our comparison feasible, we use the true value ofrin all calcula-tions. FromTable 1, we find that the LQMLE is more efficient than the NGQMLE for the cases thatεt ∼gg1 andgg0.8. This is because the LQMLE is an efficient estimator whenεt ∼gg1. For the remaining cases, the NGQMLE is more efficient than the LQMLE due to the adaption property of the NGQMLE. But the difference seems not to be very large except for very few cases.

ACKNOWLEDGMENT

The authors thank the funding support in part from Hong Kong RGC Grants (numbered HKUST641912 and 603413) and National Natural Science Foundation of China (No. 11201459).

REFERENCES

Berkes, I., and Horv´ath, L. (2004), “The Efficiency of the Estimators of the Parameters in GARCH Processes,”The Annals of Statistics, 32, 633–655. [202]

Zhu, K., and Ling, S. (2011), “Global Self-Weighted and Local Quasi-Maximum Exponential Likelihood Estimators for ARMA-GARCH/IGARCH Models,”

The Annals of Statistics, 39, 2131–2163. [202]

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Ling and Zhu: Comment 203

Table 1. Estimation results for LQMLE and NGQMLE

LQMLE NGQMLE

εt n σn a1n b1n σˆn aˆ1n bˆ1n

t20 500 Bias 0.0051 0.0402 −0.0324 0.0031 0.0348 −0.0305

RMSE 0.1016 0.3573 0.2290 0.1008 0.3472 0.2283 1000 Bias 0.0039 0.0187 ₋0.0223 0.0029 0.0162 ₋0.0209 RMSE 0.0826 0.2495 0.1932 0.0822 0.2432 0.1924

t9 500 Bias 0.0082 0.0358 −0.0418 0.0042 0.0303 −0.0354

RMSE 0.1017 0.3578 0.2294 0.1010 0.3499 0.2310 1000 Bias 0.0080 0.0157 ₋0.0315 0.0068 0.0119 ₋0.0306 RMSE 0.0782 0.2441 0.1832 0.0781 0.2397 0.1832

t6 500 Bias −0.0028 0.0500 −0.0149 −0.0071 0.0423 −0.0088

RMSE 0.1035 0.3685 0.2350 0.1049 0.3607 0.2385 1000 Bias 0.0031 0.0080 ₋0.0186 0.0023 0.0055 ₋0.0186 RMSE 0.0816 0.2501 0.1904 0.0816 0.2461 0.1906

t4 500 Bias −0.0018 0.0621 −0.0217 −0.0062 0.0532 −0.0179

RMSE 0.1064 0.4036 0.2416 0.1062 0.3794 0.2398 1000 Bias 0.0024 0.0352 ₋0.0211 ₋0.0033 0.0314 ₋0.0137 RMSE 0.0789 0.2706 0.1879 0.0789 0.2541 0.1874

t3 500 Bias −0.0042 0.1031 −0.0194 −0.0178 0.0808 −0.0058

RMSE 0.1037 0.5075 0.2414 0.1025 0.4087 0.2359 1000 Bias ₋0.0003 0.0507 ₋0.0166 ₋0.0063 0.0405 ₋0.0126 RMSE 0.0809 0.3433 0.1968 0.0806 0.2908 0.1900

gg4 500 Bias 0.0075 0.0207 −0.0351 0.0075 0.0099 −0.0341

RMSE 0.1030 0.3509 0.2318 0.0999 0.3194 0.2266 1000 Bias 0.0068 0.0251 ₋0.0315 0.0076 0.0169 ₋0.0322 RMSE 0.0832 0.2384 0.1918 0.0792 0.2204 0.1840

gg2 500 Bias 0.0013 0.0461 −0.0255 −0.0010 0.0408 −0.0209

RMSE 0.1027 0.3540 0.2323 0.1025 0.3425 0.2328 1000 Bias 0.0026 0.0393 ₋0.0223 0.0021 0.0350 ₋0.0218 RMSE 0.0825 0.2536 0.1878 0.0805 0.2428 0.1848

gg1 500 Bias −0.0024 0.0446 −0.0160 −0.0092 0.0489 −0.0071

RMSE 0.1077 0.4059 0.2429 0.1111 0.4150 0.2507 1000 Bias 0.0002 0.0240 ₋0.0165 ₋0.0017 0.0237 ₋0.0152 RMSE 0.0900 0.2649 0.2065 0.0917 0.2686 0.2098

gg0.8 500 Bias −0.0072 0.0819 −0.0117 −0.0153 0.0900 −0.0020

RMSE 0.1111 0.4370 0.2521 0.1136 0.4533 0.2564 1000 Bias ₋0.0021 0.0405 ₋0.0116 ₋0.0089 0.0456 ₋0.0006 RMSE 0.0886 0.2929 0.02047 0.0914 0.2983 0.2105

gg0.4 500 Bias −0.0106 0.2216 −0.0219 −0.0378 0.2403 0.0103

RMSE 0.1274 0.8502 0.2827 0.1276 0.7976 0.2821 1000 Bias ₋0.0035 0.1032 ₋0.0195 ₋0.0214 0.1272 0.0036 RMSE 0.1011 0.4935 0.2361 0.1021 0.4581 0.2358