*2.1. Estimating Volatility*

We incorporate two alternatives to calculate the daily volatility. The first model belongs to the family of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models originating from Engle (1982). Within the GARCH framework, the process for returns *rt* is formulated as

$$r\_t = \mu + \varepsilon\_{t\prime}$$

$$\varepsilon\_t = \sqrt{h\_t} z\_{t\prime}$$

where *μ* denotes the mean, the conditional variance is defined as *ht* = V (*rt*|F*<sup>t</sup>*−<sup>1</sup>), and the random variable *zt* follows a Skewed Student's-*t* distribution<sup>4</sup> with *zt* ∼ SkSt*<sup>ν</sup>*,*<sup>ξ</sup>* (0, 1) i.i.d. for all *t* = 1, ... , *n* (Hansen 1994). Here, F*<sup>t</sup>*−<sup>1</sup> is a sigma algebra containing all past information of returns and conditional volatilities up to time *t* − 1.

The distributional parameter for the Skewed Student's-*t* distribution, the degrees-of-freedom *ν* and the skewness *ξ*, are estimated along with the model parameters. For the conditional variance *ht*, we consider the GARCH(1,1) specification of Bollerslev (1986), which reads:

$$
\hbar h\_t = \omega + a\varepsilon\_{t-1}^2 + \beta h\_{t-1}.\tag{1}
$$

A well-known characteristic of volatility is the negative correlation with returns, also known as the leverage effect (Black 1976; Christie 1982). In order to cope with this stylized fact, we implement the Asymmetric Power ARCH (APARCH, (Ding et al. 1993)), which is defined as:

$$h\_t^{\frac{\delta}{2}} = \omega + \mathfrak{a} \left( |\varepsilon\_{t-1}| - \gamma \varepsilon\_{t-1} \right)^{\delta} + \beta h\_{t-1}^{\frac{\delta}{2}} \tag{2}$$

where *γ* ∈ (−1, 1) refers to the leverage parameter indicating whether negative or positive shocks have a larger impact on the daily volatility. For example, an estimated *γ* > 0 reveals that negative residuals increase the conditional volatility more than their positive equivalents, which is of particular interest for shocks.

We include the Fractionally Integrated GARCH (FIGARCH, (Baillie et al. 1996)) to cover the long memory effect. The standard FIGARCH(1,*d*,1) reads:

$$\begin{split} h\_{t} &= \frac{\omega}{1-\beta} + \left( 1 - \frac{\left(1 - \phi L\right)\left(1 - L\right)^{d}}{1 - \beta L} \right) \varepsilon\_{t}^{2} \\ &= \frac{\omega}{1-\beta} + \sum\_{i=1}^{\infty} \lambda\_{i} \varepsilon\_{t-i\prime}^{2} \end{split} \tag{3}$$

where

$$\begin{aligned} \lambda\_1 &= \phi - \beta - d, \\ \lambda\_i &= \beta \lambda\_{i-1} + \left(\frac{i-1-d}{i} - \phi\_1\right) \left(\frac{(i-2-d)!}{i!(1-d)!}\right), \end{aligned} \tag{4}$$

<sup>4</sup> The assumption of Skewed Student's-*t* distributed errors is justified in the Data section.

with the long memory parameter *d* and the lag operator *L*. To combine the leverage and the long memory effect, Tse (1998) proposed the Fractionally Integrated APARCH (FIAPARCH):

$$\begin{split} h\_t^{\frac{\delta}{2}} &= \frac{\omega}{1 - \beta\_1} + \left( 1 - \frac{\left( 1 - \phi\_1 L \right) \left( 1 - L \right)^d}{1 - \beta\_1 L} \right) \left( |\varepsilon\_t| - \gamma \varepsilon\_t \right)^\delta, \\ &= \frac{\omega}{1 - \beta\_1} + \sum\_{i=1}^\infty \lambda\_i \left( |\varepsilon\_{t-i}| - \gamma \varepsilon\_{t-i} \right)^\delta. \end{split} \tag{5}$$

Note that the ARCH(∞) representation in Equations (3) and (5) is carried out using the fast fractional differencing method of Klein and Walther (2017) with truncation lag 5000. All GARCH-type models introduced above are estimated with maximum-likelihood estimations (MLE), ensuring that non-negativity and stationarity conditions, if applicable, hold for each model. All parameter estimates and robust standard errors following Bollerslev and Wooldridge (1992) are available upon request.

As an alternative to the GARCH framework, we also consider the stochastic volatility framework. Stochastic volatility models belong to the family of state-space models (Sarkka 2013, ch. 4). The standard stochastic volatility (SV) model is introduced by Taylor (1986) as

$$
\tau\_t = \mu + \sqrt{h\_t} z\_{t,\*} \tag{6}
$$

$$
\log h\_t = a + \beta \log h\_{t-1} + \sigma \eta\_{t\prime} \tag{7}
$$

$$
\begin{pmatrix} z\_t \\ \eta\_t \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) \text{i.i.d. } t = 1, \ldots, n. \tag{8}
$$

The SV model contains two noise processes, {*zt*}*t* and {*ηt*}*<sup>t</sup>*, respectively accounting for the return shocks and the volatility shocks. In the SV model above, {*zt*}*t* and {*ηt*}*t* are independent.

Harvey and Shephard (1996) introduce a more general setting where the noise processes {*zt*}*t* and {*ηt*}*t* are correlated as

$$
\begin{pmatrix} z\_t \\ \eta\_{t+1} \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \right) \cdot \mathcal{N}
$$

instead of being independent as in Equation (8). The correlation coefficient *ρ* accounts for the leverage effect, defined as the negative correlation between shocks on return and volatility (i.e., *ρ* < 0). This model is called the asymmetric SV model or SV-leverage (SV-L) model.

We consider a third stochastic volatility model where the return shocks {*zt*}*t* follow Student's *t*-distribution with *ν* degrees of freedom. It allows more extreme observations than with Gaussian return shocks as the Student's *t*-distribution has heavier tails. The volatility shocks {*ηt*}*t* follow the standard Gaussian distribution. In this model, *zt* and *ηt* are independent. It is referred to as the SV-*t* model.

We end up with three different stochastic volatility models: the SV model (Gaussian and independent shocks), the SV-L model (Gaussian and correlated shocks), and the SV-*t* model (*t*-distributed return shock, Gaussian volatility shock, independent shocks). In the three models, the parameters are estimated by Bayesian inference using the Markov Chain Monte Carlo (MCMC) sampling algorithms from Chan and Grant (2016).<sup>5</sup>

Lastly, we consider the RiskMetrics approach, the historical simulation, as well as the semi-parametric filtered historical simulation, which we explain in detail in the next subsection.

<sup>5</sup> We are thankful to Joshua Chan for providing the MatLab (MathWorks, Natick, Massachusetts, United States) code for estimating the stochastic volatility models on his personal webpage joshuachan.org.
