Note on Small Sample Propert the Two-step Estimators in the AR-GARCH Models iesof

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A

Note on Small Sample Propert

the Two-step Estimators in the AR-GARCH Models

Yoshihisa Baba

ies of

I Introduction

Many papers have estimated various ARCH type models since its introduction by Engle (1982). The typical model has the following structure:

Yt=xty+6t Et/Ot-i -- N(0, ht) where ht=w+cr~t_1~aht-1

y are the parameters in the mean equation and 0=(a), a, j9) are ones in the conditional variance. Under some regularities conditions, the maximum likelihood estimator (e ) asymptotically normally distributes. (1) In many empirical works, the following two-step estimator is actually employed. First, estimate y by OLS and obtain the residuals §t.

Next, obtain the MLE CO for GARCH model, using Engle (1982) mentions the

two-step estimator should be asymptotically equivalent to the MLE. However, the small sample properties of two estimators have never been addressed in the literature. It may be worthwhile to examine the above problem since most applied works adopt the two-step estimator.

The main purpose of the present note is to examine the small sample properties of the two-step estimators for GARCH model parameters through Monte Carlo experiments. The note also examines the small sample properties of OLS estimator for the parameters in the mean equation. The rest of the present note is as follows: Section II explains the setup of the Monte Carlo experiments. Section III reports the Monte Carlo results. Section IV concludes the note.

II The Monte Carlo Design

The simple first order autoregressive process is adopted for the mean equation for the present Monte Carlo experiment.

(1) See, for example, Hansen and Lee (1995) and Lumsdaine (1996).

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50All AtVol. XXIX, No. 1.2 Yt=C+yt-1+Et

For the conditional variance equation we adopt GARCH (1, 1) model with several parameters' values below, following Lumsdaine (1995) ;

GARCH (I) GARCH (II) GARCH (III)

a0.10.250.4

0.50.650.5

Among the above GARCH models, the GARCH (III) model violates the existence of a finite fourth moment, assuming the conditional normality (see Bollerslev (1986)) . The data are generated based upon the following equations:

yt =C+pyt-i+Etht' ht=co+ (aEt-i+a) ht-i

where et is drawn from a standard normal-number generator and the first hundred observations are discarded in order to prevent the possible initial value bias. Throughout the experiments, we fix the sample size to be 200 and both C and w to be one. Each case will be replicated 500 times.

To each set of parameters 0= { C, p, cv, a, a} Et and yt with 300 observations is generated and first 100 observations are discarded. Thus, remaining Et and yt with 200 observations are used to estimate the model parameters. Three estimation methods are used to study the small sample properties of the two-step estimator. The first estimator is the Maximum Likelihood Estimator (MLE) for 0 denoted by Full MLE. The second one denoted by Two-step MLE is obtained as follows: regressing yt on constant and yt-i, we obtain lest square residuals g.t. Next we maximize the likelihood function, using et. The third one denoted by MLE3 is the Maximum Likelihood Estimator (MLE) for { co, a, a }, using et. Berndt-Hall-Hall-Hauseman (BHHH) algorithm is used for all MLE calculations. (3)

III Results of the Monte Carlo Experiment

We present simulation studies for three distinct values for p. Table 1, Table 2 and Table 3 contain results of p=0.0, 0.5 and 0.8 respectively.

In the case of p-0.0, both Full MLE and Two-step MLE for C and p behave similarly.

They show small bias and Std. Dev. and R. M. E. are similar size. The apparent efficiency gain cannot be observed, comparing Full MLE and Two-step MLE. In GARCH (III) case

(2) Engle and Ng (1994) for two-step estimator.

(3) All computations in this notes are carried out by Eviews distributed by Quantitative Micro Software.

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which corresponds to non existence of a finite fourth moment, both estimates do not distribute normally according to Jarque-Bera statistics.

In the case of p=0.5 and 0.8, the results of both Full MLE and Two-step MLE for C and p are qualitatively similar to the case of p=0.0 except Jarque-Bera statistics which are significant in most cases. Bias and Std. Dev. of both estimators are similar. Those results give some justification to the use of OLS for the mean equation with ARCH type disturbance.

In order to assess performances of Full and Two-step MLE for w, a, and a, the results of MLE3 can be used as the benchmark. We can observe some pattern about the bias of MLE3: the upward bias for w is rather large, comparing to ones for a and /3. The bias for w is generally large in GARCH (II) process and Std. Dev. is small in GRACH (III) process which is very close to IGARCH process. Jarque-Bera statistics are always significant and so 200 observations are not enough to obtain a normal distribution.

In the case of p=0.0, both Full and Two-step MLE for w, a and 8 are very similar to MLE3 across three GARCH processes and noticeable differences among those three are not found out. In the case of p =0.5 and 0.8, we have similar results to the case of p =0.0.

Full and Two-step MLE are as good as MLE3, but those are not normally distributed in the sample of 200 observations. Some caution should be needed to use hypothesis tests by t-statistics.

IV Concluding Remarks

The present note examine small sample properties of Full and Two-step MLE for AR (1) -GRACH (1, 1) processes. Because of computational difficulties, Two-step MLE is used in many empirical works in despite of that its finite sample properties are unknown. With very limited experiments, Two-step MLE is as good as Full MLE in most cases considered

in the note. Thus, the results of this note give some support to use Two-step MLE in applied works. While many empirical papers employ ARCH or GARCH models to examine volatilities of time series data, we have only a few papers which address the asymptotic and finite sample properties of MLE in those models. Much more Monte Carlo experiments should be needed to assess various properties of MLE for GARCH processes.

References

[ 1 ] Bollerslev, T. (1986) "Generalized Autoregressive Conditional Heteroscedasticity", Journal of Econometrics, 51, 307-327.

[ 2 ] Engle, R. F. (1982) "Autoregressive Conditioinal Hereroscedasticity with Estimates of the Variance of the United Kingdom Inflation", Econometrica, 50, 987-1007.

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52 [3]

[4]

[5]

[6]

PRi1 Pi lift a 'A prVol. XXIX, No. 1.2 Engle, R. F. and V. K. Ng (1993) "Measuring and Testing the Impact of News on Volatility", Journal of Finance, 48, 1749-1778.

Lee, Sang-Won and Bruce E. Hansen (1994) "Asymptotic Theory for the GARCH (1,1) Quasi- Maximum Likelihood Estimator", Econometric Theory, 10, 29-52.

Lumsdaine, Robin L. (1995) "Finite-Sample Properties of the Maximum Likelihood Estimator in GARCH (1,1) and IGARCH (1,1) Models: A Monte Carlo Investigation", Journal of Business and Economic Statistic, 13, 1-10.

Lumsdaine, Robin L. (1996) "Consistency and Asymptotic Normality of the Quasi-Maximum Likelihood Estimator in IGARCH (1,1) and Convariance Stationary GARCH (1,1) Models", Econometrica, 64, 575-596.

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A. GAECH (I ) Full MLE

Table. 1 p = 0.0

C p ^a, Q

Mean 1.0063 --0 .0043 1.0561 0.0698 0.5020

Std. Dev. 0.1448 0.0762 1.0003 0.1137 0.4497

R. M. E. 0.1448 0.0763 1.0009 0.1176 0.4493

Skewness ^0.0568 ^{--0 .0951} ^1.5097 ^0.2446 ^--1 ^.0083

Kurtosis 2.8565 3.3768 5.1699 2.5187 3.5786

Jarque-Bera ^0.6974 ^3.7125 228.0306 9.8104 91.6943

Probability 0.7056 0.1563 0.0000 0.0074 0.0000

Two-step MLE

C _p co a Q

Mean 1.0043 --0 .0041 1.0401 ^0.0647 ^0.5141

Std. Dev. ^0.1410 ^0.0760 ^1.0198 ^0.1085 ^0.4531

R. M. E. 0.1409 0.0761 1.0196 0.1140 0.4528

Skewness 0.0581 --0 .1027 1.7003 0.2269 -1 .1568

Kurtosis 2.6908 3.1699 6.0192 2.4054 4.0442

Jarque-Bera ^2.2724 ^1.4812 ^430.8375 11.6567 134.2367

Probability ^0.3210 ^0.4768 0.0000 0.0029 0.0000

MLE 3

C p co a

Mean 1.0000 0.0000 1.1362 0.0738 0.4701

Std. Dev. 0.0000 0.0000 1.0941 0.1103 0.4745

R. M. E. n.a. n.a. 1.1015 0.1132 0.4750

Skewness n.a. n.a. 1.5649 0.1687 --1 .0318

Kurtosis n.a. n.a. 5.1511 2.4201 3.5397

Jarque-Bera ^n.a. ^n.a. ^300.4876 ^9.3764 ^94.7848

Probability n.a. n.a. 0.0000 0.0092 0.0000

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B. GAECH (II ) Full MLE

E1 'ldfi Alt Vol. XXIX, No. P2

C _p co a le

Mean 0.9973 --0 .0083 1.2420 0.2368 0.6297

Std. Dev. 0.2084 0.0777 0.9434 0.1039 0.1619

R. M. E. 0.2082 0.8120 0.9731 0.1047 0.1630

Skewness 0.0664 --0 .0352 2.7608 0.0743 -1 .3425

Kurtosis 2.7972 2.9105 14.9798 3.8260 7.3039

Jarque-Bera ^1.2238 ^0.2700 ^3625.0852 ^14.6751 ^536.1119

Probability 0.5423 0.8737 0.0000 0.0007 0.0000

Two-step MLE

C p ^w ^a

Mean 0.9888 --0 .0022 1.2621 0.2278 0.6339

Std. Dev. 0.2601 0.0982 0.9841 0.1006 0.1720

R M. E. 0.2601 0.8081 1.0174 0.1029 0.1726

Skewness --0 .0831 --0 .0088 2.9564 0.0519 -2 .1689

Kurtosis 3.2220 3.1674 16.1603 3.7696 14.5788

Jarque-Bera ^1.6021 ^0.5904 ^4336.5780 ^12.5637 ^3185.1175

Probability 0.4489 0.7444 0.0000 0.0019 0.0000

MLE 3

C p co a a

Mean 1.0000 0.0000 1.3242 0.2369 0.6187

Std. Dev. 0.0000 0.0000 1.2081 0.1029 0.1953

R. M. E. n.a. n.a. 1.2497 0.1036 0.1976

Skewness n.a. n.a. 4.6052 0.2121 —3 .1977

Kurtosis n.a. n.a. 36.6538 4.3016 22.7250

Jarque-Bera ^n.a. ^n.a. ^25362.6351 ^39.0421 ^8957.9066

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C. GAECH (III) Full MLE

C p U) a a

Mean 1.0026 --0 .0055 1.1542 0.3905 0.4788

Std. Dev. 0.1762 0.0826 0.6284 0.1278 0.1551

R. M. E. 0.1761 0.8097 0.6464 0.1280 0.1564

Skewness 0.3305 0.1466 2.0180 0.0301 --0 .8273

Kurtosis 3.4227 2.8674 9.6165 3.3070 5.8469

Jarque-Bera 12.8268 ^2.1570 1251.4149 ^2.0384 225.8837

Probability 0.0016 0.3401 0.0000 0.3609 0.0000

Two-step MLE

C p ^w ^a a

Mean 1.0213 --0 .0135 1.1639 0.3781 0.4900

Std. Dev. 0.2517 0.1166 0.6120 0.1239 0.1475

R. M. E. 0.2515 0.1165 0.6115 0.1238 0.1473

Skewness 0.1850 0.1308 1.9211 0.1047 --0 .7965

Kurtosis 3.6446 3.7389 9.5651 3.0781 5.5558

Jarque-Bera ^11.5090 ^12.8011 ^1205.4949 ^1.0401 ^188.9470

Probability 0.0032 0.0017 0.0000 0.5945 0.0000

MLE 3

C _to CD a Q

Mean 1.0000 0.0000 1.1909 0.3913 0.4736

Std. Dev. 0.0000 0.0000 0.6504 0.1252 0.1552

R. M. E. n.a. n.a. 0.6499 0.1250 0.1551

Skewness n.a. n.a. 2.0503 0.1360 --0 .8499

Kurtosis n.a. n.a. 10.0211 3.5016 5.8756

Jarque-Bera ^n.a. ^n.a. ^1377.2992 ^6.7821 ^232.4668

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Table. 2 p = 0.5

Vol. XXIX, No. 1.2

C p ⁰⁾ ^a a

Mean 1.0253 0.4900 1.0743 0.0675 0.4983

Std. Dev. 0.1766 0.0664 1.0954 0.1128 0.4787

R. M. E. 0.1782 0.3170 1.0968 0.1173 0.4782

Skewness 0.2002 --0 .1903 1.5878 0.3552 --1 .0993

Kurtosis 3.3736 3.2695 5.2486 2.7832 3.7786

Jarque-Bera ^6.2467 4.5309 315.4218 11.4942 113.3408

Probability 0.0440 0.1038 0.0000 0.0032 0.0000

Two-step MLE

C p a a

Mean 1.0290 0.4891 1.0872 0.0654 0.4963

Std. Dev. 0.1726 0.0650 1.1061 0.1083 0.4775

R. M. E. 0.1748 0.3176 1.1085 0.1136 0.4771

Skewness 0.1523 --0 .2092 1.5749 0.3721 -1 .1144

Kurtosis 3.4509 3.0951 5.1372 2.9090 3.8412

Jarque-Bera ^6.1684 ^3.8355 ^301.8590 ^11.7097 ^118.2325

Probability 0.0458 0.1469 0.0000 0.0029 0.0000

MLE 3

C A ^a 3

Mean 1.0000 0.5000 1.0917 0.0719 0.4899

Std. Dev. 0.0000 0.0000 1.0669 0.1102 0.4694

R. M. E. n.a. n.a. 1.0698 0.1137 0.4690

Skewness ^n.a. ^n.a. ^1.5710 0.2724 --1

.0965

Kurtosis n.a. n.a. 5.4020 2.7745 3.8213

Jarque-Bera ^n.a. ^n.a. ^325.8643 ^7.2414 ^114.2492

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B. GAECH (II) Full MLE

C p cu a 13

Mean 1.0187 0.4915 1.3879 0.2474 0.5987

Std. Dev. 0.2440 0.0688 1.2323 ^0.1107 0.1979

R. M. E. 0.2445 0.3160 1.2907 0.1107 0.2042

Skewness 0.4622 --0 .3786 4.3917 0.4116 -2 .2446

Kurtosis 3.4358 3.2777 31.4300 3.5294 12.6477

Jarque-Bera ^21.7582 ^13.5492 ^18446.1350 ^19.9596 ^2358.9955

Probability 0.0000 0.0011 0.0000 0.0000 0.0000

Two-step MLE

C to a

Mean 1.0298 0.4896 1.4000 0.2362 0.6061

Std. Dev. 0.2903 0.0815 1.2634 0.1049 0.2003

R. M. E. 0.2915 0.3209 1.3241 0.1057 0.2049

Skewness 0.4986 --0 .3609 4.2608 0.4532 -2 .4067

Kurtosis 3.8246 3.3467 29.0190 3.7822 13.1936

Jarque-Bera ^34.8797 ^13.3582 ^15616.8001 ^29.8634 ^2647.4435

Probability 0.0000 0.0013 0.0000 0.0000 0.0000

MLE 3

C _A m a 13

Mean 1.0000 0.5000 1.3864 0.2457 0.5999

Std. Dev. 0.0000 0.0000 1.1828 0.1077 0.1986

R. M. E. n.a. n.a. 1.2432 0.1077 0.2046

Skewness ^n.a. ^n.a. ^4.7426 ^0.3851 ^{—2 .6646}

Kurtosis n.a. n.a. 38.0789 3.5565 16.1191

Jarque-Bera ^n.a. ^n.a. ^27510.2877 ^18.8064 ^4177.2795

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Fri ,1 TO pi Vol. XXIX, No. 1.2

C p w a ie

Mean 1.0153 0.4897 1.1988 0.4075 0.4579

Std. Dev. 0.1885 0.0639 0.5678 0.1252 0.1357

R. M. E. 0.1890 0.3168 0.6010 0.1253 0.1420

Skewness 0.2337 --0 .0249 1.1235 0.3470 --0 .3157

Kurtosis 3.0838 3.2708 4.9783 3.3603 4.2635

Jarque-Bera ^4.6970 ^1.5791 ^186.7219 ^12.7392 ^41.5645

Probability 0.0955 0.4540 0.0000 0.0017 0.0000

Two-step MLE

C p ^a a

Mean 1.0300 0.4833 1.2147 0.3894 0.4710

Std. Dev. 0.3026 0.1047 0.5867 0.1176 0.1367

R. M. E. 0.3038 0.3335 0.6241 0.1179 0.1396

Skewness 0.4280 --0 .5786 1.6712 0.2245 --0.8544

Kurtosis 4.2080 4.3132 10.1713 3.0315 7.9938

Jarque-Bera ^45.6671 ^63.8230 ^1304.1532 ^4.2195 ^580.3629

Probability 0.0000 0.0000 0.0000 0.1213 0.0000

MLE 3

C P a Q

Mean 1.0000 0.5000 1.2524 0.4077 0.4511

Std. Dev. 0.0000 0.0000 0.8176 0.1248 0.1538

R. M. E. n.a. n.a. 0.8549 0.1250 0.1612

Skewness n.a. n.a. 7.5159 0.3186 --2

.0761

Kurtosis n.a. n.a. 105.6776 3.4038 18.6789

Jarque-Bera ^n.a. ^n.a. 224346.5370 11.8577 5480.5785

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Table. 3 p = 0.8

C p U) a Q

Mean 1.0790 0.7841 1.0950 0.0864 0.4761

Std. Dev. 0.2674 0.0482 1.0564 0.1163 0.4579

R. M. E. 0.2785 0.0507 1.0596 0.1170 0.4580

Skewness 0.6110 --0 .5527 1.7098 0.2400 -1

.0026

Kurtosis 4.0216 3.5111 6.3145 2.6182 3.6638

Jarque-Bera ^52.8483 ^30.8952 ^472.4981 ^7.8360 ^92.9551

Probability 0.0000 0.0000 0.0000 0.0199 0.0000

Two-step MLE

C _p 0) a Q

Mean 1.0844 0.7828 1.1300 0.0843 0.4606

Std. Dev. 0.2623 0.0472 1.0577 0.1100 0.4657

R. M. E. 0.2753 0.0502 1.0646 0.1110 0.4669

Skewness 0.6095 --0 .5643 1.5716 0.1867 --1 .0478

Kurtosis 3.9840 3.3741 5.4730 2.4871 3.7882

Jarque-Bera ^51.1358 ^29.4514 ^333.2442 ^8.3863 ^104.4259

Probability 0.0000 0.0000 0.0000 0.0151 0.0000

MLE 3

C p co a a

Mean 1.0000 0.8000 1.1210 0.0857 0.4678

Std. Dev. 0.0000 0.0000 1.0514 0.1099 0.4601

R. M. E. n.a. n.a. 1.0572 0.1108 0.4608

Skewness n.a. n.a. 1.5217 0.1511 —1 .0107

Kurtosis n.a. n.a. 5.2371 2.4242 3.7172

Jarque-Bera ^n.a. ^n.a. ^297.2423 ^8.8093 ^95.8351

Probability ^n.a. ^n.a. ^0.0000 ^0.0122 0.0000

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B. GAECH (II ) Full MLE

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C _p U) a Q

Mean 1.0950 0.7811 1.3644 0.2411 0.6053

Std. Dev. 0.2891 0.0482 1.1069 0.1040 0.1859

R. M. E. 0.3040 0.0517 1.1643 0.1043 0.1910

Skewness 0.3847 --0

.5494 2.7151 0.0628 -1 .3554

Kurtosis 3.1150 3.3326 13.6766 3.8290 6.4087

Jarque-Bera ^12.6075 ^27.4558 ^2989.0672 ^14.6482 ^395.1608

Probability 0.0018 0.0000 0.0000 0.0007 0.0000

Two-step MLE

C P a a

Mean 1.1190 0.7759 1.3752 0.2323 0.6115

Std. Dev. 0.3509 0.0566 1.1266 0.0992 0.1829

R. M. E. 0.3702 0.0615 1.1864 0.1007 0.1867

Skewness 0.1366 --0 .5833 2.9347 --0 .0357 -1

.3317

Kurtosis 2.9434 3.7505 15.8055 3.4808 6.3343

Jarque-Bera ^1.6216 ^40.0888 ^4133.9493 ^4.9212 ^379.3971

Probability 0.4445 0.0000 0.0000 0.0854 0.0000

MLE 3

C _p co a a

Mean 1.0000 0.8000 1.3739 0.2372 0.6076

Std. Dev. 0.0000 0.0000 1.1646 0.1021 0.1933

R. M. E. n.a. n.a. 1.2220 0.1028 0.1977

Skewness n.a. n.a. 3.3828 --0 _0662 --2 .15I1

Kurtosis n.a. n.a. 20.8015 3.6761 12.4493

Jarque-Bera ^n.a. ^n.a. ^7555.5438 ^9.8881 ^2245.8071

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C _p a a

Mean 1.0636 0.7870 1.1519 0.3924 0.4749

Std. Dev. 0.2698 0.0439 0.5815 0.1304 0.1498

R. M. E. 0.2769 0.0457 0.6005 0.1305 0.1518

Skewness 0.4575 --0 .4186 1.8059 0.1713 --0 .9088

Kurtosis 3.5813 3.0973 9.4369 3.2166 6.3178

Jarque-Bera ^24.4780 ^14.7985 ^1134.9673 ^3.4222 ^298.1567

Probability 0.0000 0.0006 0.0000 0.1807 0.0000

Two-step MLE

C _to a Q

Mean 1.1317 0.7727 1.1760 0.3765 0.4844

Std. Dev. 0.3943 0.0661 0.5864 0.1253 0.1569

R. M. E. 0.4153 0.0715 0.6117 0.1274 0.1575

Skewness 0.7786 --0 .8822 1.7519 0.0769 - 1 .7806

Kurtosis 4.6814 4.7819 9.1874 3.3935 15.2833

Jarque-Bera ^109.4186 ^131.0084 ^1053.3470 ^3.7192 ^3407.5081

Probability 0.0000 0.0000 0.0000 0.1557 0.0000

MLE 3

C p ^a a

Mean 1.0000 0.8000 1.1487 0.3869 0.4820

Std. Dev. 0.0000 0.0000 0.5660 0.1285 0.1454

R. M. E. n.a. n.a. 0.5847 0.1291 0.1463

Skewness n.a. n.a. 1.5186 0.1729 --0 .6461

Kurtosis n.a. n.a. 6.9347 3.4520 5.2434

Jarque-Bera ^n.a. ^n.a. ^514.7110 ^6.7466 ^139.6337