Score-driven currency exchange rate seasonality as applied to the Guatemalan Quetzal/US Dollar

3.1 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.

3.2 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 \(\lambda _{0}\).

3.3 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.

3.4 Statistical inference

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 \(C_{\lambda }<1\).

5.1 Statistical performance

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We use the following metrics to compare statistical performance: LL, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Hannan-Quinn Criterion (HQC). According to AIC, BIC and HQC, the in-sample statistical performance of the DCS-Skew-Gen-t model is superior to that of all the alternatives of this paper (Table 2). We also perform a Likelihood-Ratio (LR) test for non-nested models (Vuong 1989). In the LR test, we estimate the linear regression \(d_{t}=c+\epsilon _{t}\), where \(d_{t}\) is the difference between the log-densities of two models for day t. We estimate this equation by using the OLS-HAC (Ordinary Least Squares - Heteroscedasticity and Autocorrelation Consistent) estimator (Newey and West 1987). In Table 2, we report three different LR test results: (1) LR1 compares the LLs of all alternatives with that of the DCS-Skew-Gen-t model (i.e. the model with the highest LL estimate). According to the results, the LL of the DCS-Skew-Gen-t model is significantly higher than the LLs of the alternative models. (2) LR2 compares the LLs of all alternatives with that of the standard financial time series model (i.e. the model with the lowest LL estimate). According to the results, the LLs of all DCS models are significantly higher than the LL of the standard financial time series model. (3) LR3 compares the LLs of those DCS models that undertake trimming in the location equation (i.e. the recent DCS-t model and the new DCS-Skew-Gen-t model), and also compares the LLs of those DCS models that undertake Winsorizing in the location equation (i.e. the recent DCS-EGB2 model and the new DCS-NIG model). According to the results, the LL of the DCS-Skew-Gen-t model is significantly higher than the LL of DCS-t model, while the LLs of DCS-EGB2 and DCS-NIG models do not differ significantly.

The properties of the scaled score functions and score functions that update the local level and log-scale of \(p_{t}\) are important with respect to the likelihood-based performance of alternative DCS models. In the following, we present those properties in the empirical analysis that can be related to the mathematical details of those updating terms (Sect. 3.3). The updating terms \(u_{\mu ,t}\) and \(u_{\lambda ,t}\) of the DCS-t, DCS-Skew-Gen-t, DCS-EGB2 and DCS-NIG models are presented in Fig. 2a–d, respectively, as functions of \(\epsilon _{t}\). We evaluate \(u_{\mu ,t}\) and \(u_{\lambda ,t}\) by using the ML estimates of the shape parameters, and for \(\lambda _{t}\) we use its unconditional mean estimate \({\hat{\omega }}/(1-{\hat{\beta }})\). We also present in Fig. 2e, f the updating terms of the local level \(\mu _{t}\) and the conditional variance \(\lambda _{t}\) equations for the standard financial time series model. For the evaluation of the updating term of \(\lambda _{t}\), we use the ML estimate of its unconditional mean: \({\hat{\omega }}/(1-{\hat{\alpha }}-{\hat{\beta }})\).

For the DCS-t and DCS-Skew-Gen-t models, \(u_{\mu ,t}\) undertakes a smooth form of trimming (Fig. 2a). For the greater part of the support of the probability distribution, the DCS-Skew-Gen-t model discounts more observations than the DCS-t model (the only exception is for a small negative interval in the central part of the distribution) (Fig. 2a). Furthermore, for the DCS-t and DCS-Skew-Gen-t models, \(u_{\lambda ,t}\) undertakes a smooth form of Winsorizing (Fig. 2b). In the central part of the distribution, observations are discounted in similar ways for the DCS-t and DCS-Skew-Gen-t models (Fig. 2b). In the extreme parts of the distribution, observations are discounted more for the DCS-Skew-Gen-t model than for the DCS-t model (Fig. 2b). All LL-based metrics of Table 2 suggest that the discounting of extreme values for the DCS-Skew-Gen-t model is more effective than the discounting of extreme values for the DCS-t model.

For the DCS-EGB2 and DCS-NIG models, \(u_{\mu ,t}\) undertakes a smooth form of Winsorizing (Fig. 2c), and \(u_{\lambda ,t}\) increases linearly as \(|\epsilon _{t}| \rightarrow \infty \) (Fig. 2d). We find that, for both \(u_{\mu ,t}\) and \(u_{\lambda ,t}\), observations are discounted more for the DCS-EGB2 model than for the DCS-NIG model (Fig. 2c, d). Moreover, for both the DCS-EGB2 and DCS-NIG models, observations are discounted differently with respect to the left and right tails of the distribution (i.e. observations in the right tail are discounted more than observations in the left tail by both \(u_{\mu ,t}\) and \(u_{\lambda ,t}\)) (Fig. 2c, d). The AIC, BIC and HQC metrics presented in Table 2 suggest that the discounting of extreme values for the DCS-NIG model is more effective than the discounting of extreme values for the DCS-EGB2 model.

With respect to the updating terms of conditional mean \(\mu _{t}\) and conditional variance \(\lambda _{t}\) for the standard financial time series model, in Fig. 2e, f we present that extreme values in the noise \(\epsilon _{t}\) are transformed according to linear and quadratic functions, respectively, for the empirical GTQ/USD exchange rate dataset. All LL-based metrics of Table 2 suggest that the discounting of extreme values for the DCS models is more effective than the transformation of extreme values for the standard financial time series model.

5.2 Stochastic seasonality component

For the GTQ/USD exchange rate, significant stochastic annual seasonality \(s_{t}\) estimates are shown in Fig. 3a–d for the DCS-t, DCS-Skew-Gen-t, DCS-EGB2 and DCS-NIG models, respectively. We also present those seasonality \(s_{t}\) estimates for the standard financial time series model in Fig. 3e. With respect to the economic significance of seasonality effects, for the highest local maximum and lowest local minimum points of \(s_{t}\), we estimate approximately \(+2\%\) and \(-1.5\%\), respectively, for \(s_{t}/p_{t}\). The amplitude of seasonality is time-varying. However, for the greater part of the sample period, with respect to the local maximum and local minimum values of \(s_{t}\), we estimate at least \(+\,0.8\%\) and \(-\,0.8\%\), respectively, for \(s_{t}/p_{t}\).

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The presence of the annual seasonality component is against the efficiency of the GTQ/USD exchange rate market. The currency market does not eliminate the GTQ/USD seasonality, because the bid and ask exchange rates for clients, that are offered by financial institutions in Guatemala, are such that it is impossible to obtain profits based on the seasonal movements. As an example, we refer to the bid and ask exchange rates offered by Banco Industrial for 12 November 2018. With respect to total assets in 31 December 2017, Banco Industrial is the largest bank in Guatemala (source: Superintendency of Banks, Guatemala). The corresponding bid and ask prices are 7.60 GTQ/USD and 7.80 GTQ/USD, respectively. According to those prices, the bid-ask spread to the mean GTQ/USD ratio is 2.6%, which is higher than the seasonality amplitude that is estimated for any year of the sample period. This example for the relative bid-ask spread is also representative for other Guatemalan financial institutions, for any year of the sample period.

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Firstly, in Table 3, we present the evolution of total exports from Guatemala and total imports to Guatemala, for the period of 1993–2016. The growth rate of total exports from Guatemala decreases over time: the mean growth rates of total exports for (R1), (R2) and (R3) are 15.1%, 7.8% and 3.4%, respectively (Table 3). This reduction in the growth rate of total exports suggests a decreasing amplitude of the annual seasonality for periods (R1)–(R3).

Secondly, the relative importance of total exports, with respect to total currency inflows and outflows, decreases for the data window. In Fig. 4, we present the relative importance of the following foreign currency movements for (R1), (R2) and (R3): (1) total exports from Guatemala (Fig. 4a); (2) total imports to Guatemala (Fig. 4b); (3) receipt of loans to Guatemala (Fig. 4c); (4) payment of loans from Guatemala (Fig. 4d); (5) remittance payments to Guatemala (Fig. 4e). For all cases, we compute relative importance with respect to the sum of total inflows and total outflows of foreign currency, and we estimate average relative importance separately for each month. We find that the relative importance of total exports, on average, significantly decreases from (R1) to (R2) and (R3) (Fig. 4a). Furthermore, we also find that the relative importance of loans and remittance payments that do not have a significant seasonality component, on average, significantly increases from (R1) to (R3) (Fig. 4c–e). These results also support the reducing amplitude of GTQ/USD seasonality for the data window.

Thirdly, a further explanation for the decreasing amplitude of the annual seasonality component is the reduction of the relative importance of agricultural product exports within total exports. In Table 4, we present the export income from coffee, sugar, banana and cardamom, which, as aforementioned, are the main agricultural export products of Guatemala. The relative importance of these products reduces significantly during the period of 1994–2016.

These results suggest that the stochastic seasonality component of the GTQ/USD currency exchange rate is significant, both from the statistical and economic points of view, for the period of 1994–2008. The results also suggest that the amplitude of this stochastic seasonality component has decreased for the period of 2009–2017, due to the reduced relative importance of agricultural exports of Guatemala.