Главная Коллекция "Revolution" Экономика и экономическая теория Measuring and forecasting volatility of financial assets

Measuring and forecasting volatility of financial assets

Market analysis and assess regulation policies. Pre-crisis and post-crisis windows definition. Forecast comparison for standalone models. Rolling regression with dynamic forecast for models. Realized Volatility, Bipower Variation. Combination of models.

Рубрика Экономика и экономическая теория
Вид дипломная работа
Язык английский
Дата добавления 28.08.2016
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Statistics

Period

1

5

10

15

RMSE

Pre-crisis

0.054

0.063

0.064

0.066

Crisis

0.063

0.078

0.084

0.089

Total

0.238

0.320

0.384

0.424

MAE

Pre-crisis

0.032

0.040

0.041

0.045

Crisis

0.039

0.049

0.053

0.053

Total

0.076

0.090

0.103

0.112

MAPE

Pre-crisis

49.54

60.31

63.35

76.35

Crisis

41.80

55.79

63.57

67.88

Total

35.35

40.04

46.12

50.30

TIC

Pre-crisis

0.296

0.303

0.308

0.314

Crisis

0.241

0.287

0.303

0.326

Total

0.402

0.470

0.500

0.546

Table 26. HAR-RV-CJ (logarithms) forecasts for 1, 5, 10 and 15 minute frequency

Statistics

Period

1

5

10

15

RMSE

Pre-crisis

0.053

0.062

0.064

0.065

Crisis

0.060

0.074

0.080

0.084

Total

0.241

0.326

0.396

0.436

MAE

Pre-crisis

0.029

0.036

0.038

0.039

Crisis

0.034

0.042

0.045

0.044

Total

0.076

0.092

0.103

0.112

MAPE

Pre-crisis

39.33

47.64

52.82

56.15

Crisis

33.87

43.13

49.18

51.24

Total

35.98

39.00

42.62

47.09

TIC

Pre-crisis

0.305

0.314

0.323

0.332

Crisis

0.240

0.286

0.305

0.329

Total

0.420

0.499

0.551

0.591

5.3.5 Forecast comparison for standalone models

As the result of previous analysis, models HAR-RV, HAR-RR, HAR-RV-J and HAR-RV-CJ are viewed only at 1-minute frequency as the best models. As Realized Range and Realized Volatility are different volatility estimators, models that were trained to forecast those values (ex. HAR-RV that forecasts daily Realized Volatility and HAR-RR that forecasts daily Realized Range) cannot be compared by RMSE or MAE criteria. Suitability of RR and RV estimators for Russian market is out of scope in this paper.

GARCH models' performance can be analysed only compared to the values that are observed, for example Realized Volatility. Due to the past analysis, forecasts, provided by GARCH model is compared to Realized Volatility estimator computed at 1-minute frequency as having lower errors for all HAR-RV models in scope of the analysis. To be more specific, forecasted values (see 4.2.1) will be compared to observed values of realized daily volatility computed at 1-minute frequency.

From Table 27, that reflects models' performance during “pre-crisis” period it can be concluded, that HAR-RR model modifications outperform other models, especially HAR-RR (sq. roots) - HAR_RR_SQ and HAR-RR (logarithms) - HAR_RR_LN, that have lowest MAPE and TIC values over other models. GARCH (1, 1) has highest values almost in all error statistics except for TIC. Logarithm modifications of models outperform “standard” models by MAPE criteria, but have worse performance if comparison is made by TIC statistics.

Table 27. Forecast performance in pre-crisis period

Model

RMSE

Grade

MAE

Grade

MAPE

Grade

TIC

Grade

HAR_RV

0.059968

12

0.04082

12

76.18657

12

0.305162

9

HAR_RV_SQ

0.056613

9

0.033509

9

52.95236

9

0.310605

12

HAR_RV_LN

0.055647

8

0.03064

6

43.29083

5

0.320729

13

HAR_RR

0.007423

3

0.005495

3

47.69785

7

0.204858

1

HAR_RR_SQ

0.007178

2

0.005022

2

38.60555

2

0.206079

2

HAR_RR_LN

0.007163

1

0.00488

1

35.02418

1

0.211956

3

HAR_RV_J

0.057548

10

0.039502

10

72.26324

10

0.293912

4

HAR_RV_J_SQ

0.053908

6

0.031508

7

47.59028

6

0.296108

7

HAR_RV_J_LN

0.053684

5

0.029093

4

39.64342

4

0.308098

10

HAR_RV_CJ

0.057997

11

0.040471

11

76.06275

11

0.295463

5

HAR_RV_CJ_SQ

0.054027

7

0.032179

8

49.54101

8

0.295556

6

HAR_RV_CJ_LN

0.053397

4

0.029193

5

39.33363

3

0.304883

8

GARCH (1, 1)

0.067409

13

0.055535

13

113.2253

13

0.308708

11

Table 28 provides performance measures during period of crisis. It confirms exceptional performance of HAR-RR (logarithms) model. However, relatively low values of MAPE and TIC criteria are obtained for logarithm and sq. root modifications of HAR-RV-CJ and HAR-RV-J models on contrary to “pre-crisis” period, where HAR-RV-CJ and HAR-RV-J models without modifications showed lower MAPE statistics. “Standard” HAR-RV model shows poor performance as in the “pre-crisis” period.

Table 28. Forecast performance in crisis period

Model

RMSE

Grade

MAE

Grade

MAPE

Grade

TIC

Grade

HAR_RV

0.075596

10

0.058374

10

73.5695

12

0.264563

12

HAR_RV_SQ

0.062738

9

0.038683

7

42.89954

8

0.243256

8

HAR_RV_LN

0.060165

6

0.034047

4

34.34736

3

0.244348

9

HAR_RR

0.013055

3

0.010358

3

54.01801

9

0.206676

3

HAR_RR_SQ

0.011199

2

0.007846

2

36.24699

5

0.190381

1

HAR_RR_LN

0.010841

1

0.007316

1

31.614

1

0.190447

2

HAR_RV_J

0.075652

11

0.059355

11

73.32912

11

0.262968

10

HAR_RV_J_SQ

0.062274

7

0.039065

9

42.37048

7

0.239914

4

HAR_RV_J_LN

0.059789

5

0.034447

6

34.36391

4

0.241255

7

HAR_RV_CJ

0.07605

13

0.059817

13

73.24186

10

0.264065

11

HAR_RV_CJ_SQ

0.062639

8

0.039014

8

41.79518

6

0.240522

6

HAR_RV_CJ_LN

0.059775

4

0.034254

5

33.86782

2

0.239951

5

GARCH (1, 1)

0.076019

12

0.059496

12

77.4244

13

0.266131

13

Table 29 shows the results of evaluation models' performance over the total period. Models were trained during “pre-crisis” period in the country and tested during “crisis” period. HAR-RR model's modifications have lower MAPE and TIC values compared to other models. HAR-RV model has relatively high values of criteria and supports findings for “crisis” and “pre-crisis” performance evaluations. GARCH (1, 1) model shows the worst performance that is consistent with other periods in scope. Sq. root modifications of HAR-RV-J and HAR-RV-CJ shows at least not worst performance then logarithm modifications of the same models, consequently HAR-RV-CJ outperform HAR-RV-J for this testing case. For the total period, sq. root modifications of HAR-RV-J and HAR-RV-CJ show lower TIC and higher MAPE statistics than logarithm modifications of respective models.

Table 29. Forecast performance in total period

Model

RMSE

Grade

MAE

Grade

MAPE

Grade

TIC

Grade

HAR_RV

0.243069

10

0.079298

10

39.96402

10

0.410372

8

HAR_RV_SQ

0.244494

11

0.076162

6

35.15096

5

0.434102

11

HAR_RV_LN

0.251882

12

0.076088

5

32.37465

3

0.474824

12

HAR_RR

0.045633

1

0.015972

3

33.12327

4

0.343787

1

HAR_RR_SQ

0.046574

2

0.015837

2

31.16515

2

0.365328

2

HAR_RR_LN

0.049136

3

0.015714

1

29.47063

1

0.40797

7

HAR_RV_J

0.240854

8

0.081868

12

43.00764

12

0.390105

3

HAR_RV_J_SQ

0.238926

5

0.079975

11

40.03568

11

0.393319

4

HAR_RV_J_LN

0.241707

9

0.078384

9

37.75072

9

0.422281

10

HAR_RV_CJ

0.24037

6

0.077032

8

36.87273

8

0.403189

6

HAR_RV_CJ_SQ

0.238294

4

0.075755

4

35.34756

6

0.402367

5

HAR_RV_CJ_LN

0.240815

7

0.076293

7

35.98127

7

0.420326

9

GARCH (1, 1)

0.291607

13

0.113634

13

68.46508

13

0.582375

13

Model Confidence Set test was done for every period in scope. Results are as follows:

Table 30. Models' ranks during different periods

Period

Model

Pre-crisis

Crisis

Total

GARCH (1,1)

-

-

-

HAR-RV

-

-

-

HAR-RV (sq. roots)

-

-

-

HAR-RV (ln)

3

-

-

HAR-RV-J

-

-

5

HAR-RV-J(sq. roots)

-

-

2

HAR-RV-J(ln)

2

2

6

HAR-RV-CJ

-

-

3

HAR-RV-CJ(sq. roots)

-

-

1

HAR-RV-CJ(ln)

1

1

4

If the value for the model in Table 30 is “-” then it is treated as eliminated during the process of Model Confidence Set building. It can be interpreted, that models, that were selected (have valid rank) are different from eliminated models at least at 5% significance level. However, models that were selected cannot be distinguished from each other at the same confidence level. It proofs conclusions stated above about outperformance of logarithm modifications of models on “pre-crisis” and “crisis” periods and better performance (still not statistically significant at 5% level) of square root modifications for total period.

4.4.1

5.3.6 Rolling regression with dynamic forecast for models

1 step ahead static forecasts presented in 5.3.1 - 5.3.4 can help in comparing models, but out-of-sample forecasts are not always done based on the single regression with constant parameters that are applied for the next steps, but for continuous re-estimation of the regression on moving window. Total period will be in scope and as the first step of the process models (regressions) are estimated on train sample of the total period. Then forecasts will be made for 1 day, 1 week (5 days) and 1 month (22 days) ahead. After the forecasts are made estimation window jumps for the respective period in time and the process starts from the beginning. Forecasted values for testing period are then being tested as described in 4.3.

Window size is 985 trading days which is the size of training sample for the total period.

Results are presented in Table 31 for HAR-RR, HAR-RV, HAR-RV-J and HAR-RV-CJ models and their modifications. It is done only for 1-minute frequency estimators as they have shown lower values of errors' statistics for 1 day rolling regression forecasts. Results show that “standard” models have worse performance than modifications based on any criteria. This support the logarithm and sq. root models outperformance over “standard” modifications.

Exception is for comparison of HAR-RR model family. Logarithm modification has lower values of MAE and MAPE but the highest in RMSE and TIC.

Table 31. Errors for 1-step ahead rolling regression forecast

Model

RMSE

MAE

MAPE

TIC

HAR_RV

0.1831

0.0728

41.5980

0.3339

HAR_RV_SQ

0.1814

0.0684

35.1909

0.3460

HAR_RV_LN

0.1899

0.0685

32.4186

0.3861

HAR_RV_J

0.1814

0.0746

43.7723

0.3168

HAR_RV_J_SQ

0.1777

0.0708

38.1700

0.3175

HAR_RV_J_LN

0.1810

0.0695

35.8476

0.3466

HAR_RV_CJ

0.1845

0.0738

40.7027

0.3257

HAR_RV_CJ_SQ

0.1783

0.0693

36.3822

0.3212

HAR_RV_CJ_LN

0.1794

0.0679

34.9687

0.3421

HAR_RR

0.0451

0.0161

34.3662

0.3313

HAR_RR_SQ

0.0449

0.0155

31.7072

0.3450

HAR_RR_LN

0.0471

0.0152

29.8441

0.3879

Additionally Model Confidence Set was built for this case:

Table 32. Models' ranks for 1-step ahead forecasting with rolling regression estimation

Model

Total

HAR-RV

-

HAR-RV (sq. roots)

5

HAR-RV (ln)

-

HAR-RV-J

6

HAR-RV-J(sq. roots)

1

HAR-RV-J(ln)

4

HAR-RV-CJ

-

HAR-RV-CJ(sq. roots)

2

HAR-RV-CJ(ln)

3

These models are indistinguishable from each other in terms of performance not only at 5% level of significance but also at 1% level.

For 1-week and 1-month forecasts only HAR-RV and HAR-RR models can be evaluated as for other models time series of Jumps or Continuous components of Realized Volatility model have to be estimated. It can be seen, that performance of models in the respective family is evaluated differently by each criteria. As the result, based on error' statistics it cannot be concluded, that models perform non-equally at different forecast horizons.

Table 33. Errors for 1-week and 1-month forecasts for rolling regression

Forecast horizon

Model

RMSE

MAE

MAPE

TIC

1 WEEK

HAR_RV

0.1834

0.0727

41.6084

0.3359

HAR_RV_SQ

0.1819

0.0685

35.2075

0.3487

HAR_RV_LN

0.1904

0.0684

32.3831

0.3886

HAR_RR

0.0443

0.0156

34.2198

0.3338

HAR_RR_SQ

0.0448

0.0154

31.6325

0.3495

HAR_RR_LN

0.0473

0.0151

29.7410

0.3918

1 MONTH

HAR_RV

0.1835

0.0731

41.7657

0.3343

HAR_RV_SQ

0.1816

0.0685

35.2525

0.3464

HAR_RV_LN

0.1901

0.0685

32.4244

0.3867

HAR_RR

0.0454

0.0161

34.4266

0.3315

HAR_RR_SQ

0.0449

0.0154

31.6935

0.3453

HAR_RR_LN

0.0472

0.0152

29.8272

0.3888

5.4 Combination of models

After standalone model performance analysis, it can be concluded that HAR-RV (and its' modifications) and GARCH (1, 1) model should not be taken in account when combining model performance. Moreover, HAR-RV-CJ and HAR-RV-J (standard modifications) models will be excluded from further analysis. However, as HAR-RR models are predicting different values (Realized Range instead of Realized Volatility), it will be meaningless to include their forecasts into further analysis.

This model “portfolio” is done for the forecasts of the models obtained in 5.3.1 - 5.3.4., i.e. 1-day ahead “static” forecast was used.

To be consistent, assumption is made, that models are trained and tested over “pre-crisis” period and as they show relatively high performance, they are used in the combination. After that their combination is used during “crisis” period to evaluate performance. As “pre-crisis” valuation can be used, the following models (as the best based on errors' statistics) will form a portfolio:

· HAR-RV-J (sq. roots)

· HAR-RV-J (logarithms)

· HAR-RV-CJ (sq. roots)

· HAR-RV-CJ (logarithms)

As described in 4.2.6 equal weights method with exclusion of outliers is used and also method with dynamic weights (where weights are assigned due to past performance of the model). However there are 2 main variables for weight assignment in case of dynamic weights: value of (discount factor) and number of lags (days of estimation models' performance to compute weights). Discount factor is used only for weights assignment and not for calculation of RMSE, MAPE, TIC and MAE criteria. Table 34 shows forecast errors for equal and dynamic (with lag equals 5) weights. As it can be concluded, equal weights method does not show significant improvement from standalone models. However, combination with dynamic weights has outperformed any of the models included into the analysis in RMSE, MAE and TIC criteria. TIC criterion is the lowest among all of the models (including HAR-RR family). It can be also concluded that in this specific case, coefficient (discount factor) does not have significant influence on the outcome.

Table 34. Models' combination evaluation with equal and dynamic weights methods

Combination method

Discount factor ()

RMSE RMSE and MAE is multiplied by 1000

MAE3

MAPE

TIC

Equal weights

0.23860

0.07681

37.09038

0.40761

Dynamic weights

0.9

0.17559

0.06836

36.62318

0.33251

0.8

0.17557

0.06836

36.62197

0.33247

0.5

0.17553

0.06835

36.61277

0.33236

Model combination with dynamic weights was added to the pool of models to check for statistical significance outperformance of the models for the total period.

Table 35. Models' ranks for total period performance including model combination with dynamic weights

Model

Total

HAR-RV

-

HAR-RV (sq. roots)

-

HAR-RV (ln)

-

HAR-RV-J

6

HAR-RV-J(sq. roots)

3

HAR-RV-J(ln)

-

HAR-RV-CJ

4

HAR-RV-CJ(sq. roots)

1

HAR-RV-CJ(ln)

5

Model combination with dynamic weights

2

As it can be seen from Table 35, models' combination cannot be distinguished from HAR-RV-CJ and HAR-RV-J (“standard” and sq. root modifications) at 1% or 5% significance level according to Model Confidence Set results.

5.5 Value-at-Risk modelling

To check how models can be applied in practice, Value-at-Risk (VaR) is computed for each model based on Realized Volatility or Realized Range forecasts. VaR daily forecasts for each model are calculated. Portfolio price is set to 1 for simplicity. Losses in the portfolio are computed as daily return on the MICEX index.

Values, forecasted by HAR-RV, HAR-RV-J, HAR-RV-CJ, HAR-RR models on testing period (01.03.2014 - 31.12.2015, 461 observation, “static” forecast was used) of total sample where used to calculate daily VaR forecast, based on the model described in 4.4. Number of violations was summed up and Kupiec test was performed. Both 1% and 5% daily VaR values were computed and tested.

Table 36. Performance of Value-at-Risk models based on Volatility estimators' forecasts

Confidence level

1%

5%

Model

Number of violations

Kupiec test-statistics

Number of violations

Kupiec test-statistics

HAR_RV

5

0.032

25

0.169

HAR_RV_SQ

8

2.065

28

1.050

HAR_RV_LN

9

3.304

29

1.500

HAR_RV_J

3

0.648

24

0.041

HAR_RV_J_SQ

3

0.648

27

0.677

HAR_RV_J_LN

5

0.032

27

0.677

HAR_RV_CJ

6

0.387

26

0.382

HAR_RV_CJ_SQ

6

0.387

28

1.050

HAR_RV_CJ_LN

5

0.032

27

0.677

HAR_RR

54

172.531

93

131.378

HAR_RR_SQ

56

182.915

97

144.143

HAR_RR_LN

59

198.795

100

153.991

The threshold values of the test-statistics are 3.84 (5% significance level for the test) and 6.63 (1% level), when the value of Kupiec test-statistics is higher than the threshold, null hypothesis of valid VaR estimation is rejected.

From Table 36 it can be concluded, that for all HAR-RV, HAR-RV-J and HAR-RV-CJ models null hypothesis of valid VaR estimation cannot be rejected for testing subsample of total period. However, for HAR-RR models null hypothesis can be rejected.

5

6. Conclusion

This paper analysed various forecasting volatility models for Russian market (MICEX index). Two volatility estimators were used (Realized Range and Realized volatility). GARCH, HAR-RV, HAR-RV-J, HAR-RV-CJ, HAR-RR models and their modifications were compared in efficiency of predictions during “pre-crisis”, “crisis” and total periods. 1-minute, 5-minute, 10-minute and 15-minute ticks were used to compute volatility estimators. As results show, GARCH model has the worst performance during any period. Standard specifications of models have higher performance than sq. root or logarithm modifications.

It is shown, that weekly and monthly Realized Volatility and Jumps variables are not significant in most models during “crisis” period. However, daily values are significant during any period.

Models' performance is higher with higher frequencies that support theoretical results presented in (Barndorff-Nielsen, 2002). However it is in contrast to results for the Turkish market obtained in (Зelik, 2014).

Based on TIC and MAPE criteria HAR-RR (logarithms) is the best for forecasting volatility estimated by Realized Range. Realized Variance models with modifications also show better performance than ones without. During “pre-crisis” and “crisis” periods logarithm modifications of HAR-RV-J and HAR-RV-CJ models outperform sq. root modifications but during total period TIC criteria are lower for sq. root. Outperformance of logarithm and sq. root modifications support findings of (Зelik, 2014) and (Andersen T. G., 2003). At 5% significance level best Realized Volatility predicting models during “pre-crisis” period are logarithm modifications of HAR-RV, HAR-RV-J and HAR-RV-CJ models, during “crisis” period - logarithm modifications of HAR-RV-J and HAR-RV-CJ models and during total period - all modifications of HAR-RV-J and HAR-RV-CJ models.

1-day ahead forecasts for models with rolling regression show that at 1% significance level 6 models: HAR-RV (sq. roots), HAR-RV-J (“standard”, sq. roots, logarithms), HAR-RV-CJ (sq. roots, logarithms) have the same performance.

Combination of best performance models with equal weights does not show outperformance over best standalone models according to errors' statistics. However, when using dynamic weights, based on the past performance errors' statistics for such combination is lower than for single models. Difference of models' combination and best standalone models performances are not statistically significant.

Daily VaR models (both 1% and 5%) based on HAR-RV, HAR-RV-J and HAR-RV-CJ models' forecasts show appropriate forecasting results (based on Kupiec test) when compared with actual returns on the MICEX index.

6

7. References

Andersen, T. B. (2001). The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96, 42-55.

Andersen, T. G. (2003). Modelling and forecasting realized volatility. Econometrica 71, 579-625.

Andersen, T. G. (2003). Some like it smooth and some like it rough: untangling continuous and jump components in measuring modelling and forecasting asset return volatility. CFS Working Paper, No:2003/35.

Barndorff, N. O. (2004). Power and bipower variation with stochastic volatility and jumps. J.Financ. Econ. 2, 1-37.

Barndorff-Nielsen, O. S. (2002). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society Series B 64, 253-280.

Зelik, H. E. (2014). Volatility Forecasting using high frequency data: Evidence from Turkish stock markets. Economic Modelling, Volume 36.

Dimitrios P. Louzisa, S. X.-S. (2014). Realized volatility models and alternative Value-at-Risk prediction strategies. Economic Modelling,Volume 40, 101-116.

Hansen, P. R. (2011). The Model confidence set. Econometrica 79, 453-497.

Kim Christensena, M. P. (2007). Realized range-based estimation of integrated variance. Journal of Econometrics, Volume 141, Issue 2, 323-349.

Kurmaю Akdoрan, S. B. (2012). Short-term Inflation Forecasting Models For Turkey and a Forecast Combination analysis. Working paper 12/09.

Martin Martens, D. v. (2007). Measuring volatility with the realized range. Journal of Econometrics, Volume 138, Issue 1, 181-207.

7

8. Appendix

Table 37. Summary of RV, BV and Jump estimator for 1-minute frequency

Variable

Period

Mean

Maximum

Minimum

St.dev

Skewness

Kurtosis

RV_daily

Pre-crisis

0.00014

0.00219

0.00001

0.00020

5.20

36.53

Crisis

0.00019

0.00360

0.00003

0.00029

8.17

84.99

Total

0.00016

0.00360

0.00001

0.00024

7.42

80.77

RV_weekly

Pre-crisis

0.00071

0.00791

0.00007

0.00083

4.14

22.40

Crisis

0.00095

0.00823

0.00025

0.00100

4.32

23.16

Total

0.00080

0.00823

0.00007

0.00091

4.25

23.48

RV_monthly

Pre-crisis

0.00314

0.01412

0.00062

0.00285

2.22

4.40

Crisis

0.00418

0.01537

0.00142

0.00306

2.13

4.05

Total

0.00354

0.01537

0.00062

0.00298

2.12

4.12

BV_daily

Pre-crisis

0.00004

0.00066

0.00000

0.00006

6.26

52.36

Crisis

0.00006

0.00121

0.00001

0.00009

7.81

83.07

Total

0.00005

0.00121

0.00000

0.00007

7.68

88.06

BV_weekly

Pre-crisis

0.00020

0.00210

0.00002

0.00023

4.09

21.79

Crisis

0.00031

0.00295

0.00008

0.00033

4.87

30.86

Total

0.00024

0.00295

0.00002

0.00028

4.72

31.87

BV_monthly

Pre-crisis

0.00088

0.00393

0.00015

0.00080

2.15

4.16

Crisis

0.00138

0.00559

0.00044

0.00104

2.47

6.17

Total

0.00107

0.00559

0.00015

0.00093

2.37

6.39

J_daily

Pre-crisis

0.00010

0.00154

0.00000

0.00015

5.15

34.83

Crisis

0.00013

0.00294

0.00002

0.00021

8.88

102.22

Total

0.00011

0.00294

0.00000

0.00017

7.70

89.15

J_weekly

Pre-crisis

0.00051

0.00593

0.00004

0.00061

4.20

23.21

Crisis

0.00064

0.00529

0.00016

0.00068

4.14

20.53

Total

0.00056

0.00593

0.00004

0.00064

4.16

21.95

J_monthly

Pre-crisis

0.00226

0.01035

0.00044

0.00206

2.24

4.53

Crisis

0.00280

0.00980

0.00096

0.00205

2.02

3.39

Total

0.00247

0.01035

0.00044

0.00207

2.10

3.87

J_monthly

Pre-crisis

0.00004

0.00066

0.00000

0.00006

6.26

52.36

Crisis

0.00006

0.00121

0.00001

0.00009

7.81

83.07

Total

0.00005

0.00121

0.00000

0.00007

7.68

88.06

C_weekly

Pre-crisis

0.00020

0.00210

0.00002

0.00023

4.09

21.79

Crisis

0.00031

0.00295

0.00008

0.00033

4.87

30.86

Total

0.00024

0.00295

0.00002

0.00028

4.72

31.87

C_monthly

Pre-crisis

0.00088

0.00393

0.00015

0.00080

2.15

4.16

Crisis

0.00138

0.00559

0.00044

0.00104

2.47

6.17

Total

0.00107

0.00559

0.00015

0.00093

2.37

6.39

Table 38. Summary of RV, BV and Jump estimator for 5-minute frequency

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Variable

Period

Mean

Maximum

Minimum

St.dev

Skewness

Kurtosis

RV_daily

Pre-crisis

0.00017

0.00219

0.00001

0.00026

6.57

61.14

Crisis

0.00020

0.00360

0.00003

0.00037

9.50

109.98

Total

0.00018

0.00360

0.00001

0.00031

8.78

107.12

RV_weekly

Pre-crisis

0.00084

0.00791

0.00008

0.00105

4.58

27.21

Crisis

0.00101

0.00823

0.00022

0.00123

4.80

27.30

Total

0.00091

0.00823

0.00008

0.00113

4.72

27.87

RV_monthly

Pre-crisis

0.00371

0.01412

0.00082

0.00354

2.37

5.28

Crisis

0.00445

0.01537

0.00143

0.00361

2.27

4.61

Total

0.00400

0.01537

0.00082

0.00359

2.30

4.89

BV_daily

Pre-crisis

0.00005

0.00066

0.00000

0.00009

8.63

106.71

Crisis

0.00007

0.00121

0.00001

0.00011

9.94

128.94

Total

0.00006

0.00121

0.00000

0.00010

9.63

129.83

BV_weekly

Pre-crisis

0.00026

0.00210

0.00003

0.00034

4.86

30.49

Crisis

0.00034

0.00295

0.00008

0.00042

5.66

39.96

Total

0.00029

0.00295

0.00003

0.00037

5.35

37.63

BV_monthly

Pre-crisis

0.00115

0.00393

0.00022

0.00110

2.33

5.04

Crisis

0.00150

0.00559

0.00047

0.00125

2.64

6.90

Total

0.00129

0.00559

0.00022

0.00117

2.47

6.22

J_daily

Pre-crisis

0.00012

0.00154

0.00000

0.00018

5.96

47.62

Crisis

0.00013

0.00294

0.00002

0.00026

10.37

133.90

Total

0.00012

0.00294

0.00000

0.00022

9.31

126.62

J_weekly

Pre-crisis

0.00058

0.00593

0.00005

0.00072

4.47

26.09

Crisis

0.00067

0.00529

0.00014

0.00083

4.60

24.30

Total

0.00062

0.00593

0.00005

0.00077

4.56

25.64

J_monthly

Pre-crisis

0.00256

0.01035

0.00056

0.00245

2.40

5.49

Crisis

0.00295

0.00980

0.00095

0.00239

2.13

3.82

Total

0.00271

0.01035

0.00056

0.00243

2.27

4.72

J_monthly

Pre-crisis

0.00005

0.00066

0.00000

0.00009

8.63

106.71

Crisis

0.00007

0.00121

0.00001

0.00011

9.94

128.94

Total

0.00006

0.00121

0.00000

0.00010

9.63

129.83

C_weekly

Pre-crisis

0.00026

0.00210

0.00003

0.00034

4.86

30.49

Crisis

0.00034

0.00295

0.00008

0.00042

5.66

39.96