EXTENDED ABSTRACT
INTRODUCTION
In this study, in order to examine and compare two modeling approaches in econometrics in terms of prediction accuracy, the time series of Bitcoin prices with high-frequency data (5-minute daily Data) has been selected for the period from 2015 to 2023. Bitcoin, as the most popular and well-known digital currency, has become an important asset for many investors in financial markets and even ordinary people in recent years. Therefore, extensive studies are being conducted to discover the price structure and variations of this asset over time. The modeling approach using Mixed-data sampling (MIDAS) has been proposed as a new approach in modeling in recent years and, given the significant limitations in econometric modeling, it is being studied more than ever. Prediction approaches are divided into two groups: quantitative and qualitative approaches. In the field of econometrics, there are causal and non-causal regression patterns, which include their autoregression models with conditional heteroskedasticity and autoregressive integrated moving average from causal methods in the domain of economic variable prediction (Brockwell et al., 2002). These models also faced limitations, especially when dealing with data with different time frequencies. Researchers aiming to model time series variables with different time frequencies need to first synchronize all variables and then incorporate them into the model. There are two approaches for this: harmonizing variable frequencies towards higher time frequencies or harmonizing towards lower time frequencies. The first method is called temporal aggregation, and the second method is called temporal disaggregation (Bayat & Nofouresti, 1394). In most cases, the application of these methods results in reduced model efficiency and loss of useful information. The mixed data sampling approach, developed by Ghysels (2004), is an innovative and novel approach in modeling economic variables. In this study, traditional approaches are tested against the introduced novel approach for prediction using information from a time series. The main question posed is whether traditional econometric methods provide better models in terms of lower error and higher accuracy compared to novel approaches or not?
 
METHODOLOGY
This study compares the prediction accuracy of ARIMA-GARCH and modeling using the mixed data frequency approach with different time frequencies (MIDAS). To examine the validity of the two mentioned approaches, daily 5-minute intraday time series data of Bitcoin returns were collected and an ARIMA model was implemented for Bitcoin returns (both daily and weekly). To select the orders of p and q in the ARIMA model, a set of models with different p and q orders is estimated while considering visual inspection of autocorrelation and partial autocorrelation functions. Information criteria are then calculated for that set of models. The best ARIMA model in this section is the one with the lowest Akaike information criterion (AIC) value. In the next step, considering the base GARCH (1,1) model, each of the daily and weekly Bitcoin return time series is estimated in three modes: sGARCH, eGARCH, and gjrGARCH. Additionally, in each of the estimated models, five distribution modes are considered: normal distribution, Cholesky normal distribution, Student’s t-distribution, Cholesky Student’s t-distribution, and generalized error distribution. In total, 15 models are estimated for each time series at this stage. In order to prepare the data for comparing the prediction accuracy of different models, the daily data (3024 data points) and weekly data (432 weekly data points) have been divided into two sections. Daily returns from the beginning of 2015 until the end of 2022 (2528 data points) were used for model estimation, and from the beginning of 2023 until 08/05/2023 (496 data points) were used for prediction. Additionally, for weekly returns, the data up to the end of 2022 (413 weekly data points) were used in the estimated model, and the data from 2023 were excluded to assess prediction accuracy. Using the novel approach of modeling mixed data with different time frequencies, the daily and weekly Bitcoin return time series are examined and modeled. For this purpose, the daily and weekly return data, constructed from intraday (5-minute) returns, are utilized. Considering that cryptocurrency markets operate 7 days a week, the modeling in this section involves aligning the higher-frequency data (5-minute intraday returns) as explanatory variables with the lower-frequency data (daily returns) as dependent variables. Additionally, a correspondence of m=7 is defined to link the daily data (as explanatory variables) with the weekly data (as dependent variables). The commonly used weighting functions in the MIDAS framework include Almon, exponential Almon, and beta functions. All three weighting functions are separately incorporated into the model, and the results are used for predicting daily and weekly return time series. The estimations are performed using the midas_r software package, developed by Ghysels and colleagues, which automatically selects optimal initial values and breakpoint counts. Given that the ultimate goal of this study is to assess the predictive power of the model, the best model with the lowest Akaike information criterion (AIC) value at each stage is automatically chosen by the software.
 
FINDINGS
The main objective of this study is to examine two approaches in time series modeling for the purpose of determining the optimal model for prediction. The optimal model is selected based on minimizing error and maximizing accuracy. To perform predictions, the collected data range is divided into two parts: training and testing. Both selected models are estimated using the training data. Subsequently, using the estimated model, time series values are predicted for the testing section and compared with the actual values. To ensure consistency in the prediction time range, all data related to the year 2023 are excluded from the sample, and both approaches are estimated for the period from the beginning of 2015 to the end of 2022. Predictions for the ARIMA-GARCH family models are conducted in three modes: sGARCH, eGARCH, and gjrGARCH, each with five distribution types: normal distribution, Cholesky normal distribution, Student’s t-distribution, Cholesky Student’s t-distribution, and generalized error distribution. In the modeling section with different frequency data, predictions are made for three weighting functions: Almon, exponential Almon, and beta, based on the rolling window approach and recursive pattern. In total, six models are estimated and predicted in this section. The results of this stage indicate that for the daily return time series, the estimated model using the MIDAS framework with the Almon weighting function outperforms the ARIMA-GARCH family models with different distributions and the MIDAS models with beta and exponential Almon weighting functions in terms of prediction accuracy. For the weekly return time series, the MIDAS model with the exponential Almon weighting function and the rolling window prediction approach demonstrates complete superiority. All evaluation metrics for prediction accuracy favor this approach over the other methods introduced in this study
 
CONCLUSION
In this study, two approaches in time series modeling were examined in terms of prediction accuracy. The traditional econometric approach, which has been used for many years in time series modeling, was compared to a novel approach designed to address a significant issue in economic modeling: the need for consistent time frequencies within a regression model. In this study, all possible scenarios for both approaches were investigated and tested, including various existing models, common error distributions, and weighting functions used in previous studies. Additionally, high-frequency data (5-minute intraday data) were utilized for calculating returns and modeling. In general, the results of this section indicate that the prediction accuracy of MIDAS regression is higher compared to ARIMA-GARCH family models, and selecting an appropriate weighting function can have an impact on improving the prediction accuracy of MIDAS regression. The main complexity in modeling with the MIDAS approach lies in choosing the appropriate number of lags for the high-frequency variable and selecting a suitable weighting function, both of which significantly affect the presentation of more accurate predictions.