Abstract
The paper provides a comparative empirical study of predictability of cryptocurrency returns and prices using econometrically justified robust inference methods. We present robust econometric analysis of predictive regressions incorporating factors, which were suggested by Liu, Y., & Tsyvinski, A. (2018). Risks and returns of cryptocurrency. NBER working paper no. 24877; Liu, Y., & Tsyvinski, A. (2021). Risks and returns of cryptocurrency. The Review of Financial Studies, 34(6), 2689–2727, as useful predictors for cryptocurrency returns, including cryptocurrency momentum, stock market factors, acceptance of Bitcoin, and Google trends measure of investors’ attention. Due to inherent heterogeneity and dependence properties of returns and other time series in financial and crypto markets, we provide the analysis of the predictive regressions using both heteroskedasticity and autocorrelation consistent (HAC) standarderrors and also the recently developed
1 Introduction
This paper focuses on how different factors predict the returns of cryptocurrencies. We provide a comparative analysis of predictability of cryptocurrency returns using econometrically justified robust inference methods that account for inherent heterogeneity and dependence in the time series dealt with. We focus on five predictive models in the study by Liu and Tsyvinski [29,30] who provide the first detailed empirical analysis and estimates of predictive regressions for cryptocurrency returns, with the assessment of significance of their coefficients based on i.i.d. standard errors in the former working paper and the use of heteroskedasticity and autocorrelation consistent (HAC) NeweyWest standard errors in some of the predictive models in its latter recently published version.
We provide a study of the robustness of the empirical analysis of the predictive models by using both HAC inference based on NeweyWest standard errors [35] (see also [3] for HAC inference approaches) and the newly developed
As is well known, many important economic and financial variables, including returns’ time series, are characterized by inherent heterogeneity, autocorrelation and nonlinear dependence, volatility clustering, excess kurtosis, heavy tailedness, and outliers, as in the case of GARCH or ARGARCHtype models for financial and cryptocurrency returns (see, among others, the review and discussion in [4,8,9,13,14, 15,20,26,33,42]). Naturally, as compared to developed financial and economic markets dealt with in the most of the literature, the properties of heterogeneity, volatility clustering, nonlinear dependence, and heavy tailedness are typically even more pronounced in emerging and cryptocurrency markets that are subject to more frequent and more pronounced external and internal shocks.
It is important to use inference methods that account for autocorrelation and heterogeneity in their analysis (see, among others, [20], and references therein). In general, the i.i.d. standard errors are not valid under heterogeneity, autocorrelation or dependence, and hence so are the results of ttests since standard errors may be underestimated or overestimated.
The most widely used approach in econometrics, economics, and finance to conduct asymptotically valid inference under heterogeneous and dependent data is that based on consistent variance estimators. In the context of time series, asymptotically valid inference is traditionally based on the popular HAC standard errors with the adjustments for serial correlation and dependence derived by [3,35].^{[1]}
This paper focuses on econometrically justified inference in predictive regressions for cryptocurrency returns and prices using robust methods. Due to inherent heterogeneity and dependence properties of returns and other time series in financial and crypto markets, we provide the analysis of the predictive regressions using (HAC) NeweyWest standard errors [35].
While widely used and quite general, HAC inference procedures often have poor finite sample properties, especially when observations exhibit pervasive and pronounced correlations (see, among others, the discussion in [22,23], Section 3.3 in [20], and references therein). Motivated by this, we further present the analysis of the predictive regressions using recently developed tstatistic robust inference approaches [22,23]. Robust tests in the approaches are based on tstatistics in group estimates of model parameters and do not require consistent estimation of their limiting variances, as shown in HAC methods. In the context of predictive regressions, robust large sample inference is conducted as follows: the time series is partitioned into
One should also note that the moment assumptions for the validity of HAC inference methods, including the approaches based on NeweyWest standard errors, such as the often imposed condition that the regressors and the regression errors have finite eighth moments (see Section 16.4 in [46]), are typically too restrictive for predictive regressions for financial returns. For example, according to many studies in the literature, the distributions of financial returns are heavy tailed with tail indices smaller than four, thus implying infinite fourth moments and, in the case of emerging markets, even tail indices smaller than 2 are not uncommon (see the review in Section 1.2 in [20] and the references therein). On the other hand,
These conclusions suggest that the
We provide comparisons of robust predictive regression estimates between different cryptocurrencies and their corresponding risk and factor exposures. We further contrast the results from the robust predictive regression analysis with the conclusions implied by the i.i.d. standard errors. In general, the number of significant factors disappears or decreases as we use more robust ttests, and the tstatistic robust inference approaches appear to perform better than the ttests based on HAC standard errors in terms of pointing out interpretable economic conclusions. In particular, this is observed for predictive regressions incorporating cryptocurrency momentum (see Section 3.1) that indicates that cryptocurrency returns exhibit the stylized fact of the absence of linear autocorrelations and linear dependence similar to financial returns (see [13] for a review and discussion of this and other key stylized facts for financial markets). In addition, the aforementioned property is confirmed for predictive regressions for Bitcoin returns incorporating pricetoacceptance ratio (an analog of pricetodividend ratio for stocks, see Section 3.3) that points to apparent/potential absence of intrinsic/fundamental value of Bitcoin, in contrast to financial assets with pricing models incorporating dividends. According to the results in the paper, the aforementioned disappearance of significance under more robust
To our knowledge, this paper is the first to apply several econometrically justified robust inference approaches in examining the predictability of cryptocurrency returns, including
The results in this paper point to the advantages of complementing empirical analyses of dependent and heterogeneous economic and financial data using HAC and other (e.g., clustered) consistent standard errors by the simpletoimplement tstatistic robust inference approaches that do not require consistent estimation of liming variances of estimators dealt with. The analysis in the paper further emphasizes that statistical conclusions from econometrically justified robust tests are different from those implied by i.i.d. standard errors.
The main goal of the paper is to emphasize the importance of the use of robust inference approaches in the analysis of economic and financial data affected by the problems of heterogeneity and dependence, and predictive regressions for cryptocurrency returns are mainly used for illustration and comparisons of conclusions implied by different inference methods.
At the same time, the results and robust inference methods used in this paper may be helpful in further analysis of predictive models for cryptocurrency markets and the development of economic models for them. The results in this paper further point out to several similarities between cryptocurrency and financial markets. The empirical applications of robust inference methods in the paper may inspire future researches on robust tests of crypto markets’ efficiency or the robust analysis of stylized facts of cryptocurrency returns.
Many papers in the literature have focused on the analysis of empirical properties of cryptocurrency markets. [1] emphasizes the importance of using correct data sources in empirical analyses of crypto markets, especially in the case of time series of returns and prices of crypto market indices. It is widely agreed that cryptocurrencies suffer from much higher volatility compared to the traditional currencies (see, among others, the review and discussion in Ch. 8 in [2,4,5,8,14,52]). The Bitcoin market efficiency is discussed by [34,48,50]. [11] presents an econometric analysis of bubbles in the Bitcoin market using the methodology developed in [39,40]. According to the analysis in [6], cryptocurrencies are highly exposed to tail risk within crypto markets but not in other global markets. Nguyen et al. [36] show that the right tail risk among cryptocurrencies is more pronounced as compared to the left one. Trimborn and Härdle [47] proposed the construction of an index that quickly reacts to cryptocurrency market changes.
Several authors have also focused on the analysis of predictability of cryptocurrency prices and returns. Poyser [41] summarized factors that drive the price of cryptocurrency into two categories: internal factors and external factors. Sovbetov [45] used cointegration models and tests to identify factors that influence the prices of five major cryptocurrencies. Lintilhac and Tourin [28] applied cointegration models in the analysis of pairs trading strategies in the bitcoin markets.
Liu and Tsyvinski [29,30] provided the first detailed empirical analysis and estimates of predictive regressions for cryptocurrency returns that incorporate several traditional and crypto risk factors, including foreign exchange rates, prices of metal commodities, momentum, and measures of investors’ attention. Among other results, the authors propose, for the first time in the literature, an analog of the price to dividend ratio factor for cryptocurrencies, with, e.g., the intrinsic/fundamental value of Bitcoin proxied by the number of Bitcoin wallet users. The analysis in working paper Liu and Tsyvinski [29] is based on i.i.d. and (i.i.d.) bootstrapped standard errors, while the assessment of the significance of the coefficients in some of the predictive regressions in its recently published version [30] is based on HAC NeweyWest standard errors. Based on the empirical analysis, [29,30] reach the conclusion that cryptocurrency returns appear to be less affected by factors related to traditional asset classes, but appear to be predictable by momentum and investors’ attention.^{[3]}
Liu et al. [31] provided estimates of factor models for the crosssection of returns for a large set of cryptocurrencies and concluded that the returns are explained by three cryptospecific factors, namely, market, size, and momentum.^{[4]} [7,27,32] documented large differences in bitcoin prices across exchanges located in different countries, and for different fiat currency pairs. [32] argued that these bitcoin discounts are explained by capital market segmentation, capital controls, and weak financial institutions. [27] pointed to the importance of market inefficiencies. Focusing on the most reputable exchanges, [7] showed that the large bitcoin price differences in them are due to costly arbitrage and idiosyncratic risks. Among other results, [7] provided estimates, with NeweyWest HAC standard errors, for the threefactor model for cryptocurrency returns in [31] incorporating a crypto market, size, and momentum factors (see Table 4 in [7]). The authors further relate idiosyncratic risk measured as residuals from the estimated threefactor regressions to a set of portfolio characteristics and find that it is higher for portfolios containing exchangecurrency pairs with higher liquidity and execution risks.
This paper is organized as follows. The data used in the analysis are described in Section 2. Section 3 presents the results of the empirical analysis of predictive regressions for cryptocurrency returns and factors considered. Section 4 makes some concluding remarks. Appendix provides several tables on the empirical analysis referred to in the paper.
2 Data
The data frequency is specified throughout the paper. Sections 3.1 and 3.2 deal with predictive regressions for daily cryptocurrency returns using daily data on the returns and stock market factors. Section 3.3 provides estimation results in predictive regressions for weekly Bitcoin returns on the proxy for acceptance of Bitcoin given by the ratio of the spot price of Bitcoin to the number of wallet users (the data on the latter number is updated weekly). Section 3.4 provides the estimates for predictive regressions of weekly cryptocurrency returns on the proxies for weekly investors’ attention measured using the data on Google trends. Weekly cryptocurrency returns are calculated using the data on daily cryptocurrency prices and returns.
We use Yahoo Finance as the main data source for cryptocurrency prices.^{[5]} The time interval for Bitcoin is from 01/01/2011 to 31/05/2018, for Ripple is from 04/08/2013 to 31/05/2018, and for Ethereum is from 07/08/2015 to 31/05/2018.^{[6]}
As in [29,30], the CAPM [44], FamaFrench threefactor [17], Carhart fourfactor [10], FamaFrench fivefactor [18], and FamaFrench sixfactor [19] data are obtained from Kenneth French’s website.^{[7]} The data on the number of Bitcoin wallet users are from blockchain.info.^{[8]} Further, as in [29,30], the frequencies of words “bitcoin,” “ripple,” and “ethereum” searched on Google are measured by Google Trends.^{[9]}
3 Empirical analysis
3.1 Cryptocurrency momentum
The analysis of cryptocurrency momentum is based on the daily returns of Bitcoin, Ripple, and Ethereum.^{[10]} The tables on the empirical results in this and subsequent sections provide the estimates of the coefficients of predictive regressions considered (e.g., the regression of daily cryptocurrency returns on their lags in the momentum analysis) and the HAC
In the
One would expect the significance of a predictive regression coefficient to decrease when using more robust tests (e.g., HAC
Tables 1–3 present the estimation results for regressions of future daily cryptocurrency returns (









(1)  (2)  (3)  (4)  (5)  (6)  (7)  


0.082  0.081 


0.032 

i.i.d. 





1.408 

HAC 





1.088 


0.596  0.692  1.579 


0.609  0.562 

1.286  0.039  1.568 



0.372 









(1)  (2)  (3)  (4)  (5)  (6)  (7)  


0.243 

0.111 

0.121 

i.i.d. 







HAC 



1.179 

1.483 



2.466 

0.683 

1.011 



1.018  0.001  0.411 

1.219 










(1)  (2)  (3)  (4)  (5)  (6)  (7)  

0.038 



0.081 

0.032 
i.i.d.  1.189 



1.893  0.511  0.474 
HAC  0.950 





0.781 

1.909 

0.327  0.003  0.946 



1.346 

0.153  0.064  0.772 


In line with the above, overall, HAC
In contrast, interestingly, HAC
According to the most of the results for the
3.2 Stock market factors
Similar to [29,30], in analogy to the analysis of predictability of daily stock excess returns, we consider CAPM, FamaFrench threefactor, Carhart fourfactor, FamaFrench fivefactor, and FamaFrench sixfactor models. All data used are daily. We regress daily excess cryptocurrency return on the stock market factors mentioned in these five models. We subtract the riskfree returns (1month US Treasury bill rate) from Bitcoin, Ehthereum, and Ripple returns in order to obtain the excess returns. In total, there are six stock market factors considered: excess return on the stock market (Mkt.RF), market capitalization (SMB), value premium (HML), momentum (MOM), profitability (RMW), and investment (CMA). Detailed constructions of these factors could be found in [10,17,18, 19,44].
The estimates and the analysis of significance for the corresponding predictive regressions for Bitcoin returns are summarized in Table 4. The estimates and significance analysis for predictive regressions for Ripple and Ethereum are provided in Tables A1 and A2 in Appendix. In Table 4 and Tables A1 and A2, the
Excess Bitcoin return  

CAPM  3F  4F  5F  6F  
Mkt.RF 





[0.932]  [0.270]  [0.271]  [0.049]  [0.052]  
(

(

(

(

(








SMB  0.586  0.545  0.597  0.546  
[1.488]  [1.454]  [1.214]  [1.174]  
(0.947)  (0.860)  (0.914)  (0.820)  
{1.099}  {0.969}  {0.994}  {0.906}  
HML  0.167  0.023 



[1.132]  [0.971]  [0.646]  [0.523]  
(0.446)  (0.063)  (

(


{0.808}  {0.488} 



MOM 



[

[


(

(





RMW  0.184  0.162  
[

[


(0.325)  (0.294)  
{0.231}  {0.364}  
CMA  1.260  1.322  
[0.247]  [0.262]  
(1.902)  (2.102


{1.175}  {1.284}  
Alpha  0.009  0.009  0.009  0.009  0.009 
[

[

[

[

[


(

(

(

(

(


{2.393}  {2.383}  {2.387}  {2.450}  {2.437} 
The excess return is given by the difference between the Bitcoin return and the riskfree return. The values in the first row of each stock factor are coefficients of each stock factor. The
According to the tables, all alphas appear to be strongly significant for the three cryptocurrencies, and nearly none of the factors are significant, even when i.i.d. standard errors are used.^{[13]} Although the riskfree market returns are not statistically significant for these cryptocurrencies, we notice that the coefficients on them are negative in all models for the three cryptocurrencies considered. The overall conclusion using HAC inference methods is that Bitcoin, Ethereum, and Ripple prices are indifferent to changes in the stock market.
Remarkably, momentum (MOM) factors in FamaFrench fourfactor models are strongly significant for Bitcoin and Ethereum returns according to the
Importantly, for Ripple, one observes a decrease in the significance of the value (HML) factor in FamaFrench five and sixfactor models compared to the threefactor model according to the
The decrease in the significance of HML in fivefactor model according to
3.3 Acceptance of Bitcoin
Liu and Tsyvinski [29,30] proxy the intrinsic/fundamental value of Bitcoin by the number of Bitcoin wallet users and use the ratio of the Bitcoin price to the number of wallet users as an analogue of the pricetodividend ratio in predictive regressions for financial returns. This is motivated by the interpretation of the pricetodividend ratio as a measure of the gap between an asset’s market and intrinsic/fundamental values (see [29,30] for the discussion).^{[14]} In this section, we present the analysis of robust predictive regressions for Bitcoin returns with the same pricetodividend proxy used as a regressor. More precisely, the results are provided for robust inference approaches applied to regressions of weekly Bitcoin returns on the lagged ratio of the spot price of Bitcoin to the number of wallet users (the data on the latter number are updated weekly). It is natural to view the number of Bitcoin wallet users as a measure of acceptance of Bitcoin and to refer to the latter ratio as pricetoacceptance ratio, denoted by
We regress the future weekly Bitcoin returns (









(1)  (2)  (3)  (4)  (5)  (6)  (7)  

0.113  0.001  0.207  0.163  0.140  0.303 

i.i.d.  0.506  0.006  0.927  0.729  0.626  1.357 

HAC  7.606

0.078  12.882

9.958

8.156

17.862



1.584  1.023  0.669  0.589  0.503  0.504  0.503 
The significance of the pricetoacceptance proxy decreases or disappears when using the more robust
3.4 Google trends
Similar to [29,30], to measure how investors’ attention affects the returns of Bitcoin, we use the Google Trends of words “bitcoin,” “ripple,” and “ethereum” as proxies. In this section, we consider the weekly returns of the three cryptocurrencies. In order to maintain the consistency of Google Trends, we use the last day of the previous 180day interval as the first day of the new 180day interval, calculate the ratio of that overlapped day’s Google Trend, and use the ratio to normalize the new interval. Then, we calculate the average of every sevenday Google Trends and use these averages as proxies for weekly investors’ attention.^{[16]}
^{,}
^{[17]} We regress the future weekly cryptocurrency return (









(1)  (2)  (3)  (4)  (5)  (6)  (7)  
Google Trend  0.0001  0.0001  0.0001  0.0001  0.0001  0.0001  0.00005 
i.i.d.  6.572

5.625

4.813

4.140

3.570

3.062

2.669

HAC  0.587  0.455  0.352  0.285  0.256  0.244  0.231 

1.920  1.499  1.226  1.069  0.990  1.028  1.063 









(1)  (2)  (3)  (4)  (5)  (6)  (7)  
Google Trend  0.001  0.0004  0.0004  0.0003  0.0002  0.0002  0.0002 
i.i.d.  8.959

7.476

6.125

4.933

3.907

3.104

2.616

HAC  1.041  0.764  0.698  0.675  0.547  0.417  0.314 

2.533  2.062  1.718  0.325 












(1)  (2)  (3)  (4)  (5)  (6)  (7)  
Google Trend  0.0002  0.0002  0.0001  0.0001  0.00001 


i.i.d.  3.594

2.559

1.567  0.729  0.076 


HAC  0.739  0.598  0.413  0.212  0.028 



1.487  1.309  1.106  0.914  0.753  0.629  0.562 
The
However, the statistical significance of Google Trends disappears when HAC
4 Conclusion
In order to examine the predictability in cryptocurrency markets, this paper analyses four different models on cryptocurrency returns and three different
According to HAC
When
For the two Bitcoinspecific factors, pricetoacceptance ratio and the Google trend, we do not find any significance at all for weekly return prediction using the
Throughout the paper, we discuss the advantages and limitations of different approaches to robust inference under heterogeneity and dependence in data. Overall, the tstatistic robust inference approaches appear to perform better than the HAC ttests in terms of pointing out interpret able economic conclusions.
The results and conclusions in the paper emphasize the necessity in the use of econometrically justified inference methods that account for autocorrelation and heterogeneity in observations. They further emphasize the usefulness of
Further research on the topic may focus on the development of pricing models for cryptocurrencies incorporating the factors that appear to have predictive power for crypto prices and returns. It would also be of interest to provide econometrically justified and robust analysis of predictive regressions incorporating further factors, such as, importantly, the measures of liquidity, network security, and nonfundamental uncertainty that enter valuation models for cryptocurrencies (see [37,38]). It would also be of interest to consider applications of twosample
Acknowledgments
We thank two anonymous referees, Andrea Buraschi, Emiliano Pagnotta, Artem Prokhorov and Johan Walden and the participants at the 1st Inaugural International Conference on Econometrics and Business Analytics (iCEBA) and the seminars at CEBA, St. Petersburg State University, and the joint meeting of the 2nd Workshop in Applied Econometrics and the VII International Conference on Modern Econometric Tools and Applications (Higher School of Economics, Moscow and Nizhny Novgorod, Russia) for helpful comments. Rustam Ibragimov gratefully acknowledges support provided by the Russian Foundation for Basic Research, Project No. 2001000960.

Conflict of interest: The authors state no conflict of interest.
Appendix
Excess Ripple return  

CAPM  3F  4F  5F  6F  
Mkt.RF 





[1.213]  [1.104]  [1.139]  [0.927]  [0.974]  
(

(

(

(

(


{0.154}  {

{

{0.111}  {


SMB  0.191  0.210 



[0.822]  [0.912]  [0.318]  [0.434]  
(0.221)  (0.234)  (

(


{

{

{

{


HML 





[0.068]  [0.254]  [

[


(

(

(

(


{

{

{

{


MOM  0.001  0.001  
[0.509]  [0.609]  
(0.141)  (0.255)  
{

{0.158}  
RMW 



[

[


(

(


{

{


CMA  1.562  1.589  
[1.180]  [1.217]  
(1.212)  (1.261)  
{1.610}  {1.701}  
Alpha  0.015  0.015  0.015  0.015  0.015 
[

[

[

[

[


(

(

(

(

(


{

{

{

{

{

The excess Ripple returns are defined as the difference between Ripple returns and the riskfree rate. The values in the first row of each stock factor are coefficients of each stock factor. The
Excess Ethereum return  

CAPM  3F  4F  5F  6F  
Mkt.RF 





[0.029]  [0.288]  [0.516]  [0.298]  [0.590]  
(

(

(

(

(


{0.005}  {

{

{

{


SMB 

0.093 



[

[

[

[


(

(0.120)  (

(


{0.163}  {

{0.116}  {


HML 





[

[0.009]  [

[


(

(

(

(


{

{

{

{


MOM  0.003  0.004  
[2.518]  [2.640]  
(0.634)  (0.720)  
{

{


RMW 



[

[


(

(


{

{


CMA  1.567  1.677  
[1.029]  [1.301]  
(1.142)  (1.181)  
{

{


Alpha  0.013  0.013  0.013  0.014  0.014 
[

[

[

[

[


(

(

(

(

(


{2.478}  {2.259}  {2.327}  {2.364}  {2.374} 
The excess Ethereum returns are defined as the difference between Ethereum returns and the riskfree rate. The values in the first row of each stock factor are coefficients of each stock factor. The
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