Analysis of Optimal Conditional Heteroskedasticity Model

abridgment of Optimal qualified Heteroskedasticity Model uprise Recently cryptocurrency grocerys endure seen an immense growth. Bitcoin is angio gosin converting enzyme of the or so pop cryptocurrencies noticeing for the highest parcel out of tout ensemble cryptocurrency grocerys, put away though it save remains quite a indecipherable whether it resembles to a bulkyer ex tennert than to a currency, a commodity or an summation. Previous query has shown that Bitcoin is often employ for enthronization purposes, a fact that suggests the immenseness of analysing its excit qualification. In this expression, we contemplate the optimum qualified heteroskedasticity sham, not al angiotensin converting enzyme in footing of skinnyness-of- decease, that if alike in base of operations of guessing exploit, an empyrean which has been on a lower floorexplored in the carapace of Bitcoin. harmonise to the passs, the optimal qualified heteroskedastic ity lay that drop fit the serial is not the same as the one that sack up suppose it better. As molding G sozzled do in Bitcoin securities industry effectively is pivotal for appropriate portfolio perplexity, our results rat help investors and intimately some early(a) determination clear uprs crystallise more than certified endings. Keywords Bitcoin, Cryptocurrencies, GARCH, Volatility, foretell JEL categorization C22, C5, G1 1. Introduction some(prenominal) everywhere the lastly hardly a(prenominal) years, the compendium of Bitcoin has skeletal a passel of twain frequent and academic attention. Bitcoin is the occur one implementation of a concept c everyed cryptocurrency, which was head start nominated in 1998 by Wei Dai on the cypherpunks mailing call, suggesting the radical of a unfermented form of money that physical exercises secret writing to incorporate its creation and transactions, preferably than a primeval conditi onity, but the first base Bitcoin judicial admission was produce in 2009 in a cryptography mailing list by Satoshi Nakamoto ( Bitcoin.org 2017 ). The market of cryptocurrencies has big unmistakably with Bitcoin creation weighed the most famous cryptocurrency, with an estimated market capitalisation of $ 19.6 million (coinmarketcap.com accessed on eighth March 2017), which shortly accounts for around 84.4% of the list estimated cryptocurrency capitalisation. An over take care of Bitcoin bath be prepare in, e.g., Becker et al. (2013), Dwyer (2015), Frisby (2014), Bhme et al. (2015) and Selgin (2015). hence, Bitcoin is completely briefly introduced here. It has been preceding(prenominal)ly argued that Bitcoin sh atomic number 18s m each elements of currencies. However, novel fluctuations in Bitcoin scathes (see Figure 1) do resulted in capricious irritability inframining the constituent Bitcoin plays as a unit of measurement of account (Cheah and Fry 2 015), trance users begin necessitate Bitcoin not only as a currency but likewise for investment funds purposes. In fact, sunrise(prenominal) users tend to wad Bitcoin on a wild investment intention basis and sustain let out intention to imprecate on the profound ne twainrk as connotes for stipendiary goods or work (Glaser et al. 2014). The Bitcoin market is gum olibanum highly unfit at present, and w hence Bitcoin may be mostly use as an asset rather than a currency (Baek and Elbeck 2015 Dyhrberg 2016a). muchover, recent studies welcome examined the hedge capabilities of the Bitcoin (see, e.g., Dyhrberg (2016a, b), justifying the view of it as an asset, as considerably as the world power of antithetic exchanges in the harm husking process of Bitcoin (Brandvold et al. 2015), trance it has overly been previously shown that cryptocurrency markets sh be some stylised teaching- ground facts with separate markets, e.g., a vulnerability to specul ative bubbles (Cheah and Fry 2015). Consequently, Bitcoin has a place in the fiscal markets and in portfolio management (Dyhrberg 2016a). Bitcoin has make up great challenges and opportunities for insurance policy makers, economists, entrepreneurs, and consumers since its introduction (Dyhrberg 2016b), dapple Bitcoin outlay excitability seems to be a major engross for most of the widely distributed public at this time (Bouoiyour and Selmi 2016). As a result, th low mugvass Bitcoin footing excitability is of high importance. future(a) the extensive publications on manikin financial asset monetary cherishs development the family of mouth Autoregressive qualified Heteroskedasticity (GARCH) pretences, late in that location has besides been an increased please in simulationling Bitcoin pass judgment irritability using confusable methods. Previous studies slang apply divers(prenominal) types of GARCH exercises when examining the Bitcoin hurt capriciou sness.For example, the undecomposable GARCH seat has been utilise by Glaser et al. (2014), Gronwald (2014) and Dyhrberg (2016a). On the other hand, other studies nominate considered extensions to the GARCH fashion amaze in straddle to accept asymmetries in Bitcoin monetary value volatility. For instance, the exponential function GARCH (EGARCH) manakin has been use by Dyhrberg (2016a) and Bouoiyour and Selmi (2015, 2016), the sceptre GARCH (TGARCH) ( GJR-GARCH ) specimen has been use by Dyhrberg (2016b), Bouoiyour and Selmi (2015, 2016) and Bouri et al. (2017), head the irregular actor ARCH (APARCH) and division with Multiple Threshold-GARCH (CMT-GARCH) poses pay off been utilise by Bouoiyour and Selmi (2015, 2016). Nevertheless, it is rather unclear which qualified heteroskedasticity mildew should be apply when accounting the Bitcoin footing volatility. Previous studies of the Bitcoin footing volatility assume focused primarily on the use of a case-by -case conditional heteroskedasticity object lesson, without bedvas distinguishable GARCH-type personates , though , with the only exceptions being the studies of Bouoiyour and Selmi (2015, 2016), which have rub PK1 the sample into assorted sub- uttermosts, though , and the knowledge of Katsiampa (2017/forthcomng?), which has not considered the risk- devolve relationships, though PK2 . In addition, minute attention has been paid to vaticination the volatility of the Bitcoin prices. To the out come of the authors familiarity only the study of Bouoiyour and Selmi (2016) has examined the anticipate mathematical operation of the CMT-GARCH and APARCH theoretical accounts, but no study has comp argond the prophetic ability of different GARCH mouldings with regards to Bitcoin. Consequently, we aim to append to the literature by investigating which conditional heteroskedasticity poseur can describe and forecast the Bitcoin prices better. The remainder of the hold is organised as follows The beside section presents the deterrent examples employed in this study. The selective development and methodology used in the study argon discussed in the troika section, fleck the ordinal section lucubrate our verifiable results. Finally, the conclusions skeletal and the implications argon presented in section tailfin. 2. Models In this section, the works used in this research argon introduced. The manikins consist of an Autoregressive dumbfound for the conditional destinespirited and a first- direct GARCH-type or a GARCH-in- inculpate-type warning for the conditional diversion 1 , as follows , , , where is the Bitcoin price return on solar day , is the faulting term, is a etiolated noise process, is the conditional meter difference of opinion, and hence is the conditional difference. When is vanquish crimson to cryptograph, the resulting personate is the autoregressive poser with a GARCH-type condi tion for the conditional partition, term when is different from zero a GARCH-in- baseborn-type spec for the conditional variance is obtained. Adding the standard deviation to the take to be equation measures the risk and helps with the realisation and measurement of any risk-return relationship. The conventional GARCH(1,1) feigning is represented by , with , and . The GARCH toughie (Bollerslev 1986) is doubtlessly one of the most popular specimens for describing the conditional variance of financial returns. Nevertheless, since its introduction, there have been proposed many extensions of the GARCH framework and there have been a attracter of advances in molding the conditional variance. Hence in this study, we in like manner consider fin extensions to the linear GARCH model, viz. the EGARCH model of Nelson (1991), the TGARCH model introduced by Glosten et al. (1993), the APARCH model proposed by dong et al. (1993), the Component GARCH (CGARCH) model of Engle and Lee (1999) and the crooked CGARCH (ACGARCH) model. All these models pee-pee examples of extensions of the simple GARCH model and have seek to describe the conditional variance more accurately. Moreover, compared with the simple GARCH model, the EGARCH, TGARCH and APARCH models surrender for different volatility responses to opposite markers of the previous shocks. More specifically, the EGARCH model is delimitate as , and considers the unsymmetrical volatility responses to proscribe news, that is , and peremptory news, , as prone by the sign of , if is different from zero. The TGARCH model is assumption by , where is the indicator function, with if and 0 otherwise, suggesting that irrefutable shocks and interdict shocks have once again different effectuate on the volatility, if is different from 0. On the other hand, the APARCH model is specify as , where , , , and . This model imposes a Box-Cox power transformation of the conditional standard deviation process and the asymmetric absolute residuals (Ding et al. 1993). Furthermore, in phone line with the GARCH model, the conditional variance of which shows imagine reversion to , which is a constant for all time, the CGARCH model allows for twain a long run component of conditional variance, , which is time vary and slowly mean-reverting, and a short component, , and is delimitate as . Christoffersen et al. (2008) show that by including some(prenominal) a short-run and a long run component allows the CGARCH model to outperform the GARCH model. Finally, the Asymmetric Component GARCH (ACGARCH) model admits the CGARCH model with the TGARCH model, introducing asymmetric do in the transitory equation, and takes the future(a) form , where is a dummy protean which evinces negative shocks, slice positive value of suggest the charge of transitory leverage make in the conditional variance. 3. info and methodology T he information consists of routine shutdown prices for the Bitcoin Coindesk proponent from 19 th July 2010 to 10 th January 2017. The affection sample covers the period between 19 th July 2017 and 31 st celestial latitude 2017 direct to a add up number of 2357 observations, enchantment the remaining ten observations are used in the presage sample. The Bitcoin CoinDesk Index is listed in USD and the data are publicly lendable online at http//www.coindesk.com/price. The data are converted to natural logarithms, and indeed the returns are defined as , where is the logarithmic Bitcoin price top executive change and is the daily Bitcoin price magnate on day . Figures 1 and 2 illustrate the Bitcoin prices and price returns, some(prenominal)(prenominal)ly, in the inclination period. We start the empirical analysis by producing descriptive statistics for the Bitcoin price returns, term the increase Dickey-Fuller (ADF) and Phillips-Perron (PP) unit- res olve turn outs are excessively performed to examine the stationarity of the returns. As will be seen in the next section, the results show that the series is stationary. In order to choose the outgo model in call of appointment to data, terce information criteria, namely Akaike (AIC), Bayesian (BIC) and Hannan-Quinn (HQ), are employed. For given data sets, all of these information criteria consider twain how good the accommodate of the model is and how many debates there are in the model, rewarding a better alteration and penalising an increased number of logical arguments. The take up-loved model is the one with the respective minimum quantity value. However, since model pickaxe is often not only based on a models goodness-of-fit to data, but to a fault on prognostication surgical process, it is important to withal watch the models prophetic ability, as a better fitting model does not always lam to better forecasts. Hence, the go around model specification in price of forecasting is selected fit in to the Root entail Squared foretelling fallacy (RMSE), Mean Absolute Forecasting Error (MAE) and Mean Absolute section Forecasting Error (MAPE), all of which are used as measures of forecasting murder. Although the RMSE is one of the most commonly used measures of prognostic ability, the additional measures have been used in order to operate the results. 2 The models forecasting performance is evaluated based on out-of-sample forecasts, and model pickax is examined in hurt of twain multi-step-ahead and five-fold 1-step-ahead forecasting. The preferred model is the one with the terminal set of the measures of prophetic ability. Fig. 1 . routine closing prices of the Coindesk Bitcoin Index (US Dollars). Fig. 2 . free-and-easy Bitcoin price returns. 4. Results instrument panel 1 reports the descriptive statistics for the daily returns of the Bitcoin price index. The daily amount closing return is positive and correspond to 0.5805% with a standard deviation of 0.0606. Moreover, the returns are positively skewed, indicating that it is more likely to come rangy positive returns, and leptokurtic as a result of substantial excess kurtosis. The Jarque-Bera (JB) screen confirms the departure from northward, p clump of ground the results of the ARCH(5) see for conditional heteroskedasticity show shew of ARCH effects in the returns of the Bitcoin price index. Therefore the Autoregressive model for the conditional mean needs to be combined with an Autoregressive Conditional Heteroskedasticity process to model the conditional variance. It can be observe that the ARCH effects can similarly be sight from Figure 2 where large (small) price changes tend to be followed by large (small) price changes over time. Furthermore, the results from both the increase Dickey-Fuller and Phillips-Perron unit root tests indicate that stationarity is ensured. display board 1. descriptive statistics and unit roots tests. table A Descriptive statistics Observations 2357 Mean 0.005805 median(a) 0.000741 Maximum 0.528947 lower limit -0.388309 Std. Dev. 0.060607 Skewness 0.873024 Kurtosis 15.64823 JB 16010.55*** ARCH(5) 56.56059*** plank B unit of measurement root test statistics ADF -46.90888*** PP -47.56848*** Note *** indicates the rejection of the cryptograph hypotheses at the 1% level. Next, the inclination results of the GARCH-type models are discussed. The conditional mean equation acknowledges a constant and an autoregressive term, while the conditional variance is modelled by unhomogeneous competing GARCH models. The model parameters are estimated by using the maximum likelihood approach nether the Gaussian distribution. circuit card 2 presents the attachment results of each model. These include the model parameter estimates, the log-likelihood set and the three information criteria values. In addition, the ARCH(5) test to check whether the conditional heteroskedasticity is eliminated and the Ljung-Box test for autocorrelation with 10 lags employ to form residuals, as comfortably as the Jarque-Bera (JB) test of normality of the residuals have been used as diagnostic tests, the results of which are likewise account in control panel 2. According to the results, both the AIC and HQ information criteria select the AR(1)-ACGARCH(1,1) model as the preferred model in price of fitting to data, followed by the AR(1)-CGARCH(1,1)-M and AR(1)-CGARCH(1,1) models, suggesting the important type of having both long-run and short-run components of conditional variance. The log-likelihood is also maximised under the AR(1)-ACGARCH(1,1) model. On the other hand, the preferred model according to the BIC is the AR(1)-CGARCH(1,1) , followed by the AR(1)-ACGARCH(1,1) model. The latter result could be explained, though, by the fact that the BIC penalises more a high number of model parameters, and hence the pickaxe of the AR(1)-ACGARCH(1,1) model seems appropriate. It can also be noticed that for the AR(1)-ACGARCH(1,1) model all the parameter estimates are statistically significant. Moreover, the results of the ARCH(5) and tests applied to the squared residuals of the AR(1)-ACGARCH(1,1) model indicate that the selected AR(1)-ACGARCH(1,1) model with Gaussian distribution is powerful specified because the hypotheses of no remaining ARCH effects and no autocorrelation cannot be rejected. Furthermore, notwithstanding the fact that the residuals still depart from normality, the value of the Jarque-Bera statistic associated with the residuals of the AR(1)-ACGARCH(1,1) model is more lower than the match value for the defenseless returns. Consequently, the AR-ACGARCH model seems to be effectual to descri be the volatility of the returns of the Bitcoin price index. This result seems to be logical with the study of Bouoiyour and Selmi (2016) PK3 who found that the opera hat model for the period from December 2010 to December 2014 is the CMT-GARCH model, which also includes both transitory and long-lived components as well as thresholds think to positive and negative shocks. With regards to the out-of-sample forecasting performance, the five- and ten-day-ahead forecasts as well as the five and ten 1-day-ahead forecasts of the twelve competing GARCH-type models were generated. We then(prenominal) compared the models forecasting performance based on the three mean loss functions (RMSE, MAE and MAPE). Table 3 reports the obtained results, while the bold song indicate the crush model in impairment of forecast accuracy. An interesting purpose is that boilers suit the information criteria for model natural extract in terms of goodness-of-fit do not agree with the measures o f prognosticative ability. Even though the minimum RMSE values of the 10-step-ahead and ten 1-step-ahead forecasts were both given for the AR-CGARCH model, a result which is logical with the Bayesian study Criterion, the results of the other two measures of predictive ability (MAE and MAPE) showed that there are other models that perform better than the AR-ACGARCH and AR-CGARCH models when it comes to forecasting. More specifically, the minimum RMSE values of the 5-step-ahead and five 1-step-ahead forecasts were both given for the AR-EGARCH-M model. On the other hand, the final MAE and MAPE values of the 5- and 10-step-ahead forecasting as well as those of the five 1-step-ahead forecasting were all given for the AR-EGARCH model. The terminal MAE value of the ten 1-step-ahead forecasting was also given for the AR-EGARCH model, while the lowest MAPE value of the ten 1-step-ahead forecasting was given for the AR-APARCH-M model. In summary, according to our estimation results the AR-ACGARCH model is preferred to the other competing models in terms of volatility estimates for the returns. However, the preferred model in terms of forecasting is overall the AR-EGARCH. This result is of the essence(p) for portfolio management and decision making in general by individuals who use Bitcoin for speculative purposes. Finally, it should be historied that the model parameters were estimated under the Student- t and GED distributions as well, but as there was no improvement in either the goodness-of-fit or forecasting performance, the results are not reported here. 3 This is in contrast with the results of the study of Bouri et al. (2017) who found that the TGARCH(1,1) model under the GED density is the beat out fit. 5. Conclusions Over the last few years cryptocurrency markets have grown to a great extent, with Bitcoin having attracted a lot of attention from both the public and researchers. This article aimed to offer a discussion into Bitcoin price v olatility by selecting an optimal GARCH-type model in terms of both goodness-of-fit to data and forecasting performance chosen among several extensions. It was found that even though the top hat model in terms of goodness-of-fit is the AR-ACGARCH, a result which is consistent with previous studies PK4 , with regards to forecasting performance the best model seems to be overall the AR-EGARCH. Consequently, if the physical object is to find the best model in terms of predictive ability, model selection based on information criteria only might not be adequate. As Bitcoin can combine some of the advantages of both commodities and currencies in the financial markets (Dyhrberg 2016a), it can be a useful tool for portfolio analysis and risk management. Hence, individuals in portfolio and risk management need to get a more detailed view of the Bitcoin price volatility. Our results may thus have important implications principally for investors but also for other decision makers, such a s policymakers, as they can enable them to make more informed decisions.