Volatility-forecasting-using-high-frequency-data-Evidence-from-stock-markets_2014_Economic-Modelling

Volatility forecasting using high frequency data:Evidence from stock markets ☆
Sibel Çelik a ,⁎,Hüseyin Ergin b
a Dumlupinar University,School of Applied Sciences,Turkey b
Dumlupinar University,Business Administration,Turkey
a b s t r a c t
a r t i c l e i n f o Article history:
Accepted 24September 2013JEL classi fication:C22G00
Keywords:Volatility
Realized volatility High frequency data Price jumps
The paper aims to suggest the best volatility forecasting model for stock markets in Turkey.The findings of this paper support the superiority of high frequency based volatility forecasting models over traditional GARCH models.MIDAS and HAR-RV-CJ models are found to be the best among high frequency based volatility forecasting models.Moreover,MIDAS model performs better in crisis period.The findings of paper are important for financial institutions,investors and policy makers.
©2013Elsevier B.V.All rights reserved.
1.Introduction
Volatility plays an important role in theoretical and practical applica-tions in finance.The availability of high frequency data brings a new dimension to volatility modeling and forecasting of returns on financial assets.First and foremost,nonparametric estimation of volatility of asset returns becomes feasible and so modeling and forecasting volatility of asset returns has been a focus for researchers in the literature (Andersen and Bollerslev,1998;Andersen et al.,2001,2003b ,2007;Corsi,2004;Engle and Gallo,2006;Ghysels et al.,2004,2005,2006a,b;Hansen et al.,2010;Shephard and Sheppard,2010).The empirical find-ings of existing studies support the superiority of high frequency based volatility models to popular GARCH models and stochastic volatility models in the literature (Andersen et al.,2003b ).Besides,earlier studies point to importance of allowing for discontinuities (jumps)in volatility models and pricing derivatives (Andersen et al.,2002;Chernov et al.,2003).Availability of high frequency data is also a turning point in order to distinguishing jump from continuous part of price process.Empirical findings from recent studies show that incorporating the jumps to volatility models increase the forecasting performance of models supporting the earlier evidence (Andersen et al.,2003b,2007).
This paper aims to suggest the best volatility forecasting model in stock markets in Turkey.For this purpose,first,we analyze the data generating process and calculate the high frequency based volatility and examine the return and volatility characteristics.Second,we propose the best volatility forecasting model by comparing different volatility forecasting models.
In doing so,the paper will contribute to the literature in terms of filling five main gaps.First,it suggests the best volatility forecasting model from the alternatives including high frequency-based models and traditional GARCH models.Second,it reveals the forecasting performance of volatility models during the periods of structural change.Because,recent studies in the literature indicate that financial crisis affect the volatility dynamics deeply (Dungey et al.,2011).Third,it analyses forecasting performance of volatility in stock futures markets rather than spot markets.There are three reasons for usage of stock futures markets in this study.Firstly,there are findings in the literature that futures markets respond to new information faster than spot markets (Stoll and Whaley,1990).Secondly,using futures contracts rather than spot indexes re-duces nonsynchronous trading problems (Wu et al.,2005).Thirdly,using futures contracts provides additional evidence to the existing literature on spot markets (Wu et al.,2005).Fourth,it compares the findings at different frequencies to inference about optimal fre-quency since the sampling selection is important for high frequency data based studies.Because,while higher sampling frequency may cause bias in realized volatility,lower sampling frequency may cause information st,it contributes to literature in terms of presenting evidence from an Emerging Market.
Economic Modelling 36(2014)176–190
☆This paper is based on my doctoral dissertation “Volatility Forecasting in Stock Markets:Evidence From High Frequency Data of Istanbul Stock Exchange ”which was completed at Dumlupinar University,in 2012.
⁎Corresponding author at:Dumlupinar University,School of Applied Sciences,Insurance and Risk Management Department,Turkey.Tel.:+902742652031x4664.
E-mail address:sibelcelik1@ (S.
Çelik).0264-9993/$–see front matter ©2013Elsevier B.V.All rights reserved.
/10.1016/j.econmod.2013.09.038
Contents lists available at ScienceDirect
Economic Modelling
j ou r n a l h o m e p a ge :w ww.e l s e v i e r.c o m /l oc a t e /e c mo d
The paper proceeds as follows.Section2introduces dataset of the paper.Section3explains the methodologies used in the paper. Section4summarizes the empiricalfindings.Section5concludes the paper.
2.Data
The dataset comprises of ISE-30index futures data at intradaily and daily frequency from04.02.2005to30.04.2010.1We generate new data sampled at1-minute interval,5-minute interval,10-minute interval and15-minute interval.The number of intraday observations of ISE-30index future are502,101,51and34,respectively.
Careful data cleaning is one of the most important point in volatility estimation from high frequency data.The importance of cleaning of high frequency data is emphasized in the literature (Brownless and Gallo,2006;Dacorogna et al.,2001;Hansen and Lunde,2006).
In this paper,we used following steps for data cleaning process.
1.We delete entries which related to weekends.
2.We delete entries of public holidays,which is announced by Istanbul
Stock Exchange and Turkish Derivatives Exchange.
3.We delete entries when the Stock Exchanges do not trade full days.
4.We delete entries which is not common for Istanbul Stock Exchange
and Turkish Derivatives Exchange from04.02.2005to30.04.2010.
3.Methodology
3.1.Methodologies for volatility modeling and data analysis
3.1.1.GARCH model
The GARCH models are as follows:
r t¼
ffiffiffiffiffi
h t
q
εtð1Þ
h t¼α0þ
X q
i¼1αi r2t−iþ
X p
j¼1
βj h t−jð2Þ
p≥0,q N0,α0N0,αi≥0∀i≥1,i=1,……..p,βj≥0∀j≥1.
3.1.2.Continuous-jump diffusion process
The continuous-time jump diffusion process traditionally used in asset pricingfinance is expressed as:
dp t¼μdtþσtðÞdW tðÞþγtðÞdq tðÞ0≤t≤Tð3Þwhereμt is a continuous and locally bounded variation process,σt is stochastic volatility process,W t denotes a Standard Brownian motion, dq t is a counting process with dq t=1,corresponding to a jump at time t and dq t=0otherwise with jump intensityψ(t),andγ(t)refers to the size of jumps.The quadratic variation for the cumulative return process,r(t)=p(t)−p(0)is given by,
r;r ½
t ¼
Z t
σ2sðÞdsþ
X
0⊲s≤t
γ2sðÞ:ð4Þ
Quadratic variation consists of∫
t
0σ2sðÞds continuous and∑
0⊲s≤t
γ2sðÞ,
jump components.In the absence of jumps,the second term in the right will not exist and quadratic variation will equal to integrated volatility (Andersen et al.,2003a).3.1.3.Realized volatility,bipower variation and jumps
Let theθ—period returns be denoted by,r t,θ=p(t)−p(t−θ).We define daily realized volatility by the summing corresponding1/θhigh frequency intradaily squared returns as follows:
RV tþ1θðÞ¼
X1=θ
j¼1
r tþj:θ;θ2:ð5Þ
In the absence of jumps,realized volatility is the consistent estimate of the integrated volatility.However,in the presence of jumps,we need more powerful measurement.Barndorff et al.(2004)introduce the volatility measurement which is powerful in the case of jumps called bipower variation(henceforth:BV).BV is defined as follows:
BV tþ1θðÞ¼μ1−2
X1=θ
j¼2
r tþj
r tþj−1
ðÞ
ð6Þ
μ1−2≅0:7979in Eq.(6).
While the realized volatility consists of both continuous and jump components,BV only includes continuous component.Thus,jump component may be consistently estimated by,
RV tþ1θðÞ−BV tþ1θðÞ¼
X
t≺s≤tþ1
γ2sðÞ:ð7Þ
To prevent the right hand-side of Eq.(7)from becoming negative, we impose non-negativity truncation on the jump measurements.
J tþ1θðÞ¼max RV tþ1θðÞ−BV tþ1θðÞ;0
ÂÃ
ð8ÞContinuous component is given in Eq.(9).
BV tþ1θðÞ¼RV tþ1θðÞ−J tþ1θðÞð9Þ3.2.Methodologies for volatility forecasting
3.2.1.GARCH model
To evaluate the forecasting performance of GARCH model,first we es-timate Eqs.(1)and(2).Let h tþ1G denote the predicted value for h t.The forecast error for the GARCH model for the observation t+1is computed as RV tþ1−h tþ1G based on the existing literature(Alper et al.,2009).
3.2.2.HAR-RV model
HAR-RV model is introduced by Corsi(2004)and denoted as,
RV tþ1¼β0þβD RV tþβW RV t−5;tþβM RV t−22;tþεtþ1ð10Þt=1,2,3…..T.RV t,RV t−5and RV t−22mark daily,weekly and monthly realized volatility respectively.Multi-period realized volatility compo-nents such as weekly and monthly realized volatility is calculated as, RV t;tþh¼h−1RV tþ1þRV tþ2þ::::þRV tþh
ÂÃ
ð11Þh¼1;2;…RV t;tþ1≡RV tþ1:ð12Þ
In this paper,we take h=5and h=22as the weekly and monthly volatility,respectively.Andersen et al.(2003b)state that the distribu-tion of standard deviation and logarithmic form of realized volatility are close to normal than original form and so using these proxies increases performance of volatility forecasting.Therefore,we estimate standard deviation and logarithmic form of Eqs.(11)and(12).
RV tþ1
ÀÁ1=2
¼β0þβD RV t
ðÞ1=2þβW RV t−5;t
1=2
þβM RV t−22;t
1=2
þεtþ1ð13Þ
1ISE-30index futures data were taken from Turkish Derivatives Exchange.177
S.Çelik,H.Ergin/Economic Modelling36(2014)176–190
log RV t þ1ÀÁ
¼β0þβD log RV t ðÞþβW log RV t −5;t
þβM log RV t −22;t
þεt þ1
ð14Þ
3.2.3.HAR-RV-J model
HAR-RV-J model is developed by Andersen et al.(2003b)by including jump component in HAR-RV model.Daily HAR-RV-J model is expressed in Eq.(15),
RV t ;t þ1¼β0þβD RV t þβW RV t −5;t þβM RV t −22;t þβj J t þεt ;t þ1:
ð15Þ
Logarithmic and standard deviation form of HAR-RV-J model is given in Eq.(16)and (17).RV t þ1
ÀÁ1=2
¼β0þβD RV t ðÞ1=2
þβW RV t −5;t
1=2
þβM RV t −22;t 1=2
βj J t ðÞ1=2
þεt þ1
ð16Þ
log RV t þ1ÀÁ
¼β0þβD log RV t ðÞþβW log RV t −5;t
þβM log RV t −22;t
þβj log J t þ1ðÞþεt þ1
ð17Þ
3.2.
4.HAR-RV-CJ model
Andersen et al.(2007)develop HAR-RV-CJ model by including jump and continuous components separately in HAR-RV model.Daily HAR-RV-CJ model is stated as follows,
RV t ;t þ1¼β0þβCD C t þβCW C t −5;t þβCM C t −22;t þβjD J t þβJW J t −5;t
þβJM J t −22;t þεt ;t þ1:ð18ÞMulti period jump and continuous components are calculated as in
Eqs.(19)and (20),J t ;t þh ¼h
−1J t þ1þJ t þ2þ::::::þj t þh
Â
Ã
ð19ÞC t ;t þh ¼h
−1C t þ1þC t þ2þ::::::þC t þh Â
Ã
:
ð20Þ
Logarithmic and standard deviation form of HAR-RV-CJ model is given in Eqs.(21)and (22).
RV t ;t þ1
1=2¼β0þβCD C t ðÞ1=2
þβCW C t −5;t 1=2
þβCM C t −22;t 1=2þβjD J t ðÞ1=2
þβJW J t −5;t
1=2
þβJM J t −22;t 1=2
þεt ;t þ1ð21Þ
log RV t ;t þ1 ¼β0þβCD log C t ðÞþβCW log C t −5;t
þβCM log C t −22;t
þβjD log J t þ1ðÞ
þβJW log J t −5;t þ1 þβJM log J t −22;t þ1
þεt ;t þ1
ð22Þ
3.2.5.MIDAS (mixed data sampling)model
MIDAS model is introduced by Ghysels et al.(2004,2005,2006a,b).Univariate MIDAS linear regression is given in Eq.(23),
Y t ¼δ0þδ1
X k max k ¼0
B k ;θðÞX
m ðÞ
t −k =m þεt :½23
Y t and X (m )are one-dimensional processes,B (k ,θ)is polynomial
weighting function depending on k and θparameter and X t m ðÞis sampled m times more frequent than Y t .For example,if t denotes a
22-day monthly sampling and m =22,model (23)shows a MIDAS regression of monthly data (Y t )on past k max daily data (X t )RV t þ1;t ¼δ0þδ1
X k max k ¼0
B k ;θðÞRV
m ðÞ
t −k =m
þεt ð24Þ
m =1,k max =50and t refers to daily observations.
Ghysels et al.(2006a)suggest various alternatives for B (k ,θ)polyno-mial.In this paper,we focus on beta polynomial following Ghysels et al.(2006a).B (k ,θ),is denoted as in Eq.(25).B k ;θðÞ¼f k =k max ;θ0;θ1
ÀÁX k max k ¼1f k =k max
;θ0;θ1ÀÁð25Þ
and,f x ;θ0;θ1ðÞ¼
x θ0−11−x ðÞθ1−1Γθ0þθ1ðÞ
Γθ0ðÞΓθ1ðÞ
:
ð26Þ
Γ(.)is gamma function.In beta function,we restrict our attention to θ0=1and estimate θ1N 1.
3.2.6.Realized GARCH model
Realized GARCH model which is introduced by Hansen et al.(2010)can be expressed as in Eqs.(27),(28)and (29).r t ¼
ffiffiffiffiffih t q z t
ð27Þh t ¼w þβh t −1þγx t −1ð28Þx t ¼ξþϕh t þτz t ðÞþu t
ð29Þ
r t is return;z t ~iid (0,1),u t e iid 0;σu 2ÀÁ
,τ(z )is leverage function,h t =var(r t |F t −1),and F t =σ(r t ,x t ,r t −1,x t −1,.....).
RGARCH model is estimated with maximum likelihood method as GARCH model.Log likelihood function is given in Eq.(30).‘r ;x ;θðÞ¼−
1X n t ¼1
log h t ðÞþr t 2=h t þlog σu 2 þu t 2=σu 2
h i :ð30Þ
We evaluate the forecasting performance of RGARCH model as in GARCH model.
3.3.Evaluation of forecasting performance
We use mean squared error (MSE),mean absolute error (MAE),mean absolute percentage error (MAPE)and Theil's U statistic (TIC)to evaluate performance of volatility forecasting models.Calculation of the loss functions is as follows;RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN
−1X N i ¼1
RV t ;t þH −R ^V t ;t þH 2v u u t ð31Þ
MAE ¼N
−1
X N i ¼1
RV t ;t þH −R ^V t ;t þH
ð32Þ
MAPE ¼N −1X N i ¼1
RV t ;t þH −R ^V t ;t þH RV t ;t þH
ð33Þ
178S.Çelik,H.Ergin /Economic Modelling 36(2014)176–190
TIC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N i ¼1
RV t ;t þH −R ^V t ;t þH
2v u u t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N i ¼1
RV t ;t þH 2v u u t þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N i ¼1
R ^V t ;t þH
2v u u t :ð34Þ
RV t ,t +H and R ^V
t ;t þH denote actual and predicted values of realized volatility,respectively.N is the number of observation.In addition to these loss functions,we also use Mincer and Zarnowitz regression (1969)in this paper.Mincer –Zarnowitz regression is given in Eq.(35).
RV t ;t þH ¼a þb R ^V t ;t þH þu t ;t þH
ð35Þ
In the regression model,null hypothesis is formed as “a and b equal
to 0”.If the forecasting is unbiased,a and b coef ficients must equal to 0and 1respectively and coef ficient of b must be signi ficant.4.Empirical findings
The empirical findings are categorized under four sub-sections.•We use six different models (GARCH,HAR-RV,HAR-RV-J,HAR-RV-CJ,MIDAS,RGARCH)to determine the best volatility forecasting model.
•We present the findings of different transformations of volatility (stan-dard deviation,logarithmic)following Andersen et al.(1999,2000)and Andersen et al.(2001).2
•We compare the volatility forecasting performance at different frequencies to inference about optimal sampling frequency (1min,5min,10min and 15min).
•We compare the volatility forecasting performance for different
sample periods to examine the impact of financial crisis on volatility structure (pre-crisis period,crisis period and total period).3Tables 1,2,3and 4present summary statistics of the variables 4at 1-minute,5-minute,10-minute and 15-minute frequencies respective-ly.According to return statistics,as expected the mean returns are higher in pre-crisis period for all frequencies.However,there are no consistent information about the maximum and minimum values.The standard deviation of returns is higher in pre-crisis period for most of the frequencies.While returns have positive skewness in pre-crisis period,it is negative in crisis period for most of the frequencies.LB statistics show the evidence of autocorrelation between return series.Mean values of realized volatility and jump statistics are higher in pre-crisis period,5however mean values of bipower variation are higher in crisis period for most of the frequencies.
Variables have kurtosis greater than 3except log(RV)supporting leptokurtic distribution.The higher the frequency,skewness and kurto-sis degree of returns also increase.The distribution of standard devia-tion and logarithmic transformation of variables are close to normal.
Table 5shows the summary statistics of conditional variance series of GARCH(1,1)estimation.The mean returns are higher in pre-crisis period.Maximum and minimum values of returns are appeared in crisis period.The standard deviation of returns is higher in crisis period.Re-turn series did not distribute normal.The mean conditional variance is higher in crisis period with the value of 0.0005.The conditional variance series is rightly skewed and has leptokurtic distribution.Appendices
2
We examine the standard deviation and logarithmic transformation of realized vola-tility,jumps statistics since Andersen et al.(1999,2000)and Andersen et al.(2001)indi-cate that the distribution of standard deviation and logarithmic transformations are close to normal and using these proxies increase performance of volatility forecasting.
3
We examine the impact of 2007global crisis on volatility characteristics.In the litera-ture,there are some findings that global financial crisis give the first signal with announce-ment of the problems with the hedge funds of Bear Stearns in July 2007.Some papers use 17July 2007as a starting date of the global financial crisis (Dungey,2009).Following these findings,we determine three sub-periods (from 04.02.2005to 16.07.2007is pre crisis pe-riod,from 17.07.2007to 30.04.2010is crisis period and from 17.07.2007to 30.04.2010is total period).4
Return,realized volatility,standard deviation and logarithmic transformation of real-ized volatility,jumps,standard deviation and logarithmic transformation of jumps and bipower variation.5
Turkish Derivative Exchange has started operating on February 2005and pre-crisis period includes this period.In the first few days,trading volume is low and there is more time difference between instantaneous price quotations.For this reason,realized volatility or price jumps may increase in pre-crisis period.
Table 1
Summary statistics for ISE −30index futures at 1minute frequency.Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LB
Return
Pre-crisis period 1.91000.1185−0.11750.00360.056269.727134,906Crisis period 0.33400.1071−0.10800.0044−0.048759.854577,165Total period 1.08000.1185−0.11750.0040−0.013764.8631115,010RV
Pre-crisis period 0.00060.1006 4.33000.0120 4.422627.88391887.80Crisis period 0.00970.263710.6000.0195 5.960658.6426776.67Total period 0.00830.2637 4.33000.0165 6.165466.35392008.80RV 1/2
Pre-crisis period 0.06480.31720.00650.0503 1.80407.56731939.60Crisis period 0.07510.51360.01020.0643 1.94338.81892143.70Total period 0.07020.51360.00650.0584 1.98199.14684117.40log(RV)
Pre-crisis period −6.0277−2.2960−10.0473 1.5362−0.1492 2.49181646.00Crisis period −5.8363−1.3325−9.1513 1.65110.0407 2.13533490.50Total period −5.9263−1.3325−10.0473
1.6004−0.0255
2.31535114.80J
Pre-crisis period 278.4000.04410.00000.0050 3.734620.70281438.30Crisis period 175.3000.03870.00000.0035 4.6203 4.620332.6315Total period 223.8000.04410.00000.0043 4.194826.18642584.70J 1/2
Pre-crisis period 0.04070.21010.00000.0336 1.7732 6.59631830.30Crisis period 0.03070.19680.00000.0284 1.88457.55042033.20Total period 0.03540.21010.00000.0313 1.84657.18163982.60log(J +1)
Pre-crisis period 0.00270.04320.00000.0049 3.692220.23241451.40Crisis period 0.00170.03800.00000.0035 4.566931.8685929.74Total period 0.00220.04320.00000.0043 4.146525.57822609.70BV
Pre-crisis period 395.0000.07950.0000844.300 4.984233.88221777.30Crisis period 803.6000.22507.30001640.00 6.326564.7405699.36Total period
611.400
0.2250
0.0000
1340.80
6.9436
82.1317
1795.60
Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung –Box (1979)Q test.Mean returns are multiplied by 106and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by 105.
179
S.Çelik,H.Ergin /Economic Modelling 36(2014)176–190
1–18present the results of GARCH,HAR-RV,HAR-RV-J,HAR-RV-CJ, MIDAS and RGARCH estimations,lost functions and Mincer Zarnowitz test.Prior evidence shows that the performance of volatility models is the best at15-minute frequency sampling for ISE-30index futures since lost functions are minimum at15-minute frequency.Therefore, in this section,we only compare the performance of different volatility models using15-minute frequency as a base for ISE-30index futures rather than comparison of different frequencies.
Tables6and7present the comparison of forecasting performance of models in pre-crisis and crisis period for ISE-30index futures.In pre-crisis period,RMSE,MAE and TIC functions indicate that GARCH model has the worst forecasting performance.Different from the RMSE, MAE and TIC functions,MAPE function support that GARCH and RGARCH models have the best forecasting performance.The best model is controversial for the pre-crisis period.RMSE supports the superiority of MIDAS1/2model,MAE functions support the superiority of MIDAS log model,according to MAPE function,GARCH model is the best,and TIC function supports the superiority of HAR-RV-CJ model.When we eval-uate all loss functions together,MIDAS1/2and MIDAS log models seem to be the best models.Then,HAR-RV-CJ1/2,MIDAS,HAR-RV-CJ AND HAR-RV-J1/2models perform well,respectively.Thefindings of HAR-RV, HAR-RV-J and HAR-RV-CJ models support that including jump
Table2
Summary Statistics for ISE-30index futures at5-minute frequency.
Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LB
Getiri Pre-crisis period9.49000.0980−0.09870.00500.043444.61377845.80 Crisis period 1.66000.1056−0.10410.0048−0.115343.646710292.00
Total period 5.34000.1056−0.10410.0049−0.035844.208817940.00 RV Pre-crisis period0.00250.0412 3.25000.0044 3.813721.90291232.10 Crisis period0.00230.0651 6.39000.0046 5.917859.1149578.34
Total period0.00240.0651 3.25000.0045 4.992443.11311675.10 RV1/2Pre-crisis period0.04020.20300.00570.0306 1.85787.00601634.40 Crisis period0.03810.25530.00790.0302 2.17099.83241590.00
Total period0.03910.25530.00570.0304 2.01828.43503213.60 log(RV)Pre-crisis period−6.9039−3.1888−10.3342 1.37870.1349 2.61381464.40 Crisis period−7.0161−2.7304−9.6581 1.34920.4119 2.43062465.60
Total period−6.9634−2.7304−10.3342 1.36380.2792 2.50223847.20 J Pre-crisis period80.0000.01650.00000.0018 4.578029.11271835.30 Crisis period37.3000.00920.00000.0008 5.117639.0344328.33
Total period57.8000.01650.00000.0013 5.596444.63463207.60 J1/2Pre-crisis period0.02040.12870.00000.0198 2.18988.84651950.50 Crisis period0.01400.09640.00000.0132 2.06438.8297816.22
Total period0.01700.12870.00000.0169 2.374710.63753328.50 log(J+1)Pre-crisis period0.00080.01640.00000.0017 4.558828.87451840.70 Crisis period0.00030.00920.00000.0008 5.103438.8103329.55
Total period0.00050.01640.00000.0013 5.571044.22803213.90 BV Pre-crisis period176.4000.04390.0000365.000 5.719148.5876683.49 Crisis period200.0000.0653 4.5000418.0007.125586.0992462.04
Total period189.0000.06530.0000394.000 6.650274.74101076.40
Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung–Box(1979)Q test.Mean returns are multiplied by106 and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by105.
Table3
Summary statistics for ISE-30index futures at10-minute frequency.
Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LB
Getiri Pre-crisis period18.8000.0980−0.08200.00570.177432.72843738.50 Crisis period 3.28000.1010−0.10330.0053−0.400842.59323266.00
Total period10.6000.1010−0.10330.0055−0.096837.43296961.50
Pre-crisis period0.00170.0234 2.31000.0029 3.445717.38321621.90 RV Crisis period0.00140.0271 5.55000.0027 4.647631.6358499.54 Total period0.00150.0271 2.31000.0028 4.018723.84162014.90
Pre-crisis period0.03280.15300.00480.0250 1.8493 6.59321875.60 RV1/2Crisis period0.03080.16480.00740.0228 2.18809.14871246.40 Total period0.03180.16480.00480.0239 2.01807.77933137.30
Pre-crisis period−7.2965−3.7541−10.6756 1.34680.2134 2.73751393.20 log(RV)Crisis period−7.3606−3.6050−9.7991 1.22020.5178 2.72401901.00 Total period−7.3305−3.6050−10.6756 1.28120.3605 2.74873234.00
Pre-crisis period51.6000.01490.00000.0012 5.868249.57951260.70 J Crisis period30.0000.00840.00000.0006 5.559745.5093235.43 Total period40.1000.01490.00000.0010 6.557864.93791944.80
Pre-crisis period0.01580.12230.00000.0162 2.397510.88001425.90 J1/2Crisis period0.01230.09170.00000.0121 2.24449.9725467.14 Total period0.01390.12230.00000.0143 2.469911.78152112.00
Pre-crisis period0.00050.01480.00000.0012 5.841649.13721266.60 log(J+1)Crisis period0.00020.00830.00000.0006 5.545545.2689235.76 Total period0.00040.01480.00000.0010 6.526364.29861951.10
Pre-crisis period121.0000.0278 1.0800236.600 5.085640.7553799.06 BV Crisis period118.0000.0229 3.8800225.100 4.816233.6960464.41 Total period120.0000.0278 1.0800230.000 4.958337.47171235.90
Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung–Box(1979)Q test.Mean returns are multiplied by106 and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by105.
180S.Çelik,H.Ergin/Economic Modelling36(2014)176–190
component in model increase the forecasting performance.In general,GARCH and RGARCH models have the worst forecasting performance.
In crisis period,GARCH model has the worst forecasting perfor-mance according to RMSE and TIC.It is not clear which model is the best,however MIDAS log and MIDAS 1/2models perform well than others.Than HAR-RV-CJ,HAR-RV-J and HAR-RV models follow MIDAS log and MIDAS 1/2models.Both in pre-crisis and crisis period,high
frequency based volatility models have better forecasting performance than traditional GARCH model.5.Conclusion
This paper aims to suggest the best volatility forecasting model for stock markets in Turkey.For this purpose,first we analyze the data
Table 4
Summary statistics for ISE-30index futures at 15-minute frequency.Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LB
Getiri
Pre-crisis period 28.2000.0971−0.10130.0059−0.115332.71252276.40Crisis period 5.13000.0773−0.07810.0056−0.367324.92381269.00Total period 16.0000.0971−0.10130.0058−0.238529.09403459.00Pre-crisis period 0.00120.01980.40000.0020 3.977024.33541100.00RV
Crisis period 0.00100.0196 3.60000.0018 4.339529.8961691.43Total period 0.00110.01980.40000.0019 4.170627.09391814.40Pre-crisis period 0.02810.14080.00200.0206 1.89737.31491493.60Crisis period 0.02730.14010.00600.0185 1.96888.03101315.70RV 1/2
Total period 0.02770.14080.00200.0195 1.94287.72142824.50Pre-crisis period −7.5796−3.9195−12.4292 1.31410.1007 3.01891050.90Crisis period −7.5594−3.9298−10.2319 1.16060.4454 2.61091627.60log(RV)
Total period −7.5689−3.9195−12.4292
1.23470.2497
2.90112596.20Pre-crisis period 38.2000.01520.00000.00097.891398.2244638.07Crisis period 20.9000.00340.00000.0004 4.210324.154229
3.98J
Total period 29.0000.01520.00000.00079.1269144.26301233.20Pre-crisis period 0.01350.12340.00000.0140 2.426612.3670832.71Crisis period 0.01050.05900.00000.0098 1.81747.3270334.84J 1/2
Total period 0.01190.12340.00000.0121 2.428013.02061352.40Pre-crisis period 0.00030.01510.00000.00097.840797.0506643.18Crisis period 0.00020.00340.00000.0004 4.206924.1184294.13log(J +1)
Total period 0.00020.01510.00000.00079.0609142.28051241.00Pre-crisis period 85.0000.01610.3000158.000 4.504430.4697571.91Crisis period 89.1000.0200 3.3000159.200 5.152343.3425485.57BV
Total period
87.200
0.0200
0.3000
158.700
4.8501
37.3844
1042.70
Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung –Box (1979)Q test.Mean returns are multiplied by 106and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by 105.
Table 5
Daily GARCH(1,1)estimations.Variable Sampling frequency Mean Maximum Minimum S.dev Skewness Kurtosis LB
Return
Pre-crisis period 987.0000.0667−0.08180.0164−0.1892 4.75245312.20Crisis period 180.0000.0965−0.09970.0245−0.0108 5.047817.4420Total period 558.0000.0965−0.09970.0211−0.0720 5.691621.7370GARCH(1,1)
Pre-crisis period 0.00020.00090.00000.0001 1.58717.11342631.9Crisis period 0.00050.00230.00020.0003 1.9888 6.75545090.7Total period
0.0004
0.0024
0.0000
0.0003
2.6075
10.8636
10124.0
Note:Mean returns are multiplied by 106.
Table 6
Comparison of models in pre-crisis period for ISE-30index futures.RMSE
MAE
MAPE
TIC
Statistic
Order Statistic Order Statistic Order Statistic Order GARCH 2.252014GARCH 1.002014GARCH 96.68461GARCH 0.845014HAR-RV 1.46008HAR-RV 0.724012HAR-RV 269.061114HAR-RV 0.42733HAR-RV-1/2 1.46419HAR-RV-1/20.66407HAR-RV-1/2185.740010HAR-RV-1/20.45566HAR-RV-LOG 1.543212HAR-RV-LOG 0.67309HAR-RV-LOG 143.28006HAR-RV-LOG 0.540110HAR-RV-J 1.44606HAR-RV-J 0.714011HAR-RV-J 265.805912HAR-RV-J 0.42122HAR-RV-J 1/2 1.45437HAR-RV-J 1/20.65506HAR-RV-J 1/2184.04118HAR-RV-J 1/20.45115HAR-RV-J LOG 1.538711HAR-RV-J LOG 0.66808HAR-RV-J LOG 143.19675HAR-RV-J LOG 0.540811HAR-RV-CJ 1.43504HAR-RV-CJ 0.705010HAR-RV-CJ 264.001011HAR-RV-CJ 0.41671HAR-RV-CJ 1/2 1.43905HAR-RV-CJ 1/20.64804HAR-RV-CJ 1/2185.42719HAR-RV-CJ 1/20.44424HAR-RV-CJ LOG 1.505210HAR-RV-CJ LOG 0.65105HAR-RV-CJ LOG 139.29024HAR-RV-CJ LOG 0.50039MIDAS 1.31422MIDAS 0.64103MIDAS 267.139413MIDAS 0.46027MIDAS 1/2 1.30141MIDAS 1/20.57862MIDAS 1/2183.84307MIDAS 1/20.48618MIDAS LOG 1.36673MIDAS LOG 0.57521MIDAS LOG 137.68483MIDAS LOG 0.569712RGARCH
1.9965
13
RGARCH 0.8660
13
RGARCH 98.3694
2
RGARCH 0.6959
13
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