Forecasting Bitcoin returns: Econometric time series analysis vs. machine learning

Berger, Theo; Koubová, Jana Article — Published Version Forecasting Bitcoin returns: Econometric time series analysis vs. machine learning Journal of Forecasting Provided in Cooperation with: John Wiley & Sons Suggested Citation: Berger, Theo; Koubová, Jana (2024) : Forecasting Bitcoin returns: Econometric time series analysis vs. machine learning, Journal of Forecasting, ISSN 1099-131X, Wiley, Hoboken, NJ, Vol. 43, Iss. 7, pp. 2904-2916, https://doi.org/10.1002/for.3165 This Version is available at: https://hdl.handle.net/10419/306196 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/4.0/ RESEARCH ARTICLE Forecasting Bitcoin returns: Econometric time series analysis vs. machine learning Theo Berger 1,2 | Jana Koubov a 3 1 Department of Business and Computer Science, University of Applied Sciences Hannover, Hanover, Germany 2 Department of Business and Administration, University of Bremen, Bremen, Germany 3 Sparkassen Rating und Risikosysteme, Berlin, Germany Correspondence Theo Berger, Department of Business and Computer Science, University of Applied Sciences Hannover, Ricklinger Stadtweg 120, D-30459 Hannover, Germany. Email: [email protected] Abstract We study the statistical properties of the Bitcoin return series and provide a thorough forecasting exercise. Also, we calibrate state-of-the-art machine learning techniques and compare the results with econometric time series models. The empirical assessment provides evidence that the application of machine learning techniques outperforms econometric benchmarks in terms of forecasting precision for both inand out-of-sample forecasts. We find that both deep learning architectures as well as complex layers, such as LSTM, do not increase the precision of daily forecasts. Specifically, a simple recurrent neural network describes a sensible choice for forecasting daily return series. KEYWORDS forecasting, machine learning, risk measurement, time series analysis JEL CLASSIFICATION C33, C58, G17, G23 1|INTRODUCTION Finding the adequate methodological approach to forecast individual financial return series describes a staggering task. In order to achieve precise out-of-sample forecasts, it is necessary to understand the properties of the underlying time series. In this vein, the statistical properties of financial time series have been widely discussed and the application of Autoregressive Moving Average (ARMA) Models is widely accepted (Berger & Gencay, 2018; Halbleib & Pohlmeier, 2012). Due to steadily growing computational power, as well as increasing data availability, forecasting economic time series via machine learning techniques describes a novel string of research. As discussed by Kraus et al. (2020), machine learning is less restrictive regarding the assumptions on the underlying data and current machine learning approaches can adjust to properties of economic time series individually. Therefore machine learning describes a fruitful alternative to econometric modelling. Although machine learning techniques are characterized as black boxes, recent studies provide empirical evidence that machine learning achieves higher forecasting precision than widely accepted econometric approaches. Gu et al. (2020) provide a thorough empirical assessment and discuss competing machine-learning techniques applied to economic data sets. As a result, adequately calibrated machine learning approaches outperform interpretable econometric models in terms of forecasting accuracy. Feng et al. (2020) confirm these results for stock returns, Longo et al. (2022) for GDP forecasts, and Makridakis et al. (2018) for various economic data sets. Received: 5 June 2023 Revised: 28 March 2024 Accepted: 19 May 2024 DOI: 10.1002/for.3165 This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2024 The Author(s). Journal of Forecasting published by John Wiley & Sons Ltd. 2904 Journal of Forecasting. 2024;43:2904–2916. wileyonlinelibrary.com/journal/for This study adds to this string of literature, and we focus on forecasts for innovative economic return series, namely Bitcoin returns. As described in Alessandretti et al. (2018) and Tandon et al. (2019), Bitcoins belong to the asset class of cryptocurrencies and are characterized by higher volatility than classical currencies. Furthermore, as cryptocurrencies are not controlled by national central banks, Bitcoins are exposed to different economic determinants and hence historic return series can exhibit different time series characteristics. Alessandretti et al. (2018) provide empirical evidence that the application of neural networks to time series predictions is a fruitful approach, especially when it comes to predicting Bitcoins. Also, Lahmiri and Bekiros (2019) provide empirical evidence that forecasting Bitcoin return series via machine learning outperforms typical econometric benchmarks, such as ARMA models. Typically, recent studies assess forecasting performance via MAE (Mean Absolute Error) or RMSE (Root Mean Squared Error) to discuss the precision of competing machine learning models (see Jang & Lee, 2018; Lahmiri & Bekiros, 2019; Phaladisailoed & Numnonda, 2018; Tandon et al., 2019). Our empirical assessment provides a thorough forecasting study with an exclusive focus on daily Bitcoin prices and the contribution of our study is twofold. First, we provide an economic assessment of statistical time series properties of daily Bitcoin prices to assess if the identified characteristics are in line with the underlying assumptions of econometric benchmark models. Second, we apply state-of-the-art machine learning approaches and analyze the performance of deep learning network architectures and complex recurrent neural network (RNN) layers, namely recurrent Long ShortTerm Memory (LSTM). Also, we compare the forecasting performance of both econometric time series models and machine learning techniques to discuss state-of-the-art machine learning techniques against solid econometric benchmarks. Specifically, in addition to the widely used ARMA(1,1) approach, we also take into account conditional volatility clustering via generalized autoregressive conditional heteroscedasticity (GARCH) and assess a variety of competing ARMA-GARCH approaches. The remainder of this paper is structured as follows. Section 2provides the relevant literature, Section 3gives an overview of the methodology, and Section 4presents the investigated data. The results of the empirical assessment are in Section 5and Section 6concludes. 2|LITERATURE REVIEW The string of literature, dealing with time series forecasts of cryptocurrencies can be separated into two substrings. One string deals with the application of black-box machine learning approaches and one deals with interpretable econometric approaches. 2.1 |Machine Learning and Bitcoins An extensive study on machine learning-based forecasts is presented by Alessandretti et al. (2018). The authors study the performance of three models and predict daily cryptocurrency prices of 1,681 currencies from time period between November 11th 2015 and April 24th 2018. Two of the applied approaches are based on gradientboosting decision trees and one is based on recurrent Long Short-Term Memory (LSTM) neural networks. Then, investment portfolios are built based on the predictions, and the comparison of their performance in terms of return on investment is assessed. The authors provide evidence that all models outperform the baseline “simple moving average”approach. The authors find that the recurrent LSTM approach is characterized by superior forecasting performance. Also, Jang and Lee (2018) provide an empirical horse race between Bayesian Neural Networks (BNNs) with competing linear and non-linear benchmark models to predict Bitcoin price processes. They cover the daily data from September 11th 2011 to August 22th 2017. As a result, BNNs perform well in predicting Bitcoin price time series and explaining relevant volatility components of historic prices. Based on log-transformed market prices and volatility processes, the authors provide experimental evidence that the predictive performance of BNNs outperforms competing benchmark methods. Karasu et al. (2018) also assess Bitcoin price predictions via machine learning using daily data between January 2012 and December 2018. The authors study Support Vector Machines (SVM) and linear regressions to assess daily closing prices. As a result, SVM models are characterized by higher forecasting precision than linear regression models. Lahmiri and Bekiros (2019)studytheapplicationof sophisticated machine learning applications for cryptocurrency prediction for the period between July 10th 2010 and October 1st 2018 and find that the accuracy of more complex RNN layers, namely Long Short-Term Memory (LSTM), significantly increases forecasting precision. As a benchmark, the authors apply a generalized neural regression architecture benchmark. In this vein, Muniye (2020) compares two deep learning techniques, LSTM and Gated Recurrent Unit (GRU) using daily data between January 1th 2014 and February 20th 2018. The results suggest that the GRU model describes an adequate approach for predicting Bitcoin prices as it requires BERGER and KOUBOV ´ A2905 less compilation time than LSTM. Phaladisailoed and Numnonda (2018) provide similar conclusion based on high-frequency, 1-minute interval trading data from January 1th 2012 to January 8th 2018. The authors compare the performance of competing regression models with RNNs with both LSTM and GRU layers and find that GRU results in superior forecasting accuracy. Also, Tandon et al. (2019) provide a thorough study on Bitcoin price predictions for the period between 2013 and 2019 and compare RNN with LSTM with 10-fold cross validation, linear regression and random forests. The results indicate that cross-validation in RNN with LSTM contributes to the improvement of the efficiency of the model for Bitcoin prediction. The proposed RNN with LSTM using 10-fold cross validation leads to significantly lower MAE (mean absolute error) than random forest and linear regression models. 2.2 |Econometric Time Series Analysis and Bitcoins In the context of econometric modeling, Chu et al. (2017) provide an extensive study on GARCH modeling dealing with the seven most popular cryptocurrencies using data between June 22nd 2014 and May 17th 2017. They find that IGARCH and GJR-GARCH models result in superior model fit for conditional volatility modeling. Among the assessed GARCH-type models, it is the IGARCH(1,1) model that provides the best fit for Bitcoins. Troster et al. (2019) compare the GARCH and GAS model's forecasting precision to predict the conditional mean and volatility of the Bitcoin return series. They cover the daily data from July 19th 2010 to April 16th 2018. The authors take fat tails into account and demonstrate that the assumption of normally distributed returns gets outperformed by the heavy-tailed GAS approach, measured via Value-at-Risk (VaR) forecasts. By fitting nonlinear econometric models to historical data, Fig a-Talamanca and Patacca (2019) study the relative impact of attention measures on both the mean and the variance of Bitcoin returns using data between January 1th 2012 and December 31th 2017. The authors study competing models belonging to the family of ARMA(p,q)-X (E)GARCH(1,1)-X nonlinear models and include attention-related explanatory variables. The results provide evidence that attention measures have a significant impact on the conditional mean and conditional variance of Bitcoin returns, especially when attention is measured in terms of trading volume. The authors assess the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) as well as forecast performance. 2.3 |Machine Learning and Econometric Time Series Analysis Shen et al. (2021) provide a pioneering study to compare both econometric and machine learning approaches. The authors study GARCH models and RNN with GRU layers to forecast Bitcoin's return volatility and VaR figures on data between April 30th 2013 and May 21th 2021. The authors provide empirical evidence that RNN outperforms GARCH and EWMA in terms of average forecasting performance. Furthermore, RNN shows poor performance in VaR forecasting, indicating that econometric models outperform machine learning in dealing with extreme volatility. Furthermore, this study suggests an alternative method of Bitcoin volatility analysis and finds that machine learning methods perform well in less volatile financial market conditions. McNally et al. (2018) compare both econometric and machine learning techniques and compare Bayesian optimized RNN, LSTM, and ARIMA models for predicting historic market prices of Bitcoins using data between August 19th 2013 and July 19th 2016. The authors demonstrate that non-linear deep learning methods outperform the ARIMA prediction. Also, Cortez et al. (2021) compare the predictions of the ARMA-GARCH model with the K-Nearest Neighbour approach (KNN) applied to market liquidity in cryptocurrencies from February 9th 2018 to February 8th 2019. The authors provide empirical evidence that KNN approaches outperform ARMA and GARCH models in predicting the log rates of the bid-ask spreads. Furthermore, the authors demonstrate that, compared to the ARMA and GARCH models, the KNN approach is more effective at capturing the short-term market liquidity of cryptocurrencies. Further studies that confirm the superiority of Neural Networks when it comes to predicting Bitcoin prices are given by Alessandretti et al. (2018), Jang and Lee (2018), Lahmiri and Bekiros (2019), Phaladisailoed and Numnonda (2018), and Tandon et al. (2019). Most of the papers apply MAE or RMSE to assess the performance of applied models, see Tandon et al. (2019). Furthermore, there are also studies that investigate other features, which could have an effect on the behavior of Bitcoin (Balcilar et al., 2017;Fig  a-Talamanca & Patacca, 2019;Saad&Mohaisen,2018;Sin&Wang,2017; Velankar et al., 2018). As well, Taskaya-Temizel and Casey (2005) also discuss the idea of hybrid approaches and combine time series models with machine learning models. 3|METHODOLOGY In this section, we introduce the notation used throughout the paper, define the desirable properties of 2906 BERGER and KOUBOV ´ A econometric time series models and recurrent neural networks, specify long short-term memory, and present the performance metrics. 3.1 |Econometric Time Series Models In order to adequately capture the relevant characteristics of Bitcoin returns, we apply an autoregressive moving average (ARMA) approach to model the conditional mean of the daily return series. This approach describes a combination of the autoregressive model with plags and the moving average model with qlags. Furthermore, we also take into account time-varying conditional volatility and autoregressive volatility clustering and apply the generalized autoregressive conditional heteroscedasticity (GARCH) model to the squared residuals of the ARMA approach, with mand nlags, respectively. Then the ARMA(p,q)-GARCH(m,n) model is defined as follows: yt¼X p i¼1 ϕiytiþX q j¼1 θjutjþut, with :utiid D 0,σ2 t  , ð1Þ σ2 t¼α0þα1u2 t1þα2u2 t2þ…þαmu2 tmþβ1σ2 t1 þβ2σ2 t2þ…þβnσ2 tn, ð2Þ with: α0>0,αi≥0i¼1,…,mðÞ,βj≥0j¼1,…,nðÞand P max m,nðÞ i¼1 αiþβi ðÞ<1:Where utis white noise (has zero mean, constant variance σ2, and is uncorrelated in time) and the time series ytare the Bitcoin returns at time t. Also, mdescribes the order of the ARCH terms and nthe order of the GARCH terms. The residuals utare characterized by distribution D0,σ2 ðÞ. We will assess both a Gaussian and a t-distribution. Also, we apply a naive benchmark, which is simply the return from the previous period. Further, this framework allows us to also assess competing approaches, that are nested in equation (2), namely ARMA(p,q), ARMA(p,q)-GARCH(m,n), and different parameterizations of p,q,m , and n. The parameters are estimated via the maximum likelihood method. 3.2 |Recurrent Neural Networks Our baseline machine-learning approach is a simple RNN. In comparison to typical Feed-Forward-Networks (FFNs), RNNs include backward connections and therefore this approach is predestined to capture timedependent features of a time series. Also, by stacking more than one RNN layer, we are able to study Deep RNNs. Let the input and output at time step tbe described by Xtand yt, the weights for the hidden layer by Whx and bias by bx. The information from the previous time step is ht1with weights Whh and bias bh. Then, in order to calculate the hidden state at the time step ht, that is the information that will be passed to tþ1, is described as follows: ht¼fWht1,Xt ðÞ, with : t¼1,…,Mand h0¼0: ð3Þ The hidden state at time tcan be described as a function with parameters Wof the previous hidden state and the input at time t. In our study, we apply the standard hyperbolic tangent function (tanh) and fW. The amount of tuples of the train set is M. Then, we apply RNN and stacked RNNs which are then defined as follows: ht¼tanh Whhht1þWhxXtþbh ðÞ,ð4Þ yt¼tanh Wyhhtþby  , with : t¼1,…,M: ð5Þ For instance, for a Deep RNN with 2 hidden layers, 5 neurons, and a forecasting horizon of 10 days, the output of the last hidden layer (htfor time step t) is a matrix 105, as it contains the results of each neuron. The output layer is a simple layer and transforms this matrix into 101 vector of forecasts with the weights Wyh,as described in equation 5. From this, the number of neurons can also be seen as a dimension of the hidden state of one layer. In this paper, we study 1, 2, and 3 hidden simple RNN layers in combination with 1, 5, and 10 units and we refer to Géron (2020) for a thorough introduction to RNNs and machine learning. 3.2.1 | Long Short-Term Memory In addition to simple RNN, we also apply RNN with long short-term memory (LSTM) layer. LSTM was introduced by Hochreiter and Schmidhuer (1997) in order to overcome the problem of long-term dependency data causing a vanishing gradient within a simple RNN framework. In comparison to RNN, the training of LSTM converges BERGER and KOUBOV ´ A2907 faster and the identification of long-term dependencies in the data is possible. In contradiction to RNN, where every layer gets 2 inputs, at the time step t:Xtand the output of the previous time step t1, namely ht1, LSTM adds an additional input to each layer. That is, the current long-term memory of the network, known as the cell state ct1and the hidden state htbecomes the short-term state, see Géron (2020). The parameterization of the LSTM approach is given as follows: ft¼σWfxXtþWfhht1ðÞþbf  ð6Þ gt¼tanh WgxXtþWghht1ðÞþbg  ð7Þ it¼σWixXtþWihht1ðÞþbi ðÞð8Þ ot¼σWoxXtþWohht1ðÞht1ðÞþbo ðÞð9Þ ct¼ftOct1þitOgtð10Þ yt¼ht¼otOtanh ct ðÞ ð11Þ The first step in the LSTM cell is the Forget Gate (ft) as described in equation (6). Through this gate, certain information is deleted (forgotten) and the output of the Forget Gate (ftNct1) is then added to the output of the Input Gate (itNgt). The result for ctgets stored without further transformation as described in equation (10). Hence, at each time step, irrelevant information is discarded and relevant historical information is added, see equation (11). The long-term state ctis copied after this addition and passed through the tanh function. As described in Yu et al. (2019). The number of neurons per LSTM layer denotes the dimensions of the hidden state and of the output state of one layer. Similar to the RNN framework, a dense layer with 1, 5, and 10 neurons is used as an Output Layer for the RNN with LSTM. In order to train the networks, we apply Adam (Adaptive Moment Estimation) optimization. As described in Géron (2020) Adam optimization combines the idea of Momentum Optimization and RMSProp. 3.3 |Performance Metrics In addition to the assessment of autocorrelated residual and partial autocorrelation functions, we draw on existing studies and assess the precision of out-of-sample forecasts via both root mean squared error (RMSE) and mean absolute errors (MAE). The metrics are calculated as follows: RMSET B¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PM i¼1PT j¼1yB,forecast ij yB,true ij  2 MT v u u t,ð12Þ MAET B¼PM i¼1PT j¼1jyB,forecast ij yB,true ij j MT:ð13Þ Where Mdenotes the number of tuples, Bthe batch size, 100, 250, or 500 days, and Tthe number of days that describe the forecasting horizon, 1, 5, or 10 days. Furthermore, in our study, naive forecasting deals as a benchmark for both approaches. That is, the last day of the training sample is used as the forecast for the next consecutive day. Furthermore, in order to assess competing neural nets, we apply a naive fully connected neural net as an additional benchmark. 4|DATA The data starts with the first available market price of Bitcoin, April 28th 2013, and ranges until December 12th in 2021. 1 The graphical representation of the prices and logarithmic returns is displayed in Figure 1. Figure 1illustrates both daily market prices and daily log returns over time. Table 1presents the descriptive statistics of the data. We find that daily returns are characterized by a standard deviation of 4.1698 and a mean of 0.1884. These results are in line with Cunha and Silva (2020). However, in comparison to other currencies, as reported by Mignot and Westerhoff (2024), daily Bitcoin returns are characterized by higher volatility. 2 Table 2gives the Ljung-Box (LB) test statistics to assess autocorrelation of the logarithmic returns and squared logarithmic returns, the Augmented DickeyFuller (ADF) test to investigate stationarity, and the Jarque-Bera (JB) test to study normality of the logarithmic returns. The ADF Test shows, that the time series of Bitcoin daily prices is not stationary. As a result, logarithmic returns are created (hereafter referred to only as returns). As can be seen in Table 2, the Bitcoin daily returns are stationary. As discussed in Box et al. (1994), we set the number of investigated lags for the Ljung-Box test to 20. The standard deviation of Bitcoin returns is described as 4.1698 and, as indicated by the LB test statistic is characterized by statistically significant volatility 1 Data source is wwww.coinpaprika.com and the applied data set is also available upon request to the authors. 2 For an in-depth discussion on stylized facts of exchange rates, we refer to Guillaume et al. (1997) and Grauwe and Grimaldi (2006). 2908 BERGER and KOUBOV ´ A clusters. However, as illustrated in Figure 2the autocorrelation of returns decays quickly. These findings are consistent with Hu et al. (2019), Momtaz (2021), and Zhang et al. (2018) who report similar findings on volatility and volatility clusters. Furthermore, as presented by Zhang et al. (2018) and Borri (2019), based on the JB test, we can confirm that the distribution of Bitcoin returns is non-normal and exhibits fat tails, a stylized fact that is well-known from the stock market and can also be seen in Figure 3. As described above, Bitcoin returns are in line with stylized facts of financial return series, as also shown in the literature. The facts that are shown in this paper are high volatility, volatility clustering, and fat tails. In order to assess forecasting accuracy, we separate the data into overlapping tuples comprising 110, 260, and 510 days. Hence, the training set of the tuple consists of 100, 250, and 500 days, the additional 10 days are used for assessing the goodness of a forecast. All in all, there are 3,032 tuples of 110 days, 2,882 tuples of 260 days, and 2,632 tuples of 510 days for T¼100,250 , and 500 and 10 days ahead forecast. An example of this separation is displayed in Figure 4, the underlying data for the machine learning exercise is separated into train, validation, and test data sets. Following the study from Shen et al. (2021), the tuples are characterized as follows. The train dataset contains the first 70% of the tuples, the validation data set the next consecutive 20% and the test data set the remaining 10%. Then, based on the trained model, the train and validation data set describe the in-sample data, and the test data is the out-of-sample data set. As illustrated in Figure 4, the data can then be split into different data sets. In our study, we assess training and validation sets comprising T¼100,250 , and 500 days and study forecast of 1,5 and 10 days ahead. FIGURE 1 Bitcoin price and returns (in %) in time. TABLE 1 Descriptive statistics. Bitcoin daily prices Bitcoin daily returns n 3145.00 3143.00 Mean 8818.48 0.19 Std 14568.24 4.17 Min 67.81 43.37 25% 442.82 1.37 50% 3410.45 0.21 75% 9239.90 1.90 Max 67617.02 28.71 Note: The table presents descriptive statistics for the daily market prices and return series of Bitcoins from April 28, 2013 to December 12, 2021. The amount of observations is n, Min and Max are the minimum and maximum value of each variable. 25%, 50% and 75% are the respective quartiles of the empirical return distribution and Std is the standard deviation of the mean. TABLE 2 Ljung-Box, ADF and Jarque-Bera tests. Ljung-Box test statistic P-value Bitcoin returns 53.70 0.00 Bitcoin squared returns 356.65 0.00 ADF test statistic P-value Bitcoin prices 0.39 0.91 Bitcoin returns 14.71 0.00 Jarque-Bera test statistic P-value Bitcoin returns 21.02 0.00 Note: This table provides the results of the Ljung-Box (LB), Augmented Dickey Fuller (ADF) and Jarque-Bera (JB) test. The Ljung-Box (LB) test is performed with 20 lags with H0: The data is independently distributed and H1: The data is not independently distributed, and exhibits serial correlation. The Augmented Dickey Fuller (ADF) test assesses H0: The series is non-stationary. H1: The series is stationary. And the hypothesis of the Jarque-Bera (JB) test test are H0: The series is normally distributed. H1: The series follows a non-normal distribution. For all test, the p-value is presented. BERGER and KOUBOV ´ A2909 FIGURE 2 Autocorrelation function and partial autocorrelation function of daily returns and squared Returns. The upper graphs visualize the Autocorrelation function (ACF) of daily Bitcoin returns (left) and squared Bitcoin returns (right). The lower graphs provide information on the partial autocorrelation for Bitcoin returns (left) and squared Bitcoin returns (right). FIGURE 3 Histograms of Bitcoin Returns (in %) and comparison with Normal Distribution. FIGURE 4 Separating the data into tuples for recurrent neural nets (RNN). Each tuple contains training data (yellow) and labeled forecasting data (red). The data is a time series and is described by consecutive daily Bitcoin returns. In order to train a neural net, 90% of the tuples are used to train and validate the neural net, and 10% are applied to test the results. The training and test data is split into 70% training and 20% testing data the test data presents the basis for the evaluation of out-of-sample forecasts. 2910 BERGER and KOUBOV ´ A 5|EMPIRICAL ASSESSMENT In this section, we discuss the empirical results of our forecasting study. Due to the fact, that we compare two competing methodological approaches, namely econometric modeling and machine learning, we provide a discussion on the optimal parameterization of both approaches, econometric and machine learning, TABLE 3 AIC values for ARMA p,qðÞand GARCH m,nðÞmodel. ARMA Gaussian distribution p=q012345 111,046.28 11,044.39 11,043.19 11,046.69 11,047.48 11,046.82 211,044.72 11,043.24 11,044.73 11,076.25 11,052.37 11,073.12 311,043.90 11,047.88 11,050.74 11,051.88 11,067.92 11,067.21 411,044.79 11,048.39 11,053.40 11,035.87 11,044.69 11,067.93 511,045.94 11,048.40 11,051.41 11,077.58 11,035.10 11,035.78 t-distribution p=q012345 112,055.74 12,054.43 12,055.72 12,054.39 12,056.16 12,086.41 212,055.07 12,055.66 12,058.30 12,085.20 12,089.96 12,083.32 312,054.64 12,054.46 12,089.46 12,060.73 12,090.63 12,088.63 412,054.49 12,056.17 12,085.86 12,090.29 12,094.27 12,089.88 512,053.56 12,088.41 12,056.28 12,092.36 12,092.34 12,091.47 GARCH m=n012345 112237.04 12547.45 12529.13 12551.13 12549.13 12531.89 212310.91 12543.04 12527.54 12548.28 12538.42 12531.98 312361.34 12525.42 12526.80 12548.17 12529.90 12525.40 412421.43 12539.38 12473.29 12524.98 12534.34 12543.49 512452.35 12536.82 12539.46 12536.74 12144.82 12538.43 Note: This table presents the AIC criterion for different combinations of lags under the assumption of Gaussian and t-distributed residuals. TABLE 4 Forecasting performance: Econometric approach. Naive benchmark ARMA-GARCH RMSE 1 day 5 days 10 days 1 day 5 days 10 days 100 days 0.0585 0.0578 0.0576 0.0427 0.0431 0.0427 250 days 0.0567 0.0558 0.0556 0.0405 0.0402 0.0401 500 days 0.0563 0.0555 0.0554 0.0398 0.0401 0.0408 Mean 0.0572 0.0564 0.0562 0.0410 0.0411 0.0412 MAE 1 day 5 days 10 days 1 day 5 days 10 days 100 days 0.0391 0.0391 0.0391 0.0284 0.0282 0.0279 250 days 0.0382 0.0381 0.0381 0.0265 0.0263 0.0262 500 days 0.0380 0.0379 0.0379 0.0259 0.0260 0.0260 Mean 0.0384 0.0384 0.0384 0.0269 0.0268 0.0267 Note: This table presents the out-of-sample forecasting performance of the naive benchmark is the return from the previous period and the applied ARMA(4,4)-GARCH(1,3) model. The forecasting horizon is 1 day, 5 days, and 10 days and the metrics are the root mean squared errors (RMSE) and mean absolute error (MAE). BERGER and KOUBOV ´ A2911