Application of Hybrid LSTAR-GARCH Model with Expected Taill Loss in Predicting the Price Movement of Bitcoin Cryptocurrency against Rupiah Currency

ABSTRACT


INTRODUCTION
Cryptocurrencies garnered global attention toward the close of 2017, primarily due to Bitcoin being one of the digital currencies with an exchange rate exceeding 250 million Indonesia rupiahs for asingle unit.Bitcoin, cherished by many, stands as the foremost and most popular cryptocurrency globally, particularly resonating with the millennial generation.Shaid and Idris(2022) expond that Bitcoin, launched in January 2009, operates as a decentralized digital currenxy.The focal point of their research lies in the predicament of price volality, which undergoes daily fluctuations.Thus, the necessity for a mathematical model to prognosticate the future value of the Bitcoin cryptocurrency evident.
As time goes by, the development of technology and science continues to advance.The same applies to the field of learning time series data.The most commonly used time series model is the Box-Jenkins method.The model generated by the Box-Jenkins method is a linear model, but not all financial time series are liner (Tsay, 2005).Smooth Transition Autoregressive (STAR) is an extension of the autoregressive model for nonlinear time series data.According to Terasvirta (1994), STAR models include exponential STAR (ESTAR) and logistic STAR (LSTAR) models.In this case, the LSTAR method is used.The Logistic Smoothing Transition Autoregressive (LSTAR) model is a time series model that can be applied to data that follows a nonlinear model.Nonlinear time series models can be found in data that has fluctuations.The GARCH model is an improvement of the ARCH model where the volatility depends on yesterday's daily value along with the previous volatility value.
Besides being able to provide benefits, bitcoin investment also contains an element of risk.To find out the value of risk, the Expected Tail Loss (ETL) risk measurement tool or often also called Conditional Value at Risk (CVaR) can be used.ETL is the average of tail losses or losses that exceed VaR at a certain confidence level.

RESULT AND ANALYSIS 1. Data Description
A graph of daily bitcoin closing price data for the period April 2022 to April 2023 can be seen in Figure 1.
Figure 1 shows that the closing price of bitcoin is volatile over time.At some point, the closing price of bitcoin decreases and then increases again.This can be caused by price fluctuations.

Stationarity Testing
Data stationarity was tested using the Augmented Dickey Fuller (ADF) test.  1, it can be seen that the ADF test statistic value is -1.7794 with a p-value of 0.67>. (0,05) which means accepting H0 that there is a unit root or the data is not stationary.Because the data is not Figure 2 shows that the data is stationary in terms of mean and variance.To ensure this, the Augmented Dickey Fuller (ADF) test will be conducted.2, it can be seen that the ADF test statistic value is -7.7699 with a p-value of 0.001 <0.001, which means rejecting H , namely there is no unit root or stationary data. (0,05) which means rejecting H0 that there is no unit root or stationary data.

Model Identification
After the stationary assumption is met, a temporary model will be formed by looking at the ACF and PACF plots.Furthermore, the ARMA model parameters are estimated using the Least Square method.The model parameter estimation results can be seen in table 3 below.Based on table 3, the parameters of the AR(1), MA(1) models are significant because they have a probability less than that of the AR(2), ARIMA(1,1), and ARMA(2,1) models. = 0,05.While AR(2), ARIMA(1,1), and ARMA(2,1) are not significant because they have a probability of more than  = 0,05.

Best Model Selection
To determine the best model, it can be seen from the significant parameters that meet the statistical model test and have the smallest AIC, the smaller the AIC value the better the model.Based on Table 3   Based on the results obtained, the p-value is 0.0014 which means less than  = 0,05 so that bitcoin price data has a nonlinear pattern.

Selection of Transition Variables and Transition Functions of STAR Model
After proving nonlinear, the next step is the selection of the form of the transition function.The selection of the transition function is done by testing the hypothesized sequence of parameters  as follows.H03 :  3 = 0 H02 :  2 = 0| 3 = 0 H01 :  1 = 0| 3 =  2 = 0 with conditions: 1.If  2 ≠ 0 then the model used is the LSTAR model 2. If  3 = 0 but  2 ≠ 0 then the ESTAR model 3.If  3 = 0 and  2 = 0 but :  1 ≠ 0 then the model is LSTAR and if  1 = 0 then the model is ESTAR.-0.0037543 0.942472 Table 6 shows that the parameters  2 is not significantly not equal to zero because it has a p-value (<2.2e-16).So the transition function that should be chosen is the LSTAR function.

LSTAR Model
The estimation results of the LSTAR(1,1) model using the Nonlinear Least Square (NLS) method are approximated by the Gauss-Newton iteration as follows.The test criterion is to reject H0 if the p-value is smaller than  = 0,05.Table 8 shows that the pvalue of 0.3577 >  (0,05) so that H0 is accepted or it can be said that there is no autocorrelation in the residual model.The test criterion is to reject H0 if the p-value <0.05. (0,05).In Table 9, the chi-square p-value is 0.0026 < . (0,05) so  0 is rejected or it can be said that there is a heteroscedasticity effect in the LSTAR model.

Modeling with LSTAR-GARCH
The identification of the LSTAR-GARCH model is done by looking at the ACF and PACF correlograms of the squared residuals of the LSTAR model.The ACF and PACF correlogram results are seen in Figure 3.5 below.
The ACF correlogram of squared residuals shows that the cut off is at lag 1.Similarly, the PACF correlogram of squared residuals cuts off at lag 1.So the temporary conjecture based on the results of the ACF and PACF correlograms is the GARCH (0,1), GARCH (1,0), GARCH (1,1) model.In general, the LSTAR-GARCH model is in the form: with   ′  is the LSTAR model a. GARCH (0,1) Model Estimation: with   ′  as the LSTAR model.
The p-value of each parameter = 0.0000 <0.0000. = 0,05 so it can be said that the model is significant.
c. GARCH (1,1) model estimation with   ′  as the LSTAR model The p-value of each parameter = 0.0000 <0.0000. = 0,05 so it can be said that the model is significant.Next, the overfitting stage is carried out to compare several parameters that have been estimated by paying attention to significant parameter values and having the smallest AIC and SIC values.The next period forecast calculation, which is the forecast for the next 15 days, is presented in the following table.In the calculation of ETL, Cornish-Fisher expansion is used.The level of confidence used to calculate ETL is 99% with a return of 366 transaction days.The calculation results can be seen in the following table.16 above, the Expected Taill Loss (CVaR) value is -0.06784, which means that if an investment of Rp. 100,000,000 is made with a 95% confidence level, the maximum loss that can occur borne by investors is Rp.99,999.99 in a predicted time of one day.

CONCLUSIOON
From this research, the following conclusions were drawn: 1.The most appropriate LSTAR model to model the price of bitcoin against the rupiah is: This research uses the type of applied research, because it uses data based on historical data.The form of data used in this research is in the form of time series which is secondary data, where the author accesses bitcoin price data online through Yahoo Finance https://finance.yahoo.com.The data taken in this study is daily bitcoin price data in the period April 1, 2022 to April 1, 2023.The variables used in this study are bitcoin close price, bitcoin price return, and bitcoin volume.The stages in the data analysis process in this study include: 1.Data description 2. Stationary test with the results of the \unitroot test as well as the results of the ACF and PACF test 3. Testing the best AR model on bitcoin closing price data 4. Modeling with LSTAR In this LSTAR modeling, a nonlinearity test is carried out with a white test, and a transition test is also carried out to determine whether the transition function is correct.5. Model parameter estimation 6. Modeling with Hybrid LSTAR-GARCH 7. Forecasting with Hybrid LSTAR-GARCH Forecasting bitcoin closing price data with the LSTAR-GARCH hybrid method.8. Calculating the MSE value using variance forecasting and actual data 9. Estimating and calculating the Expected Taill Loss value of predicted bitcoin closing price returns 10.Interpretation of Expected Taill Loss value

Figure 1
Figure 1 Plot of Bitcoin Closing Price Data

Figure 2
Figure 2 Plot of Bitcoin Price Return

Table 3
the MA (1) model is the best model because the model parameters are significant, fulfill the diagnostic model test and have the smallest AIC value of 16.23158.So the MA(1) model is written as   = −0.983344−1+6. Statistical TestBefore modeling LSTAR-GARCH, it is necessary to conduct an ARCH statistical test with the null hypothesis stating that there is no residual autocorrelation in the model.0 is rejected if   <   5%   = 0,05

Table 6
Regression of Transition Variables  −1

Table 14
Forecasting results of the LSTAR-GARCH model for the periodApril 2-16, 2023The measure of the accuracy of the forecasting value is seen from the Mean Absolute Percentage Error (MAPE).MAPE describes the average residual of the forecasting results with the actual value.The smaller the MAPE value, the better the model obtained.The MAPE value for each model is presented in table13below.