Transcript How long does the river remember...?

Academy of Economic Studies
Doctoral School of Finance and Banking- DOFIN
Exploring Dual Long Memory in Returns and
Volatility across the Central and Eastern European
stock markets
Msc. Student: Mihaela Sandu
Supervisor: PhD.Professor Moisă Altăr
Bucharest, July 2009
Dissertation paper outline
• Importance of long memory
• Aims of the paper
• Data & Methodology
• Nonparametric & semiparametric aproaches
• Parametric approach: ARFIMA / FIGARCH
• Structural breaks
• The joint ARFIMA-FIGARCH model
• Model distributions
• Empirical results
•Conclusions and further improvements
• References
Long memory
• Contradicts the EMH weak-form by allowing investors and portfolio managers to
make prediction and to construct speculative strategies
•The price of an asset determined in an efficient market should follow a martingale
process in which each price change is unaffected by its predecessor and has no
memory . Pricing derivative securities with martingale methods may not be
appropriate if the underlying continuous stochastic processes exhibit long memory
• Implications to assest allocation decisions and risk management
Aims of the paper
• To ivestigate the presence of long memory in stock returns via non-, semiand parametric techniques
• To distinguish between long memory and structural breaks within return
series
• To found evidences of dual long memory processes within CEE emerging
stock markets
Short vs. Long memory processes
ρh   Cr
h
0  r 1
vs.
ρh~Ch2 d 1

 ρh  
 ρh  
fxω
fxω~Cω2d

h 1
h
h 1
ω0
The Data
• Six indices representing five CEE emerging stock markets: BET, BET-FI,
SOFIX, BUX, WIG, PX
• Daily closing stock prices transformed into continuously compounded
returns
• The estimations and tests were performed in R version 2.9.0.
• For estimating ARFIMA-FIGARCH model, the Ox Console version 5.10,
together with the G@rch Console 4.2 were used.
Methodology
•
Unit root tests:
ADF
st 
KPSS statistic:
•
t
e
i 1
i
k
k

1 
R / S (n)  Max ( X j  X n )  Min ( X j  X n )
sn 
j 1
j 1

Rescaled range statistic
1
1
2
sn    ( X j  X n ) 2 
n j

•
Wavelet based estimator
•
Log-periodogram estimator (GPH)
•
ARFIMA model:


lnI (  )   0  1 ln 4 sin 2  
 2

(L)(1  L) d ( yt  )  (L)ut
(k  d ) Lk
(1  L)  
k 0 (d )(k  1)

d

   

Methodology
•
FIGARCH model:
1  Lt2   [1  L L1 l 2 ]t2
 t2 
•
Model distributions:
LNorm
LSkSt
•

  L  t2
1   L 

1 T
   ln2   ln( t2 )  zt2
2 t 1

 ( sz t  m) 2 2 It  
  v 1
 1 T 
v 1
 2 
2
 T ln 
k 
  ln    ln[ (v  2)]  ln
  ln(s)   ln( t )  (1  v) ln 1 
2
2
2
k

1
/
k
2
v

2






t

1



 

Pearson goodness-of-fit test
g
ni  Eni 2
i 1
Eni
P( g )  
Empirical results
 Unit root tests
ADF
BET
BET-FI
SOFIX
Return
-43.07788
-39.85883 -29.567
Squared return
-21.98689
-6.619556 -8.61359
Absolute Return
-20.79984
-9.647021 -7.90783
critical values: -2.567 (1%); -1.941 (5%); -1.616 (10%)
BUX
-25.3538
-7.83846
-9.66439
WIG
-48.0174
-15.1813
-9.9316
PX
-37.5705
-7.50629
-7.27383
KPSS intercept
BET
BET-FI
SOFIX
BUX
WIG
PX
Return
0.407412
0.71574
1.127404 0.180745 0.200012 0.244626
Squared return
0.573653
0.752302 1.052643 0.438187 0.903673 0.524725
Absolute Return
0.723473
1.029525 0.906252 0.550384 1.139445 0.475664
critical values: 0.739 (1%); 0.463 (5%); 0.347 (10%)
KPSS trend and intercept
BET
BET-FI
SOFIX
BUX
WIG
PX
Return
0.402853
0.100405 0.306984 0.17435 0.197787 0.234271
Squared return
0.576017
0.344872 0.604863 0.399608 0.666785 0.267768
Absolute Return
0.724169
0.468449 0.752958 0.506399 0.845108 0.412056
critical values: 0.216 (1%); 0.146 (5%); 0.119 (10%)
•For all indices we can reject the null of a I(1) process, as well as the null of I(0)
process
Nonparametric and semiparametric estimates
BET
Returns
Squared returns
Absolute returns
SOFIX
Returns
Squared returns
Absolute returns
WIG
Returns
Squared returns
Absolute returns
R/S H
0.6263291
0.7694657
0.8325581
R/S H
0.4805514
0.6632112
0.7128511
R/S H
0.6368706
0.7688137
0.8064734
Wavelet H
d (GPH)
0.4960724 0.1505057
0.5149534 0.3544141
0.614134 0.3796489
Wavelet H
d (GPH)
0.4407017 0.3027533
0.5394242 0.3865293
0.7029066 0.4445772
Wavelet H
d (GPH)
0.5623532 0.02013889
0.754955
0.303066
0.7746495 0.3800708
BET-FI
Returns
Squared returns
Absolute returns
BUX
Returns
Squared returns
Absolute returns
PX
Returns
Squared returns
Absolute returns
R/S H
0.7625203
0.655354
0.6909146
R/S H
0.5993203
0.7575655
0.8440155
R/S H
0.5964553
0.6827839
0.7325632
Wavelet H
d (GPH)
0.51634837
0.1842962
0.5797175
0.4405869
0.6645153
0.4339933
Wavelet H
d (GPH)
0.4874384 -0.03379396
0.7539982
0.3612529
0.7708724
0.4659885
Wavelet H
d (GPH)
0.524891
0.1023029
0.738836
0.3164356
0.7920493
0.4963583
• For most of the indices, the estimates indicate the presence of long memory in returns,
squared and absolute returns
• In case of SOFIX, the estimate of H using R/S and wavelet analysis indicate no long
memory in return series.
 Parametric estimates - ARFIMA model
Following Cheung(1993), we estimate different specifications of the ARFIMA (p, ξ, q) with
p,q=0:2 for each return series. The Akaike’s information Criterion (AIC), is used to choose
the best model that describes the data.
Model
BET
ARFIMA
(0,ξ,1)
BET-FI
ARFIMA
(0,ξ,0)
Ф1
-
Ф2
-
ξ
θ1
θ2
ln(L)
SIC
AIC
Skewness
Excess kurt.
J-B
Q(20)
SOFIX
BUX
WIG
PX
ARFIMA ARFIMA ARFIMA ARFIMA
(1,ξ,1)
(1,ξ,2)
(0,ξ,2)
(1,ξ,2)
-0.9034
0.52359
0.38472
(0.0000)
(0.0014)
(0.0061)
-
0.0461
(0.0048)
-0.17412
(0.0000)
-
0.1096
(0.0000)
-
0.0713
(0.0000)
-0.850
(0.0000)
-
-5862
4.0735
4.0694
-0.2621
2.7822
3915.92
95.6889
-5042
4.8101
4.8081
0.1727
2.1083
2189.28
26.2347
-4376
4.1411
4.1331
-0.5017
22.4701
54780.01
39.9162
0.0814 0.03135**
0.1026
(0.0000)
(0.0671) (0.0000)
0.53144
-0.0663 0.40374
(0.0007)
(0.0017) (0.0035)
0.06745
0.02975 0.09132
(0.0000)
(0.0823) (0.0000)
-5643
-5208
-5189
4.0755
3.7312
3.7418
4.0670
3.7248
3.7333
-0.1618
-0.2378
1.7411
5.6850
0.1972
8.0143
8442.15
1177.87 14939.26
94.0865
33.8216 67.5781
Ln(L) is the value of the maximized Gaussian Likelihood; AIC is the Akaike information criteria; the Q(20) is the
Ljung-Box test statistic with 20 degrees of freedom based on the standardized residuals
 Parametric estimates - ARFIMA model
• the long memory parameter ξ significantly differs from zero for all return series (for
WIG at 5% level of significance)
• the results seem to confirm the idea that long memory is a property of emerging markets
rather than developed markets.
• the standardized residuals display skewness and excess kurtosis, the departure from
normality beeing also confirmed by the J-B statistic
• Q-statistic indicate that the residuals are not independent, except for BET-FI and WIG ,
for which we cannot reject the null of independent residuals
 Testing for structural breaks
We use the Supremum F test proposed by Andrews and the methodology of Bai and
Perron for detecting structural breaks in return series
BET
BET-FI
SOFIX
F statistic
15.4789
20.0107
32.4592
p-value
0.001986 0.0002238
0.0000
Breakpoint at obs.no.
2440
1657
1746
Breakdate
7/23/2007 7/24/2007 10/30/2007
BUX
6.6669
0.12
-
WIG
PX
9.8167 8.0598
0.02872*0.06421**
2347
2329
7/6/2007 7/9/2007
• for BET, BET-FI, SOFIX and WIG the breakpoint corespond to the historical maximum
value of the index.
•For BUX, the F statistic indicate that the null hypothesis of no structural break cannot be
rejected
• we further split the sample in two subsamples depending on the breakdate, and we
reestimate all the procedures for each subsample
 Subsamples technique – non and semiparametric procedures
Full sample
BET
0.6107111
R/S Hurst Exponent
Wavelet Estimator for H 0.5090784
0.157134
GPH estimator
Full sample
BET-FI
0.7625203
R/S Hurst Exponent
Wavelet Estimator for H 0.5163484
0.1842962
GPH estimator
Full sample
SOFIX
0.4805514
R/S Hurst Exponent
Wavelet Estimator for H 0.4407017
0.3027533
GPH estimator
Full sample
WIG
0.6368706
R/S Hurst Exponent
Wavelet Estimator for H 0.5623532
0.02013889
GPH estimator
Full sample
PX
0.5964553
R/S Hurst Exponent
0.524891
Wavelet Estimator for H
0.1023029
GPH estimator
Before StrBreak
0.6232049
0.496072447
0.1518671
Before StrBreak
0.7571819
0.6916477
0.03527433
Before StrBreak
0.4152389
0.4328664
0.1097595
Before StrBreak
0.6276843
0.5611866
0.06955524
Before StrBreak
0.5932412
0.5214645
0.104628
After StrBreak
0.590718
0.4388235
0.1496232
After StrBreak
0.598512
0.6628132
0.2535763
After StrBreak
0.4356528
0.6404247
0.5111778
After StrBreak
0.7175956
0.7650836
0.02715736
After StrBreak
0.7381239
0.5052401
0.07740972
For most of the series, the subsamples appear to keep the full sample properties
For SOFIX although the Hurst exponent is still below 0.5 for each subsample , indicating no long
memory properties, the log-periodogram estimate indicate a significant value for ξ on the second
subsample. We therefore examine the ARFIMA estimates on each subsample in order to conclude
upon the reliability of the initially findings.
 Subsamples technique – ARFIMA estimates
BET
ARFIMA(0,ξ,1)
ξ
p-value
BET-FI
ARFIMA(0,ξ,0)
ξ
p-value
SOFIX
ARFIMA(1,ξ,1)
ξ
p-value
WIG
ARFIMA(0,ξ,2)
ξ
p-value
PX
ARFIMA(1,ξ,2)
ξ
p-value
Full sample
0.04656
0.0059
Full sample
0.1096
0.0000
Full sample
0.0713
0.0000
Full sample
0.03135
0.0671
Full sample
0.10255
0.0000
Before structural
break
0.04479
(0.00758)
After structural
break
0.003575
(0.798)
Before structural
break
After structural
break
0.07605
0.0000
Before structural
break
0.00004583
(0.998)
Before structural
break
0.00004583
(0.998657)
Before structural
break
0.05806
0.0000
0.1294
0.0000
After structural
break
0.15008
(0.000692)
After structural
break
0.05557
0.0000
After structural
break
0.07736
0.0000
For BET, BET-FI and PX the estimate of fractional parameter is significant for
both subsamples
In case of SOFIX and WIG it can be clearly observed that the long memory
patterns of the full sample are based in fact only on the second subsample, after
the structural break
ARFIMA-FIGARCH for the Romanian stock market indices
ARFIMA(0,ξ,1)-GARCH(1,d,1) ARFIMA(0,ξ,1)-FIGARCH(1,d,1)
BET
Skewed
Student t
Normal
Skewed
Student t
Normal
0.120116
(0.002)
0.038559**
(0.0902)
0.150723
0.0000
0.138226
(0.0051)
0.227037
0.00000
0.752619
0.00000
-
0.08669*
(0.0387)
0.049464*
(0.022)
0.13304
0.0000
ν
-
ln(k)
-5390.6
3.746309
32.6493**
(0.0263764)
23.2455
(0.1813325)
2.0222*
(0.0725)
13.69
(0.1875974)
145.5949
0.0000
5.16013
0.0000
0.025942
(0.3042)
-5291.51
3.678938
33.4769**
(0.021166)
33.4769
(0.021166)
1.5052
(0.1847)
10.00
(0.4401308)
44.7966
(0.91426)
-5364.9
3.729169
34.4892**
(0.0160801)
13.914
(0.7346793)
0.33442
(0.8923)
4.59
(0.9168096)
131.3915
(0.000001)
0.076813
(0.0559)
0.049475*
(0.0262)
0.134828
0.0000
0.300451
0.0000
0.42907
(0.0984)
0.54033
(0.044)
0.371215
0.0000
5.59537
0.0000
0.029179
(0.2343)
-5272.30
3.666299
32.0277 **
(0.0310313)
15.616
(0.6193279)
0.33497
(0.892)
3.53
(0.9659432)
39.5901
(0.975501)
0.979656
0.982682
1.111378
0.9694
0.752619
0.713049
0.645352
0.54033
μ
Ф1
Ф2
ξ
θ1
θ2
ώ
α1
α2
β1
β2
d
ln(L)
AIC
Q(20)
Qs(20)
ARCH(5)
RBD(10)
P(60)
Σαi+Σβi
Σβi
0.173226
(0.0033)
0.269633
0.0000
0.713049
0.0000
-
0.103417
(0.0041)
0.0322
(0.1665)
0.156281
0.0000
0.095609
(0.0402)
0.466026
0.0003
0.645352
0.00000
0.519482
0.0000
-
ARFIMA(0,ξ,0)-GARCH(1,1)ARFIMA(0,ξ,0)-FIGARCH(1,1)
Skewed
Skewed
Normal
Normal
Student t
Student t
0.1170
0.1310
0.1135
0.1315
μ
(0.1462)
(0.0715)
(0.154)
(0.0677)
Ф1
Ф2
ξ
0.1006
0.0852
0.0983
0.0828
0.0000
0.0001
0
0.0002
θ1
θ2
ώ
0.148166
0.191158
0.179273
0.240682
(0.0073)
(0.0086)
(0.0137)
0.0288
α1
0.177186
0.217455
0.161053
0.227043
0.00000
0.0000
0.1388
(0.0548)
α2
β1
0.81906
0.785537
0.708688
0.562969
0.00000
0.0000
0.00000
(0.0006)
β2
d
0
0
0.755778
0.595596
0.0006
0.0000
ν
5.205205
5.536404
0.0000
0.0000
ln(k)
0.077417
0.081156
(0.0052)
(0.0033)
ln(L)
-4688.2
-4618.66
-4686.0
-4613.20
AIC
4.476064
4.411697
4.474999
4.407442
Q(20)
24.8211
31.6462**
25.1964
32.2579**
[0.2083615]
[0.0472168] [0.1940193]
[0.0406264]
Qs(20)
10.622
12.7388
9.19695
9.98524
[0.9097031]
[0.8068488] [0.9550042]
[0.9323872]
ARCH(5)
0.55115
0.76162
0.38078
0.51003
[0.7376]
[0.5775]
[0.8622]
[0.7689]
RBD(10)
3.46
4.35
2.43
1.93
[0.9683062]
[0.9303439] [0.9918248]
[0.9968829]
P(60)
136.9056
53.5866
139.5951
55.4750
0.0000
(0.674567)
(0.000001)
(0.606217)
BET-FI
Σαi+Σβi
Σβi
0.996246
1.002992
0.869741
0.790012
0.81906
0.785537
0.708688
0.562969
ARFIMA-FIGARCH- Remarks
• the sum of the estimates of α1 and β1 in the ARFIMA–GARCH model is very close to one,
indicating that the volatility process is highly persistent
• the estimates of β1 in the GARCH model are very high, suggesting a strong autoregressive
component in the conditional variance process
•in the ARFIMA–FIGARCH model, the estimates of both long memory parameters ξ and d
are significantly different from zero
•the results indicate that the β1 estimates are lower in the FIGARCH than those of in the
GARCH model.
•according to the AIC, the FIGARCH models fit the return series better than the GARCH
models
• P(60) test statistics reconfirm the relevance of skewed Student-t
ARFIMA-FIGARCH estimates for PX and BUX
PX
ξ
d
P(60)
BUX
ξ
d
P(60)
ARFIMA(1,ξ,2)-GARCH(1,1) ARFIMA(1,ξ,2)-FIGARCH(1,d,1)
Skewed
Skewed
Normal
Normal
Student t
Student t
0.1609
0.1753
0.1554
0.1759
(0.0095)
(0.0016)
(0.0143)
(0.0013)
0.702972
0.60617
(0.0000)
(0.0000)
61.7239
58.7045
88.5967
56.1596
(0.1675)
(0.1866)
(0.0009)
(0.2244)
ARFIMA(1,ξ,2)-GARCH(1,1) ARFIMA(1,ξ,2)-FIGARCH(1,d,1)
Skewed
Skewed
Normal
Normal
Student t
Student t
-0.0283
0.0640
0.0600
0.067143**
(0.0251)
(0.0589)
(0.0659)
(0.0589)
0
0.462322
0.455978
(0.0000)
(0.0000)
90.3605
66.0320
65.7296
64.1739
(0.0006)
(0.0527)
(0.0671)
(0.0592)
FIGARCH estimates for WIG and SOFIX
SOFIX
d
P(60)
FIGARCH(1,d,1)
Normal Skewed Student t
0.541028
0.568614
0.0000
0.0000
285.5021
74.4403*
0.0000
(0.084777)
WIG
d
P(60)
FIGARCH(1,d,1)
Normal
Skewed Student t
0.47143
0.493638
0.0000
0.0000
104.473
68.619*
(5E-05)
(0.061)
Conclusions and further improvements
• The tests and estimated models show evidence of dual long memory in Romanian, Czech
Republic and Hungarian stock markets, while Bulgarian and Poland’s markets show
evidence of long memory in volatility.
• The results support the idea that the detection of long memory properties in emerging
markes is more likely than in developed markets, having implications in portfolio
diversification, speculative strategies and risk management.
•However, one should use various methods and techniques when investingating the presence
of long memory, due to the sensitivity of the results to the selected estimation method.
•Structural breaks and regime shifts can significantly affect the results. Therefore, one
should use such techniques designed to account for these processes which could induce to a
short memory process similar patterns with a long memory process.
• Further research could be conducted using the models developed by Baillie and Morana
(2007,2009), namely Adaptive-FIGARCH and Adaptive-ARFIMA, and their generalisation
for dual long memory processes, the A2-ARFIMA-FIGARCH model, beeing designed to
take into account for both long memory and structural change in the conditional mean and
variance.
References
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