07350015%2E2014%2E907059
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 LING
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong, China (maling@ust.hk)
Ke ZHU
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, HaiDian District, Beijing, China
(mazkxaa@gmail.com)
1.
DISCUSSIONS
Congratulation to authors because of their interesting estimation approach. In this discussion, we compare the finite performance of the non-Gaussian quasi maximum likelihood estimator (NGQMLE) with that of the Laplacian QMLE
(LQMLE). Suppose that the data sample {yt }nt=1 is generated from their GARCH(p, q) model with θ = (σ, γ ′ )′ , where
γ = (a1 , . . . , ap , b1 , . . . , bq ). To introduce the LQMLE, we first
reparameterize their models as
xt = v˜t ε˜ t ,
v˜t2 = ω +
(1.1)
p
i=1
2
+
αi xt−i
q
2
,
βj v˜t−j
(1.2)
j =1
where ε˜ t = εt /r, v˜t = (rσ )vt , ω = r 2 σ 2 , αi = ai (rσ )2 ,
βj = bj , and r = E|εt |. Furthermore, we denote θ˜ =
(ω, α1 , . . . , αp , β1 , . . . , βq ) be the unknown parameter of models (1.1)–(1.2). By assuming that ε˜ t follows the standard Laplace
distribution, the log-likelihood function (ignoring some constants) can be written as
n
|xt |
1
˜
˜
,
log v˜t (θ) +
Ln (θ) =
n t=1
v˜t (θ˜ )
where v˜t (θ˜ ) satisfies the following iteration:
˜
v˜t2 (θ)
=ω+
p
2
αi xt−i
i=1
+
q
2
˜
βj v˜t−j
(θ).
j =1
Then, the LQMLE is defined as
√
θ is θn = (σn , a1n , . . . , apn , b1n , . . . , bqn ), where σn = w˜ n /r,
ain = α˜ in /ω˜ n , and bj 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 innovations εt are chosen
as Student’s t and generalized Gaussian distributions such that
Eεt = 0 and var(εt ) = 1. Table 1 reports the sample bias and
root mean square error (RMSE) of each estimator. To make our
comparison feasible, we use the true value of r in all calculations. From Table 1, we find that the LQMLE is more efficient
than the NGQMLE for the cases that εt ∼ gg1 and gg0.8 . This
is because the LQMLE is an efficient estimator when εt ∼ 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]
θ˜n = arg min Ln (θ˜ ),
θ˜
see, for example, Berkes and Horv´ath (2004) and Zhu and
Ling (2011). Unlike the NGQMLE, the LQMLE requires that
E|˜εt | = 1 for its identification, and only needs the finite second moment of ε˜ t for its asymptotically normal distribution.
In view of the relationship between θ and θ˜ , the LQMLE of
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.907059
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:46 11 January 2016
Ling and Zhu: Comment
203
Table 1. Estimation results for LQMLE and NGQMLE
LQMLE
εt
n
t20
500
1000
t9
500
1000
t6
500
1000
t4
500
1000
t3
500
1000
gg4
500
1000
gg2
500
1000
gg1
500
1000
gg0.8
500
1000
gg0.4
500
1000
σn
NGQMLE
a1n
b1n
σˆ n
aˆ 1n
bˆ1n
Bias
RMSE
Bias
RMSE
0.0051
0.1016
0.0039
0.0826
0.0402
0.3573
0.0187
0.2495
−0.0324
0.2290
−0.0223
0.1932
0.0031
0.1008
0.0029
0.0822
0.0348
0.3472
0.0162
0.2432
−0.0305
0.2283
−0.0209
0.1924
Bias
RMSE
Bias
RMSE
0.0082
0.1017
0.0080
0.0782
0.0358
0.3578
0.0157
0.2441
−0.0418
0.2294
−0.0315
0.1832
0.0042
0.1010
0.0068
0.0781
0.0303
0.3499
0.0119
0.2397
−0.0354
0.2310
−0.0306
0.1832
Bias
RMSE
Bias
RMSE
−0.0028
0.1035
0.0031
0.0816
0.0500
0.3685
0.0080
0.2501
−0.0149
0.2350
−0.0186
0.1904
−0.0071
0.1049
0.0023
0.0816
0.0423
0.3607
0.0055
0.2461
−0.0088
0.2385
−0.0186
0.1906
Bias
RMSE
Bias
RMSE
−0.0018
0.1064
0.0024
0.0789
0.0621
0.4036
0.0352
0.2706
−0.0217
0.2416
−0.0211
0.1879
−0.0062
0.1062
−0.0033
0.0789
0.0532
0.3794
0.0314
0.2541
−0.0179
0.2398
−0.0137
0.1874
Bias
RMSE
Bias
RMSE
−0.0042
0.1037
−0.0003
0.0809
0.1031
0.5075
0.0507
0.3433
−0.0194
0.2414
−0.0166
0.1968
−0.0178
0.1025
−0.0063
0.0806
0.0808
0.4087
0.0405
0.2908
−0.0058
0.2359
−0.0126
0.1900
Bias
RMSE
Bias
RMSE
0.0075
0.1030
0.0068
0.0832
0.0207
0.3509
0.0251
0.2384
−0.0351
0.2318
−0.0315
0.1918
0.0075
0.0999
0.0076
0.0792
0.0099
0.3194
0.0169
0.2204
−0.0341
0.2266
−0.0322
0.1840
Bias
RMSE
Bias
RMSE
0.0013
0.1027
0.0026
0.0825
0.0461
0.3540
0.0393
0.2536
−0.0255
0.2323
−0.0223
0.1878
−0.0010
0.1025
0.0021
0.0805
0.0408
0.3425
0.0350
0.2428
−0.0209
0.2328
−0.0218
0.1848
Bias
RMSE
Bias
RMSE
−0.0024
0.1077
0.0002
0.0900
0.0446
0.4059
0.0240
0.2649
−0.0160
0.2429
−0.0165
0.2065
−0.0092
0.1111
−0.0017
0.0917
0.0489
0.4150
0.0237
0.2686
−0.0071
0.2507
−0.0152
0.2098
Bias
RMSE
Bias
RMSE
−0.0072
0.1111
−0.0021
0.0886
0.0819
0.4370
0.0405
0.2929
−0.0117
0.2521
−0.0116
0.02047
−0.0153
0.1136
−0.0089
0.0914
0.0900
0.4533
0.0456
0.2983
−0.0020
0.2564
−0.0006
0.2105
Bias
RMSE
Bias
RMSE
−0.0106
0.1274
−0.0035
0.1011
0.2216
0.8502
0.1032
0.4935
−0.0219
0.2827
−0.0195
0.2361
−0.0378
0.1276
−0.0214
0.1021
0.2403
0.7976
0.1272
0.4581
0.0103
0.2821
0.0036
0.2358
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.
Submit your article to this journal
Article views: 67
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RA JA ALI HA JI
TANJUNGPINANG, KEPULAUAN RIAU]
Date: 11 January 2016, At: 20:46
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:46 11 January 2016
202
Journal of Business & Economic Statistics, April 2014
Comment
Shiqing LING
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong, China (maling@ust.hk)
Ke ZHU
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, HaiDian District, Beijing, China
(mazkxaa@gmail.com)
1.
DISCUSSIONS
Congratulation to authors because of their interesting estimation approach. In this discussion, we compare the finite performance of the non-Gaussian quasi maximum likelihood estimator (NGQMLE) with that of the Laplacian QMLE
(LQMLE). Suppose that the data sample {yt }nt=1 is generated from their GARCH(p, q) model with θ = (σ, γ ′ )′ , where
γ = (a1 , . . . , ap , b1 , . . . , bq ). To introduce the LQMLE, we first
reparameterize their models as
xt = v˜t ε˜ t ,
v˜t2 = ω +
(1.1)
p
i=1
2
+
αi xt−i
q
2
,
βj v˜t−j
(1.2)
j =1
where ε˜ t = εt /r, v˜t = (rσ )vt , ω = r 2 σ 2 , αi = ai (rσ )2 ,
βj = bj , and r = E|εt |. Furthermore, we denote θ˜ =
(ω, α1 , . . . , αp , β1 , . . . , βq ) be the unknown parameter of models (1.1)–(1.2). By assuming that ε˜ t follows the standard Laplace
distribution, the log-likelihood function (ignoring some constants) can be written as
n
|xt |
1
˜
˜
,
log v˜t (θ) +
Ln (θ) =
n t=1
v˜t (θ˜ )
where v˜t (θ˜ ) satisfies the following iteration:
˜
v˜t2 (θ)
=ω+
p
2
αi xt−i
i=1
+
q
2
˜
βj v˜t−j
(θ).
j =1
Then, the LQMLE is defined as
√
θ is θn = (σn , a1n , . . . , apn , b1n , . . . , bqn ), where σn = w˜ n /r,
ain = α˜ in /ω˜ n , and bj 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 innovations εt are chosen
as Student’s t and generalized Gaussian distributions such that
Eεt = 0 and var(εt ) = 1. Table 1 reports the sample bias and
root mean square error (RMSE) of each estimator. To make our
comparison feasible, we use the true value of r in all calculations. From Table 1, we find that the LQMLE is more efficient
than the NGQMLE for the cases that εt ∼ gg1 and gg0.8 . This
is because the LQMLE is an efficient estimator when εt ∼ 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]
θ˜n = arg min Ln (θ˜ ),
θ˜
see, for example, Berkes and Horv´ath (2004) and Zhu and
Ling (2011). Unlike the NGQMLE, the LQMLE requires that
E|˜εt | = 1 for its identification, and only needs the finite second moment of ε˜ t for its asymptotically normal distribution.
In view of the relationship between θ and θ˜ , the LQMLE of
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.907059
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:46 11 January 2016
Ling and Zhu: Comment
203
Table 1. Estimation results for LQMLE and NGQMLE
LQMLE
εt
n
t20
500
1000
t9
500
1000
t6
500
1000
t4
500
1000
t3
500
1000
gg4
500
1000
gg2
500
1000
gg1
500
1000
gg0.8
500
1000
gg0.4
500
1000
σn
NGQMLE
a1n
b1n
σˆ n
aˆ 1n
bˆ1n
Bias
RMSE
Bias
RMSE
0.0051
0.1016
0.0039
0.0826
0.0402
0.3573
0.0187
0.2495
−0.0324
0.2290
−0.0223
0.1932
0.0031
0.1008
0.0029
0.0822
0.0348
0.3472
0.0162
0.2432
−0.0305
0.2283
−0.0209
0.1924
Bias
RMSE
Bias
RMSE
0.0082
0.1017
0.0080
0.0782
0.0358
0.3578
0.0157
0.2441
−0.0418
0.2294
−0.0315
0.1832
0.0042
0.1010
0.0068
0.0781
0.0303
0.3499
0.0119
0.2397
−0.0354
0.2310
−0.0306
0.1832
Bias
RMSE
Bias
RMSE
−0.0028
0.1035
0.0031
0.0816
0.0500
0.3685
0.0080
0.2501
−0.0149
0.2350
−0.0186
0.1904
−0.0071
0.1049
0.0023
0.0816
0.0423
0.3607
0.0055
0.2461
−0.0088
0.2385
−0.0186
0.1906
Bias
RMSE
Bias
RMSE
−0.0018
0.1064
0.0024
0.0789
0.0621
0.4036
0.0352
0.2706
−0.0217
0.2416
−0.0211
0.1879
−0.0062
0.1062
−0.0033
0.0789
0.0532
0.3794
0.0314
0.2541
−0.0179
0.2398
−0.0137
0.1874
Bias
RMSE
Bias
RMSE
−0.0042
0.1037
−0.0003
0.0809
0.1031
0.5075
0.0507
0.3433
−0.0194
0.2414
−0.0166
0.1968
−0.0178
0.1025
−0.0063
0.0806
0.0808
0.4087
0.0405
0.2908
−0.0058
0.2359
−0.0126
0.1900
Bias
RMSE
Bias
RMSE
0.0075
0.1030
0.0068
0.0832
0.0207
0.3509
0.0251
0.2384
−0.0351
0.2318
−0.0315
0.1918
0.0075
0.0999
0.0076
0.0792
0.0099
0.3194
0.0169
0.2204
−0.0341
0.2266
−0.0322
0.1840
Bias
RMSE
Bias
RMSE
0.0013
0.1027
0.0026
0.0825
0.0461
0.3540
0.0393
0.2536
−0.0255
0.2323
−0.0223
0.1878
−0.0010
0.1025
0.0021
0.0805
0.0408
0.3425
0.0350
0.2428
−0.0209
0.2328
−0.0218
0.1848
Bias
RMSE
Bias
RMSE
−0.0024
0.1077
0.0002
0.0900
0.0446
0.4059
0.0240
0.2649
−0.0160
0.2429
−0.0165
0.2065
−0.0092
0.1111
−0.0017
0.0917
0.0489
0.4150
0.0237
0.2686
−0.0071
0.2507
−0.0152
0.2098
Bias
RMSE
Bias
RMSE
−0.0072
0.1111
−0.0021
0.0886
0.0819
0.4370
0.0405
0.2929
−0.0117
0.2521
−0.0116
0.02047
−0.0153
0.1136
−0.0089
0.0914
0.0900
0.4533
0.0456
0.2983
−0.0020
0.2564
−0.0006
0.2105
Bias
RMSE
Bias
RMSE
−0.0106
0.1274
−0.0035
0.1011
0.2216
0.8502
0.1032
0.4935
−0.0219
0.2827
−0.0195
0.2361
−0.0378
0.1276
−0.0214
0.1021
0.2403
0.7976
0.1272
0.4581
0.0103
0.2821
0.0036
0.2358