Univariate GARCH models with realized variance

Univariate GARCH models with realized variance

By Carl Börjesson & Ossian Löhnn

Department of Statistics Uppsala University

Supervisor: Lars Forsberg

2019

Abstract

This essay investigates how realized variance affects the GARCH-models (GARCH, EGARCH, GJRGARCH) when added as an external regressor. The GARCH models are estimated with three different distributions; Normal-, Student’s t- and Normal inverse gaussian distribution. The results are ambiguous - the models with realized variance improves the model fit, but when applied to forecasting, the models with realized variance are performing similar Value at Risk predictions compared to the models without realized variance.

Keywords​: ​GARCH, EGARCH, GJRGARCH, external regressor, realized variance, volatility, Value at Risk, nig, Normal inverse gaussian, std, Student’s t distribution, norm, Normal distribution, rugarch, rolling forecast.

Acknowledgements

We would like to thank our supervisor Lars Forsberg for his assistance during the entire process of this thesis, especially for his help during rough periods in the creative process. We would also like to thank the R-community and the authors of the packages used to make this thesis possible.

1. Introduction 5

2.1 Economic Theory 6

2.1.2 Volatility 6

2.1.3 Realized Variance 6

2.2 Statistical Theory 7

2.2.1 The ARCH model 7

2.2.2 The Generalized ARCH model 8

2.2.3 GARCH(1, 1) 9

2.2.4 Extensions of the GARCH model 9

2.2.5 EGARCH(1, 1) 9

2.2.6 GJR-GARCH(1, 1) 10

2.2.7 GARCH with Realized Variance 10

2.3 Distributions 11

2.3.1 The Normal Distribution 11

2.3.2 The Student’s t-distribution 11

2.3.3 The Normal Inverse Gaussian Distribution (NIG) 12

2.4 Diagnostics and Evaluation 12

2.4.1 Normality test 12

2.4.2 Akaike Information Criterion (AIC) 13

2.5 Forecast Evaluation 13

2.5.1 MAE 13

2.5.2 RMSE 14

2.6 Value at Risk 14

2.6.1 Backtesting the Value at Risk & Kupiec’s test 14

3. Empirical Analysis 16

3.1 Data 16

3.1.1 Data evaluation 17

3.1.2 Realized Variance 18

3.1.3 Descriptive Statistics 20

3.1.4 Result of normality test 20

3.2 Model estimation 21

4. Forecast evaluation 23

5.Value at Risk Forecasting & Backtesting 25

6. Result Discussion 27

7. Conclusion 28

8. Further Studies 29

9. Appendix 30

9.1 Appendix I - Tables and figures 30

9.1.1 Squared returns and Realized variance 30

9.1.2 Estimated models 33

9.2 Appendix II - RStudio - Rugarch 37

References 38

1. Introduction

Risk management envelops all financial management as information regarding the potential loss of an investment is vital for investors. Investments in an asset, a portfolio or any other financial derivative is characterized by a trade-off between returns and risk. Higher risk can lead to a potential higher return. The importance of calculating risk accurately cannot be understated as a miscalculation may lead to an unexpected loss. There are multiple ways to evaluate and calculate financial risk, where volatility is one of the most commonly used measurements to get an overview of the fluctuations in an asset.

It is widely known that time series of financial data are heteroscedastic and not homoscedastic; they suffer from volatility clustering, i.e. large fluctuations are followed by large fluctuations and small fluctuations are followed by small fluctuations, which was observed by Mandelbrot in 1963. The lack of models available to handle heteroscedasticity in data lead to the development of the ARCH-model by Engle (Autoregressive Conditional Heteroscedastic model) in the publication Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation ​in 1982. Further research in the field lead to the development of the GARCH-model by Bollerslev in 1986, first published in the research paper ​Generalized ​Autoregressive Conditional Heteroscedasticity.​The possibility to account for heteroskedasticity when modelling financial time-series through the models developed by Engle and Bollerslev, have deepened the research in the field and ​a real-life application for the use of GARCH-models is risk management (Andersen et al, 2007).

Value at Risk is a measurement of investment risk where volatility is a key component in the estimation. Banks, investment firms and other financial institutions are obliged to report their VaR estimations since the introduction of the Basel III framework to account for minimum capital requirements. However, the methodology to estimate VaR using GARCH-models and in extension minimum capital requirements differs since separate methods leads to different outcomes. Applying GARCH-models for volatility forecasts and Value at Risk estimation also presents another issue. The family of ARCH-models treats volatility as a latent unobservable variable. To account for this Andersen and Bollerslev proposed realized variance as an alternative measure. Realized variance is an “observable” measure of the volatility in financial returns and previous studies have shown that implementing realized variance by using intraday data to estimate the conditional variance in GARCH-models could provide a more accurate forecast performance​ (Wong et al, 2016).

The purpose of this study is to evaluate the addition of realized variance to different GARCH processes, and to compare these models to the GARCH models without realized variance.

The comparison will be done by both evaluating the model fit, the forecasting performance and Value at Risk predictions.

2. Theoretical framework

The following section will cover the economic and statistical theory of the study. The economic theory will cover volatility in financial data and its attributes and the statistical theory will include the tests, models and statistical concepts used in the paper. Lastly Value at Risk will be covered, its application on financial data and backtesting procedures.

2.1 Economic Theory

2.1.2 Volatility

Volatility in financial data can be defined as the standard deviation or fluctuations of an asset.

Volatility is widely used in many areas in empirical finance. Pricing securities and estimating market risk are two examples where volatility is used. Investors needs to know how the fluctuations in an asset vary over time to calculate risk and thus correctly predicting volatility is essential. Complicating the use of volatility in modelling is the asymmetry of positive and negative returns and the attribute of volatility clustering. The returns of financial asset respond differently to negative and positive shocks were negative shocks have a larger impact than positive ones. The attribute of volatility clustering implies serial dependency in periods of high or low volatility, that is periods of high volatility or low volatility tend to cluster together. The assumption that the error term is independently distributed does not hold for time series of financial data and in reality, this means that periods of high or low changes in prices or returns persist for a period of time. (Ruey, 2005)

2.1.3 Realized Variance

Most commonly when modelling volatility is the use of daily data. When using intraday data frequencies for modelling and forecasting volatility the concept of realized variance needs to be covered. Realized variance is the sum of squared returns over a given period, usually for intraday data the sum of differentiated squared returns over 1 day is used for predictions.

Realized variance is an effective method of easily summarizing intraday variance to daily variance which is used for modelling. (Andersen et al, 2001) (Frijns & Margaritis, 2008). The realized variance of period ​t​ is computed as

, og(p ) log(p )

r_t = l _t − _t−1 (1)

, ealised V ariance for period t RV_t = ∑^M

j=1

r²_{t, j} = R

where r_t is the returns, p_t is the asset price at time p​t​, _t−1 is the asset price one time period before and​ M ​is the number of intraday observations.

2.2 Statistical Theory

2.2.1 The ARCH model

The simplest ordinary least square model (OLS) assumes that the expected value of all the squared residuals is the same at any given point, that the error terms are homoscedastic. With the purpose of estimating a model that would account for non-homoscedastic data, i.e. that the uncertainty could change over time, Engle (1982) created the first conditional heteroskedasticity model, so called the autoregressive conditional heteroskedasticity (ARCH) model. The ARCH model treats the heteroskedasticity as variance to be modeled.

The ARCH model turned out to be of great interest, particular with financial applications.

This new tool captured the tendency of financial variables to move between high and low volatility and became an essential part of modern asset pricing theory and practice (Orskaug, 2009).

The ARCH model assumes that the variance of the error term depends on its lagged value, i.e. the previous conditional variance of the error term. The mean process of ARCH models is formulated as

μ ε ,

r_t= + _t t = 1 , ..., T , (2)

where ​μ​ is the mean of the process ​r​t​, ​ε​t​​is its residual and ​T​ is the number of observations.

The residual, ​ε​t​, can be expressed as ε_t = σ_{t t}z , where ​z​t​ ~ iid​.

The conditional variance of the ARCH process is modeled as ,

ε .. ε

σ²_t = α₀+ α_{1 t−1}² + . + α_q ²_t−q + λ_{1 t−1}x (3) is an external regressor where >0, ​α​0​>0 ​and ​α​i 0, i>0.

λ₁ λ₁ ≥

This stochastic process functions as a predictor for the average size of the upcoming residuals. If ​z​t ​is assumed to be ​N(0,1)​, i.e. have zero mean and a variance of one, the conditional variance of ​r​t​ is the same as the variance of σ²_t丨t−1.

Even though the ARCH model had a great impact on the financial theory, it has its weaknesses. One is that it assumes that positive and negative shocks have the same response of the volatility of a time series. This is contrary to the reality since it is very common that the price of a financial asset responds in different ways to positive and negative shocks, which is shown by Paul (2007). Another weakness of the ARCH model is that it requires estimations of a large number of parameters due to the high order of the stochastic process needed for the

purpose of catching the dynamic of the conditional variance. This weakness is solved by the generalized ARCH (GARCH) model described in the next section.

2.2.2 The Generalized ARCH model

Bollerslev (1986) and Taylor (1986) proposed, independently of each other, the Generalized ARCH model that later would replace the ARCH model in applications (Teräsvirta, 2006).

The GARCH model allows for both a longer memory and a more flexible lag structure than the ARCH model (Bollerslev 1986) which makes the GARCH model to be more successful in predicting volatilities than the ARCH model.

In the GARCH model, a limited number of lags of conditional variances is replacing the unlimited number of lags of the residuals. This simplifies the lag structure as well as the estimation process of the parameters of the ARCH model.

A general notation of the GARCH model is GARCH( ​p, q​), where ​p represents the number of lags of the squared returns to be included in the model and ​q stands for the number of lags of variances to be included in the model. A GARCH(​0, q​) is thereby simply an ARCH(​q​) model.

The GARCH(​p, q​) model is specified as

r_t= μ_t+ ε_t (4)

σ ε_t= z_{t t}

ε .. ε σ .. σ x ..λ x

σ_t² = α₀+ α₁ ²_t−1+ . + α_{q t−q}² + β₁ ²_t−1 + . + β_p ²_t−p + λ_{1 t−1}+ . _{p t−p} where;

:

r_t log return of an asset at time ​t​.

:

μ_t expected value of the conditional log return of an asset at time ​t​.

:

ε_t mean corrected return of an asset at time ​t​.

:

σ_t the conditional variance at time ​t​.

:

z_t a sequence of random variables of specified distribution. If normal distribution, N(0, 1), E

[

]

[

]

:

α , , ..,₀ α₁ . α_q ARCH parameter of the GARCH-model.

:

β , ..,₁ . β_p GARCH parameter of the model.

: , .., λ

λ₁ . _p External regressor parameter.

: ,

p q order of the GARCH model.

A property of the GARCH model that can be seen as a weakness is that the model cannot recognize if there is a positive or a negative movement on the market, since the mean corrected returns, , are squared (as seen in equation (4) above).ε_t

2.2.3 GARCH(1, 1)

The most popular GARCH model in applications has been the GARCH(1, 1) model, where p​=​q​=1 (Teräsvirta, 2006). There are several advantages of using the GARCH(1, 1) ​model order. Primarily, it is fast, easy to estimate and commonly known to produce accurate results (Andersen & Bollerslev, 1998).

The GARCH(1, 1) means that the model has one lag of the squared return and one lag of the conditional variance, meaning that the conditional variance of the GARCH(1, 1) model is specified as

.

ε σ x

σ²_t = α₀+ α₁ ²_t−1+ β₁ ²_t−1+ λ_{1 t−1} (5)

Following, the notations of the GARCH(1, 1) will be used, since it is the model order used in this thesis.

2.2.4 Extensions of the GARCH model

Apart from the GARCH model, that is described above, this paper will examine two extensions of the standard GARCH model.

2.2.5 EGARCH(1, 1)

The exponential GARCH (EGARCH) model is an extended version of the GARCH model that utilizes the logarithmic values of the conditional variance. The EGARCH was developed to capture asymmetric effects of positive and negative returns when modelling and forecasting (Nelson, 1991). In the EGARCH(1, 1), ​a​t​have the same representation as before with the conditional variance now defined as

,

n(σ ) ε ( ε ( ε )) ln(σ ) x

l _t² = α₀+ α_{1 t−1}+ γ₁ |

| ^t−1|

| − E |

| ^t−1|

| + β^{1 2}t−t + λ_{1 t−1} (6) where the coefficient α₁ captures the sign effect and γ₁the size effect.

To demonstrate the capability of the EGARCH model to account for asymmetrical effects of positive and negative returns, consider the function g defined as

.

(ε ) ε ( ε ( ε ))

g _t = α_{1 t−1}+ γ₁ |

| ^t−1|

| − E |

| ^t−1|

|

The function can be rewritten with an indicator function, ​I​, as

,

(ε ) ) ε I( ε ) ) ε I( ε ) E( ε )

g _t = (α₁+ γ_{1 t t} > 0 + (α₁− γ_{1 t t} < 0 − γ₁ |

| ^t−1|

|

where the asymmetrical effect of positive and negative returns is noticeable. Positive shocks have an impact by (α₁+ γ₁) and negative shocks have an impact by (α₁− γ₁) on the logarithm of the conditional variance. With this model, negative price shocks have a larger impact than positive shocks.

2.2.6 GJR-GARCH(1, 1)

The GJR-GARCH (Glosten, Jagannathan and Runkle -GARCH) is an alternative way of taking the asymmetric effects of positive and negative price shocks into account. Like the EGARCH, the GJR-GARCH was developed to account for volatility clustering in financial data and handle asymmetry in the returns (Glosten et al, 1993). In the GJR-GARCH model, ​a​t

have the same representation as before and the conditional variance is given as ,

ε (1 [ε ]) ε I[ε ]) σ x

σ_t² = α₀+ α₁ ²_t−1 − I _t−1 > 0 + γ₁ ²_{t−1 t−1} > 0 + β₁ ²_t−1+ λ_{1 t−1} (7) where α₀> 0, α₁≥ 0, β₁ ≥ 0, γ₁ ≥ 0, and ​I​ is an indicator function.

As in the case with the EGARCH model, the asymmetrical effect of positive and negative shocks can be demonstrated with the function

,

(ε ) ε (1 I[ ε ]) ε I[ ε ]

g _t = α₁ ²_t−1 − _t−1> 0 + γ₁ ²_{t−1 t} > 0

where the asymmetrical effects can be seen. Positive shocks have an impact on the conditional variance with γ₁, while negative shocks have an impact on the conditional variance withα₁. Normally α₁ > γ₁ which means that a negative shock has a larger impact than positive shocks, as with the EGARCH.

2.2.7 GARCH with Realized Variance

In this thesis, the realized variance (RV, ​covered in section 2.1.3) will be added to the conditional variance equation of the various GARCH(1, 1) models as an external regressor.

This means that our models now take the following form:

GARCH(1, 1):

ε σ RV

σ_t² = α₀+ α_{1 t−1}² + β₁ ²_t−1+ λ_{1 t−1} (8) EGARCH(1, 1):

n(σ ) ε ( ε ( ε )) ln(σ ) RV

l _t² = α₀ + α_{1 t−1}+ γ₁ |

| ^t−1|

| − E |

| ^t−1|

| + β^{1 2}t−1 + λ_{1 t−1} (9)

GJRGARCH(1, 1):

ε (1 [ε ]) ε I[ε ]) σ RV

σ_t² = α₀+ α₁ ²_t−1 − I _t−1 > 0 + γ₁ ²_{t−1 t−1} > 0 + β₁ ²_t−1+ λ_{1 t−1} (10) where RV_t−1 is the realized variance of one day before. The variable ​RV ​is added to all GARCH-models evaluated and for each model three different frequencies of the realized daily variance are used as external regressor, 15-minute, 30-minute and 1-hour intraday data.

When put in as external regressor, the realized variance is used to estimate σ_t² by using to explain the conditional variance for the next period.

V

R _t−1

2.3 Distributions

As mentioned in the notation of the error term, ,z_t follows a specified distribution. The density function of ε_t will differ depending on which distribution that the error term is assumed to follow.

In this paper, three different distributions will be applied to the error term when estimating the different GARCH-models and the results will be compared.

2.3.1 The Normal Distribution

The normal distribution, although debated as non-sufficient for modelling financial data is still one of the most widely used distributions for financial modelling (​Costa et al, 2005).

The error term is normally distributed if the probability density function is given by

(z )

f

=

e

√

− 2σ2 (z −μ)t 2

(11)

where is the mean and μ σ² is the variance.

2.3.2 The Student’s t-distribution

The student’s t-distribution has the property of heavier tails than the normal distribution, depending upon which degree of freedom that is used. The student’s t distribution is very common in the field of financial modelling since stock returns often have larger tails than the normal distribution.

The error term is student’s t-distributed if the probability density function is given by

(z ) (1 )

f

=

σ (v−2) + 12

Γ(^{σ (v − 2)}₂ )

2

√

+

−^{σ (v−2)+1}₂

2

(12)

where is the gamma function and Γ ​v​ the degrees of freedom and ​v > 2​.

2.3.3 The Normal Inverse Gaussian Distribution (NIG)

The Normal Inverse Gaussian Distribution is a subclass of the generalized hyperbolic distribution. The NIG-distribution is versatile and allows for modelling leptokurtic and skewed processes with heavier tails than the normal distribution. (​Önalan & Ömer, 2010) The error term is Normal Inverse Gaussian distributed z_t ~ ​NIG(z , , , , )_t α β μ δ if the probability density function is given by

(z , , , , ) e

f

α β μ δ =

π

√

αδK (α₁

√

where μ is the mean, α is the tail heaviness, β is the skewness term and δ is the scale parameter.

K​j denotes a modified Bessel function. The conditions for the parameters are .

≤ α ≤ α, μ ∈ R, 0 0 | | , β < δ

2.4 Diagnostics and Evaluation

2.4.1 Normality test

Since financial time-series data tend to suffer from heteroskedasticity or volatility-clustering, statistical testing is needed to confirm that the data display the aspects associated with financial time series. Several different tests for examining normality exists. To test a univariate time-series the Jarque-Bera test of normality is used to evaluate if the logarithmic returns contains skewness or excess kurtosis that is significantly different from zero (Cryer &

Chan, 2008). The null hypothesis of the test is that the excess kurtosis and skewness is equal to zero while the alternative hypothesis is that either the excess kurtosis or skewness is not equal to zero. Testing of the skewness and kurtosis is made since the normal distribution have specific attributes, the normal distribution has a skewness equal to zero and a kurtosis equal to three. This means that accepting the null hypothesis implies that the data is normally distributed and failing to reject the alternative hypothesis implies that the data is not normally distributed.

Formally the Jarque-Bera test of normality is expressed as

,

B (S (K

J = ^{n −1+1} ₆ ²+ ₄¹ − 3) )² (14)

where ​S ​= Skewness = ,

1/n ∑ⁿ(x −x)

i=1 i 3

(1/n ∑ⁿ(x −x) )

i=1 i 3 3/2

K​ = excess kurtosis = ^{1/n ∑(x −x)}

n

i=1 i 4

(1/n ∑ⁿ(x −x) )

i=1 i 2 2

and the null hypothesis is H₀ : S = K = 0 against the alternative hypothesis,

≠0 or K≠0 H_a : S

2.4.2 Akaike Information Criterion (AIC)

Before proceeding with the Value at Risk, model estimations and their corresponding tests a statistical measure for model fit is needed for evaluation. When evaluating the model fit of the GARCH-models, the Akaike Information Criterion (AIC) will be used.

The Akaike information criteria methodology is based on maximizing the likelihood-function and is used to compare different models where the model(s) with the lowest information criteria best fits the data (Burnham et al, 2004). Formally the AIC is computed as

IC ×LLF 2m,

A = − 2 + (15)

where​ LLF​ is the log-likelihood function,​ m​ the number of parameters and ​N​ the sample size.

2.5 Forecast Evaluation

To evaluate how well the forecasts performs, the forecast of the conditional variance is compared to their paired actual values on a validation dataset. The conditional variance is forecasted with a rolling one day forecast, meaning that the models used for forecasts are refitted each day. The forecast is performed over a 200-day period from 2018/07/06 to 2019/04/24. Two of the most common statistical measures will be used to evaluate the forecast, the MAE and RMSE values.

2.5.1 MAE

The first measurement used to compare the estimated models is the Mean Absolute Error.

The MAE measures the average absolute error, that is how close the predicted conditional correlations are to the real conditional correlations (Matsuura & Wilmott, 2005). The MAE is calculated as

AE

M =

y −y

∑^T

t=1

|

|

=

e

∑^T

t=1

|

|

(17)

where yˆ_i is the predicted value, y_i the observed value and ​T the sample size. The model with the lowest MAE-value should perform the best when performing the forecast.

2.5.2 RMSE

The second measurement used to compare the models is the Root Mean Square Error. The RMSE uses the same approach as the MAE but does not compare absolute values. Instead of the average absolute error the standard deviation of the residuals is calculated (Matsuura, Wilmott 2005) as

MSE

R = √

where yˆ_t is the predicted value, y_t the observed value and T the sample size. As with the MAE, a lower value is preferred where 0 indicates that the model perfectly predicts/fit the data.

2.6 Value at Risk

Value at risk is a statistic measurement that quantifies and measures the worst loss over a period time for an asset, portfolio or security, that will not be exceeded given a specific confidence level, hence VaR is a measurement of risk for an investment. Formally, VaR for an asset can be expressed as

,

(1 ) (α)

V aR_asset − α = − σ_asset× q_asset (19)

where V aR_asset is the VaR for the asset, α = the confidence level, σ_asset = the volatility of the asset

and q_asset(α) = the standardized distribution(Cao et al. 2010).

2.6.1 Backtesting the Value at Risk & Kupiec’s test

To evaluate Value at Risk a method for testing the accuracy of the predictions is needed. A common method of testing VaR accuracy is to summarize the number of VaR violations i.e.

the number occasions where the estimated loss of returns is greater than the specified confidence level. The number of violations is then divided by the number of expected violations. This is the violation ratio which is formally expressed as

iolation ratio V R

V = = _q^x , (20)

where ​x is the number of actual violations and ​q ​the number of expected violations. A violation rate equal to 1 indicates that the models perfectly predict the VaR. If the ​VR is larger than 1 the model underestimates the risk and if the ​VR is less than 1 the model overestimates the risk. Either of these options indicate that the model performs poorly and optimally a failure rate closer to 1 is preferred.

To statistically test if the number of violations matches the actual VaR this paper will use Kupiec’s test. Kupiec’s test is a likelihood-ratio test of the binomial distribution which examines if the number of VaR violations exceeds the proportion specified by the given confidence level. (Kupiec, 1995)

The likelihood ratio is formally presented through the following equation:

R ln ln

L = − 2

[

]

[

]

The test is performed under the null hypothesis that p = pˆ = x/N and the alternative hypothesis that ≠ pp ˆ = x/N.If the null hypothesis is accepted the number of VaR violations correspond with the given confidence level and if the null hypothesis is rejected in favor for the alternative hypothesis the number of VaR violations exceed the acceptance of an accurate prediction.

3. Empirical Analysis

The following section will cover the data, analysis of the data, model estimation and evaluation. The estimations performed in this paper was done in R, mainly using the rugarch package and other existing packages as well as computations and code written by the authors (Ghalanos, 2018).

3.1 Data

The data for all the assets was collected from Finam Holdings, a Russian financial services company. The data is from 2015/01/02 to 2019/04/24 giving 1081 daily closing prices, resulting in 1080 daily return values. As the data is provided with 26 price observations per day (every 15th minute), a total of 28106 observations per stock are provided.

The in-sample period used to estimate the various GARCH models, from 2015/01/02 to 2018/07/06, consists of 800 daily returns, and the out-sample period used to validate the forecasts, from 2018/07/09 to 2019/04/24, consists of 200 daily returns.

Figure 1 shows the closing rates of Microsoft, Disney and Bank of America for the full period, 2015/01/02 to 2019/04/24.

Figure 1 - ​Daily closing price for Microsoft, Disney and Bank of America

3.1.1 Data evaluation

Figure 2 to 4 shows the daily logarithmic returns of the three stocks. All three of the financial time series shows signs that they are suffering from heteroscedasticity as the volatility clusters during periods of high and low fluctuations. All three assets show similar patterns of volatility with Disney and Microsoft occasionally spiking in their logarithmic returns. The data seems to contain the characteristics of financial time series, but further testing is still needed to statistically confirm the data characteristics.

Figure 2 - ​Returns Microsoft

Figure 3 - ​Returns Bank of America

Figure 4 - ​Returns Disney

3.1.2 Realized Variance

As the data provides 26 observations (every 15 minutes) of the stock price every day, the realized variance calculated from different return-frequencies can be used as external regressor in order to find how higher frequencies affect the model fit and forecasts.

The chosen frequencies that will be used as external regressors in the GARCH models in this thesis are;

RV15, where the daily realized variance is the sum of the squared intraday returns given every 15 minutes.

RV30, where the daily realized variance is the sum of the squared intraday returns given every 30 minutes.

RV1h, where the daily realized variance is the sum of the squared intraday returns given every hour.

By investigating the plots below, all of the realized variance plots are following the same pattern as the daily squared return. The plots below are data of the Microsoft stock - plots of the other assets can be found in the appendix

(Figure 9-16).

Figure 5 -​ Squared Returns Microsoft

Figure 6 - ​RV1h Microsoft

Figure 7 - ​RV30min Microsoft

Figure 8 - ​RV15min Microsoft

3.1.3 Descriptive Statistics

By observing Table 1, it seems like the mean of the returns for Bank of America and Disney is similar while Microsoft have a higher mean. The standard deviation for all stocks is quite similar with some variation between the assets. The minimum and maximum values for Microsoft and Disney are very similar while Bank of America have a lower spread in the logarithmic returns, which also can be observed in Figure 3. Furthermore, the skewness and kurtosis of the three assets show signs of non-normality and Microsoft and Disney show evidence of leptokurtic distributions. To investigate this further, a test of normality is performed.

Table 1 ​- Descriptive statistics for Microsoft, Bank of America & Disney

3.1.4 Result of normality test

Evaluating the results of Jarque Bera test in Table 2, confirms that the time series are not normally distributed. P-values for the Jarque-Bera test of the logarithmic returns are all below

0,05. Therefore, the null hypothesis, that the assets are normally distributed, is rejected in favor for the alternative hypothesis, that the assets are not normally distributed.

Table 2 - ​Results of Jarque Bera test

The logarithmic returns of the univariate time-series contain the classic characteristics of financial data i.e they are not normally distributed. To account for this, in modelling and forecasting, this thesis will use the student’s t-distribution and the normal inverse gaussian distribution in addition to the normal distribution to allow for heavier tails.

3.2 Model estimation

The estimated GARCH-models for each stock can be found in table 10-12 in the appendix.

By examine the tables, the effect that the external regressor has on the model can be seen. As realized variance (RV) is added to the models as an external regressor λ)( , the ARCH α)( , GARCH β)( and asymmetry γ)( terms become non-significant in most of the cases. Table 3 shows this result.

In the cases where the external regressor is non-significant, α + β < 1 , fulfilling the basic stationary criteria for GARCH-models, as table 4 shows.

Table 3​ - Estimated Parameters Standard GARCH model with normal distribution. Disney.

External regressor significant. (Full table in appendix)

Table 4​ - Estimated Parameters of Standard GARCH model with student t distribution.

Disney. External regressor not significant. (Full table in appendix)

As Table 3 and 4 above shows, the addition of the external regressor to the model also affects the model fit and the Akaike information criteria. The models, where the external regressor ​is significant, have a lower AIC compared to the AIC of the same model without ​RV ​as external regressor. This is the case for every model - if the external regressor is significant, the AIC is significantly lower than comparable model without external regressor.

In the cases where the external regressor is non-significant, the model produces equal AIC values as the model without an external regressor.

4. Forecast evaluation

As previously mentioned in section 4.1, the forecasts are performed over a 200-period from 2018/07/09 to 2019/04/24. A rolling forecast for each day have been used, meaning that the model refits every day.

The MAE and RMSE for the different model forecast, presented in table 5 to 7, indicates that the difference between the values for all models are very small. The marked values in the tables below indicate a non-significant external regressor in the model. As can be seen, the models with a non-significant external regressors produce equal MAE and RMSE values compared to the GARCH without the external regressor. The RMSE and MAE values corresponding to a model where the external regressor is non-significant, are marked in grey.

In the models with a significant external regressor the RMSE and MAE values are in most cases lower when the realized variance is included in the models, although the difference, as previously stated is very small. By looking at Table 5-7 there seems to be some evidence that the GARCH-models with hourly sampled realized variance produces the lowest values.

However, there are exceptions for all assets and distributions.

Table 5​ - RMSE and MAE for Microsoft

Table 6​ - RMSE and MAE for Bank of America

Table 7​ - RMSE and MAE for Disney

5.Value at Risk Forecasting & Backtesting

The Value at Risk is computed by using the 200-day forecast, on a 95% confidence level on three assets Microsoft, Disney and Bank of America. The forecast is performed through the use of the ​EGARCH(1, 1)​, ​GJR-GARCH(1,1) and ​GARCH(1,1)​-models with three different distributions for each model on six different assets. The expected number of violations for each Value at Risk forecast is 5% and this means that the results in table 8 is benchmarked against 5%. For example: The EGARCH forecast for Microsoft with the student’s t-distribution with 15-minute ​RV ​as an external regressor ​have 11,5% VaR violations (expected violations = 5%) and a p-value for Kupiec’s test (described in section 2.6.1) of 0,000 meaning that the model overestimated the risk, underperformed in the forecasting procedure and therefore is not valid for accurate Value at Risk predictions.

Table 8 - ​VaR Violations and Kupiec test for Microsoft, Bank of America and Disney

Overall, the models performed well when used for Value at Risk predictions. The majority of the models centered around a 5% VaR violation level and have a p-value for Kupiec’s test larger than 0,05 meaning that the null-hypothesis of an accurate Value at Risk prediction is accepted. Depending on the underlying asset/time series no model performed significantly better with the exception of the EGARCH-student’s t-distribution for Microsoft which provided an inaccurate forecast for the model with ​RV15min​.

To validate the results for the three assets mainly covered in this paper (Microsoft, Disney &

Bank of America) a 200-day forecast and Value at Risk prediction during the same time

period was performed on three additional assets: The Dow Jones Industrial Average, Apple Inc and Citigroup. The results (Table 9) when predicting and backtesting Value at Risk for the additional assets are similar to the results of the main assets with all the models centering around 5% VaR violations and p-values larger than 0,05 for Kupiecs-test.

Table 9 - ​VaR Violations and Kupiec test for Dow Jones, Apple Inc and Citigroup

6. Result Discussion

In the cases where the realized variance is added to the model and is significant, the parameter β becomes insignificant in most of the cases. As β is the parameter for _σ² , the

t−1

previous conditional variance, it seems like ​RV​, the realized variance, overtakes the explanatory power and that the model uses the realized variance instead of the conditional variance in order to estimate the upcoming conditional variance.

Even though the addition of realized variance as external regressor seems to improve the model fit, according to the AIC-value, compared to the models without realized variance the Value at Risk forecast is not better than the daily model predictions where both the GARCH with and without realized variance​predictions produces accurate results. Interestingly, when the realized variance parameter is significant, the AIC values is lowest for the hourly-sampled realized variance, and not the most frequent sampled realized variance. This implies that even though “more” information is provided regarding the realized variance, the model fit does not improve. It is reasonable to believe that more frequent data may contain noise which negatively affects the estimation, and therefore the more frequent sampled realized variance may not be reliable to model daily realized variance which leads to the hourly sampled realized variance producing a better model fit.

7. Conclusion

The purpose of this study was to study the effects of realized variance calculated with different sampling frequencies on different GARCH-processes. The estimation of the model, the forecast performance and the Value at Risk predictions have been investigated and analyzed. Estimation and forecasts were made with the use of three different GARCH-models (EGARCH, GJR-GARCH and GARCH) with three different distribution applications.

From the results in this paper it is shown that the models where realized variance is included as a significant variable produces a better model fit compared to the use of daily data according to the methodology of the Akaike information criteria. When the realized variance parameter is significant, the AIC values is lowest for the hourly-sampled realized variance, and not the most frequent sampled realized variance.

Regarding the forecasting performance, the GARCH models with realized variance, when significant, seems to produce slightly better RMSE and MAE values in most of the cases. As the case with the AIC values, the hourly sampled realized variance provides the best values in most of the cases. With that said, the differences in the RMSE and MAE values are very small and produces ambiguous results. Therefore, there is no clear evidence that GARCH models with realized variance produces better forecasts than GARCH models without realized variance.

When it comes to predicting Value at Risk, all models - with and without realized variance - performed accurate predictions. Almost no models made inaccurate predictions according to Kupiec’s test since the p-values were larger than 0,05. By validating the result of the VaR predictions further by applying it on three additional assets the conclusion of the results from the VaR predictions is that neither the GARCH models with- and without realized variance produces a more accurate prediction compared to each other.

8. Further Studies

Regarding further studies of GARCH-processes and the use of intraday data would be to use the application of Value at Risk predictions computed in this study and extended it to the area of Multivariate GARCH-models as this might prove to be more applicable to real life scenarios. As investments often are made in portfolios, funds and other securities were there are several univariate time-series of assets simultaneously affecting the outcome. The study may also be improved upon by evaluating other GARCH-models beyond the ones covered in this study.

A second area of interest for further studies would be to apply the models to more volatile assets and test the accuracy of the different GARCH-processes under more uncertain conditions. Since this paper made use of major assets and blue-chip stocks, an evaluation of more volatile assets may provide additional perspectives of the GARCH-model and its Value at Risk application.

In addition to applying the GARCH-model on more volatile assets another potential extension of this study would be to look at the effects of periods of market stress and financial crises on the forecasts as this was ​consciously ​excluded in this study.

Lastly, to further develop this study, less and more frequent sampling frequencies of realized variance should be evaluated with the goal of finding a optimal frequency to use for estimation of the daily variance.

9. Appendix

In the Appendix, additional tables and figures together with a brief description of the rugarch package of RStudio can be found.

9.1 Appendix I - Tables and figures

9.1.1 Squared returns and Realized variance

Below are the remaining plots from section 3.1.2. By examine the plots, the realized variance seems to follow the same pattern as the squared returns.

Figure 9 - ​Squared Returns Bank of America

Figure 10 - ​RV1h Bank of America

Figure 11 - ​RV30min Bank of America

Figure 12 - ​RV15min Bank of America

Figure 13 - ​Squared Returns Disney

Figure 14 - ​RV1h Disney

Figure 15 - ​RV30min Disney

​

Figure 16 - ​RV15min Disney

9.1.2 Estimated models

In the coming pages, the complete tables of the model estimations are provided. In section 3.2, a sub-table from these tables are provided.

9.2 Appendix II - RStudio - Rugarch

The rugarch package (Ghalanos, 2018) helps us model the GARCH models that is used in this thesis.

To construct a GARCH model we first have to specify which model we want. This is done with ugarchspec.

garchspec <-  ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1),   external.regressors = NULL),  

mean.model = list(armaOrder = c(0, 0)), distribution.model = "norm") 

First, we specify the variance model which is done by choosing which GARCH model to use.

In this example “sGarch” is used, which gives us the standard GARCH. To specify the exponential GARCH model, “eGARCH” is used and “gjrGARCH” is used for the gjr-GARCH model.

The model order and external regressor is also included in the variance model. In our thesis, the different frequencies of realized variance is added as an external regressor in the variance model.

Secondly, we specify the mean model, by choosing arma order and which distribution we assume for the process. In this thesis the Standard normal distribution, “norm”, Student’s t distribution, “std”, and Normal inverse gaussian distribution, “nig”, have been used.

When the model is specified, we can fit a model to the data we have. This is done with ugarchfit. The estimations of parameters in the specified model will be maximum likelihood estimates.

garchfit <- ugarchfit(spec = garchspec, data = log.returns,   out.sample = 200, solver = 'hybrid') 

Where garchspec is the specified model and log.returns are the returns that are used to estimate the model. By “​out.sample = 200 ​”, we leave 200 observations out of the model, as these are the observations that will be used for forecasting. In this thesis we are using a rolling forecast, meaning that the model will refit every day.

garchroll <-  ugarchroll(spec = garchspec, data=log.returns, n.ahead = 1,  forecast.length = 200, refit.every = 1,  

calculate.VaR = TRUE, VaR.alpha =0.05) 

By specifying “​calculate.VaR = TRUE ​”, the ugarchroll calculates the VaR violation rate at the specified alpha, which in this case is 5%.

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