True or spurious long memory in the cryptocurrency markets: evidence from a multivariate test and other Whittle estimation methods

Abstract

This paper applies a new proposed multivariate score-type test against spurious long memory to a group of cryptocurrency market returns. The test statistic developed by Sibbertsen et al. (J Econ 203(1): 33–49, 2018) is based on the multivariate local Whittle likelihood function and is proven to be consistent against the alternative two cases of random level shifts and smooth trends. We apply the test to the returns, absolute returns, and modified absolute returns. Overall, the recently developed test statistic fails to reject the null hypothesis of true long memory for most cryptocurrencies, except for the Stellar market. Therefore, applying the new test statistic supports the argument that the long memory in the cryptocurrency markets is real and is not a spurious one. Our results are further supported by applying other consistent local Whittle methods that allow for the estimation of the memory parameter by accounting for the presence of perturbations or low-frequency contaminations.

Appendix: The local whittle estimator

The Whittle estimator is initially proposed by Künsch (1987) and then modified by Robinson (1995). It is used to estimate the long-memory parameter d. Ford ∈ (-1/2, 1/2), it is given by:

$$ \hat{d}_{{{\text{LW}}}} = \arg \min_{d} \left[ {\log \hat{G}(d) - \frac{2d}{m}\sum\limits_{j = 1}^{m} {\log \lambda_{j} } } \right] $$

where $\hat{G}(d) = \frac{1}{m}\sum\nolimits_{j = 1}^{m} {\lambda_{j}^{2d} I_{T} ,y(\lambda_{j} ),I_{T} ,y} (\lambda_{j} ) = (2\pi T)^{ - 1} \left| {\sum\nolimits_{t = 1}^{T} {y_{t} e^{{it\lambda_{j} }} } } \right|^{2}$ is the periodogram of the series y_t and $\lambda_{j} = (\frac{2\pi j}{T})$ represent the Fourier frequencies.

Hou and Perron (2014) suggest a modified version and call it the modified local Whittle estimator. The local Whittle estimator is consistent in the case of two scenarios: the presence or absence of mean shifts and is given by:

$$ \hat{d}_{mLW} = \arg \min_{d,\theta } \left[ {\log \hat{G}(d,\theta ) + \frac{1}{m}\sum\limits_{j = 1}^{m} {\log \lambda^{ - 2d}_{j} + \theta \lambda_{j}^{ - 2} /T} } \right], $$

where $\hat{G}(d,\theta ) = \frac{1}{m}\sum\nolimits_{j = 1}^{m} {\frac{{I_{T} ,y(\lambda_{j} )}}{{\lambda_{j}^{ - 2d} + \theta \lambda_{j}^{ - 2} /T}}}$ where θ reflects the scaling of the shifts. Hou and Perron (2014) prove the consistency and asymptotic normality of the modified local Whittle estimator for the case when m > T^5/9.

Frederiksen et al. (2012) fit the following two polynomials: $\log \phi_{y} \left( \lambda \right) \simeq \log G + h_{y} \left( {\theta_{y} ,\lambda } \right)$ and $\log \phi_{w} \left( \lambda \right) \simeq \log G + \log \theta_{p} + h_{w} \left( {\theta_{w} ,\lambda } \right)$ to approximate the logarithms of $\phi_{y} \left( \lambda \right)$ and $\phi_{w} \left( \lambda \right)$ in$f_{Z} \left( \lambda \right) = \phi_{y} \left( \lambda \right)\lambda^{ - 2d} + \phi_{w} \left( \lambda \right) + \phi_{u} \left( \lambda \right)\lambda^{ - 2} /T$. Here $\theta = \left( {\theta^{\prime}_{y} ,\theta_{p} ,\theta^{\prime}_{w} } \right)^{^{\prime}}$, $\theta_{p} = \phi_{w} \left( 0 \right)/\phi_{y} \left( 0 \right)$ is the long-run signal-to-noise ratio,${h}_{a}\left({\theta }_{a,}\lambda \right)=\sum_{r=1}^{{R}_{a}}{\theta }_{a,r}{\lambda }^{2r}$, and$a \epsilon \left\{y,w\right\}$. Then, ${\widehat{G}}_{\mathrm{LPWN}}=\frac{1}{m}\sum_{j=1}^{m}\frac{{\lambda }_{j}^{2d}{I}_{\mathcal{z}}({\lambda }_{j})}{{g}_{\mathrm{LPWN}}(d,\theta ,{\lambda }_{j})}$. Then, ${g}_{\mathrm{LPWN}}\left(d,\theta ,\lambda \right)=\mathrm{exp}\left({h}_{y}\left({\theta }_{y},\lambda \right)\right)+{\theta }_{\rho }{\lambda }^{2d}\mathrm{exp}({h}_{w}\left({\theta }_{w},\lambda \right))$ and the estimator is given by:

$$\left({\widehat{d}}_{\mathrm{LPWN}},\widehat{\theta }\right)=\mathop{\mathrm{arg min}}\limits_{\mathit{d\epsilon }\left[{d}_{1},{d}_{2}\right],\mathit{\theta \epsilon }\Theta }{R}_{\mathrm{LPWN}}(d,\theta ),$$

where $0<{d}_{1}<{d}_{2}<1$ is assumed to be stationary, and $\Theta$ is a compact and convex set in ${\mathbb{R}}^{{R}_{y}}\times (0,\infty )\times {\mathbb{R}}^{{R}_{w}}$