# A Study on Cryptocurrency Log-Return Price Prediction Using Multivariate Time-Series Model

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Feature Selection

#### Gini Impurity

#### 2.2. Traditional Time-Series Analysis Methods

#### 2.2.1. ARCH

#### 2.2.2. GARCH

#### 2.2.3. ARIMA

#### 2.3. Deep Neural Network

#### 2.3.1. RNN

#### 2.3.2. LSTM

#### 2.3.3. GRU

## 3. Data Analysis

#### 3.1. Data Description

#### 3.1.1. Data Collection

#### 3.1.2. Preprocessing

#### 3.2. Feature Selection

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Cryptocurrency | Test Statistic | p Value |
---|---|---|

Bitcoin | 0.145652 | 0.1000 |

Ethereum | 0.37252 | 0.0890 |

Binance coin | 0.193457 | 0.0100 |

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**Figure 1.**Structure of RNN. RNN has a recurrent structure and consists of multiple recurrent nodes [25].

**Figure 2.**Structure of LSTM. LSTM is similar to RNN, but long-term and short-term memory cells are added [31].

**Figure 3.**Structure of the GRU. GRU is a simplified form of LSTM and has a structure that uses less computation [34].

**Figure 4.**Graph of change in transaction amount of selected major cryptocurrencies (Bitcoin, Ethereum, Binance Coin). From 31 May 2018 to 31 May 2022, there are 1462 timepoints.

**Figure 5.**Log transformation and min-max normalization application result graph for selected cryptocurrencies.

**Figure 6.**Feature importance extraction results. (

**a**) Extracting feature importance for Bitcoin; (

**b**) extracting feature importance for Ethereum; (

**c**) extracting feature importance for Binance Coin.

**Figure 7.**ACF for cryptocurrencies. All three figures appear similar. In this study, 2-time leg units are considered as the main factor.

**Table 1.**Descriptive statistics for the 11 collected cryptocurrencies. The unit of cryptocurrency transaction is USD. The mean, quartile, skewness, and kurtosis of each cryptocurrency are shown.

BTC | ETH | BNB | XLM | ADA | XRP | IOT | QTU | EOS | LTC | NEO | |
---|---|---|---|---|---|---|---|---|---|---|---|

Count | 1462 | 1462 | 1462 | 1462 | 1462 | 1462 | 1462 | 1462 | 1462 | 1462 | 1462 |

Mean | 21,416.9 | 1148.6 | 147.1 | 0.2 | 0.5 | 0.5 | 0.6 | 5.2 | 4.0 | 98.1 | 22.7 |

Std. | 18,576.8 | 1327.1 | 194.6 | 0.1 | 0.7 | 0.3 | 0.5 | 4.3 | 1.9 | 61.3 | 18.3 |

Min | 3211.7 | 83.8 | 4.5 | 0.0 | 0.0 | 0.1 | 0.1 | 1.0 | 1.2 | 23.1 | 5.4 |

25% | 7197.5 | 187.1 | 15.4 | 0.1 | 0.1 | 0.3 | 0.3 | 2.1 | 2.6 | 51.7 | 10.2 |

50% | 10,253.8 | 347.6 | 25.3 | 0.1 | 0.1 | 0.4 | 0.4 | 3.2 | 3.5 | 75.8 | 16.8 |

75% | 38,670.0 | 2159.4 | 321.9 | 0.3 | 1.1 | 0.6 | 0.9 | 7.0 | 4.9 | 133.8 | 28.8 |

Max | 67,525.8 | 4808.0 | 676.2 | 0.7 | 3.0 | 1.8 | 2.5 | 27.4 | 14.7 | 387.8 | 122.8 |

Skewness. | 0.82 | 1.07 | 1.07 | 1.16 | 1.38 | 1.38 | 1.27 | 1.61 | 1.96 | 1.31 | 2.17 |

Kurtosis. | −0.87 | −0.31 | −0.41 | 1.13 | 0.88 | 1.42 | 0.88 | 2.53 | 5.69 | 1.69 | 6.15 |

**Table 2.**Cryptocurrency volatility prediction results by ARCH (1) and GARCH (1, 1). Statistics on the ARCH (1) and GARCH (1) coefficients of each cryptocurrency are shown.

Models | Mu | Omega | Alpha | Beta | |||||
---|---|---|---|---|---|---|---|---|---|

Coef. | p | Coef. | p | Coef. | p | Coef. | p | ||

XLM | ARCH (1) | 0.45 | 0.000 *** | 0.00 | 0.000 *** | 0.30 | 0.000 *** | - | - |

GARCH (1, 1) | 0.45 | 0.000 *** | 0.00 | 0.000 *** | 0.24 | 0.000 *** | 0.63 | 0.000 *** | |

ADA | ARCH (1) | 0.65 | 0.000 *** | 0.00 | 0.000 *** | 0.12 | 0.000 *** | - | - |

GARCH (1, 1) | 0.65 | 0.000 *** | 0.00 | 0.000 *** | 0.11 | 0.000 *** | 0.80 | 0.000 *** | |

XRP | ARCH (1) | 0.54 | 0.000 *** | 0.00 | 0.000 *** | 0.70 | 0.000 *** | - | - |

GARCH (1, 1) | 0.54 | 0.000 *** | 0.00 | 0.04 | 0.27 | 0.000 *** | 0.64 | 0.000 *** | |

BNB | ARCH (1) | 0.52 | 0.000 *** | 0.00 | 0.000 *** | 0.20 | 0.000 *** | - | - |

GARCH (1, 1) | 0.52 | 0.000 *** | 0.00 | 0.000 *** | 0.15 | 0.000 *** | 0.82 | 0.000 *** | |

IOT | ARCH (1) | 0.65 | 0.000 *** | 0.00 | 0.000 *** | 0.09 | 0.124 | - | - |

GARCH (1, 1) | 0.65 | 0.000 *** | 0.00 | 0.103 | 0.11 | 0.001 ** | 0.86 | 0.000 *** | |

QTU | ARCH (1) | 0.61 | 0.000 *** | 0.00 | 0.000 *** | 0.19 | 0.02 | - | - |

GARCH (1, 1) | 0.61 | 0.000 *** | 0.00 | 0.311 | 0.09 | 0.179 | 0.85 | 0.000 *** | |

EOS | ARCH (1) | 0.55 | 0.000 *** | 0.00 | 0.000 *** | 0.16 | 0.02 | - | - |

GARCH (1, 1) | 0.55 | 0.000 *** | 0.00 | 0.000 *** | 0.07 | 0.000 *** | 0.88 | 0.000 *** | |

LTC | ARCH (1) | 0.65 | 0.000*** | 0.00 | 0.000 *** | 0.10 | 0.07 | - | - |

GARCH (1, 1) | 0.65 | 0.000 *** | 0.00 | 0.000 *** | 0.07 | 0.001 ** | 0.87 | 0.000 *** | |

ETH | ARCH (1) | 0.72 | 0.000 *** | 0.00 | 0.000 *** | 0.04 | 0.000 *** | - | - |

GARCH (1, 1) | 0.72 | 0.000 *** | 0.00 | 0.000 *** | 0.08 | 0.04 | 0.86 | 0.000 *** | |

NEO | ARCH (1) | 0.66 | 0.000 *** | 0.00 | 0.000 *** | 0.13 | 0.000 *** | - | - |

GARCH (1, 1) | 0.66 | 0.000 *** | 0.554 | 00.21 | 0.11 | 0.03 | 0.80 | 0.000 *** | |

BTC | ARCH (1) | 0.74 | 0.000 *** | 0.00 | 0.000 *** | 0.03 | 0.198 | - | - |

GARCH (1, 1) | 0.74 | 0.000 *** | 0.00 | 0.000 *** | 0.07 | 0.07 | 0.85 | 0.000 *** |

**Table 3.**Description of features. Describes the features used in the analysis and indicates whether they are independent or dependent.

Features | Description | Dependent Features |
---|---|---|

Daily closing prices of cryptocurrencies converted to log-returns | These are the features that convert the daily closing price of the cryptocurrency used in this analysis into log-return price. It consists of a total of 11 and is named ‘cryptocurrency Close’. Among them, 3 features are used as dependent features, and the rest are used as independent features. | BTC_Close ETH_Close BNB_Close |

Daily volatility of cryptocurrencies derived with ARCH (1) | Features converted from ARCH (1) volatility analysis for cryptocurrency used in this analysis. It consists of a total of 11 and is named ‘cryptocurrency ARCH’. All features are used as independent features. | - |

Daily volatility of cryptocurrencies derived with GARCH (1, 1) | Features converted from GARCH (1, 1) volatility analysis for cryptocurrency used in this analysis. It consists of a total of 11 and is named ‘cryptocurrency GARCH’. All features are used as independent features. | - |

**Table 4.**Detailed architecture for each model. It consists of a time-series neural network layer and a dense layer.

Model | Composition of Layers |
---|---|

Architecture 1 | RNN (32)/LSTM (32)/GRU (32) + dense (64-32-16-8-1) |

Architecture 2 | RNN (32)/LSTM (32)/GRU (32) + dense (32-16-8-1) |

Architecture 3 | RNN (32)/LSTM (32)/GRU (32) + dense (16-8-4-1) |

Architecture 4 | RNN (32)/LSTM (32)/GRU (32) + dense (16-8-1) |

Architecture 5 | RNN (32)/LSTM (32)/GRU (32) + dense (64-1) |

Architecture 6 | RNN (32)/LSTM (32)/GRU (32) + dense (16-1) |

Activation | Linear |

Loss | Mean squared error |

Optimizer | Adam |

**Table 5.**Log-return price prediction result of Bitcoin validation data by ARIMA and artificial neural network-based model.

Methods | MAE | MSE | RMSE | |
---|---|---|---|---|

ARIMA (2, 1, 0) | 0.0422 | 0.0028 | 0.0532 | |

Architecture 1 | RNN | 0.0377 | 0.0024 | 0.0492 |

LSTM | 0.0383 | 0.0025 | 0.0502 | |

GRU | 0.0378 | 0.0025 | 0.0504 | |

Architecture 2 | RNN | 0.0376 | 0.0024 | 0.0491 |

LSTM | 0.0381 | 0.0024 | 0.0497 | |

GRU | 0.0391 | 0.0026 | 0.0509 | |

Architecture 3 | RNN | 0.0382 | 0.0025 | 0.0497 |

LSTM | 0.0383 | 0.0025 | 0.0501 | |

GRU | 0.0382 | 0.0025 | 0.0500 | |

Architecture 4 | RNN | 0.0376 | 0.0024 | 0.0491 |

LSTM | 0.0382 | 0.0025 | 0.0496 | |

GRU | 0.0377 | 0.0025 | 0.0497 | |

Architecture 5 | RNN | 0.0374 | 0.0024 | 0.0491 |

LSTM | 0.0381 | 0.0024 | 0.0494 | |

GRU | 0.0381 | 0.0025 | 0.0496 | |

Architecture 6 | RNN | 0.0377 | 0.0025 | 0.0495 |

LSTM | 0.0379 | 0.0025 | 0.0498 | |

GRU | 0.0384 | 0.0024 | 0.0492 |

**Table 6.**Log-return price prediction result of Ethereum validation data by ARIMA and artificial neural network-based model.

Methods | MAE | MSE | RMSE | |
---|---|---|---|---|

ARIMA (2, 1, 0) | 0.0442 | 0.0033 | 0.0575 | |

Architecture 1 | RNN | 0.0400 | 0.0024 | 0.0486 |

LSTM | 0.0400 | 0.0024 | 0.0488 | |

GRU | 0.0399 | 0.0024 | 0.0488 | |

Architecture 2 | RNN | 0.0402 | 0.0024 | 0.0490 |

LSTM | 0.0420 | 0.0027 | 0.0521 | |

GRU | 0.0400 | 0.0033 | 0.0575 | |

Architecture 3 | RNN | 0.0401 | 0.0024 | 0.0489 |

LSTM | 0.0415 | 0.0026 | 0.0506 | |

GRU | 0.0418 | 0.0026 | 0.0506 | |

Architecture 4 | RNN | 0.0397 | 0.0024 | 0.0486 |

LSTM | 0.0417 | 0.0025 | 0.0497 | |

GRU | 0.0411 | 0.0025 | 0.0497 | |

Architecture 5 | RNN | 0.0396 | 0.0024 | 0.0487 |

LSTM | 0.0396 | 0.0024 | 0.0485 | |

GRU | 0.0407 | 0.0025 | 0.0495 | |

Architecture 6 | RNN | 0.0400 | 0.0024 | 0.0490 |

LSTM | 0.0413 | 0.0025 | 0.0500 | |

GRU | 0.0393 | 0.0026 | 0.0486 |

**Table 7.**Log-return price prediction result of Binance Coin validation data by ARIMA and artificial neural network-based model.

Methods | MAE | MSE | RMSE | |
---|---|---|---|---|

ARIMA (2, 1, 0) | 0.0293 | 0.0016 | 0.0395 | |

Architecture 1 | RNN | 0.0252 | 0.0013 | 0.0357 |

LSTM | 0.0262 | 0.0014 | 0.0369 | |

GRU | 0.0264 | 0.0013 | 0.0365 | |

Architecture 2 | RNN | 0.0252 | 0.0013 | 0.0353 |

LSTM | 0.0259 | 0.0012 | 0.0352 | |

GRU | 0.0254 | 0.0012 | 0.0352 | |

Architecture 3 | RNN | 0.0251 | 0.0012 | 0.0353 |

LSTM | 0.0261 | 0.0013 | 0.0363 | |

GRU | 0.0252 | 0.0013 | 0.0354 | |

Architecture 4 | RNN | 0.0254 | 0.0013 | 0.0354 |

LSTM | 0.0266 | 0.0013 | 0.0361 | |

GRU | 0.0257 | 0.0013 | 0.0363 | |

Architecture 5 | RNN | 0.0251 | 0.0013 | 0.0355 |

LSTM | 0.0261 | 0.0013 | 0.0361 | |

GRU | 0.0257 | 0.0013 | 0.0357 | |

Architecture 6 | RNN | 0.0251 | 0.0013 | 0.0355 |

LSTM | 0.0261 | 0.0013 | 0.0362 | |

GRU | 0.0257 | 0.0013 | 0.0357 |

**Table 8.**Log-return price prediction result of selected cryptocurrency test data by ARIMA and artificial neural network-based model. The artificial neural network takes the structure of Architecture 6.

Cryptocurrency/Methods | MAE | MSE | RMSE | |
---|---|---|---|---|

Bitcoin | ARIMA | 0.0422 | 0.0028 | 0.0532 |

RNN | 0.0378 | 0.0026 | 0.0506 | |

LSTM | 0.0385 | 0.0026 | 0.0512 | |

GRU | 0.0379 | 0.0026 | 0.0507 | |

Ethereum | ARIMA | 0.0464 | 0.0034 | 0.0586 |

RNN | 0.0423 | 0.0027 | 0.0524 | |

LSTM | 0.0421 | 0.0028 | 0.0527 | |

GRU | 0.0417 | 0.0026 | 0.0510 | |

Binance coin | ARIMA | 0.0340 | 0.0020 | 0.0450 |

RNN | 0.0289 | 0.0016 | 0.0401 | |

LSTM | 0.0290 | 0.0016 | 0.0406 | |

GRU | 0.0297 | 0.0016 | 0.0406 |

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**MDPI and ACS Style**

Sung, S.-H.; Kim, J.-M.; Park, B.-K.; Kim, S.
A Study on Cryptocurrency Log-Return Price Prediction Using Multivariate Time-Series Model. *Axioms* **2022**, *11*, 448.
https://doi.org/10.3390/axioms11090448

**AMA Style**

Sung S-H, Kim J-M, Park B-K, Kim S.
A Study on Cryptocurrency Log-Return Price Prediction Using Multivariate Time-Series Model. *Axioms*. 2022; 11(9):448.
https://doi.org/10.3390/axioms11090448

**Chicago/Turabian Style**

Sung, Sang-Ha, Jong-Min Kim, Byung-Kwon Park, and Sangjin Kim.
2022. "A Study on Cryptocurrency Log-Return Price Prediction Using Multivariate Time-Series Model" *Axioms* 11, no. 9: 448.
https://doi.org/10.3390/axioms11090448