Mean Reversion in Stock Index Futures - A Nonlinear Analysis
Nonlinear dynamics: evidence for Bucharest Stock Exchange
description
Transcript of Nonlinear dynamics: evidence for Bucharest Stock Exchange
Nonlinear dynamics: evidence for Bucharest Stock Exchange
Dissertation paper:
Anca Svoronos(Merdescu)
Goals
To analyse a good volatility model by its ability to capture “stylized facts”
To analyse changes in models behavior with respect to temporal aggregation
To perform an empirical evidence for Bucharest Stock Exchange using its reference index BET
Introduction
The finding of nonlinear dynamics in financial time series dates back to the works of Mandelbrot and Fama in the 1960’s:
- Mandelbrot first noted in 1963 that “large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes”
- Fama developed the efficient-market hypothesis (EMH) – which asserts that financial markets are “informationally efficient”
Volatility Models
GARCH models
Engle (1982)
Bollerslev (1986)
Nelson (1991)
Glosten, Jagannathan and Runkle (1993)
Markov regime switching model
Hamilton (1989)
GARCH models
GARCH
(p,q)
TARCH
(p,q)
EGARCH
(p,q)
p
j jtj
q
i itit
ttt
hrheqiance
hreqmean
11
20:.var
:.
p
j ttjtj
q
i itit drhrheqiance1 1
211
20:.var
it
itp
j jtj
it
itq
i ith
rh
h
rheqiance
110 log
2log:.var
Markov switching model
);N(~: 2i ittt SriS
State
The model assumes the existence of an unobserved variable denoted:
;1,0iwhere
tttt SSr ][ 1010
The conditional mean and variance are defined:
;)( 10 tt SS ;)( 10 tt SS
The transition (=conditional) probabilities are :
qSSob
pSSob
tt
tt
1)01(Pr
)11(Pr
1
1
qSSob
pSSob
tt
tt
)00(Pr
1)10(Pr
1
1
The maximum likelihood will estimate the following vector containing six parameters:
),,,,,( 02
012
1 qp
, t is i.i.d N(0,1).
Data Description
Data series: BET stock index Time length: Jan 3rd, 2001 – March 4th, 2009 2131 daily returns: ]ln[ln*100 1 ttt PPr
Daily closing prices of BET index
2001 2002 2003 2004 2005 2006 2007 20080
2000
4000
6000
8000
10000
12000
Statistical properties of the returns
Non-normal distribution
-13.2 -9.9 -6.6 -3.3 0.0 3.3 6.6 9.9
0
50
100
150
200
250
300
350
Mean 0.059859
Median 0.000000
Maximum 10.09070
Minimum -13.11680
Std. Dev. 1.697205
Skewness -0.676328
Kurtosis 10.29213
Jarque-Bera 4881.677
Probability 0.000000
Observations 2130Histogram of BET returns
Statistical properties of the returns
Heteroscedasticity
BET squared returns series
2001 2002 2003 2004 2005 2006 2007 20080
25
50
75
100
125
150
175BET return series
2001 2002 2003 2004 2005 2006 2007 2008-15
-10
-5
0
5
10
15
Statistical properties of the returns
Autocorrelation
- High serial dependence in returns
- The Ljung-Box statistic for 20 lags is 85,75 (0.000)
Daily BET returns correlogram
-0.05
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
AC
PAC
Daily BET sq returns correlogram
-0.1
0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
AC
PAC
-The Ljung-Box statistic for 20 lags is 1442,6 (0.000)
-LM (1): 260,61
=> BET index returns exhibit ARCH effects
6.37
*2
01.0;20
2
RnLM
Statistical properties of the returns
• BDS independence test (Brocht, Dechert, Scheinkman)of the null hypotheses that time series is independently and identically distributed, is a general test for identifying nonlinear dependence(m=5, ε=0,7)
Dimension BDS Statistic Std. Error z-Statistic Prob.
2 0.035426 0.002149 16.48861 0.0000
3 0.063786 0.003416 18.67311 0.0000
4 0.081721 0.004070 20.07747 0.0000
5 0.090652 0.004246 21.35227 0.0000
• The results presented above show a rejection of the independence hypothesis for all embedding dimensions m
Statistical properties of the returns
Stationarity: Unit root tests for BET return series
ADF Test Statistic -40.43444 1% Critical Value* -3.433224 5% Critical Value -2.862696 10% Critical Value -2.567431
*MacKinnon critical values for rejection of hypothesis of a unit root.
PP Test Statistic -40.51466 1% Critical Value* -3.433224 5% Critical Value -2.862696 10% Critical Value -2.567431
*MacKinnon critical values for rejection of hypothesis of a unit root.
Models specification (daily data)
Model 1:TARCH (1, 1)
Model 2: GARCH (1,1)
Model 3: EGARCH (1,1)
Model 4: Markov Switching (MS)
dDuahauaah
urbbr
tttt
ttt
21413
2110
121
dhauaah
urbrbrbrbbr
ttt
tttttt
122
110
195144113121
dhahuah
uaah
urbrbrbrbbr
ttt
t
tt
tttttt
113112
1
110
195144113121
)log()/(2
)log(
tttt
tttt
SSSS
SSr
1010
1010
)();()(
][)(
Model Estimates
Variable Coeff T-stat Signif
1. Intercept (b1) 0.111933 3.394027 0.0007
2. AR(1) (b2) 0.140159 5.858505 0.0000
3. Constant (a0) 0.263106 11.750979 0.0000
4. ARCH (a1) 0.169324 8.292505 0.0000
5.Asymmetric coeff (a4) 0.066380** 2.297879 0.0215
6. GARCH (a3) 0.671665 32.13333 0.0000
7. Dummy(π) 26.46984 3.061015 0.0022
Q(20) st res 28.953** - 0.027
Q(20) st sq res 19.120** - 0.124
SIC 3.519069 - -
Model 1 – TGARCH(1,1)
*Denotes significance at the 1% level of significance**Denotes significance at the 5% level of significance
- Null hypothesis of BDS is not rejected at any significance level
- The standardized squared residuals are serially uncorrelated both at 5% and 1% significance level
- Volatility persistence given by is 0,874179 < 1, implying a half life volatility of about 8 days
- > 0 therefore we could stress that a leverage effect exists but testing the null hypothesis of = 0 at 1% level of significance we find that the shock is symmetric
=> a symmetric model specification should be tested
2/431 aaa
4a
4a
Dimension BDS Statistic Std. Error z-Statistic Probab
2 -0.000325 0.001966 -0.165132 0.8688
3 -0.000515 0.003116 -0.165165 0.8688
4 -0.002476 0.003701 -0.669060 0/5035
5 -0.004216 0.003848 -1,095643 0.2732
BDS test
Model Estimates
Variable Coeff T-stat Signif
1. Intercept (b1) 0.10058 3.639956 0.0003
2. AR(1) (b2) 0.140762 5.925356 0.0000
3. AR(11) (b3) 0.033611 2.028312 0.0425
4. AR(14) (b4) 0.024985 1.444312 0.1487
5. AR(19) (b5) 0.032554 1.681572 0.0927
6. Constant (a0) 0.257882 11.59915 0.0000
7. ARCH (a1) 0.205343 11.71204 0.0000
8. GARCH (a2) 0.671857 32.11108 0.0000
9. Dummy (π) 26.55657 3.016097 0.0026
Q(20) st res 19.044** - 0.519
Q(20) st sq res 20.531** - 0.425
SIC 3.518409 - -
Model 2 – GARCH(1,1)
• Null hypothesis of BDS is accepted at any significance level for all 5 dimensions;
•The standardized squared residuals are serially uncorrelated at both significance level of 5% and 1%
• Volatility persistence is 0,8772 < 1, implying a half life volatility of about 8 days, similar to the one implied by Model 1
*Denotes significance at the 1% level of significance**Denotes significance at the 5% level of significance
Dimension BDS Statistic Std. Error z-Statistic Prob.
2 -0.000941 0.001990 -0.472662 0.6365
3 -0.001732 0.003155 -0.548993 0.5830
4 -0.003770 0.003748 -1.005762 0.3145
5 -0.005712 0.003898 -1.465476 0.1428
BDS test
0
20
40
60
80
100
120
2001 2002 2003 2004 2005 2006 2007 2008
VAR_MODEL2
Model Estimates
Variable Coeff T-stat Signif
1. Intercept (b1) 0.049671 2.099358 0.0358
2. AR(1) (b2) 0.124025 5.44934 0.0000
3. AR(11) (b3) 0.037543 2.428243 0.0152
4. AR(14) (b4) 0.026551 1.634159 0.1022
5. AR(19) (b5) 0.015345 0.949694 0.3423
3. Constant (a0) -0.28373 -10.5953 0.0000
4. ARCH (a1) 0.53286 10.4155 0.0000
5. Asymmetric coeff (a2) -0.06249**
-2.05898 0.0395
6. GARCH (a3) 0.857612 42.09454 0.0000
7. Dummy coeff (π) 1.728375 4.90853 0.0000
Q(20) st res 18.731** 0.539
Q(20) st sq res 13.912** 0.835
SIC 3.5308
Model 3 – EGARCH(1,1)
Dimension BDS Statistic Std. Error z-Statistic Prob.
2 -0.002040 0.001939 -1.052297 0.2927
3 -0.004246 0.003073 -1.381835 0.1670
4 -0.006847 0.003650 -1.875949 0.0607
5 -0.009025 0.003795 -2.378192 0.0174
BDS test
- Null hypothesis of BDS is being rejected by dimension m=5 and m=4 if using a significance level of 5%(1,64) and by m=5 for 1%(2,33);
- The standardized squared residuals are serially uncorrelated both at 5% and 1% significance level
- Volatility persistence given by is 0,857612 < 1, implying a half life volatility of about 8 days
- < 0 therefore we can stress a leverage effect exists although testing the null hypothesis of = 0 at 1% level of significance we find that the shock is still symmetric
*Denotes significance at the 1% level of significance**Denotes significance at the 5% level of significance
3a
2a
2a
Model estimates
Model 3 – Markov Switching
Variable Coeff T-stat Signif
1. Mean State 1(a01) 0.161229411 5.42759 0.00000006
2. Variance State 1 (σ1) 0.961946011 19.43944 0.00000000
3. Mean State 2 (a02) -0.206468087
-1.41064 0.15834962
4. Variance State 2 (σ2) 2.813250016 13.61520 0.00000000
5. Matrix of Markov transition probabilities
0.03526796 0.90738168
-
0.96473204 0.09261833
-
SIC 18576
-Both probabilities are quite small which means neither regime is too persistent – there is no evidence for “long swings” hypothesis
-We find slight asymmetry in the persistence of the regimes – upward moves are short and sharp (a01 is positive and p11 is small) and downwards moves could be gradual and drawn out (a02 negative and p22 larger)
-The ML estimates associate state 1 with a 0,16% daily increase while in state 2 the stock index falls by -0,2% with considerably more variability in state 2 than in state 1
-SIC value is significantly higher than the values estimated with GARCH models
Evidence for lower frequencies
Monthly data (99 observations)
0
2000
4000
6000
8000
10000
12000
2001 2002 2003 2004 2005 2006 2007 2008
R_D
-30
-20
-10
0
10
20
30
40
2001 2002 2003 2004 2005 2006 2007 2008
R_D_M
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera
Prob.
2.734560 2.482586 29.,76911 -27.07954 8.433922 -0.135998 5.164572 19.43403 0.000
Monthly closing prices for BET
Autocorrelation at lag 1 0.005 Signif. 0.958
Q(20) 4.7020 Signif. 1.0000
LM(1) 0.470581
Signif. 0.9582
Q(20) for squares 12.434 Signif. 0.900
Monthly returns for BET
=> There are no significant evidence of dynamics
Model estimation
GARCH Models failed to converge (see Appendix 3)
Markov Switching models
Variable Coeff T-stat Signif
1. Mean State 1(a01) 2.97641612 3.09876 0.00194335
2. Variance State 1 (σ1) 6.21076790 5.97818 0.00000000
3. Mean State 2 (a02) -3.44338062 -0.67761 0.49801854
4. Variance State 2 (σ2) 15.96667883
6.31980 0.00000000
5. Matrix of Markov transition probabilities
0.06728487 0.83141057
0.93271513 0.16858943
SIC 978
-Two states are again high mean/lower volatility and low mean/higher volatility
-p22 is larger than p11 which means regime 2 should be slightly more persistent – again there is no evidence for “long swings” hypothesis
-again we find asymmetry in the persistence of the regimes
-The ML estimates associate state 1 with an approx 3% monthly increase while in state 2 the stock index falls by -3,5% with considerably more variability in state 2 than in state 1
-In general, the characteristics of the regimes are still present at a monthly frequency in contrast with GARCH
Concluding remarks
If judging from the behavior of residuals, out of the GARCH models, GARCH (1,1) is the model of choice.
Compared with Markov Switching by SIC value we find GARCH(1,1) superior
Considering temporal aggregation, we find that GARCH models fail to converge while Markov Switching model still shows power
Further research:
-forecast ability of both models
Bibliography
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Bibliography
Appendix 1
BDS test for TARCH (1,1)
Appendix 1BDS test for GARCH (1,1)
Appendix 1BDS test for EGARCH (1,1)
Appendix 2
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-4 -2 0 2 4 6
Series: Standardized ResidualsSample 1/05/2001 3/04/2009Observations 2129
Mean -0.010894Median -0.070050Maximum 6.133064Minimum -5.456463Std. Dev. 1.000382Skewness 0.121775Kurtosis 5.351825
Jarque-Bera 495.9148Probability 0.000000
Residuals histogram following GARCH(1,1)
Appendix 2
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-4 -2 0 2 4 6
Series: Standardized ResidualsSample 1/31/2001 3/04/2009Observations 2111
Mean -0.022092Median -0.067989Maximum 6.107907Minimum -5.369981Std. Dev. 1.000232Skewness 0.094679Kurtosis 5.355370
Jarque-Bera 491.1263Probability 0.000000
Residuals histogram following GARCH(1,1)
Appendix 2
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-4 -2 0 2 4 6
Series: Standardized ResidualsSample 1/31/2001 3/04/2009Observations 2111
Mean -0.035377Median -0.080966Maximum 6.110862Minimum -5.199899Std. Dev. 0.999736Skewness 0.187264Kurtosis 5.592336
Jarque-Bera 603.4363Probability 0.000000
Residuals histogram following EGARCH(1,1)
Appendix 3GARCH(1,1) on monthly data