DCS models with local level and seasonality
The DCS models of this paper are formulated as: \(p_{t}=\mu _{t}+s_{t}+v_{t}=\mu _{t}+s_{t}+\exp (\lambda _{t})\epsilon _{t}\) for days \(t=1,\ldots ,T\), where T is the number of observations. The model includes three score-driven components: stochastic local level component \(\mu _{t}\), stochastic annual seasonality component \(s_{t}\), and irregular component \(v_{t}\). The irregular component is the product of a dynamic scale parameter \(\exp (\lambda _{t})\) and a standardized error term \(\epsilon _{t}\).
For \(\epsilon _{t}\), we use the Student’s-t, Skew-Gen-t, EGB2 and NIG distributions (we present the corresponding density functions in Sect. 3.3). For these probability distributions, the updating terms of the DCS equations either trim or Winsorize the extreme observations, or transform them according to a linear function. Due to these transformations, the DCS models for the GTQ/USD currency exchange rate of the present paper are robust to extreme observations.
Firstly, the local level component \(\mu _{t}=\mu _{t-1}+\delta u_{\mu ,t-1}\) is updated by the scaled score function \(u_{\mu ,t}\) with respect to \(\mu _{t}\) (\(u_{\mu ,t}\) is defined in Sect. 3.3). We initialize \(\mu _{t}\) by using the first observation \(p_{1}\). As an alternative, we also consider the use of parameter \(\mu _{0}\) to initialize \(\mu _{t}\). We obtain very similar results for both cases, thus, in this paper we only report results for \(\mu _{1}=p_{1}\). With respect to these alternatives of initialization, we refer to the work of Harvey (2013, p. 76).
Secondly, the annual seasonality component is \(s_{t}=D'_{t}\rho _{t}=(D_{\text {Jan},t},D_{\text {Feb},t},\ldots ,D_{\text {Dec},t})'\rho _{t}\), where the monthly dummies \(D_{j,t}\) with \(j\in \{\text {Jan},\ldots ,\text {Dec}\}\) select an element from the \(12 \times 1\) vector of dynamic variables \(\rho _{t}\). The vector \(\rho _{t}\) is formulated as \(\rho _{t}=\rho _{t-1}+\gamma _{t} u_{\mu ,t-1}\). Vector \(\rho _{t}\) is updated by the scaled score function \(u_{\mu ,t}\) with respect to \(\mu _{t}\) (Sect. 3.3), and \(u_{\mu ,t}\) is multiplied by the \(12 \times 1\) vector of dynamic parameters \(\gamma _{t}\). Each element of the \(\gamma _{t}\) vector is given by \(\gamma _{jt}=\gamma _{j}\) for \(D_{jt}=1\) and \(\gamma _{jt}=-\gamma _{j}/(12-1)\) for \(D_{jt}=0\), where \(\gamma _{j}\) with \(j\in \{\text {Jan},\ldots ,\text {Dec}\}\) are seasonality parameters to be estimated. This specification ensures that the sum of the seasonality parameters is zero, hence, \(s_{t}\) has mean zero and it is effectively separated from \(\mu _{t}\).
We initialize \(\rho _{t}\) by estimating the equation \(p_{t}=a+b t+c_{\text {Jan}} D_{\text {Jan},t}+\cdots +c_{\text {Dec}} D_{\text {Dec},t}+\epsilon _{t}\), under the restriction \(c_{\text {Jan}}+\cdots +c_{\text {Dec}}=0\). Due to this restriction multicollinearity is avoided, thus, all parameters are identified in the equation. For the estimation, we use data from the first year of the full data window (i.e. the first 259 observations of the GTQ/USD exchange rate sample from year 1994), and we estimate the parameters by using the Non-linear Least Squares (NLS) method. The initial values of \(\rho _{t}\) are the NLS estimates of \((c_{\text {Jan}},\ldots ,c_{\text {Dec}})'\). With respect to this method of initialization, we refer to the work of Harvey (2013, p. 80).
In the DCS models of this paper, the same scaled score function updates both the local level and the seasonality components. Therefore, the local level and seasonality shocks are correlated. The DCS models with stochastic local level and stochastic seasonality of this paper are alternatives to the recent Unobserved Components Model (UCM) of Hindrayanto et al. (2018) that uses correlated shocks for the local level and seasonality components.
Thirdly, we model the time-varying scale of the irregular component \(v_{t}\) by using the DCS-EGARCH(1,1) model \(\lambda _{t}=\omega +\beta \lambda _{t-1}+\alpha u_{\lambda ,t-1}\), which is updated by the score function \(u_{\lambda ,t}\) with respect to \(\lambda _{t}\) (\(u_{\lambda ,t}\) is defined in Sect. 3.3). DCS-EGARCH models for the Student’s-t, Skew-Gen-t, EGB2 and NIG distributions are named Beta-t-EGARCH (Harvey and Chakravarty 2008), Skew-Gen-t-EGARCH (Harvey and Lange 2017), EGB2-EGARCH (Caivano and Harvey 2014) and NIG-EGARCH (Blazsek et al. 2018c), respectively. We initialize \(\lambda _{t}\) by using parameter \(\lambda _{0}\). As an alternative, we also consider DCS-EGARCH with leverage effects (Harvey 2013). However, we find that the parameter that measures leverage effects is not significantly different from zero for the GTQ/USD dataset that is used in this paper.
Motivated by the works of Dacorogna et al. (1993) and Andersen and Bollerslev (1998), we also consider a seasonality component in volatility. We add the seasonality component \({\tilde{s}}_{t}\) into scale, as follows: \(p_{t}=\mu _{t}+s_{t}+v_{t}=\mu _{t}+s_{t}+\exp (\lambda _{t}+{\tilde{s}}_{t})\epsilon _{t}\), where we use the Student’s t, Skew-Gen-t, EGB2 and NIG distributions for \(\epsilon _{t}\). The seasonality component is specified as \({\tilde{s}}_{t}=D'_{t}{\tilde{\rho }}_{t}=(D_{\text {Jan},t},D_{\text {Feb},t},\ldots ,D_{\text {Dec},t})'{\tilde{\rho }}_{t}\) and \({\tilde{\rho }}_{t}={\tilde{\rho }}_{t-1}+\kappa _{t} u_{\lambda ,t-1}\). Each element of the \(\kappa _{t}\) vector is given by \(\kappa _{jt}=\kappa _{j}\) for \(D_{jt}=1\) and \(\kappa _{jt}=-\kappa _{j}/(12-1)\) for \(D_{jt}=0\), where \(\kappa _{j}\) with \(j\in \{\text {Jan},\ldots ,\text {Dec}\}\) are parameters to be estimated. This specification ensures that the sum of the seasonality parameters is zero, hence, \({\tilde{s}}_{t}\) has mean zero and it is effectively separated from \(\lambda _{t}\). We do not report results for these extended DCS specifications with seasonal volatility, because the ML estimator does not converge to an optimum for the GTQ/USD currency exchange rate dataset of the present paper. Nevertheless, this specification may be helpful in future applications for currency exchange rates involving stochastic seasonality and extreme observations.
Standard financial time series model with local level and seasonality
The standard financial time series model is formulated as: \(p_{t}=\mu _{t}+s_{t}+v_{t}=\mu _{t}+s_{t}+\lambda ^{1/2}_{t}\epsilon _{t}\) for days \(t=1,\ldots ,T\). We use the same notation for the local level, seasonality and irregular components as for the DCS models, and for the error term we use \(\epsilon _{t} \sim N(0,1)\).
Firstly, the local level component is \(\mu _{t}=\mu _{t-1}+\delta v_{t-1}\) (our motivation for using this updating term is outlined in Sect. 3.3). We initialize \(\mu _{t}\) by using the first observation \(p_{1}\). As an alternative, we also consider the use of parameter \(\mu _{0}\) to initialize \(\mu _{t}\). We obtain very similar results for both cases, thus, in this paper we only report results for \(\mu _{1}=p_{1}\).
Secondly, the annual seasonality component is \(s_{t}=D'_{t}\rho _{t}=(D_{\text {Jan},t},D_{\text {Feb},t},\ldots ,D_{\text {Dec},t})'\rho _{t}\), where the monthly dummies \(D_{j,t}\) with \(j\in \{\text {Jan},\ldots ,\text {Dec}\}\) select an element from the \(12 \times 1\) vector of dynamic variables \(\rho _{t}\). The vector \(\rho _{t}\) is formulated as \(\rho _{t}=\rho _{t-1}+\gamma _{t} v_{t-1}\) (Sect. 3.3). We initialize \(\rho _{t}\) in the same way as for the DCS models. Each element of the \(\gamma _{t}\) vector is given by \(\gamma _{jt}=\gamma _{j}\) for \(D_{jt}=1\) and \(\gamma _{jt}=-\gamma _{j}/(12-1)\) for \(D_{jt}=0\), where \(\gamma _{j}\) with \(j\in \{\text {Jan},\ldots ,\text {Dec}\}\) are seasonality parameters to be estimated. This specification ensures that \(s_{t}\) is centred at zero.
Thirdly, we model the conditional variance of \(v_{t}\) by using the classic GARCH(1,1) specification \(\lambda _{t}=\omega +\beta \lambda _{t-1}+\alpha v_{t-1}^{2}\). We initialize \(\lambda _{t}\) by using parameter \(\lambda _{0}\).
Conditional densities, score functions and updating terms
We use four probability distributions for \(\epsilon _{t}\) in the DCS models. In this section, for each alternative, we present the log conditional density of \(p_{t}\), and the score functions \(u_{\mu ,t}\) and \(u_{\lambda ,t}\). Furthermore, in this section we also present the log conditional density of \(p_{t}\) and the properties of the updating terms of \(\mu _{t}\) and \(\lambda _{t}\) for the standard financial time series model.
Firstly, \(\epsilon _{t} \sim t[0,1,\exp (\nu )+2]\) is the Student’s t-distribution, where \(\nu \in {\mathbb {R}}\) influences tail-thickness. The degrees of freedom \(\exp (\nu )+2\) parameter specification ensures finite conditional variance for \(p_{t}\). The log conditional density of \(p_{t}\) is
$$\begin{aligned} \ln f(p_{t}|p_{1},\ldots ,p_{t-1})= & {} \ln \Gamma \left[ \frac{\exp (\nu )+3}{2}\right] -\ln \Gamma \left[ \frac{\exp (\nu )+2}{2}\right] \nonumber \\&-\frac{\ln (\pi )+\ln [\exp (\nu )+2]}{2}-\lambda _{t} \nonumber \\&-\frac{\exp (\nu )+3}{2}\ln \left\{ 1+\frac{\epsilon _{t}^{2}}{\exp (\nu )+2}\right\} \end{aligned}$$
(1)
where \(\Gamma (x)\) is the gamma function. The score function with respect to \(\mu _{t}\) is given by
$$\begin{aligned} \frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \mu _{t}}= & {} \frac{[\exp (\nu )+2]\exp (\lambda _{t})\epsilon _{t}}{\epsilon _{t}^{2}+\exp (\nu )+2}\times \frac{\exp (\nu )+3}{[\exp (\nu )+2]\exp (2\lambda _{t})}\nonumber \\= & {} u_{\mu ,t} \times \frac{\exp (\nu )+3}{[\exp (\nu )+2]\exp (2\lambda _{t})} \end{aligned}$$
(2)
where the scaled score function \(u_{\mu ,t}\) is defined according to the last equality. The \(u_{\mu ,t}\) term trims extreme observations, because \(u_{\mu ,t} \rightarrow _{p} 0\) when \(|\epsilon _{t}| \rightarrow \infty \). The discounting that is undertaken by \(u_{\mu ,t}\) is identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results). The updating term \(v_{t}\) that is used in the local level and stochastic seasonality equations of the standard financial time series model is a limiting special case of the scaled score function \(u_{\mu ,t}\), because:
$$\begin{aligned} u_{\mu ,t}=\frac{[\exp (\nu )+2]\exp (\lambda _{t})\epsilon _{t}}{\epsilon _{t}^{2}+\exp (\nu )+2}= \frac{\exp (\lambda _{t})\epsilon _{t}}{[\exp (\nu )+2]^{-1}\epsilon _{t}^{2}+1} \longrightarrow _{p} \exp (\lambda _{t})\epsilon _{t}=v_{t} \end{aligned}$$
(3)
as \(\nu \rightarrow \infty \). Related to this, we also note that under the same limit \(t[0,1,\exp (\nu )+2] \rightarrow _{d} N(0,1)\), i.e. the standardized error term of the standard financial time series model is obtained. The score function \(u_{\lambda ,t}\) is
$$\begin{aligned} u_{\lambda ,t}=\frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \lambda _{t}} =\frac{[\exp (\nu )+3]\epsilon _{t}^{2}}{\exp (\nu )+2+\epsilon _{t}^{2}}-1 \end{aligned}$$
(4)
The updating term \(u_{\lambda ,t}\) Winsorizes extreme observations, because \(u_{\lambda ,t} \rightarrow _{p} c\) (\(c>0\) is a real number) when \(|\epsilon _{t}| \rightarrow \infty \). The discounting that is undertaken by \(u_{\lambda ,t}\) is identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results). We also show the limiting case for \(u_{\lambda ,t}\) when \(\nu \rightarrow \infty \):
$$\begin{aligned} u_{\lambda ,t}=\frac{[\exp (\nu )+3]\epsilon _{t}^{2}}{\exp (\nu )+2+\epsilon _{t}^{2}}-1 \longrightarrow _{p} \epsilon _{t}^{2}-1=\exp (-2\lambda _{t})v_{t}^{2}-1 \end{aligned}$$
(5)
as \(\nu \rightarrow \infty \). The last equality shows that \(u_{\lambda ,t}\) performs a quadratic transformation of \(v_{t}\) for the limiting case, as per the conditional variance equation in the standard financial time series model.
Secondly, \(\epsilon _{t} \sim \text {Skew-Gen-}t[0,1,\text {tanh}(\tau ),\exp (\nu )+2,\exp (\eta )]\) (McDonald and Michelfelder 2017), where \(\text {tanh}(x)\) is the hyperbolic tangent function, and \(\tau \in {\mathbb {R}}\), \(\nu \in {\mathbb {R}}\) and \(\eta \in {\mathbb {R}}\) influence the asymmetry, tail-thickness and peakedness, respectively. The Skew-Get-t distribution is a generalization of the Student’s t distribution. By setting \(\text {tanh}(\tau )=0\) and \(\exp (\eta )=2\), Skew-Get-t coincides with Student’s t. The degrees of freedom \(\exp (\nu )+2\) specification ensures finite conditional variance for \(p_{t}\), as for the Student’s t distribution. The log-density of \(p_{t}\) is
$$\begin{aligned}&\ln f(p_{t}|p_{1},\ldots ,p_{t-1}) = \eta -\lambda _{t}-\ln (2)-\frac{\ln [\exp (\nu )+2]}{\exp (\eta )}-\ln \Gamma \left[ \frac{\exp (\nu )+2}{\exp (\eta )}\right] \nonumber \\&\qquad -\ln \Gamma [\exp (-\eta )]+\ln \Gamma \left[ \frac{\exp (\nu )+3}{\exp (\eta )}\right] \nonumber \\&\qquad -\frac{\exp (\nu )+3}{\exp (\eta )}\ln \left\{ 1+\frac{|\epsilon _{t}|^{\exp (\eta )}}{[1+\text {tanh}(\tau )\text {sgn}(\epsilon _{t})]^{\exp (\eta )} \times [\exp (\nu )+2]} \right\} \end{aligned}$$
(6)
where \(\text {sgn}(x)\) is the signum function. The score function with respect to \(\mu _{t}\) is given by
$$\begin{aligned}&\frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \mu _{t}}=\frac{[\exp (\nu )+2]\exp (\lambda _{t})\epsilon _{t}|\epsilon _{t}|^{\exp (\eta )-2}}{|\epsilon _{t}|^{\exp (\eta )}+ [1+\text {tanh}(\tau )\text {sgn}(\epsilon _{t})]^{\exp (\eta )}[\exp (\nu )+2]}\nonumber \\&\quad \quad \quad \times \frac{\exp (\nu )+3}{[\exp (\nu )+2]\exp (2\lambda _{t})}=u_{\mu ,t} \times \frac{\exp (\nu )+3}{[\exp (\nu )+2]\exp (2\lambda _{t})} \end{aligned}$$
(7)
where the scaled score function \(u_{\mu ,t}\) is defined according to the second equality. The \(u_{\mu ,t}\) term trims extreme observations, because \(u_{\mu ,t} \rightarrow _{p} 0\) when \(|\epsilon _{t}| \rightarrow \infty \). The discounting that is undertaken by \(u_{\mu ,t}\) is not identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results). The score function \(u_{\lambda ,t}\) is
$$\begin{aligned} u_{\lambda ,t}= & {} \frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \lambda _{t}}\nonumber \\= & {} \frac{|\epsilon _{t}|^{\exp (\eta )}[\exp (\nu )+3]}{|\epsilon _{t}|^{\exp (\eta )}+[1+\text {tanh}(\tau )\text {sgn}(\epsilon _{t})]^{\exp (\eta )}[\exp (\nu )+2]}-1 \end{aligned}$$
(8)
The updating term \(u_{\lambda ,t}\) Winsorizes extreme observations, because \(u_{\lambda ,t} \rightarrow _{p} c_{1}\) when \(\epsilon _{t} \rightarrow -\infty \) and \(u_{\lambda ,t} \rightarrow _{p} c_{2}\) when \(\epsilon _{t} \rightarrow +\infty \) (\(c_{1}>0\) and \(c_{2}>0\) are real numbers). The Winsorizing that is undertaken by \(u_{\lambda ,t}\) is not identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results).
Thirdly, \(\epsilon _{t} \sim \text {EGB2}[0,1,\exp (\xi ),\exp (\zeta )]\), where \(\xi \in {\mathbb {R}}\) and \(\zeta \in {\mathbb {R}}\) influence both asymmetry and tail-thickness. The log conditional density of \(p_{t}\) is
$$\begin{aligned}&\ln f(p_{t}|p_{1},\ldots ,p_{t-1})= \exp (\xi )\epsilon _{t} -\lambda _{t} -\ln \Gamma [\exp (\xi )]-\ln \Gamma [\exp (\zeta )]\nonumber \\&\quad +\ln \Gamma [\exp (\xi )+\exp (\zeta )]-[\exp (\xi )+\exp (\zeta )]\ln [1+\exp (\epsilon _{t})] \end{aligned}$$
(9)
The score function with respect to \(\mu _{t}\) is given by
$$\begin{aligned} \frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \mu _{t}}= u_{\mu ,t} \times \{\Psi ^{(1)}[\exp (\xi )]+\Psi ^{(1)}[\exp (\zeta )]\}\exp (2\lambda _{t}) \end{aligned}$$
(10)
where the scaled score function \(u_{\mu ,t}\) is defined as:
$$\begin{aligned} u_{\mu ,t}= & {} \{\Psi ^{(1)}[\exp (\xi )] +\Psi ^{(1)}[\exp (\zeta )]\}\exp (\lambda _{t}) \nonumber \\&\left\{ [\exp (\xi )+\exp (\zeta )]\frac{\exp (\epsilon _{t})}{\exp (\epsilon _{t})+1}-\exp (\xi )\right\} \end{aligned}$$
(11)
where \(\Psi ^{(1)}(x)\) is the trigamma function. The \(u_{\mu ,t}\) term Winsorizes extreme observations, because \(u_{\mu ,t} \rightarrow _{p} c_{1}\) when \(\epsilon _{t} \rightarrow -\infty \) and \(u_{\mu ,t} \rightarrow _{p} c_{2}\) when \(\epsilon _{t} \rightarrow +\infty \) (\(c_{1}<0\) and \(c_{2}>0\) are a real numbers). The discounting that is undertaken by \(u_{\mu ,t}\) is not identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results). The score function \(u_{\lambda ,t}\) is
$$\begin{aligned} u_{\lambda ,t}=\frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \lambda _{t}}= [\exp (\xi )+\exp (\zeta )]\frac{\epsilon _{t}\exp (\epsilon _{t})}{\exp (\epsilon _{t})+1} -\exp (\xi )\epsilon _{t}-1 \end{aligned}$$
(12)
The updating term \(u_{\lambda ,t}\) transforms extreme observations according to a linear increasing function, because \(u_{\lambda ,t} \rightarrow _{p} \infty \) in a linear manner when \(|\epsilon _{t}| \rightarrow \infty \). The linear transformation that is undertaken by \(u_{\lambda ,t}\) is not identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results).
Fourthly, \(\epsilon _{t}\sim \text {NIG}[0,1,\exp (\nu ),\exp (\nu )\text {tanh}(\eta )]\), where \(\nu \in {\mathbb {R}}\) and \(\eta \in {\mathbb {R}}\) influence tail-thickness and asymmetry, respectively. The log conditional density of \(p_{t}\) is
$$\begin{aligned}&\ln f(p_{t}|p_{1},\ldots ,p_{t-1})= \nu -\lambda _{t}-\ln (\pi )+\exp (\nu )[1-\text {tanh}^{2}(\eta )]^{1/2}\nonumber \\&\quad +\exp (\nu )\text {tanh}(\eta )\epsilon _{t} +\ln K^{(1)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] -\frac{1}{2}\ln \big (1+\epsilon _{t}^{2}\big ) \end{aligned}$$
(13)
where \(K^{(1)}(x)\) is the modified Bessel function of the second kind of order 1. The score function with respect to \(\mu _{t}\) is given by
$$\begin{aligned}&\frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \mu _{t}} = -\exp (\nu -\lambda _{t})\text {tanh}(\eta )+\frac{\epsilon _{t}}{\exp (\lambda _{t})\big (1+\epsilon _{t}^{2}\big )}\nonumber \\&\qquad +\frac{\exp (\nu -\lambda _{t})\epsilon _{t}}{\sqrt{1+\epsilon _{t}^{2}}}\times \frac{K^{(0)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] +K^{(2)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] }{2K^{(1)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] } \qquad \end{aligned}$$
(14)
and the scaled score function \(u_{\mu ,t}\) is defined as:
$$\begin{aligned} u_{\mu ,t}=\frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \mu _{t}} \times \exp (2\lambda _{t}) \end{aligned}$$
(15)
where \(K^{(0)}(x)\) and \(K^{(2)}(x)\) are the modified Bessel functions of the second kind of orders 0 and 2, respectively. The \(u_{\mu ,t}\) term Winsorizes extreme observations, because \(u_{\mu ,t} \rightarrow _{p} c_{1}\) when \(\epsilon _{t} \rightarrow -\infty \) and \(u_{\mu ,t} \rightarrow _{p} c_{2}\) when \(\epsilon _{t} \rightarrow +\infty \) (\(c_{1}<0\) and \(c_{2}>0\) are real numbers). The discounting that is undertaken by \(u_{\mu ,t}\) is not identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results). The score function \(u_{\lambda ,t}\) is
$$\begin{aligned} u_{\lambda ,t}= & {} \frac{\partial \ln f(p_{t}|p_{1},\ldots ,p_{t-1})}{\partial \lambda _{t}} =-1-\exp (\nu )\text {tanh}(\eta )\epsilon _{t} +\frac{\epsilon _{t}^{2}}{1+\epsilon _{t}^{2}}\nonumber \\&+\frac{\exp (\nu )\epsilon _{t}^{2}}{\sqrt{1+\epsilon _{t}^{2}}}\times \frac{K^{(0)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] +K^{(2)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] }{2K^{(1)}\left[ \exp (\nu )\sqrt{1+\epsilon _{t}^{2}}\right] } \end{aligned}$$
(16)
The updating term \(u_{\lambda ,t}\) transforms extreme observations according to a linear increasing function, because \(u_{\lambda ,t} \rightarrow _{p} \infty \) in a linear manner when \(|\epsilon _{t}| \rightarrow \pm \infty \). The linear transformation that is undertaken by \(u_{\lambda ,t}\) is not identical for the positive and negative sides of the probability distribution (see Sect. 5 for empirical results).
Finally, we also present the log-density of \(p_{t}\) for the standard financial time series model:
$$\begin{aligned} \ln f(p_{t}|p_{1},\ldots ,p_{t-1})= -\frac{1}{2}\ln (2\pi \lambda _{t})-\frac{1}{2}\epsilon _{t}^{2} \end{aligned}$$
(17)
The updating terms of the equations \(\mu _{t}=\mu _{t-1}+\delta v_{t-1}\) and \(\lambda _{t}=\omega +\beta \lambda _{t-1}+\alpha v_{t-1}^{2}\) perform linear and quadratic transformations of \(\epsilon _{t}\), respectively. For both \(\mu _{t}\) and \(\lambda _{t}\), the updating terms go to infinity when \(|\epsilon _{t}| \rightarrow \infty \). The linear and quadratic transformations that are undertaken by the updating terms are identical for the positive and negative sides of the probability distribution. Compared to the DCS specifications, the standard financial time series model does not discount extreme observations. We highlight the fact that extreme observations are accentuated in GARCH by the quadratic transformation of shocks, possibly leading to an overestimation of volatility after extreme observations (Blazsek et al. 2018a).
Statistical inference
The DCS specifications of this paper are estimated by using the Maximum Likelihood (ML) method (see, for example, Davidson and MacKinnon 2003). The ML estimator is given by
$$\begin{aligned} {\hat{\Theta }}_{\text {ML}}=\arg \max _{\Theta }\text {LL}(p_{1},\ldots ,p_{T};\Theta ) =\arg \max _{\Theta }\sum _{t=1}^{T}\ln f(p_{t}|p_{1},\ldots ,p_{t-1};\Theta )\nonumber \\ \end{aligned}$$
(18)
where \(\Theta \) denotes the vector of parameters. We estimate the components \(\mu _{t}\), \(s_{t}\) and \(v_{t}\) jointly, under the initialization methods of \(\mu _{t}\), \(s_{t}\) and \(\lambda _{t}\) that are presented in Sect. 3.1 (see also Harvey 2013). The standard errors of parameters are estimated by using the inverse information matrix (Creal et al. 2013; Harvey 2013). For some parameters, we estimate their transformed values. We use the delta method to estimate the standard errors for those parameters (see, for example, Davidson and MacKinnon 2003).
For the DCS models of this paper, we use results from the work of Harvey (2013) for the conditions of consistency and asymptotic normality of the ML estimates. For the local level and stochastic seasonality equations, the dynamic parameters of the \(\mu _{t}\) and \(\rho _{t}\) equations are set to one, instead of being estimated. Therefore, the asymptotic properties of the ML estimator hold for those cases (Harvey 2013). With respect to the dynamic log-scale equation, we define the statistic \(C_{\lambda }= \beta ^{2}+2\beta \alpha E(\partial u_{\lambda ,t}/\partial \lambda _{t})+\alpha ^{2}E[(\partial u_{\lambda ,t}/\partial \lambda _{t})^{2}]\) (Harvey 2013). We estimate \(C_{\lambda }\) numerically for each DCS specification of the present paper. Firstly, the partial derivatives of the score function with respect to \(\lambda _{t}\) are computed numerically. Secondly, the Augmented Dickey and Fuller (1979) (hereinafter, ADF) is performed for each \(\partial u_{\lambda ,t}/\partial \lambda _{t}\) time series, in order to justify the use of the sample average estimator for the expectations. For all cases, the ADF test indicates that \(\partial u_{\lambda ,t}/\partial \lambda _{t}\) forms a covariance stationary time series. Thus, the sample average is a consistent estimator of the expected value (see, for example, Hamilton 1994). Two conditions for DCS-EGARCH(1,1) are \(|\beta |<1\) and \(C_{\lambda }<1\).
For the standard financial time series model: (1) we use the ML estimator, (2) we estimate the standard errors of parameters by using the inverse information matrix, and (3) we use the delta method for the transformed parameters. Even if the \(\epsilon _{t} \sim N(0,1)\) assumption does not hold, we still get consistent and asymptotically normal estimates of the parameters in accordance with the Quasi-ML (QML) results of Gouriéroux et al. (1984). For the local level and stochastic seasonality equations, the dynamic parameters of the \(\mu _{t}\) and \(\rho _{t}\) equations are not estimated but are set to one. Therefore, the asymptotic properties of ML hold for those cases. For GARCH, a sufficient condition for the asymptotic properties of ML is \(\alpha +\beta <1\).
Table 1 Descriptive statistics