The role of cryptocurrencies within the financial systems has been expanding rapidly in recent years among investors and institutions. It is therefore crucial to investigate this phenomenon and develop statistical methods able to capture their interrelationships, the links with other global systems and, at the same time, the serial heterogeneity. Here we introduce hidden Markov regression models for jointly estimating quantiles and expectiles of cryptocurrency returns using regime-switching copulas. The proposed approach allows us to focus on extreme returns and describe their temporal evolution by introducing time-dependent coefficients, evolving according to a latent Markov chain. Moreover to model their time-varying dependence structure, we consider elliptical copula functions defined by state-specific parameters. Maximum likelihood estimates are obtained via an expectation-maximization algorithm. The empirical analysis investigates the relationship between daily returns of five cryptocurrencies and major world market indices.
AgostoA and CafferataA (2020) Financial bubbles: a study of co-explosivity in the cryptocurrency market. Risks, 8(2), 34.
2.
AkaikeH (1998) Information theory and an extension of the maximum likelihood principle. In Selected papers of Hirotugu Akaike, pp. 199–213. Springer.
3.
AlfòM, SalvatiN and RanallliMG (2017) Finite mixtures of quantile and M-quantile regression models. Statistics and Computing, 27(2), 547–70.
4.
AssafA, BhandariA, CharifH and DemirE (2022) Multivariate long memory structure in the cryptocurrency market: the impact of COVID-19. International Review of Financial Analysis, 82, 102132.
5.
BaumLE, PetrieT, SoulesG and WeissN (1970) A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The Annals of Mathematical Statistics, 41(1), 164–71.
6.
BaurDG and McDermottTK (2010) Is gold a safe haven? International evidence. Journal of Banking & Finance, 34(8), 1886–98.
7.
BernardiM, GayraudG and PetrellaL (2015) Bayesian tail risk interdependence using quantile regression. Bayesian Analysis, 10(3), 553–603.
8.
BiernackiC, CeleuxG and GovaertG (2000) Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(7), 719–25.
9.
BorriN (2019) Conditional tail-risk in cryptocurrency markets. Journal of Empirical Finance, 50, 1–19.
10.
Bouri E Jalkh N MolnárP and RoubaudD (2017a) Bitcoin for energy commodities before and after the December 2013 crash: diversifier, hedge or safe haven? Applied Economics, 49(50), 5063–73.
11.
BouriE, MolnárP, AzziG, RoubaudD and HagforsL.I (2017b) On the hedge and safe haven properties of Bitcoin: is it really more than a diversifier? Finance Research Letters, 20, 192–98.
12.
BouriE, LuceyB and RoubaudD (2020) Cryptocurrencies and the downside risk in equity investments. Finance Research Letters, 33, 101211.
13.
CaferraR and Vidal-TomásD (2021) Who raised from the abyss? A comparison between cryptocurrency and stock market dynamics during the COVID-19 pandemic. Finance Research Letters, 43, 101954.
14.
ChenZ (1996) Conditional Lp-quantiles and their application to the testing of symmetry in non-parametric regression. Statistics & Probability Letters, 29(2), 107–15.
15.
CinerC, LuceyB and YarovayaL (2022) Determinants of cryptocurrency returns: a lasso quantile regression approach. Finance Research Letters, 49, 102990.
16.
ConlonT and McGeeR (2020) Safe haven or risky hazard? Bitcoin during the COVID-19 bear market. Finance Research Letters, 35, 101607.
17.
CorbetS, MeeganA, LarkinC, LuceyB and YarovayaL (2018) Exploring the dynamic relationships between cryptocurrencies and other financial assets. Economics Letters, 165, 28–34.
18.
CorbetS, LarkinC and LuceyB (2020) The contagion effects of the COVID-19 pandemic: evidence from gold and cryptocurrencies. Finance Research Letters, 35, 101554.
19.
CremaschiniA, PunzoA, MartellucciE and MaruottiA (2022) On stylized facts of cryptocurrencies returns and their relationship with other assets, with a focus on the impact of COVID-19. Applied Economics, 55(32), 3675–88.
20.
DasD, Le RouxCL, JanaRK and DuttaA (2020) Does Bitcoin hedge crude oil implied volatility and structural shocks? A comparison with gold, commodity and the US dollar. Finance Research Letters, 36, 101335.
21.
EngleRF and ManganelliS (2004) CAViaR: conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics, 22(4), 367–81.
22.
FarcomeniA (2012) Quantile regression for longitudinal data based on latent Markov subject-specific parameters. Statistics and Computing, 22(1), 141–52.
23.
ForoniB, MerloL and PetrellaL (2024) Expectile hidden Markov regression models for analyzing cryptocurrency returns. Statistics and Computing, 34(2), 66.
24.
GerlachR and ChenCW (2015) Bayesian expected shortfall forecasting incorporating the intraday range. Journal of Financial Econometrics, 14(1), 128–58.
25.
GiudiciP and Abu HashishI (2020) A hidden Markov model to detect regime changes in cryptoasset markets. Quality and Reliability Engineering International, 36(6), 2057–65.
26.
JoeH and XuJ (1996) The estimation method of inference functions for margins for multivariate models. URL https://api.semanticscholar.org/CorpusID:55446837.
27.
KoenkerR (2005) Quantile regression. Cambridge University Press.
28.
KoenkerR and BassettG (1978) Regression quantiles. Econometrica: Journal of the Econometric Society, 46(1), 33–50.
29.
KoenkerR, ChernozhukovV, HeX and PengL (2017) Handbook of quantile regression. CRC Press.
30.
LiuC (1997) ML estimation of the multivariate t distribution and the EM algorithm. Journal of Multivariate Analysis, 63(2), 296–312.
31.
MarianaCD, EkaputraIA and HusodoZA (2021) Are Bitcoin and Ethereum safe-havens for stocks during the COVID-19 pandemic? Finance research letters, 38, 101798.
32.
MaruottiA and PunzoA (2021) Initialization of hidden Markov and semi-Markov models: a critical evaluation of several strategies. International Statistical Review, 89(3), 447–80.
33.
MaruottiA, PunzoA and BagnatoL (2019) Hidden Markov and semi-Markov models with multivariate leptokurtic-normal components for robust modeling of daily returns series. Journal of Financial Econometrics, 17(1), 91–117.
34.
MaruottiA, PetrellaL and SpositoL (2021). Hidden semi-Markov-switching quantile regression for time series. Computational Statistics & Data Analysis, 159, 107208.
35.
MerloL, PetrellaL and RaponiV (2021) Forecasting VaR and ES using a joint quantile regression and its implications in portfolio allocation. Journal of Banking & Finance, 133, 106248.
36.
MerloL, MaruottiA, PetrellaL and PunzoA (2022) Quantile hidden semi-Markov models for multivariate time series. Statistics and Computing, 32(4), 1–22.
37.
NeweyWK and PowellJL (1987) Asymmetric least squares estimation and testing. Econometrica: Journal of the Econometric Society, 55(4), 819–47.
38.
NigriA, BarbiE and LevantesiS (2022) The relationship between longevity and lifespan variation. Statistical Methods & Applications, 31(3), 481–93.
39.
NystrupP, MadsenH and LindströmE (2017) Long memory of financial time series and hidden Markov models with time-varying parameters. Journal of Forecasting, 36(8),989–1002.
40.
ÖttingM, LangrockR and MaruottiA (2021) A copula-based multivariate hidden Markov model for modelling momentum in football. AStA Advances in Statistical Analysis, 107(1), 9–27.
41.
PennoniF, BartolucciF, ForteG and AmetranoF (2022) Exploring the dependencies among main cryptocurrency log-returns: a hidden Markov model. Economic Notes, 51(1), e12193.
42.
SchwarzG, . (1978) Estimating the dimension of a model. The Annals of Statistics, 6(2), 461–64.
43.
ShuM, SongR and ZhuW (2021) The 2021 bitcoin bubbles and crashes-detection and classification. Stats, 4(4), 950–70.
44.
SklarM (1959) Fonctions de repartition an dimensions et leurs marges. Publications de l'Institut de statistique de l'Université de Paris, 8, 229–31.
45.
TzavidisN, SalvatiN, SchmidT, FlouriE and MidouhasE (2016) Longitudinal analysis of the strengths and difficulties questionnaire scores of the Millennium Cohort Study children in England using M-quantile random-effects regression. Journal of the Royal Statistical Society: Series A (Statistics in Society), 179(2), 427–52.
46.
VisserI, RaijmakersME and MolenaarPC (2000) Confidence intervals for hidden Markov model parameters. British Journal of Mathematical and Statistical Psychology, 53(2), 317–27.
WaldmannE., SobotkaF. and KneibT. (2017). Bayesian regularisation in geoadditive expectile regression. Statistics and Computing, 27(6), 1539–53.
49.
WelchLR (2003) Hidden Markov models and the Baum-Welch algorithm. IEEE Information Theory Society Newsletter, 53(4), 10–13.
50.
WhiteH, KimT.-H. and ManganelliS. (2015). VAR for VaR: measuring tail dependence using multivariate regression quantiles. Journal of Econometrics, 187(1), 169–88.
51.
XiongJ, LiuQ and ZhaoL (2020) A new method to verify Bitcoin bubbles: based on the production cost. The North American Journal of Economics and Finance, 51, 101095.
52.
YousafI and AliS. (2021). Linkages between stock and cryptocurrency markets during the COVID-19 outbreak: an intraday analysis. The Singapore Economic Review, 1–20.
53.
YuK and MoyeedRA (2001) Bayesian quantile regression. Statistics & Probability Letters, 54(4), 437–47.
54.
ZucchiniW, MacDonaldIL and LangrockR (2016) Hidden Markov models for time series: an introduction using R.Chapman and Hall/CRC.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
