Exploring Dual Long Memory in Returns and Exploring Dual Long Memory in Returns and Volatility across the Central and Eastern European Volatility across the Central and Eastern European

stock markets stock markets

Academy of Economic StudiesAcademy of Economic StudiesDoctoral School of Finance and Banking- DOFINDoctoral School of Finance and Banking- DOFIN

Bucharest, July 2009Bucharest, July 2009

Msc. Student: Mihaela SanduMsc. Student: Mihaela SanduSupervisor: PhD.Professor MoisSupervisor: PhD.Professor Moisăă Alt Altăărr

Dissertation paper outlineDissertation paper outline

• Importance of long memory Importance of long memory

• Aims of the paperAims of the paper

• Data & MethodologyData & Methodology

• Nonparametric & semiparametric aproachesNonparametric & semiparametric aproaches

• Parametric approach: ARFIMA / FIGARCHParametric approach: ARFIMA / FIGARCH

• Structural breaksStructural breaks

• The joint ARFIMA-FIGARCH modelThe joint ARFIMA-FIGARCH model

• Model distributionsModel distributions

• Empirical resultsEmpirical results

•Conclusions and further improvementsConclusions and further improvements

• References References

Long memoryLong memory

• Contradicts the EMH weak-form by allowing investors and portfolio managers to Contradicts the EMH weak-form by allowing investors and portfolio managers to make prediction and to construct speculative strategies make prediction and to construct speculative strategies

•The price of an asset determined in an efficient market should follow a martingale 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 process in which each price change is unaffected by its predecessor and has no memory . Pricing derivative securities with martingale methods may not be memory . Pricing derivative securities with martingale methods may not be appropriate if the underlying continuous stochastic processes exhibit long memory appropriate if the underlying continuous stochastic processes exhibit long memory

• Implications to assest allocation decisions and risk managementImplications to assest allocation decisions and risk management

Aims of the paperAims of the paper

• To ivestigate the presence of long memory in stock returns via non-, semi- To ivestigate the presence of long memory in stock returns via non-, semi- and parametric techniquesand parametric techniques

• To distinguish between long memory and structural breaks within return To distinguish between long memory and structural breaks within return seriesseries

• To found evidences of dual long memory processes within CEE emerging To found evidences of dual long memory processes within CEE emerging stock marketsstock markets

Short vs. Long memory processesShort vs. Long memory processes

hCrhρ 10 r 12 d~Chhρ h

1h

hρ

1h

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ωfx dx ~Cωωf 2 0ω

vs.vs.

The DataThe Data

• Six indices representing five CEE emerging stock markets: BET, BET-FI, Six indices representing five CEE emerging stock markets: BET, BET-FI, SOFIX, BUX, WIG, PXSOFIX, BUX, WIG, PX

• Daily closing stock prices transformed into continuously compounded returns Daily closing stock prices transformed into continuously compounded returns

• The estimations and tests were performed in R version 2.9.0.The estimations and tests were performed in R version 2.9.0.

• For estimating ARFIMA-FIGARCH model, the Ox Console version 5.10, For estimating ARFIMA-FIGARCH model, the Ox Console version 5.10, together with the G@rch Console 4.2 were used.together with the G@rch Console 4.2 were used.

MethodologyMethodology• Unit root tests: Unit root tests:

ADFADF

KPSS statistic:KPSS statistic:

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• Rescaled range statisticRescaled range statistic

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• Wavelet based estimatorWavelet based estimator

• Log-periodogram estimator (GPH)Log-periodogram estimator (GPH)

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• ARFIMA model: ARFIMA model: ttd uLyLL )()()1)((

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MethodologyMethodology

• FIGARCH model: FIGARCH model:

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• Model distributions: Model distributions:

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• Pearson goodness-of-fit test Pearson goodness-of-fit test

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Empirical resultsEmpirical results Unit root testsUnit root tests

ADFBET BET-FI SOFIX BUX WIG PX

Return -43.07788 -39.85883 -29.567 -25.3538 -48.0174 -37.5705Squared return -21.98689 -6.619556 -8.61359 -7.83846 -15.1813 -7.50629Absolute Return -20.79984 -9.647021 -7.90783 -9.66439 -9.9316 -7.27383critical values: -2.567 (1%); -1.941 (5%); -1.616 (10%)

KPSS interceptBET BET-FI SOFIX BUX WIG PX

Return 0.407412 0.71574 1.127404 0.180745 0.200012 0.244626Squared return 0.573653 0.752302 1.052643 0.438187 0.903673 0.524725Absolute Return 0.723473 1.029525 0.906252 0.550384 1.139445 0.475664critical values: 0.739 (1%); 0.463 (5%); 0.347 (10%)

KPSS trend and interceptBET BET-FI SOFIX BUX WIG PX

Return 0.402853 0.100405 0.306984 0.17435 0.197787 0.234271Squared return 0.576017 0.344872 0.604863 0.399608 0.666785 0.267768Absolute Return 0.724169 0.468449 0.752958 0.506399 0.845108 0.412056critical 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) For all indices we can reject the null of a I(1) process, as well as the null of I(0) processprocess

BET R/S H Wavelet H d (GPH) BET-FI R/S H Wavelet H d (GPH)Returns 0.6263291 0.4960724 0.1505057 Returns 0.7625203 0.51634837 0.1842962Squared returns 0.7694657 0.5149534 0.3544141 Squared returns 0.655354 0.5797175 0.4405869Absolute returns 0.8325581 0.614134 0.3796489 Absolute returns 0.6909146 0.6645153 0.4339933SOFIX R/S H Wavelet H d (GPH) BUX R/S H Wavelet H d (GPH)Returns 0.4805514 0.4407017 0.3027533 Returns 0.5993203 0.4874384 -0.03379396Squared returns 0.6632112 0.5394242 0.3865293 Squared returns 0.7575655 0.7539982 0.3612529Absolute returns 0.7128511 0.7029066 0.4445772 Absolute returns 0.8440155 0.7708724 0.4659885WIG R/S H Wavelet H d (GPH) PX R/S H Wavelet H d (GPH)Returns 0.6368706 0.5623532 0.02013889 Returns 0.5964553 0.524891 0.1023029Squared returns 0.7688137 0.754955 0.303066 Squared returns 0.6827839 0.738836 0.3164356Absolute returns 0.8064734 0.7746495 0.3800708 Absolute returns 0.7325632 0.7920493 0.4963583

Nonparametric and semiparametric estimatesNonparametric and semiparametric estimates

• For most of the indices, the estimates indicate the presence of long memory in returns, For most of the indices, the estimates indicate the presence of long memory in returns, squared and absolute returnssquared and absolute returns

• In case of SOFIX, the estimate of H using R/S and wavelet analysis indicate no long In case of SOFIX, the estimate of H using R/S and wavelet analysis indicate no long memory in return series.memory in return series.

Parametric estimates - ARFIMA modelParametric estimates - ARFIMA model

BET BET-FI SOFIX BUX WIG PXARFIMA ARFIMA ARFIMA ARFIMA ARFIMA ARFIMA

(0,ξ,1) (0,ξ,0) (1,ξ,1) (1,ξ,2) (0,ξ,2) (1,ξ,2)

Ф1 - - -0.9034 0.52359 - 0.38472(0.0000) (0.0014) (0.0061)

Ф2 - - - - - -

ξ 0.0461 0.1096 0.0713 0.0814 0.03135** 0.1026(0.0048) (0.0000) (0.0000) (0.0000) (0.0671) (0.0000)

θ1 -0.17412 - -0.850 0.53144 -0.0663 0.40374(0.0000) (0.0000) (0.0007) (0.0017) (0.0035)

θ2 - - - 0.06745 0.02975 0.09132(0.0000) (0.0823) (0.0000)

ln(L) -5862 -5042 -4376 -5643 -5208 -5189SIC 4.0735 4.8101 4.1411 4.0755 3.7312 3.7418AIC 4.0694 4.8081 4.1331 4.0670 3.7248 3.7333Skewness -0.2621 0.1727 -0.5017 -0.1618 -0.2378 1.7411Excess kurt. 2.7822 2.1083 22.4701 5.6850 0.1972 8.0143J-B 3915.92 2189.28 54780.01 8442.15 1177.87 14939.26Q(20) 95.6889 26.2347 39.9162 94.0865 33.8216 67.5781

Model

Following Cheung(1993), we estimate different specifications of the ARFIMA (p, ξ, q) with 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 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. the best model that describes the data.

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 modelParametric estimates - ARFIMA model

• the long memory parameter the long memory parameter ξξ significantly differs from zero for all return series (for WIG significantly differs from zero for all return series (for WIG at 5% level of significance) at 5% level of significance)

• the results seem to confirm the idea that long memory is a property of emerging markets the results seem to confirm the idea that long memory is a property of emerging markets rather than developed markets.rather than developed markets.

• the standardized residuals display skewness and excess kurtosis, the departure from the standardized residuals display skewness and excess kurtosis, the departure from normality beeing also confirmed by the J-B statisticnormality beeing also confirmed by the J-B statistic

• Q-statistic indicate that the residuals are not independent, except for BET-FI and WIG , Q-statistic indicate that the residuals are not independent, except for BET-FI and WIG , for which we cannot reject the null of independent residualsfor which we cannot reject the null of independent residuals

Testing for structural breaksTesting for structural breaks

BET BET-FI SOFIX BUX WIG PXF statistic 15.4789 20.0107 32.4592 6.6669 9.8167 8.0598p-value 0.001986 0.0002238 0.0000 0.12 0.02872*0.06421**Breakpoint at obs.no. 2440 1657 1746 - 2347 2329Breakdate 7/23/2007 7/24/2007 10/30/2007 - 7/6/2007 7/9/2007

We use the Supremum F test proposed by Andrews and the methodology of Bai and We use the Supremum F test proposed by Andrews and the methodology of Bai and Perron for detecting structural breaks in return seriesPerron for detecting structural breaks in return series

• for BET, BET-FI, SOFIX and WIG the breakpoint corespond to the historical maximum for BET, BET-FI, SOFIX and WIG the breakpoint corespond to the historical maximum value of the index.value of the index.

•For BUX, the F statistic indicate that the null hypothesis of no structural break cannot be For BUX, the F statistic indicate that the null hypothesis of no structural break cannot be rejectedrejected

• we further split the sample in two subsamples depending on the breakdate, and we we further split the sample in two subsamples depending on the breakdate, and we reestimate all the procedures for each subsamplereestimate all the procedures for each subsample

Subsamples technique – non and semiparametric proceduresSubsamples technique – non and semiparametric procedures

BET Full sample Before StrBreak After StrBreak R/S Hurst Exponent 0.6107111 0.6232049 0.590718Wavelet Estimator for H 0.5090784 0.496072447 0.4388235GPH estimator 0.157134 0.1518671 0.1496232

BET-FI Full sample Before StrBreak After StrBreak

R/S Hurst Exponent 0.7625203 0.7571819 0.598512Wavelet Estimator for H 0.5163484 0.6916477 0.6628132GPH estimator 0.1842962 0.03527433 0.2535763

SOFIX Full sample Before StrBreak After StrBreak R/S Hurst Exponent 0.4805514 0.4152389 0.4356528Wavelet Estimator for H 0.4407017 0.4328664 0.6404247GPH estimator 0.3027533 0.1097595 0.5111778

WIG Full sample Before StrBreak After StrBreak R/S Hurst Exponent 0.6368706 0.6276843 0.7175956Wavelet Estimator for H 0.5623532 0.5611866 0.7650836GPH estimator 0.02013889 0.06955524 0.02715736

PX Full sample Before StrBreak After StrBreak R/S Hurst Exponent 0.5964553 0.5932412 0.7381239Wavelet Estimator for H 0.524891 0.5214645 0.5052401GPH estimator 0.1023029 0.104628 0.07740972

For most of the series, the subsamples appear to keep the full sample propertiesFor 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 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 memory properties, the log-periodogram estimate indicate a significant value for ξξ on the second on the second subsample. We therefore examine the ARFIMA estimates on each subsample in order to conclude subsample. We therefore examine the ARFIMA estimates on each subsample in order to conclude upon the reliability of the initially findings.upon the reliability of the initially findings.

BETARFIMA(0,ξ,1)

Full sampleBefore structural

breakAfter structural

break

ξ 0.04656 0.04479 0.003575p-value 0.0059 (0.00758) (0.798)

BET-FIARFIMA(0,ξ,0)

Full sampleBefore structural

breakAfter structural

break

ξ 0.1096 0.07605 0.1294p-value 0.0000 0.0000 0.0000SOFIX

ARFIMA(1,ξ,1)Full sample

Before structural break

After structural break

ξ 0.0713 0.00004583 0.15008p-value 0.0000 (0.998) (0.000692)

WIGARFIMA(0,ξ,2)

Full sampleBefore structural

breakAfter structural

break

ξ 0.03135 0.00004583 0.05557p-value 0.0671 (0.998657) 0.0000

PXARFIMA(1,ξ,2)

Full sampleBefore structural

breakAfter structural

break

ξ 0.10255 0.05806 0.07736p-value 0.0000 0.0000 0.0000

Subsamples technique – ARFIMA estimatesSubsamples technique – ARFIMA estimates

For BET, BET-FI and PX the estimate of fractional parameter is significant for For BET, BET-FI and PX the estimate of fractional parameter is significant for both subsamples both subsamples

In case of SOFIX and WIG it can be clearly observed that the long memory 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 patterns of the full sample are based in fact only on the second subsample, after the structural breakthe structural break

ARFIMA-FIGARCH for the Romanian stock market indices

μ 0.120116 0.08669* 0.103417 0.076813(0.002) (0.0387) (0.0041) (0.0559)

Ф1 - - - -

Ф2 - - - -ξ 0.038559** 0.049464* 0.0322 0.049475*

(0.0902) (0.022) (0.1665) (0.0262)θ1 0.150723 0.13304 0.156281 0.134828

0.0000 0.0000 0.0000 0.0000θ2 - - -ώ 0.138226 0.173226 0.095609 0.300451

(0.0051) (0.0033) (0.0402) 0.0000α1 0.227037 0.269633 0.466026 0.42907

0.00000 0.0000 0.0003 (0.0984)α2 - - - -

β1 0.752619 0.713049 0.645352 0.540330.00000 0.0000 0.00000 (0.044)

β2 - - - -d - - 0.519482 0.371215

0.0000 0.0000ν - 5.16013 - 5.59537

0.0000 0.0000ln(k) - 0.025942 - 0.029179

(0.3042) (0.2343)ln(L) -5390.6 -5291.51 -5364.9 -5272.30AIC 3.746309 3.678938 3.729169 3.666299Q(20) 32.6493** 33.4769** 34.4892** 32.0277 **

(0.0263764) (0.021166) (0.0160801) (0.0310313)Qs(20) 23.2455 33.4769 13.914 15.616

(0.1813325) (0.021166) (0.7346793) (0.6193279)ARCH(5) 2.0222* 1.5052 0.33442 0.33497

(0.0725) (0.1847) (0.8923) (0.892)RBD(10) 13.69 10.00 4.59 3.53

(0.1875974) (0.4401308) (0.9168096) (0.9659432)P(60) 145.5949 44.7966 131.3915 39.5901

0.0000 (0.91426) (0.000001) (0.975501)

Σαi+Σβi 0.979656 0.982682 1.111378 0.9694

Σβi 0.752619 0.713049 0.645352 0.54033

BET

ARFIMA(0,ξ,1)-GARCH(1,d,1) ARFIMA(0,ξ,1)-FIGARCH(1,d,1)

NormalSkewed

Student tNormal

Skewed Student t

ARFIMA(0,ξ,0)-GARCH(1,1)ARFIMA(0,ξ,0)-FIGARCH(1,1)

NormalSkewed

Student tNormal

Skewed Student t

μ 0.1170 0.1310 0.1135 0.1315(0.1462) (0.0715) (0.154) (0.0677)

Ф1 - - - -Ф2 - - - -ξ 0.1006 0.0852 0.0983 0.0828

0.0000 0.0001 0 0.0002θ1 - - - -θ2 - - - -ώ 0.148166 0.191158 0.179273 0.240682

(0.0073) (0.0086) (0.0137) 0.0288α1 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 0.0000ν - 5.205205 - 5.536404

0.0000 0.0000ln(k) - 0.077417 - 0.081156

(0.0052) (0.0033)ln(L) -4688.2 -4618.66 -4686.0 -4613.20AIC 4.476064 4.411697 4.474999 4.407442Q(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)

Σαi+Σβi 0.996246 1.002992 0.869741 0.790012

Σβi 0.81906 0.785537 0.708688 0.562969

BET-FI

ARFIMA-FIGARCH- Remarks

• the sum of the estimates of αthe sum of the estimates of α11 and β and β11 in the ARFIMA–GARCH model is very close to one, in the ARFIMA–GARCH model is very close to one,

indicating that the volatility process is highly persistentindicating that the volatility process is highly persistent

• the estimates of βthe estimates of β11 in the GARCH model are very high, suggesting a strong autoregressive in the GARCH model are very high, suggesting a strong autoregressive

component in the conditional variance processcomponent in the conditional variance process

•in the ARFIMA–FIGARCH model, the estimates of both long memory parameters ξ and d in the ARFIMA–FIGARCH model, the estimates of both long memory parameters ξ and d are significantly different from zeroare significantly different from zero

•the results indicate that the βthe results indicate that the β11 estimates are lower in the FIGARCH than those of in the estimates are lower in the FIGARCH than those of in the

GARCH model. GARCH model.

•according to the AIC, the FIGARCH models fit the return series better than the GARCH according to the AIC, the FIGARCH models fit the return series better than the GARCH modelsmodels

• P(60) test statistics reconfirm the relevance of skewed Student-tP(60) test statistics reconfirm the relevance of skewed Student-t

ξ 0.1609 0.1753 0.1554 0.1759(0.0095) (0.0016) (0.0143) (0.0013)

d - - 0.702972 0.60617(0.0000) (0.0000)

P(60) 61.7239 58.7045 88.5967 56.1596(0.1675) (0.1866) (0.0009) (0.2244)

ξ -0.0283 0.0640 0.0600 0.067143**(0.0251) (0.0589) (0.0659) (0.0589)

d - 0 0.462322 0.455978(0.0000) (0.0000)

P(60) 90.3605 66.0320 65.7296 64.1739(0.0006) (0.0527) (0.0671) (0.0592)

BUXARFIMA(1,ξ,2)-GARCH(1,1) ARFIMA(1,ξ,2)-FIGARCH(1,d,1)

NormalSkewed

Student tNormal

Skewed Student t

PXARFIMA(1,ξ,2)-GARCH(1,1) ARFIMA(1,ξ,2)-FIGARCH(1,d,1)

NormalSkewed

Student tNormal

Skewed Student t

d 0.541028 0.5686140.0000 0.0000

P(60) 285.5021 74.4403*0.0000 (0.084777)

SOFIXFIGARCH(1,d,1)

Normal Skewed Student t

ARFIMA-FIGARCH estimates for PX and BUX

FIGARCH estimates for WIG and SOFIX

d 0.47143 0.4936380.0000 0.0000

P(60) 104.473 68.619*(5E-05) (0.061)

WIGFIGARCH(1,d,1)

Normal Skewed Student t

Conclusions and further improvementsConclusions and further improvements

• The tests and estimated models show evidence of dual long memory in Romanian, Czech 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 Republic and Hungarian stock markets, while Bulgarian and Poland’s markets show evidence of long memory in volatility.evidence of long memory in volatility.

• The results support the idea that the detection of long memory properties in emerging 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 markes is more likely than in developed markets, having implications in portfolio diversification, speculative strategies and risk management.diversification, speculative strategies and risk management.

•However, one should use various methods and techniques when investingating the presence 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.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 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 use such techniques designed to account for these processes which could induce to a short memory process similar patterns with a long memory process.memory process similar patterns with a long memory process.

• Further research could be conducted using the models developed by Baillie and Morana Further research could be conducted using the models developed by Baillie and Morana (2007,2009), namely Adaptive-FIGARCH and Adaptive-ARFIMA, and their generalisation (2007,2009), namely Adaptive-FIGARCH and Adaptive-ARFIMA, and their generalisation for dual long memory processes, the Afor dual long memory processes, the A22-ARFIMA-FIGARCH model, beeing designed to -ARFIMA-FIGARCH model, beeing designed to

take into account for both long memory and structural change in the conditional mean and take into account for both long memory and structural change in the conditional mean and variance. variance.

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