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Contents

1. Introduction

2. Literature review

3. Data

3.1 Data description

3.2 Pre-crisis and post-crisis windows definition

3.2.1 Samples statistics

3.2.2 Data clearing

4. Methodology

4.1 Volatility estimators

4.1.1 Realized Volatility

4.1.2 Bipower Variation and Jumps

4.1.3 Realized Range

4.2 Models for volatility forecasting

4.2.1 GARCH models

4.2.2 HAR-RV model

4.2.3 HAR-RR model

4.2.4 HAR-RV-J model

4.2.5 HAR-RV-CJ model

4.2.6 Pull of models

4.3 Forecast comparison

4.3.1 Standard approach

4.3.2 Model Confidence Set

4.4 Value at Risk estimation

5. Results

5.1 Statistics on volatility estimators

5.1.1 Realized Volatility, Bipower Variation and Jumps

5.1.2 Realized Range, Returns

5.2 Regression estimations

5.2.1 GARCH model

5.2.2 HAR-RV models

5.2.3 HAR-RR models

5.2.4 HAR-RV-J models

5.2.5 HAR-RV-CJ models

5.3 Forecast results of standalone models

5.3.1 HAR-RV models

5.3.2 HAR-RR models

5.3.3 HAR-RV-J models

5.3.4 HAR-RV-CJ models

5.3.5 Forecast comparison for standalone models

5.3.6 Rolling regression with dynamic forecast for models

5.4 Combination of models

5.5 Value-at-Risk modelling

6. Conclusion

7. References

8. Appendix

Figure 1. Daily (1 minute) Realized Volatility time series

Figure 2. MICEX Index (Closing price) time series

Figure 3. Trading Volume of MICEX Index (Daily)

Figure 4. RVI (Russian volatility index)

Figure 5. Realized Volatility and Realized Range daily estimators (1-minute frequency) during period from 01/01/2014 till 01/01/2015

Table 1. Data summary

Table 2. GARCH (1, 1) model coefficients

Table 3. HAR-RV model coefficients for 1, 5, 10, 15 minute frequency

Table 4. HAR-RV (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Table 5. HAR-RV (logarithms) model coefficients for 1, 5, 10, 15 minute frequency

Table 6. HAR-RR model coefficients for 1, 5, 10, 15 minute frequency

Table 7. HAR-RR (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Table 8. HAR-RR (logarithms) model coefficients for 1, 5, 10, 15 minute frequency

Table 9. HAR-RV-J model coefficients for 1, 5, 10, 15 minute frequency

Table 10. HAR-RV-J (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Table 11. HAR-RV-J (logarithms) model coefficients for 1, 5, 10, 15 minute frequency

Table 12. HAR-RV-CJ model coefficients for 1, 5, 10, 15 minute frequency

Table 13. HAR-RV-CJ (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Table 14. HAR-RV-CJ (logarithms) model coefficients for 1, 5, 10, 15 minute frequency

Table 15. HAR-RV forecasts for 1, 5, 10 and 15 minute frequency

Table 16. HAR-RV (sq. roots) forecasts for 1, 5, 10 and 15 minute frequency

Table 17. HAR-RV (logarithms) forecasts for 1, 5, 10 and 15 minute frequency

Table 18. HAR-RR forecasts for 1, 5, 10 and 15 minute frequency

Table 19. HAR-RR (sq. roots) forecasts for 1, 5, 10 and 15 minute frequency

Table 20. HAR-RR (logarithms) forecasts for 1, 5, 10 and 15 minute frequency

Table 21. HAR-RV-J forecasts for 1, 5, 10 and 15 minute frequency

Table 22. HAR-RV-J (sq.roots) forecasts for 1, 5, 10 and 15 minute frequency

Table 23. HAR-RV-J (logarithms) forecasts for 1, 5, 10 and 15 minute frequency

Table 24. HAR-RV-CJ forecasts for 1, 5, 10 and 15 minute frequency

Table 25. HAR-RV-CJ (sq. roots) forecasts for 1, 5, 10 and 15 minute frequency

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

Table 27. Forecast performance in pre-crisis period

Table 28. Forecast performance in crisis period

Table 29. Forecast performance in total period

Table 30. Models' ranks during different periods

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

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

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

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

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

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

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

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

Table 39. Summary of RV, BV and Jump estimator for 10-minute frequency

Table 40. Summary of RV, BV and Jump estimator for 15-minute frequency

Table 41. Summary on RR estimator 1 minute frequency

Table 42. Summary on RR estimator 5 minute frequency

Table 43. Summary on RR estimator 10 minute frequency

Table 44. Summary on RR estimator 15 minute frequency

Table 45. Summary on Daily returns

1 

1. Introduction

Measuring and forecasting volatility of financial assets returns is important for derivatives' pricing. It is used in portfolio and risk management. Volatility forecasts are often used by regulators for market analysis and to assess regulation policies.

Precise volatility forecasts can play crucial role in creating provisions by banks as the result of regulators' requirements for valid risk management, what is important for banking industry. It is extremely important for developing banking regulatory environment in Russia.

This paper aims to find the best model to estimate volatility in the Russian market (MICEX index). Paper is focused on high-frequency models using intraday data for Realized Volatility and Realized Range estimators. It compares performance of high-frequency models with GARCH type models. Also it analyses different forecast combinations and their performance. Paper checks whether forecasts can be applied for Value-at-Risk modelling.

Different variants for model performance valuation are used to check how crisis in Russia can influence models' performance.



2. Literature review

With development of technology high frequency data became available that opened ways to volatility forecast without using additional parameters. Many articles were devoted to the topic of high-frequency volatility models and their superiority over popular stochastic volatility and GARCH models (see. (Andersen T. G., 2003)).

In theory, the Realized Volatility is not only efficient but also unbiased estimator, as is shown in (Andersen T. B., 2001), and converges to the true integrated variance when the frequency of intra-day observations is going to infinity (see. (Barndorff-Nielsen, 2002)). However, paper (Martin Martens, 2007) provides both theoretical basis for Realized Range estimator usage and its' practical efficiency over Realized variance estimator.

Moreover, some articles provide empirical evidence of outperformance of high frequency models that include “jumps” (see. (Andersen T. G., 2003), (Andersen T. G., 2003)). Realized Volatility models were compared on various markets and results confirmed high performance of models that include “jumps” (see. (Зelik, 2014)). Articles show that HAR-RV, HAR-RV-J and HAR-RV-CJ models produce better predictions when are used in square root or logarithm modifications (see. (Andersen T. G., 2003)).

Most popular errors' statistics are RMSE, MAE, MAPE and TIC that are used in most of articles (see. (Зelik, 2014)). However, this statistics cannot prove statistically significant difference between models. Model Confidence Set has been proposed to select best performing model or number of such models taking into account significance of difference in errors of forecasts (see. (Hansen, 2011)).

Article (Dimitrios P. Louzisa, 2014) examines the possibility of high frequency volatility estimators and respective models usage for Value-at-Risk. It shows that these models can be successfully applied for building Value-at-Risk models.

Paper (Kurmaю Akdoрan, 2012) that studies short-term inflation forecasts shows that standalone models can be outperformed by mix of the models and describe ways to create model combination.

2 

3 Data

3.1 Data description

For the modelling purposes intraday data was used. Data was obtained from “finam” www.finam.ru. Initially, all available information from 01/01/2010 till 01/01/2016 for every minute tick was obtained. These dates state the range including holidays and other non-trading days. Number of observations was equal to 819092. Number of trading days over that period was equal to 1501, meaning 250 trading days per year (including 251 days in 2012).

MICEX index consists of weighted sum of 50 most liquid stocks that are traded on the Moscow Stock Exchange. Weights are assigned according to capitalization. It is priced in Russian rubbles.

For every minute tick the following characteristics were obtained:

1. Ticker name (MICEXINDEXCF)

2. Date

3. Time

4. Opening price

5. Highest price

6. Lowest price

7. Closing price

8. Volume of trades

3.2 Pre-crisis and post-crisis windows definition

During the analysis, all models will be trained and tested on 3 different samples. This is done to capture the effect of crisis in Russia on performance of volatility forecasting models.

Both Realized Volatility and Realized Range were computed and as they show similar dynamics (see. Figure 5) in this paragraph qualitative analysis and comparisons are done for Realized Volatility but logic is the same for Realized Range.

Crisis in the market can be defined in different ways, however, in this paper “pre-crisis” and “crisis” periods are defined from analysis of GDP growth rates and exchange rates' crisis in Russia started in 2014.

Fundamental factors, such as GDP growth rate shows, that Russian GDP growth became stable after 2^nd quarter of 2010. The exchange rate dynamics does not show any huge changes compared to “crisis” period.

As the result, the assumption based on GDP growth rates is that the end of crisis in Russia was in the 2^nd quarter of 2010, and that it has not started before the 2^nd quarter of 2014.

The graph (see. Figure 1) provides time series of Daily (1 minute) Realized volatility (see 4.1.1). It can be seen, that there is volatility increase from 08/2011 till 11/2011 during “pre-crisis” period. Increase in Realized Volatility during this time window can be supported by relatively high volatility in exchange rates and overall turbulence in international markets (external shocks).

Figure 2 shows huge drop in prices of MICEX index that is followed by Realized Volatility increase.

In March 2014 high level of Realized volatility is presented as well as drop in GDP in the 2^nd quarter of 2014. From Figure 3 it can be seen, that in March 2014 there was a sharp increase in trading volume. Due to decrease in GDP growth rate in 2^nd quarter of 2014 and sharp increase in exchange rates, March of 2014 can be treated as a start of crisis in Russia.

Concluding previous abstracts, period from 01/04/2010 till 28/02/2014 will be called “pre-crisis” period and period from 01/03/2014 till 31/12/2015 will be called “crisis” period.

All models, described below will be trained and tested in 3 following samples:

1. Pre-crisis training (from 01/04/2010 till 31/12/2012), pre-crisis testing (from 01/01/2013 till 28/02/2014)

2. Crisis training (from 01/03/2014 till 31/05/2015), crisis testing (from 01/06/2015 till 31/12/2015)

3. Pre-crisis training (from 01/04/2010 till 28/02/2014), crisis testing(from 01/03/2014 till 31/12/2015)

Figure 1. Daily (1 minute) Realized Volatility time series

Figure 2. MICEX Index (Closing price) time series



Figure 3. Trading Volume of MICEX Index (Daily)

3.2.1 Samples statistics

As described in 3.2 there are 2 periods: “pre-crisis” and “crisis”. Data statistics is summarized in Table 1.

Table 1. Data summary

Period:	Number of trading days:	Number of minute ticks:
Pre-crisis training (01/04/2010 - 31/12/2012)	695	390892
Pre-crisis testing (01/01/2013 - 28/02/2014)	289	160835
Pre-crisis (01/04/2010 - 31/12/2012)	984	551727
Crisis training (01/03/2014 - 31/05/2015)	310	161178
Crisis testing (01/06/2015 - 31/12/2015)	151	78470
Crisis (01/03/2014 - 31/12/2015)	461	239648

3.2.2 Data clearing

After analysis of obtained intraday data it was concluded that no further operations needed. Data shows sufficient quality and no missed/irregular values were found.

4 

4. Methodology

4.1 Volatility estimators

Today there exist various volatility estimators that claim to be the best under certain conditions. Volatility cannot be directly observed in the market. However, it is traded as volatility index in different countries. For example there exist RVI (Russian volatility index) that reflect expectations of volatility during next 30 days http://moex.com/s381 - computation details.

Its' calculation is based on RTS index options prices. Implied volatility from near and next time series of RTS index options is used to calculate this index. Main difference of RTS index from MICEX index is currency, as RTS is priced in US dollars. It cannot be directly compared to Realized Range or Realized Volatility estimators computed in this paper as it is calculated for another index.

In this thesis in terms of volatility forecasting two main volatility estimators are used:

1. Realized volatility

2. Realized range

Figure 4. RVI (Russian volatility index)

4.1.1 Realized Volatility

Realized Volatility can be used as intraday volatility estimator:

Where - realized daily volatility, - periods per day, - observed price of the security/underlying.

- can be freely chosen, however such number should be sufficient. For example, if the trading day runs from 10 AM till 6 PM (8 hours), then 480 minute data is available for analysis. In case of analysis is done on minute basis, . If Realized volatility is going to be calculated on 5 minute intervals then . In most studies 1 minute and 5 minute ticks are used.

This estimator is efficient in practice, but only in case when jumps are not presented. As shown in Figure 4, for Russian markets even in “pre-crisis” period (see. 3.2) jumps in volatility are presented. To capture these jumps, Bipower Variation estimator was proposed in (Barndorff, 2004).

For forecasting models it is essential to compute not only daily Realized Volatility, but also weekly and monthly values:

Where - represents number of observations during the period of interest. For example, for obtaining weekly Realized Volatility based on 1-minute data, intraday observations are required, in case of 5-minute data, observations. Number 5 reflects number of trading days in a week. Consequently, when estimating monthly volatility intraday data from 22 trading days is used.

For further analysis, weekly Realized Volatility estimator is defined as - , and monthly - . Some literature (ex.: (Зelik, 2014)) provides different definition of weekly and monthly Realized Volatility estimator, more precisely estimators described above are divided by 5 and 22 respectively. This transformation leads to change in coefficient of regression with no other significant changes. Due to that it is possible not to make adjustment to these values.

4.1.2 Bipower Variation and Jumps

This estimator is connected to Realized Volatility estimator in sense that it helps to distinguish between jump part of volatility change and continuous part. It is calculated as follows:

Where - observed price of the security, .

Jump component is computed as follows:

As Realized Volatility captures both jump and continuous change it can be divided into to components:

As the result, Continuous component can be evaluated from previous equation as:

The same idea described for calculating weekly and monthly Realized Volatility (see. 4.1.1) is applied for obtaining weekly and monthly Continuous components, Bivariate Variation and Jumps.

4.1.3 Realized Range

Realized Range estimator uses maximum and minimum price through the intervals to obtain estimator of daily volatility:

Where - is maximum price during the i-th interval of the trading day, - minimum price during the i-th interval of the day. This estimator is proved to be efficient estimator for volatility in (Kim Christensena, 2007).

For purposes of forecasting, also 5 days and 22 days Realized Range is used:

Where - maximum price of the asset during i-th period, -minimum price of the asset during the i-th period. This estimator is used to measure weekly/monthly volatility depending on the interval of summation.

Without accounting for the coefficient , Realized Range estimator is always higher then Realized Volatility one with all other parameters being equal.

4.2 Models for volatility forecasting

Literature provides significant number of models that can be used for volatility forecasts. Following (Зelik, 2014), the following models will be used and compared.

4.2.1 GARCH models

GARCH (generalized autoregressive conditional heteroscedasticity) models are the most popular models for volatility modelling. They have a lot of extensions to capture various effects that data reflects. Researches show GARCH models do not perform well on real data samples due to various limitations. For simplicity and based on previous studies only standard GARCH family models will be examined. GARCH (p, q) model is stated as follows:

Where - error term obtained from regressing returns on the mean, - standard deviation dependent on time, - white noise. During analysis, the best model among GARCH family will be selected.

4.2.2 HAR-RV model

The model can be described as follows:

Where - stands for Realized Volatility in the next period, and other terms state for observed Realized Volatility today, latest observed weekly and monthly Realized Volatility. However, article (Andersen T. G., 2003) gives reasons that similar models that use square roots or logarithms of Realized Volatility perform better. For this reason, instead of including this model alone in the analysis, 2 more modifications of this model are included:

1. HAR-RV (sq.root)

2. HAR-RV (ln)

4.2.3 HAR-RR model

The model uses only Realized Range volatility estimator and be stated as:

Where - stands for Realized Range in the next period, and other terms stand for observed Realized Range today, latest observed weekly and monthly Realized Range. For reasons described in 4.2.2, modifications with square root and logarithm transformations are included:

1. HAR-RR (sq.root)

2. HAR-RR (ln)

However, there was found no evidence on comparative performance of these models. Due to similarity in statement of these models to HAR-RV there is possibility of outperformance of these models over “standard” HAR-RR and it will be checked in further paragraphs and will be subject to particular market data.

4.2.4 HAR-RV-J model

As described in 4.1.1 and 4.1.2 Realized Volatility captures both jumps and continuous change. Due to that HAR-RV-J model helps to capture these effects separately:

HAR-RV-J captures jump effect only partially, as it includes only daily term for Jump (see. 4.1.2). Moreover, the same as for HAR-RV and HAR-RR models, additional modifications are used:

1. HAR-RV-J (sq.root)

2. HAR-RV-J (ln)

4.2.5 HAR-RV-CJ model

To capture not only daily jump effect, Realized Volatility estimator is split into both Jump and Continuous (see. 4.1.2) components for all the variables of the HAR-RV model, it results in the following:



Following previous chapters, 2 more modifications are added for this model:

1. HAR-RV-CJ (sq.root)

2. HAR-RV-CJ (ln)

This model should capture huge shocks much better, compared to HAR-RV-J model.

4.2.6 Pull of models

Standalone models can provide good results in forecasting volatility. However, combination of models can possibly beat each model in accuracy of predictions. This effect can be observed, only if some models overestimate true values and some underestimate them. This can help in analysis of model performance.

Though this argument is valid for forecast accuracy valuation, this cannot be directly applied when models are trained, i.e. weights in models' combination can be assigned only having observed (realized) values.

The idea is as follows, when standalone models have been already estimated there are number of ways how to combine them:

1. Equally weighted “portfolio” of models

In this approach model forecasts are combined with equal weights. For example if there are N models in concern, and they forecast values: , then overall forecast is given by:

2. Equally weighted with exclusion of outliers

After training standalone models and obtaining forecasts, highest and lowest values of forecasts are treated as outliers and are not involved in forming an average value of volatility forecast. Forecasts that are left, form “portfolio” with the algorithm described in the previous bullet point.

3. Non-equal weights

When computing forecast combination of models, weights can be assigned based on past performance of these models. The logic is as follows, the better was model performance, more significant its' input should be in the overall result.

In this case model performance is measured as:

Where - performance of model i at time (current date), - estimated value, for simplicity here was stated Realized Volatility, - discount factor, - period, treated relevant for model performance evaluation.

As model performance is evaluated, weights are assigned as follows:

Where - weight of the model at the current time.

Forecast combination value is calculated as weighted average with weights computed above.

Articles (ex.: (Kurmaю Akdoрan, 2012)) provide evidence, that equally weighed forecast outperform other combinations, as this combination is more stable over time.

4.3 Forecast comparison

4.3.1 Standard approach

To find the best model or model combination for volatility forecast in Russian market (MICEX index), performance measures should be introduced. Various measures can be applied to estimate forecasts' accuracy, however, following (Зelik, 2014) four performance measures are used:

1. RMSE (root mean squared error):

2. MAE (mean absolute error)

3. MAPE (mean absolute percentage error):

4. TIC (Theil's U statistic)

Where N - is number of observations in out-of-time (testing) period, - observed (true) value of estimator, - forecast of the variable of interest.

In the equations above, or stands not for Realized Volatility, but instead it can be used for any variable of interest. Obviously, RMSE and MAE statistics should give similar results when comparing models. However, MAPE and TIC are relative statistics and show relative change and can be used for comparing performance of models over different periods and estimators.

Further this statistics will be calculated for each model over testing period (see. 3.2). For the results obtained comparison will be made.

4.3.2 Model Confidence Set

Criteria described in 4.3.1 do not provide efficient evidence of model outperformance on the out of time sample. For this reason Model Confidence Set is applied. Algorithm for application is described in (Hansen, 2011). Idea is to build a statistics that allows making a decision of best performing model or models based on statistical significance. Algorithm allows select best models with predefined confidence level which has to be set. In current paper this level is set to be 5%. Another essential parameter is loss function. The following loss function is set:

Where - observed level of the forecasted variable and - forecasted value at time t. Statistics is computed based on time series of loss functions for each model.

If states the loss of model i at time t, then

Stands for difference in losses between models i and j at time t.

Where is the number of models in the analysis. The hypothesis of Equal Predictive Ability (EPA - see. (Hansen, 2011)) can be stated as:

Where , i.e. expectation of difference between models losses across time. The following test-statistics can be formed:

Where the average loss difference between models i and j across time and - bootstrapped estimate of . Null hypothesis, presented above can be easily transformed into test statistics:

As the test statistic has non-standard distribution, distribution under null hypothesis is estimated using bootstrap (see. (Hansen, 2011)).

In case when the hypothesis is rejected, meaning that losses between some models are significant one model is excluded from the analysis by the rule:

The number of models is decreased by 1 and analysis starts from the beginning.

4.4 Value at Risk estimation

Various articles (see. (Dimitrios P. Louzisa, 2014)) provide evidence on possible use of high frequency volatility estimators to build VaR models. Generally, Value-at-Risk with confidence level is a threshold were probability of violating (having price lower then VaR value) is exactly . This measure is widely used to account for market risks in the banking sector .

The simplest modification for daily Value at Risk model using high frequency volatility estimator is:

Where - inverse of standard normal distribution function, - forecast for the volatility estimator, - is a confidence level. Here, price of the asset is set to 1 for simplicity, in other cases, the expression on the right side of the equation should be multiplied by asset price.

From the regulator's point of view, the lower the value of is the better, as more provisions are created by banks, for the same reason, banks do not want to overestimate VaR.

To measure the performance of VaR model, forecasted values of Value-at-Risk are compared to returns for each day. Statistical significance is tested with Kupiec test that has the null hypothesis of exact match of percentage of violations to the level of significance:

Where - is a test-statistics that is distributed as chi-squared under null hypothesis with 1 degree of freedom, - significance level, - number of observations, - number of VaR violations in the sample.

5. Results

5.1 Statistics on volatility estimators

5.1.1 Realized Volatility, Bipower Variation and Jumps

As can be observed from Table 37, Table 38, Table 39, Table 40 mean values of RV estimator are higher during “crisis” compared to “pre-crisis” period. Moreover, standard deviations of these series are also higher for period of crisis for all frequencies analysed. As RV can take only positive values, skewness is positive. The same logic is true for Bipower Variation, Jumps and Continuous part of Realized Volatility.

5.1.2 Realized Range, Returns

From Table 41, Table 42, Table 43, Table 44 it can be concluded, that average Realized Range estimator is higher during crisis as well as having higher volatility during the same period. That is in line with results obtained in 5.1.1.

Daily returns (see. Table 45) during “pre-crisis” period on average have negative values, it means that before crisis, market was falling slightly, however, during crisis, market has risen. This can be partially explained by significant drop in price of national currency, taking into account that index is traded in Russian Rubbles.

Data shows, that on 1-minute frequency correlation between Realised Volatility and Realized range is 84% for overall time period. As both estimators show similar dynamics, on average Realized Range is 10 times lower than Realized Volatility is. As the result, these estimators cannot be compared directly. As mentioned before, comparison of estimators is out of scope of this paper. Time series for both estimators are presented in Figure 5.



Figure 5. Realized Volatility and Realized Range daily estimators (1-minute frequency) during period from 01/01/2014 till 01/01/2015

5.2 Regression estimations

Regressions for each model are run on 3 different samples: “pre-crisis”, “crisis”, “total” which are described in 3.2. For each of these samples models are estimated on training samples and their performance is measured on respective testing samples (see. 5.3). For each of the coefficient p-value is not shown, but most important levels 1%, 5%, 10% are shown in the tables in this paragraph as *, **, *** respectively. When there are no stars following the estimated coefficient, it means, that this coefficient is not statistically significant at 10% level.

Paragraph is structured in the following way, for HAR model (HAR-RV, HAR-RR, HAR-RV-J, HAR-RV-CJ) not only standard model is estimated, but also model in square roots and logarithm forms (see. 4.2.2 - 4.2.5).

To test residuals for white noise Ljung-Box tests are done. Results show that for logarithm modifications of HAR-RV, HAR-RV-J, HAR-RV-CJ and HAR-RR for all periods of the model null hypothesis of no serial correlation cannot be rejected. For other models and their modifications null hypothesis can be rejected at least at 5% significance level.

5.2.1 GARCH model

After estimating several versions of GARCH (p, q) model family, it was concluded that the only model that satisfy necessary conditions was GARCH (1, 1) model. As it can be seen from Table 2, all coefficients are significant at least at 5% level for this model specification. The hypothesis of no serial correlation of standardized residuals cannot be rejected for each period.

Table 2. GARCH (1, 1) model coefficients

Period	(constant)
Pre_crisis	0.00000**	0.07616*	0.90797*
Crisis	0.00004**	0.09286**	0.7089*
Total	0.00000*	0.05655*	0.93027*

5.2.2 HAR-RV models

From Table 3, Table 4, Table 5 it can be concluded, that coefficient , that stands for monthly Realized volatility estimator, as well as , that stands for weekly Realized Volatility estimator are not significant at 10% level during crisis period. However, during total period, they are significant at least at 10% level (mostly at 1% significance level).

for all model modifications is taking highest values for 1-minute frequency Realized Volatility estimator, which means that higher portion estimator's deviation is explained in 1-minute frequency case. The only exception is HAR-RV model during pre-crisis and total periods, where 5-minute version has higher value of .

It can be concluded that coefficient is higher for the logarithm modification of the model for “pre-crisis”, “crisis” and total periods.

Table 3. HAR-RV model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	0.00003*	0.00004*	0.00004*	0.00005*
	Crisis	0.00008*	0.0001*	0.00012*	0.00013*
	Total	0.00002*	0.00003*	0.00003*	0.00004*
	Pre-crisis	0.4246*	0.41728*	0.40826*	0.31161*
	Crisis	0.43297*	0.41871*	0.45546*	0.40632*
	Total	0.41154*	0.40934*	0.401*	0.30822*
	Pre-crisis	0.05329*	0.05065*	0.05234*	0.06395*
	Crisis	0.02798	0.01699	0.00383	0.00589
	Total	0.05543*	0.05178*	0.05329*	0.06442*
	Pre-crisis	0.00577***	0.00606***	0.00643**	0.00564
	Crisis	0.00263	0.00341	0.0032	0.00363
	Total	0.00665*	0.00711*	0.00749*	0.00692**
	Pre-crisis	0.42641	0.39944	0.40099	0.30562
	Crisis	0.26036	0.21681	0.22234	0.18205
	Total	0.44049	0.41741	0.42096	0.32562

crisis regulation rolling regression

Table 4. HAR-RV (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	0.00148*	0.00167*	0.00168*	0.00199*
	Crisis	0.00342*	0.00386*	0.00418*	0.0045*
	Total	0.00128*	0.00139*	0.00138*	0.00162*
	Pre-crisis	0.37736*	0.40639*	0.35941*	0.32498*
	Crisis	0.51704*	0.5121*	0.52331*	0.46514*
	Total	0.35255*	0.38294*	0.34154*	0.30416*
	Pre-crisis	0.15322*	0.14046*	0.15941*	0.16204*
	Crisis	0.07646***	0.05761	0.04322	0.05417
	Total	0.15468*	0.14041*	0.1563*	0.16102*
	Pre-crisis	0.02861**	0.02799**	0.02871**	0.02902**
	Crisis	0.01182	0.01591	0.01576	0.01735
	Total	0.03482*	0.03545*	0.03679*	0.03736*
	Pre-crisis	0.53169	0.53564	0.52329	0.47304
	Crisis	0.43585	0.39483	0.37817	0.32915
	Total	0.53714	0.54507	0.53665	0.48849



Table 5. HAR-RV (logarithms) model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	-2.26484*	-2.22267*	-2.3251*	-2.58322*
	Crisis	-2.30134*	-2.55835*	-2.76142*	-2.98881*
	Total	-2.36262*	-2.31332*	-2.39179*	-2.63865*
	Pre-crisis	0.30196*	0.30212*	0.24235*	0.19464*
	Crisis	0.49991*	0.45705*	0.41973*	0.3789*
	Total	0.31099*	0.3121*	0.2625*	0.20924*
	Pre-crisis	0.40393*	0.41808*	0.4852*	0.48699*
	Crisis	0.24632*	0.24164**	0.26658*	0.27563*
	Total	0.32828*	0.32563*	0.37471*	0.38098*
	Pre-crisis	0.19034*	0.17564*	0.16581*	0.19498*
	Crisis	0.06023	0.08785	0.07921	0.0933
	Total	0.25676*	0.26405*	0.26577*	0.30047*
	Pre-crisis	0.55791	0.561	0.55255	0.50834
	Crisis	0.53065	0.48227	0.45563	0.41319
	Total	0.5478	0.5505	0.54244	0.49964

5.2.3 HAR-RR models

From Table 6, Table 7, Table 8 it can be concluded, that coefficient , that stands for monthly volatility, as well as , that stands for weekly volatility are not significant even at 10% level during crisis period. Exception is for logarithm model modification as coefficients for weekly volatility are significant at least at 5% significance level for all frequencies during period of crisis.

does not show same trends during various periods. However, during the period of crisis for each model the lower the frequency is, the lower is the value of this coefficient. Logarithm modification shows higher values of coefficient for all periods in scope of the analysis.

Table 6. HAR-RR model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	0.00000*	0.00002*	0.00002*	0.00002*
	Crisis	0.00001*	0.00006*	0.00007*	0.00008*
	Total	0.00000*	0.00001*	0.00002*	0.00002*
	Pre-crisis	0.55818*	0.60274*	0.57783*	0.57072*
	Crisis	0.60633*	0.52427*	0.49161*	0.5138*
	Total	0.55033*	0.59646*	0.57174*	0.56502*
	Pre-crisis	0.02469***	0.02144***	0.02596**	0.02456**
	Crisis	0.0069	0.01877	0.01874	0.00924
	Total	0.02634**	0.02251**	0.02704*	0.02554**
	Pre-crisis	0.00793*	0.00602**	0.00591**	0.00635**
	Crisis	0.00404	0.00287	0.00282	0.00317
	Total	0.00834*	0.00674*	0.00668*	0.00719*
	Pre-crisis	0.50799	0.52324	0.50706	0.49232
	Crisis	0.41387	0.34055	0.29931	0.29739
	Total	0.51954	0.53849	0.52361	0.51009

Table 7. HAR-RR (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	0.00050*	0.00105*	0.00115*	0.00121*
	Crisis	0.00124*	0.0026*	0.00293*	0.00314*
	Total	0.00047*	0.00092*	0.00099*	0.00103*
	Pre-crisis	0.4868*	0.55497*	0.53708*	0.52536*
	Crisis	0.63679*	0.58046*	0.57372*	0.57645*
	Total	0.46828*	0.53264*	0.51344*	0.50378*
	Pre-crisis	0.11645*	0.09708*	0.10395*	0.10473*
	Crisis	0.04888	0.06817***	0.06162	0.05103
	Total	0.12199*	0.10113*	0.10854*	0.10759*
	Pre-crisis	0.02902**	0.02338**	0.02328**	0.02487**
	Crisis	0.01323	0.00983	0.01068	0.01249
	Total	0.03078*	0.0277*	0.02782*	0.03005*
	Pre-crisis	0.60948	0.63052	0.62054	0.61047
	Crisis	0.55166	0.50509	0.48011	0.46189
	Total	0.61092	0.6379	0.62913	0.62137

Table 8. HAR-RR (logarithms) model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	-2.17651*	-1.88272*	-1.91655*	-1.94595*
	Crisis	-2.08828*	-2.07476*	-2.12584*	-2.2811*
	Total	-2.2174*	-1.94946*	-1.9923*	-2.01476*
	Pre-crisis	0.34192*	0.40497*	0.38425*	0.37401*
	Crisis	0.58147*	0.54072*	0.54218*	0.50471*
	Total	0.36002*	0.41215*	0.38789*	0.38397*
	Pre-crisis	0.43574*	0.38381*	0.40247*	0.40507*
	Crisis	0.21216**	0.24722*	0.23473*	0.25041*
	Total	0.39197*	0.32705*	0.34635*	0.33436*
	Pre-crisis	0.12749**	0.12275**	0.12361**	0.13017**
	Crisis	0.05255	0.04247	0.04493	0.05559
	Total	0.14929*	0.17283*	0.17706*	0.1936*
	Pre-crisis	0.61696	0.63958	0.6292	0.62521
	Crisis	0.60821	0.58567	0.57401	0.54518
	Total	0.60791	0.63428	0.62387	0.62052

5.2.4 HAR-RV-J models

Table 9, Table 10, Table 11 confirm the results of analysis provided in 5.2.2 concerning coefficients for weekly and monthly volatilities. Moreover it shows that coefficient of “daily jump” variable () is significant at 1% for most of low-frequency cases.

As in 5.2.2, is higher for higher frequencies. This coefficient is higher for logarithm modification of the model as in 4.2.2, 4.2.3, 4.2.4.

Table 9. HAR-RV-J model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	0.00003*	0.00004*	0.00004*	0.00005*
	Crisis	0.00008*	0.00011*	0.00012*	0.00013*
	Total	0.00002*	0.00003*	0.00003*	0.00004*
	Pre-crisis	1.24017*	0.61284*	1.19304*	-0.30222***
	Crisis	1.80537*	1.35861*	1.1831*	0.08395
	Total	1.27603*	0.65663*	1.20333*	-0.24777***
	Pre-crisis	0.03617*	0.04746*	0.0416*	0.07325*
	Crisis	0.00422	0.00096	-0.00399	0.00946
	Total	0.03692*	0.0476*	0.0418*	0.07329*
	Pre-crisis	0.00579***	0.00637**	0.00736**	0.00371
	Crisis	0.00223	0.00263	0.00266	0.004
	Total	0.00664*	0.00748*	0.00834*	0.00522***
	Pre-crisis	-0.98517*	-0.26507	-1.09395*	0.92756*
	Crisis	-1.84921*	-1.27658*	-1.03398***	0.45568
	Total	-1.04095*	-0.33365***	-1.1127*	0.83466*
	Pre-crisis	0.44211	0.40060	0.41782	0.32038
	Crisis	0.29219	0.23761	0.23093	0.18361
	Total	0.45722	0.41919	0.43795	0.33746

Table 10. HAR-RV-J (sq. roots) model coefficients for 1, 5, 10, 15 minute frequency

Coefficient	Period	1	5	10	15
(constant)	Pre-crisis	0.00123*	0.00147*	0.00152*	0.00197*
	Crisis	0.00336*	0.00384*	0.00416*	0.0045*
	Total	0.00104*	0.00121*	0.00128*	0.00163*
	Pre-crisis	1.86144*	1.32292*	1.48035*	0.69604*
	Crisis	1.9039*	1.82509*	1.73336*	0.4993
	Total	1.97563*	1.41257*	1.4413*	0.7587*
	Pre-crisis	0.09921*	0.10641*	0.12575*	0.14939*
	Crisis	0.04057	0.02904	0.02263	0.05358
	Total	0.09464*	0.10164*	0.1228*	0.14478*
	Pre-crisis	0.02377**	0.02976**	0.02872**	0.02996**
	Crisis	0.00897	0.00979	0.01223	0.01718
	Total	0.02879*	0.03644*	0.03537*	0.03782*

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