建立AR(1)-TAR-GARCH(1,1)-t模式

若是財務資料中存在資訊不對稱性(asymmetric information)，仍可考慮配適門檻 GARCH 模式(Threshold-GARCH Model, Tong, 1990)，本研究僅考慮一個門檻值(threshold value)區分波動函數為兩個片段的模式，小於門檻值為負面消息，反之，大於門檻值視為正面消息，條件變異數來自不對稱的正效應與負效應，在此以過去的衝擊a

_t ₋ ₁

當作門檻變數，a

_t ₋ ₁

>0為正面消息，a

_t ₋ ₁

<0為負面消息。

原始模式如下：

_t ⁱ

=φ

₀

+φ

₁

_t ⁱ ₋ ₁

+ψ

₁

_t ^j ₊ _m ₋ ₁

_t

_t ²

=α

₀

+α

₁

_t ² ₋ ₁

+α

₂

_t ² ₋ ₁

+(β

₀

+β

₁

_t ² ₋ ₁

+β

₂

_t ² ₋ ₁

_t ₋ ₁

>0),

, , ~ ( 0 , 1 ).

5.結論與建議

本研究實證發現美國股市對於歐洲股市存在外溢效果；並且價格具有不對稱的波動，以及呈現波動群集的現象。以美國股票市場所傳遞的負面消息與正面消息觀之，相較而言，美國股市所傳遞的負面消息確實會引起歐洲國家股市另一波重挫，先後藉由非對稱模式(3)、(4)模式皆證實了負面消息對人們的衝擊比正面消息大，較能充分反應真實資訊，是故非對稱模型解釋能力較佳。

在外溢效果上，經由實證結果證實了美國股票交易市場對於歐洲英、德、法三國皆具有波動不對稱性外溢效果。歐洲股票交易市場(英、德及法)先開盤，美國股票交易市場後開盤，歐洲股票價格經常因美國股票價格開盤後的走勢而發生逆轉；美國股票交易市場不但會影響著歐洲股票交易市場的變動，而且呈現同向變動，此與一般投資人觀察美國股票交易市場收盤後，預期歐洲股票交易市場開盤的報酬相同。

進行實證研究過程中，發現仍可進一步探討之處有，研究 GARCH 模型波動參數估計方法可從其他非線性估計方法以及貝氏估計方法，期待找出更符合真實狀況的分配型態，找出最適模式。

附錄

時間序列圖與經濟要事

由價格之時間序列圖來看，在2000 年(第 2089 筆)以前是美國科技股的榮景，由美國帶動全球的繁榮。由轉換成報酬率之時間序列圖看出有幾個幅度較大的時間點，為主要經濟要事發生點，以下介紹其主要影響股市之時間點：(詳見圖 1、2)

1997/10/28(第 1520 筆)－亞洲金融風暴是由於外資撤資，錢回流先進國家。英國漲幅很小，可能大部分資金回流美德法三國。

1998/9/1(第 1740 筆)－美國有一家 LTCM 公司，由二位諾貝爾得獎主所設立的公

司，專作避險操作，在 1998 年俄羅斯政府發生債務無法清償，幣值大貶，影響到美國

這家LTCM 公司的避險部分。由圖看出英德法也多少受影響，這件事很嚴重，它的避險

部位含很多重要的國家。LTCM 事件重創美國股市，在 9/1 先大漲，9/9 再大跌。

2000/4/17(第 2164 筆)－2000 年時發生美國科技股泡沫化(漲過頭了，沒基本面支撐股價)，造成全球股市崩跌。為美國科技股大跌前的大漲。

2002/7/24(第 2756 筆)－美國 2002 年上半年發生許多大型企業財報醜聞的事件(ex:

Enron

¹

, Worldcom

²

)，使美國股市信心低迷。而在該周，S&P500 index 跌幅 8.4%，已跌破911 事件時(跌幅 5.05%)所創下的最低紀錄。7/24 會大漲，僅代表跌深反彈。

2002/8/7(第 2766 筆)－德國在 2002 年 8/7 日公布 7 月失業人口，高達 404 萬 7 千人，

那時還有一個多月要選舉，重創德國人民的心。而隔天 2002/8/8 股市反彈(大漲)，可能是要選舉了，股市表現不穩定所致。

2003/3/13(第 2922 筆)－美伊戰爭 2003/3/20 開始，在此之前美國已在伊拉克佈軍已久，當時市場認為美國會打贏且會速戰速決。又因歐洲企業表現疲軟已久，在2003 年 3 月的經濟數據陸續出現佳績，歐洲股市在激勵之下，出現3/13 大漲的情形。

1. Enron, 安隆公司 – 世界第一大能源供應商，現代化的能源管理服務公司，2002 上半年歷經訴訟、前

任高階主管自殺、內部調查、員工資遣等等事情，下半年董事會改組，進入公司重整軌道。

2. Worldcom, 世界通信－前執行長艾伯斯（Bernard Ebbers）因作會計假帳被判刑。

500 1000 1500 2000 2500 3000 3500

2000

500 1000 1500 2000 2500 3000 3500

1000

500 1000 1500 2000 2500 3000 3500

1200

500 1000 1500 2000 2500 3000 3500

-6

500 1000 1500 2000 2500 3000 3500

-10

500 1000 1500 2000 2500 3000 3500

-8

表1. 股票市場日報酬之基本統計量

統計量 UK Germany France US

平均數 0.0222 0.0338 0.0272 0.0305

標準差 1.0301 1.4219 1.3227 1.0049

偏態係數－0.1246 －0.2457 －0.0918 －0.1083

(P 值) (0.0021) (0.0000) (0.0237) (0.0076)

超額峰態係數 3.4359 3.7169 2.7765 4.2923

(P 值) (0.0000) (0.0000) (0.0000) (0.0000)

t-統計量 1.2996 1.4368 1.2410 1.8330

(P 值) (0.1938) (0.1509) (0.2147) (0.0669)

Jarque-Bera 1803.3347 2136.0329 1176.5600 2806.7941

(P 值) (0.0000) (0.0000) (0.0000) (0.0000)

樣本數 3648 3648 3648 3648

表2. 股價與報酬之 ADF 單根檢定

US UK Germany France

Price Return Price Return Price Return Price Return

－1.2156 －61.3456 －1.5202 －38.1459 －1.3132 －61.2056 －1.1728 －59.4596 (0.6700) (0.0001) (0.5234) (0.0000) (0.6256) (0.0001) (0.6884) (0.0001)

註. ADF單根檢定法為t統計量，括號內數字代表統計量之P值。

表3. 美國股市與歐洲股市之因果關係檢定

Causal direction F-Test (P 值) Causal direction F-Test (P 值) A. One-period lagged model

US→UK 398.5280 (0.0000) UK→US 2.1590 (0.1418) US→Germany 329.1150 (0.0000) Germany→US 0.3601 (0.5485) US→France 357.9670 (0.0000) France→US 2.7354 (0.0982) B. Two-period lagged model

US→UK 204.5720 (0.0000) UK→US 2.1724 (0.1141) US→Germany 170.1660 (0.0000) Germany→US 0.5775 (0.5613) US→France 180.2850 (0.0000) France→US 2.8244 (0.0595)

註. P 值皆小於顯著水準 1%，統計上顯著，則美國股市在歐洲股市具影響力。

表4. ARCH效果檢定(ㄧ)

US UK

期數 Q P 值 LM P 值 Q P 值 LM P 值

1 159.2136 <.0001 159.1492 <.0001 185.5698 <.0001 185.5002 <.0001 2 303.1236 <.0001 250.7349 <.0001 508.9899 <.0001 419.6549 <.0001 3 431.4236 <.0001 306.1266 <.0001 790.8749 <.0001 538.2071 <.0001 4 494.3476 <.0001 315.3104 <.0001 995.6747 <.0001 575.4287 <.0001 5 585.7039 <.0001 340.4608 <.0001 1216.8840 <.0001 609.7952 <.0001 6 694.0301 <.0001 371.1963 <.0001 1451.5890 <.0001 646.8319 <.0001 7 782.4598 <.0001 386.3704 <.0001 1634.6400 <.0001 659.7976 <.0001 8 857.1458 <.0001 393.5763 <.0001 1912.7500 <.0001 703.7366 <.0001 9 932.8076 <.0001 401.6816 <.0001 2041.1580 <.0001 703.8962 <.0001 10 1014.9570 <.0001 412.6738 <.0001 2229.1030 <.0001 711.2945 <.0001 11 1084.9290 <.0001 417.9544 <.0001 2433.3900 <.0001 726.2511 <.0001 12 1156.2460 <.0001 423.0875 <.0001 2585.3160 <.0001 729.1680 <.0001

Germany France

期數 Q P 值 LM P 值 Q P 值 LM P 值

1 188.8539 <.0001 188.6921 <.0001 128.3370 <.0001 128.2717 <.0001 2 443.7097 <.0001 362.3781 <.0001 326.7672 <.0001 276.4649 <.0001 3 735.0532 <.0001 500.8482 <.0001 534.7462 <.0001 384.4938 <.0001 4 1012.6640 <.0001 586.7930 <.0001 671.3982 <.0001 418.6262 <.0001 5 1217.9210 <.0001 614.6600 <.0001 840.2916 <.0001 460.1967 <.0001 6 1469.7210 <.0001 653.7793 <.0001 1077.5680 <.0001 532.2667 <.0001 7 1748.2200 <.0001 697.7220 <.0001 1271.4630 <.0001 569.2951 <.0001 8 1997.8760 <.0001 723.8440 <.0001 1458.3420 <.0001 592.9098 <.0001 9 2194.6200 <.0001 730.4637 <.0001 1604.2240 <.0001 599.8609 <.0001 10 2356.9150 <.0001 730.9987 <.0001 1769.5330 <.0001 612.2633 <.0001 11 2555.7560 <.0001 737.0059 <.0001 1961.8030 <.0001 632.6453 <.0001 12 2732.1650 <.0001 740.6140 <.0001 2100.1580 <.0001 636.0606 <.0001

註. Q為Ljung-Box統計量，LM為Lagrange multiplier 檢定統計量，P值皆小於顯著水準1%，統計上顯 著，則資料變異為異質性。

表5. ARCH效果檢定(二)

表6. AR(1)-GARCH(1,1)-Normal 模式(1)之參數估計

參數 UK (P 値) Germany (P 値) France (P 値)

表8. AR(1)-GARCH(1,1)-t 模式(2)之參數估計

參數 UK (P 値) Germany (P 値) France (P 値) φ

0

0.0358 (0.0023) 0.0618 (0.0001) 0.0449 (0.0080) φ

1

－0.0954 (0.0000) －0.1300 (0.0000) －0.1101 (0.0000)

ψ1 0.3018 (0.0000) 0.4636 (0.0000) 0.3983 (0.0000)

α

0 0.0102 (0.0000) 0.0088 (0.0022) 0.0136 (0.0000) α

1

0.0819 (0.0000) 0.0801 (0.0000) 0.0602 (0.0002) β

1

0.9082 (0.0000) 0.9181 (0.0000) 0.9320 (0.0000) ν 12.2244 (0.0000) 9.3689 (0.0000) 11.2163 (0.0000)

1 1

/ 1 α β

α − −

1.0208 4.6452 1.7443

註. 各參數值之P値皆小於顯著水準1%，統計上皆為顯著，而後作模式之檢驗。

表9. Ljung-Box Q-統計量檢驗模式(2)

Q-統計量 UK (P 値) Germany (P 値) France (P 値) Q(5) 5.6222 (0.3447) 3.0771 (0.6881) 7.9194 (0.1607) Q(10) 13.9866 (0.1736) 10.0570 (0.4355) 12.6448 (0.2442) Q(15) 20.6146 (0.1496) 18.5968 (0.2326) 18.7063 (0.2274) Q(20) 28.3765 (0.1008) 22.2515 (0.3270) 23.4628 (0.2666) Q

²

(5) 7.1130 (0.2124) 6.2038 (0.2869) 6.7758 (0.2379) Q

²

(10) 12.2804 (0.2667) 8.5166 (0.5785) 10.0697 (0.4344) Q

²

(15) 20.4715 (0.1546) 12.4690 (0.6432) 13.2573 (0.5824) Q

²

(20) 23.5323 (0.2634) 15.4529 (0.7499) 14.8334 (0.7859)

|Q(5)| 2.3438 (0.7998) 6.5620 (0.2553) 5.6343 (0.3434)

|Q(10)| 7.2427 (0.7023) 8.7236 (0.5585) 9.1622 (0.5168)

|Q(15)| 16.8300 (0.3291) 13.5719 (0.5582) 14.1767 (0.5122)

|Q(20)| 19.6189 (0.4820) 16.1232 (0.7090) 17.1221 (0.6450)

註. Q( )‧ 為標準化殘差a~

_t

_{之Q-統計量，Q}2( )‧ 為標準化殘差平方之Q-統計量，|Q( )|‧ 為標準化殘差絕對值之Q-統計量，“ ”‧ 為落後期數，其中a~ =

_t

/σˆ

_t

表10. 波動不對稱性檢定

Market JT (TR

²

～χ

² (3)

) SBT (t 檢定) NSBT (t 檢定) PSBT (t 檢定) UK 207.9636 (0.0000) －0.4272 (0.6692) －11.3559 (0.0000) 9.5409 (0.0000) Germany 225.2042 (0.0000) 0.5118 (0.6088) －12.6869 (0.0000) 8.1773 (0.0000) France 113.3514 (0.0000) －0.3792 (0.7046) －8.8101 (0.0000) 6.0970 (0.0000)

註. 括號內數字代表各統計量之P値，皆小於顯著水準1%，統計上顯著，則資料具有不對稱性效果。

表11. AR(1)-GJR-GARCH(1,1)-t 模式(3)之參數估計

參數 UK (P 値) Germany (P 値) France (P 値) φ

1

－0.0963 (0.0000) －0.1226 (0.0000) －0.1100 (0.0000)

ψ1 0.2978 (0.0000) 0.4621 (0.0000) 0.3933 (0.0000)

α

0 0.0109 (0.0000) 0.0138 (0.0002) 0.0185 (0.0000) α

1

0.0208 (0.0015) 0.0425 (0.0000) 0.0253 (0.0008) β

1

0.9194 (0.0000) 0.9142 (0.0000) 0.9280 (0.0000)

γ

0.0985 (0.0000) 0.0762 (0.0000) 0.0721 (0.0000)

ν

12.3694 (0.0000) 10.0641 (0.0000) 11.8864 (0.0000)

註. 各參數值之P値皆小於顯著水準1%，統計上皆為顯著，而後作模式之檢驗。

表12. Ljung-Box Q-統計量檢驗模式(3)

Q-統計量 UK (P 値) Germany (P 値) France (P 値) Q(5) 6.1434 (0.2925) 2.9645 (0.7055) 9.2623 (0.0990) Q(10) 14.8810 (0.1365) 10.4126 (0.4051) 13.3829 (0.2030) Q(15) 22.1122 (0.1049) 19.8495 (0.1778) 19.8649 (0.1772) Q(20) 29.2229 (0.0835) 24.1889 (0.2342) 25.3725 (0.1876) Q

²

(5) 4.5300 (0.4759) 5.0780 (0.4064) 3.6538 (0.6003) Q

²

(10) 9.8398 (0.4547) 10.2864 (0.4157) 7.4563 (0.6818) Q

²

(15) 23.2147 (0.0797) 14.3812 (0.4968) 11.1797 (0.7398) Q

²

(20) 28.4586 (0.0990) 16.1626 (0.7065) 12.7377 (0.8884)

|Q(5)| 2.0185 (0.8466) 7.4109 (0.1918) 5.7783 (0.3284)

|Q(10)| 7.2711 (0.6996) 10.3796 (0.4078) 8.6182 (0.5687)

|Q(15)| 21.2724 (0.1283) 14.3639 (0.4981) 14.1756 (0.5123)

|Q(20)| 25.6444 (0.1779) 16.0857 (0.7113) 17.2115 (0.6392)

註. Q( )‧ 為標準化殘差a~

_t

_{之Q-統計量，Q}2( )‧ 為標準化殘差平方之Q-統計量，|Q( )|‧ 為標準化殘差絕對值之Q-統計量，“ ”‧ 為落後期數，其中a~ =

_t

/σˆ

_t

表13. AR(1)-TAR-GARCH(1,1)-t 模式(4)之參數估計

參數 UK (P 値) Germany (P 値) France (P 値) φ

0

0.0233 (0.0484) 0.0583 (0.0001) 0.0345 (0.0403) φ

1

－0.0946 (0.0000) －0.1356 (0.0000) －0.1094 (0.0000)

ψ1 0.2998 (0.0000) 0.4634 (0.0000) 0.3958 (0.0000)

α

0 0.0241 (0.0001) 0.0125 (0.0000) 0.0238 (0.0000) α

1

0.1094 (0.0000) 0.0831 (0.0000) 0.0858 (0.0000) α

2

0.8806 (0.0000) 0.9069 (0.0000) 0.9042 (0.0000) β

0

－0.0191 (0.0380) - - - - β

1

－0.0735 (0.0000) －0.0218 (0.1043) －0.0485 (0.0008) β

2

0.0539 (0.0031) 0.0218 (0.1043) 0.0310 (0.0242)

ν

11.2195 (0.0000) 9.9793 (0.0000) 12.9771 (0.0000)

註. “-“僅代表此參數不顯著，為了節約參數，所以不考慮放入模式中，而後作模式之檢驗。

表14. Ljung-Box Q-統計量檢驗模式(4)

Q-統計量 UK (P 値) Germany (P 値) France (P 値) Q(5) 6.3544 (0.2732) 3.8602 (0.5697) 9.3375 (0.0963) Q(10) 15.7006 (0.1085) 11.1954 (0.3425) 15.2999 (0.1215) Q(15) 22.4894 (0.0956) 18.8293 (0.2216) 20.1537 (0.1661) Q(20) 29.9664 (0.0704) 22.1827 (0.3307) 24.7948 (0.2094) Q

²

(5) 3.7263 (0.5895) 0.9686 (0.9651) 4.5838 (0.4687) Q

²

(10) 6.0398 (0.8119) 3.6987 (0.9599) 13.1709 (0.2143) Q

²

(15) 11.9113 (0.6857) 7.4884 (0.9427) 17.2455 (0.3044) Q

²

(20) 15.2919 (0.7595) 9.4643 (0.9769) 19.2663 (0.5046)

|Q(5)| 1.5161 (0.9112) 5.9073 (0.3153) 5.0630 (0.4082)

|Q(10)| 7.3980 (0.6874) 9.1754 (0.5155) 9.8126 (0.4571)

|Q(15)| 17.5469 (0.2872) 14.1478 (0.5144) 14.7148 (0.4722)

|Q(20)| 21.2107 (0.3848) 16.1873 (0.7049) 19.8760 (0.4657)

註. Q( )‧ 為標準化殘差a~

_t

_{之Q-統計量，Q}2( )‧ 為標準化殘差平方之Q-統計量，|Q( )|‧ 為標準化殘差絕對值之Q-統計量，“ ”‧ 為落後期數，其中a~ =

_t

/σˆ

_t

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自述

姓名:林美惠

學歷:逢甲大學統計與精算研究所計量財務組學號:M9416481

E-mail:[email protected]

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