*3.1. ADCC-GARCH Model*

The aim of this study is to examine the dynamic linkages between bitcoin and various traditional financial assets at different frequencies and to explore the risk diversification, hedging and safe-haven properties of bitcoin for each asset based on the dynamic linkages between bitcoin and them. To capture the time-varying correlation between bitcoin and other traditional financial assets, we employed the DCC-GARCH approach. The DCC-GARCH method can estimate the time-varying conditional correlation coefficient and has the advantage of portraying the dynamic relationship between variables. The DCC model was first introduced by Engle [38] to allow for time-varying correlation between variables. Cappiello et al. [39] further introduced an asymmetric version of the DCC-GARCH (i.e., ADCC-GARCH) to address the effect of asymmetric information on time-varying correlations. In this study, the ADCC model of Cappiello et al. [39] was used to model the volatility dynamics and conditional correlation between bitcoin and other assets.

Let *rt* be a *n* × 1 vector of asset returns. The AR(1) process for *rt* conditioned on the information set *It*−<sup>1</sup> can be written as follows:

$$r\_t = \mu + \varrho r\_{t-1} + \varepsilon\_t \tag{1}$$

The residuals are modeled as:

$$
\varepsilon\_t = H\_t^{1/2} z\_t \tag{2}
$$

*Ht* is the conditional covariance matrix of *rt*, and *zt* is a *n* × 1 i.i.d. random vector of errors. Engle's [38] DCC model is estimated in two steps, with the GARCH parameters estimated in the first step and the conditional correlation in the second step, where:

$$H\_t = D\_t \mathcal{R}\_t D\_t \tag{3}$$

where *Ht* is a *n* × *n* conditional covariance matrix, *Rt* is the conditional correlation matrix, and *Dt* is the diagonal matrix with time-varying standard deviations on the diagonal.

$$D\_t = \operatorname{diag} \left( h\_{1,t}^{1/2}, \dots, h\_{n,t}^{1/2} \right) \tag{4}$$

$$R\_t = \operatorname{diag} \left( q\_{1,t}^{-1/2}, \dots, q\_{n,t}^{-1/2} \right) \mathbf{Q}\_t \operatorname{diag} \left( q\_{1,t}^{-1/2}, \dots, q\_{n,t}^{-1/2} \right) \tag{5}$$

The expression for *h* is a univariate GARCH. For the GARCH(1,1) model, the elements of *Ht* can be written as follows:

$$h\_{i,t} = \omega\_i + \kappa\_i \varepsilon\_{i,t-1}^2 + \beta\_i h\_{i,t-1} \tag{6}$$

*Qt* is a symmetric positive definite matrix that can be written in the following form:

$$Q\_t = (1 - \theta\_1 - \theta\_2)\overline{Q} + \theta\_1 z\_t z\_{t-1}' + \theta\_2 Q\_{t-1} \tag{7}$$

where *Q* is the *n* × *n* unconditional correlation matrix of the standardized residuals *zi*,*<sup>t</sup>* (*zi*,*<sup>t</sup>* = *εi*,*t*/ *hi*,*t*). The parameters *θ*<sup>1</sup> and *θ*<sup>2</sup> are non-negative and are related to the exponential smoothing process used to construct the dynamic conditional correlations. The DCC model is mean-reverting as long as *θ*<sup>1</sup> + *θ*<sup>2</sup> < 1. The correlation is estimated as:

$$\rho\_{i,j,t} = \frac{q\_{i,j,t}}{\sqrt{q\_{i,j,t}q\_{j,j,t}}} \tag{8}$$

Since the above DCC model does not allow for asymmetries and asset-specific news impact parameter, Cappiello et al. [39] developed the ADCC model to incorporate asymmetric effects and asset-specific news impact. For the ADDC model, the dynamics of *Q* is of the following form:

$$Q\_l = \left(\overline{Q} - A'\overline{Q}A - B'\overline{Q}B - G'\overline{Q}^-G\right) + A'z\_{t-1}z\_{t-1}'A + B'Q\_{t-1}B + G'z\_t'z\_t'^-G \tag{9}$$

where *A*, *B* and *G* are *n* × *n* parameter matrices and *z*<sup>−</sup> *<sup>t</sup>* is the zero-threshold standardized error, which is equal to *zt* when less than zero, and zero otherwise. *Q* and *Q*<sup>−</sup> are the unconditional matrices of *zt* and *z*<sup>−</sup> *<sup>t</sup>* , respectively.
