# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

# Semiparametric GARCH models with long memory applied to value-at-risk and expected shortfall

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Need to know

- Semiparametric GARCH models with long memory are introduced.
- One-step ahead forecasts of Value at Risk and Expected Shortfall are improved.
- Model evaluation is performaned by means of a recently introduced selection criterion.
- Semiparametric GARCH models with long memory are found to be a meaningful substitute to their conventional counterparts.

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Abstract

In this paper new semiparametric generalized autoregressive conditional heteroscedasticity (GARCH) models with long memory are introduced. A multiplicative decomposition of the volatility into a conditional component and an unconditional component is assumed. The estimation of the latter is carried out by means of a data-driven local polynomial smoother. According to the revised recommendations by the Basel Committee on Banking Supervision to measure market risk in the banks’ trading books, these new semiparametric GARCH models are applied to obtain rolling one-step ahead forecasts for the value-at-risk and expected shortfall (ES) for market risk assets. Standard regulatory traffic-light tests and a newly introduced traffic-light test for the ES are carried out for all models. In addition, model performance is assessed via a recently introduced model selection criterion. The practical relevance of our proposal is demonstrated by a comparative study. Our results indicate that semiparametric long-memory GARCH models are a meaningful substitute for their conventional, parametric counterparts.

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Introduction

## 1 Introduction

Influenced by the global financial crisis of 2007–9, the Basel Committee on Banking Supervision (BCBS) published new recommendations for the measurement of market risk from the banks’ trading books by means of the Fundamental Review of the Trading Book (FRTB) framework (Basel Committee on Banking Supervision 2013). The review had become necessary because the crisis had clearly exposed the weaknesses of the value-at-risk (VaR) method (JP Morgan–Reuters 1996), which was predominantly used by banks to measure market risk. Thus, in accordance with the FRTB, the BCBS recommended replacing the VaR method by the coherent expected shortfall (ES) measure (Acerbi and Tasche 2002). In addition, the committee proposed banks employ the proven “traffic-light tests” for backtesting the model quality of the ES (Basel Committee on Banking Supervision 1996, 2016). However, backtesting is much more complex for ES than for VaR, since not only the number but also the amount by which the expected loss is exceeded is relevant. So far there has been no consensus on the most suitable model or method for forecasting VaR and ES. Nevertheless, empirical studies have revealed that long-memory generalized autoregressive conditional heteroscedasticity (LM-GARCH) models are very successful in accurately forecasting the conditional volatility of asset returns and often outperform short-memory GARCH-type models (see, for example, Giot and Laurent 2003; Degiannakis 2004; Tang and Shieh 2006; Grané and Veiga 2008; Härdle and Mungo 2008; Baillie and Morana 2009; Demiralay and Ulusoy 2014; Aloui and Ben Hamida 2015; Royer 2022).

Against this background, our paper focuses on the introduction and application of a new semiparametric long-memory GARCH (semi-LM-GARCH) model, which belongs to a general class of nonstationary volatility models as outlined in Sucarrat (2019). The most common approaches for modeling nonconstant conditional variances are the autoregressive conditional heteroscedasticity (ARCH) model proposed by Engle (1982) and its generalization, the GARCH model, introduced by Bollerslev (1986). Those models and their extensions imply exponentially decaying autocorrelations of the squared innovations and do not control for long memory in the conditional dynamics. However, Ding et al (1993), Ding and Granger (1996), Andersen and Bollerslev (1997), Andersen et al (1999) and Cotter (2005), among others, found evidence for the presence of long memory in the empirical autocorrelations in absolute or squared observations of financial time series. In analogy to the extension of the autoregressive integrated moving average (ARIMA) model (ie, the fractional ARIMA (FARIMA) model introduced by Granger and Joyeux (1980)), Baillie et al (1996) proposed the fractionally integrated GARCH (FIGARCH) model, which proved to be successful in modeling the long-term dynamics in volatility of various financial time series (see, for example, Bollerslev and Mikkelsen 1996; Tse 1998; Beine et al 2002; Baillie and Morana 2009). However, another branch of literature suggests that these long-term dynamics might partly stem from deterministic structural shifts in the unconditional variance (see, for example, Lamoureux and Lastrapes 1990; Mikosch and Stǎricǎ 2004). For instance, Beran and Ocker (2001) revealed the presence of a nonconstant deterministic scale function for some volatility series by means of the semiparametric fractional autoregressive (SEMIFAR) model, introduced by Beran and Ocker (1999). Further, Feng (2004) found that conditional heteroscedasticity and change in volatility usually occur simultaneously in financial return series. Under regular conditions, a process with conditional heteroscedasticity is covariance stationary, but a process with change in volatility is at best locally stationary. This potentially nonstationary process can be transformed into a weakly stationary process by eliminating the deterministic component from the original process, as was illustrated by Feng (2004) and by Van Bellegem and Von Sachs (2004). These authors assume that volatility is multiplicatively decomposed into a conditional component and an unconditional component and that the latter changes slowly over time. They estimate the time-varying unconditional variance by means of a kernel smoother of the squared residuals.

Both Engle and Rangel (2008) and Brownlees and Gallo (2010) apply other multiplicative decompositions, based on exponential quadratic and penalized B-splines, respectively. Mazur and Pipień (2012) introduce the almost periodically correlated GARCH (APC-GARCH). In this model the scaling function is parameterized by means of the flexible Fourier form by Gallant (1981, 1984). More recently, Amado and Teräsvirta (2014) introduced the time-varying GARCH model under the same assumption (see also Amado and Teräsvirta 2008, 2013, 2017) and underline the empirical importance of considering deterministic changes in the unconditional variance of financial return series.

In analogy to Feng (2004) we introduce various semi-LM-GARCH models. We propose to estimate the time-varying unconditional variance by means of an adapted version of the SEMIFAR algorithm (Beran and Feng 2002a) with a local polynomial estimator. Subsequently, the deterministic component is removed from the data and an LM-GARCH model is fitted to the approximately stationary residuals. Practical performance is first illustrated by the application to daily return series of 22 major stock indexes. Moreover, our proposal is applicable to modeling quantitative risk measures. This is illustrated by calculating the one-step-ahead out-of-sample forecasts of the VaR and ES at the 99% and 97.5% confidence levels with a forecast horizon of approximately one year, as required by the regulations proposed by the BCBS (see Basel Committee on Banking Supervision 2017). Our comprehensive comparison study between conventional parametric LM-GARCH models and semi-LM-GARCH models reveals that our proposals are an attractive alternative.

This paper is organized as follows. Section 2 recaps the most common long-memory GARCH models as well as a fractionally integrated version of the log-GARCH (Geweke 1986; Milhøj 1987; Pantula 1986) – the FI-log-GARCH – introduced by Feng et al (2020). The proposed models are defined in Section 3. In Section 4 the semiparametric estimation of the deterministic component is illustrated, an adaptation of the SEMIFAR algorithm is briefly described and the practical implementation of our proposals with regard to rolling forecasts is discussed. Section 5 explains the employment of our proposals for VaR and ES. Empirical results are presented in Section 6. Section 7 states our conclusions.

## 2 Modeling long memory in volatility

The existence of long memory in volatility was first discovered in the Standard & Poor’s 500 (S&P 500) daily closing index by Ding et al (1993). Prior to this time volatility models were assumed to have an exponentially decaying correlation of volatility. In the following decade, more researchers found evidence for the presence of long memory in volatility of financial asset prices, including intraday and high-frequency stock returns (see, for example, Ding and Granger 1996; Andersen and Bollerslev 1997; Andersen et al 1999; Cotter 2005). As a consequence, various LM-GARCH models were developed.

### 2.1 GARCH models with long memory

In the short-memory case, the ARCH model proposed by Engle (1982) and the GARCH model introduced by Bollerslev (1986) are well-known approaches for modeling nonconstant conditional variances. Let ${r}_{t}^{*}$, $t=1,\mathrm{\dots},n$, denote the (log) returns of a stock or financial index with $E({r}_{t}^{*})={\mu}_{{r}^{*}}$. One common representation of a $\mathrm{GARCH}(p,q)$ model is

$$\begin{array}{cc}\hfill {r}_{t}& ={\sqrt{h}}_{t}{\epsilon}_{t},\hfill \\ \\ \hfill {h}_{t}& =\omega +\sum _{i=1}^{q}{\alpha}_{i}{r}_{t-i}^{2}+\sum _{j=1}^{p}{\beta}_{j}{h}_{t-j},\hfill \end{array}\}$$ | (2.1) |

where ${r}_{t}={r}_{t}^{*}-{\mu}_{{r}^{*}}$ are the centralized returns; ${\epsilon}_{t}$ are independent and identically distributed (iid) random variables with $E({\epsilon}_{t})=0$ and $E({\epsilon}_{t}^{2})=1$; ${h}_{t}$ denotes the conditional variances; $\omega >0$; and ${\alpha}_{1},\mathrm{\dots},{\alpha}_{q},{\beta}_{1},\mathrm{\dots},{\beta}_{p}\ge 0$.

Due to increasing evidence for volatility series exhibiting long memory, an extension of the GARCH that captures this important feature, namely the FIGARCH, was proposed by Baillie et al (1996). Moreover, based on the exponential GARCH (EGARCH) proposed by Nelson (1991), Bollerslev and Mikkelsen (1996) introduced the fractionally integrated exponential GARCH (FIEGARCH), where the logarithm of the conditional variance is modeled as a fractionally integrated process. The EGARCH and FIEGARCH account for the so-called leverage effect, which usually has short-term effects on the dependence structure of the underlying process. Further, Ding et al (1993) proposed the so-called asymmetric power GARCH (APARCH) model. This controls for the power transformation of the volatility process and the asymmetric absolute residuals, in order to avoid misspecification for nonnormal data. The extension to the fractionally integrated APARCH (FIAPARCH), which combines the FIGARCH with the APARCH, was then proposed by Tse (1998).

Another model that is capable of capturing persistence in volatility is the $\mathrm{ARCH}(\mathrm{\infty})$ model introduced by Robinson (1991) and further investigated by Giraitis et al (2000), Kazakevičius and Leipus (2002) and Douc et al (2008). Moreover, in a more recent study, Royer (2022) proposes an $\mathrm{ARCH}(\mathrm{\infty})$ extension of the APARCH that accounts for conditional asymmetry in the presence of severe long memory. Its specification is very general and nests the $\mathrm{ARCH}(\mathrm{\infty})$ together with the threshold-$\mathrm{ARCH}(\mathrm{\infty})$ (see Bardet and Wintenberger 2009).

### 2.2 The FI-log-GARCH model

Feng et al (2020) proposed the FI-log-GARCH model, an extension of the log-GARCH model independently introduced by Geweke (1986), Pantula (1986) and Milhøj (1987), which represents a symmetric special case of the FIAPARCH for $\delta \to 0$. In the following a brief derivation of the FI-log-GARCH is given. Following Sucarrat et al (2016) and Francq and Sucarrat (2018), the log-GARCH is defined by (2.1) and

$$\mathrm{ln}{h}_{t}=\omega +\sum _{i=1}^{q}{\alpha}_{i}\mathrm{ln}{r}_{t-i}^{2}+\sum _{j=1}^{p}{\beta}_{j}\mathrm{ln}{h}_{t-j}.$$ | (2.2) |

As with the EGARCH, no nonnegativity constraints are needed. Define ${\mu}_{l{r}^{2}}=E(\mathrm{ln}{r}_{t}^{2})$ and ${\eta}_{t}=\mathrm{ln}{\epsilon}_{t}^{2}-{\mu}_{l{\epsilon}^{2}}$, where ${\mu}_{l{\epsilon}^{2}}=E(\mathrm{ln}{\epsilon}_{t}^{2})$. Then, following Francq and Sucarrat (2018), (2.2) can be represented as an $\mathrm{ARMA}({q}^{*},p)$ model with ${q}^{*}=\mathrm{max}(p,q)$:

$$\varphi (B)(\mathrm{ln}{r}_{t}^{2}-{\mu}_{l{r}^{2}})=\psi (B){\eta}_{t},$$ | (2.3) |

where $B$ is the backshift operator,

$$\varphi (B)=1-\sum _{i=1}^{{q}^{*}}{\alpha}_{i}{B}^{i}-\sum _{j=1}^{p}{\beta}_{j}{B}^{j}$$ |

and

$$\psi (B)=1+\sum _{j=1}^{p}{\psi}_{j}{B}^{j}=1-\sum _{j=1}^{p}{\beta}_{j}{B}^{j}.$$ |

Being analogous to the extension of the GARCH to the FIGARCH, the extension of the log-GARCH to the FI-log-GARCH is straightforward (see Baillie et al 1996; Feng et al 2020). Factorizing the left-hand side of (2.3) with the fractional differencing operator ${(1-B)}^{d}$ yields

$$\varphi (B){(1-B)}^{d}(\mathrm{ln}{r}_{t}^{2}-{\mu}_{l{r}^{2}})=\psi (B){\eta}_{t}.$$ | (2.4) |

According to Hosking (1981b) and Granger and Joyeux (1980), the fractional differencing operator can be defined as

$${(1-B)}^{d}=\sum _{k=0}^{\mathrm{\infty}}{\theta}_{k}(d){B}^{k},$$ | (2.5) |

where

$${\theta}_{k}(d)={(-1)}^{k}\frac{\mathrm{\Gamma}(d+1)}{\mathrm{\Gamma}(k+1)\mathrm{\Gamma}(d-k+1)},d\in (-0.5,0.5),$$ |

and $\mathrm{\Gamma}(\cdot )$ denotes the Gamma function. In this model $\mathrm{ln}{r}_{t}^{2}$ is assumed to follow a linear FARIMA process as introduced by Hosking (1981a) and Granger and Joyeux (1980). This model is well defined if the innovation distribution satisfies some regularity conditions, if $\varphi (z)$ and $\beta (z)$ have no common factors with all roots lying outside the unit circle and if $$. The log-GARCH is a special case with $d=0$. Let ${\alpha}_{d,i}=\psi (B)-\varphi (B){(1-B)}^{d}$. Then, the conditional volatility of the FI-log-GARCH can be obtained by

$$\mathrm{ln}{h}_{t}=\omega +\sum _{i=1}^{\mathrm{\infty}}{\alpha}_{d,i}\mathrm{ln}{r}_{t-1}^{2}+\sum _{j=1}^{p}{\beta}_{j}\mathrm{ln}{h}_{t-j},$$ | (2.6) |

where $\omega $ is some constant. Equation (2.6) is a long-memory extension of (2.2) with hyperbolically decaying coefficients ${\alpha}_{d,i}$. For a more detailed derivation and a comprehensive illustration of the properties of the FI-log-GARCH we refer the reader to Feng et al (2020).

## 3 Semiparametric extension of the FI-log-GARCH

In this section, we introduce a semiparametric extension of the FI-log-GARCH (ie, the semi-FI-log-GARCH), which belongs to a general class of nonstationary volatility models as outlined in Sucarrat (2019). We follow Feng (2004) as well as Van Bellegem and Von Sachs (2004) and assume a multiplicative decomposition of ${h}_{t}$ into a conditional component and an unconditional component under the assumption that the latter is slowly varying over time. While Feng (2004) and Van Bellegem and Von Sachs (2004) both estimate the unconditional variance by smoothing $\{{r}_{t}^{2}\}$ via a kernel estimator, we employ a local polynomial smoother.

### 3.1 The semi-FI-log-GARCH model

In order to control for a time-varying unconditional variance, we add a nonnegative smooth deterministic function $\sigma ({\tau}_{t,n})$ into (2.1) and obtain

$${r}_{t}=\sigma ({\tau}_{t,n}){\xi}_{t},$$ | (3.1) |

where ${\tau}_{t,n}=t/n$ denotes the rescaled time. Note that this specification of ${\tau}_{t,n}$ is required for the derivation of the asymptotic results of the local polynomial estimator of $\sigma ({\tau}_{t,n})$ (see, for example, Beran and Feng 2002b). In addition, $\{{\xi}_{t}\}$ is assumed to follow an FI-log-GARCH process given by

${\xi}_{t}=\sqrt{{h}_{t}}{\epsilon}_{t}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\mathrm{ln}{h}_{t}=\omega +{\displaystyle \sum _{i=1}^{\mathrm{\infty}}}{\alpha}_{d,i}\mathrm{ln}{\xi}_{t-i}^{2}+{\displaystyle \sum _{j=1}^{p}}{\beta}_{j}\mathrm{ln}{h}_{t-j}.$ | (3.2) |

Hence, (3.1) and (3.2) together represent a semi-FI-log-GARCH model. To ensure that our model is well defined we assume that $\mathrm{var}({\xi}_{t})=1$ and that $E(\mathrm{ln}{\xi}_{t}^{2})$ exists, which implies that ${\xi}_{t}\ne 0$ almost surely. Further, we have $g({\tau}_{t,n})=\mathrm{ln}{\sigma}^{2}({\tau}_{t,n})+{\mu}_{l{\xi}^{2}}$ and ${Z}_{t}=\mathrm{ln}{\xi}_{t}^{2}-{\mu}_{l{\xi}^{2}}=\mathrm{ln}{h}_{t}+{\eta}_{t}-({\mu}_{l{\xi}^{2}}-{\mu}_{l{\epsilon}^{2}})$, where ${\mu}_{l{\xi}^{2}}=E(\mathrm{ln}{\xi}_{t}^{2})$ and ${\eta}_{t}=\mathrm{ln}{\epsilon}_{t}^{2}-{\mu}_{l{\epsilon}^{2}}$. Let ${Y}_{t}=\mathrm{ln}{r}_{t}^{2}$. The logarithmic transformation of ${r}_{t}^{2}$ admits an additive regression model with a deterministic, nonparametric function and is of the form

$${Y}_{t}=g({\tau}_{t,n})+{Z}_{t},$$ | (3.3) |

where ${Z}_{t}$ follows a $\mathrm{FARIMA}({q}^{*},p)$ process given by

$$\varphi (B){(1-B)}^{d}{Z}_{t}=\psi (B){\eta}_{t}.$$ | (3.4) |

We see that the semi-FI-log-GARCH is equivalent to a SEMIFAR model (Beran and Feng 2002c; Beran and Ocker 1999) with an additional moving-average part and the restriction $q\ge p$. Thus, well-developed SEMIFAR algorithms are applicable for estimating $g({\tau}_{t,n})$ and ${Z}_{t}$. Moreover, we have ${\widehat{\sigma}}^{2}({\tau}_{t,n})={\widehat{C}}_{\sigma}\mathrm{exp}[\widehat{g}({\tau}_{t,n})]$ and ${\widehat{h}}_{t}={\widehat{C}}_{h}\mathrm{exp}({\widehat{Z}}_{t})$, where ${C}_{\sigma}=\mathrm{exp}(-{\mu}_{l{\xi}^{2}})$ and ${C}_{h}=\mathrm{exp}({\mu}_{l{\xi}^{2}}-{\mu}_{l{\epsilon}^{2}})$ can be estimated consistently by the scale-adjusted returns and through standardizing the innovations under the assumptions that $$ and $\mathrm{var}({\xi}_{t})=1$.

### 3.2 Other semi-LM-GARCH models

The conditional volatility in (3.1) can also be modeled by means of any conventional LM-GARCH under the assumption that strictly stationary solutions exist for these models. Hence, in the following we formulate three new semiparametric LM-GARCH models. The semi-FIGARCH is specified by

$${h}_{t}={\omega}^{*}+\{1-{\psi}^{-1}(B)\varphi (B){(1-B)}^{d}\}{\xi}_{t}^{2},$$ | (3.5) |

where ${\omega}^{*}=\omega {\psi}^{-1}(B)$ and with all roots of $\varphi (B)$ and $\psi (B)$ outside the unit circle. Moreover, the semi-FIEGARCH is given by

$$\mathrm{ln}{h}_{t}=\omega +{\varphi}^{-1}(B){(1-B)}^{-d}\alpha (B)g({\epsilon}_{t-1}),$$ | (3.6) |

where $\alpha (B)=1+{\sum}_{i=1}^{q}{\alpha}_{i}{B}^{i}$ and $g({\epsilon}_{t})=\mathrm{\Theta}{\epsilon}_{t}+\gamma [(|{\epsilon}_{t}|-E|{\epsilon}_{t}|)]$ is the news impact function with $\mathrm{\Theta},\gamma \in \mathbb{R}$. Subsequently, the semi-FIAPARCH is defined by

$${h}_{t}^{\delta}=\omega +\{1-{\psi}^{-1}(B)\varphi (B){(1-B)}^{d}\}{(|{\xi}_{t}|-{\gamma}_{i}{\xi}_{t})}^{\delta},$$ | (3.7) |

where $\delta >0$, $$ with $i=1,\mathrm{\dots},q$. Here, $\delta $ is a parameter to be estimated and ${\gamma}_{i}$ controls for an asymmetric response of the volatility to positive and negative shocks. Note that, for the FI-log-GARCH, the long-memory parameter is not affected by a power or logarithmic transformation (see Surgailis and Viano 2002; Feng et al 2020, Lemma 2). However, this is yet to be proven for other LM-GARCH models. Nonetheless, estimation of the deterministic component for models (3.5)–(3.7) is carried out analogously to the semi-FI-log-GARCH under the assumptions that $\{{\xi}_{t}^{2}\}$ is a loglinear process and that $d$ is not affected by the logarithmic transformation.

### 3.3 Related approaches

Several previous studies employ a similar methodology to decompose ${h}_{t}$ and estimate $\sigma ({\tau}_{t,n})$. Engle and Rangel (2008) and Brownlees and Gallo (2010) initially applied other multiplicative decompositions based on exponential quadratic and penalized B-splines, respectively. Mazur and Pipień (2012) proposed to parameterize the deterministic component by means of the flexible Fourier form. Moreover, Amado and Teräsvirta (2008, 2013, 2014, 2017) introduced the multiplicative time-varying GARCH model, in which the deterministic component is modeled via generalized logistic transition functions. Further, Engle et al (2013) suggest the use of mixed-data sampling (MIDAS) in order to decompose the volatility into a short-run component and a long-run component and propose the GARCH-MIDAS model. More recently, Zhang (2019) introduced the Box–Cox semi-GARCH model; they suggest estimating the deterministic scale function from the Box–Cox transformed series ${|{r}_{t}|}^{\lambda}$ instead of ${r}_{t}^{2}$.

Our paper fills a gap in the literature, since most of the aforementioned studies do not address the persistence in the stochastic component (ie, ${h}_{t}$). In addition, we propose to estimate the deterministic component (ie, $\sigma ({\tau}_{t,n})$) via a local polynomial smoother within the scope of an adapted version of the SEMIFAR algorithm.^{1}^{1} 1 See Beran and Feng (2002a) and Letmathe et al (2021) for the original and adapted versions of the algorithm, respectively.

## 4 Estimation and practical implementation

The SEMIFAR model introduced by Beran and Feng (2002c) is capable of simultaneously identifying a deterministic scale and short- and long-range dependencies. The estimation process is carried out in two steps: namely, the nonparametric estimation of the deterministic component and the parametric estimation of the parameters that determine the short- and long-range dependencies as well as integer differencing. Further, Beran et al (2015) proposed the exponential SEMIFAR (ESEMIFAR) model under the assumption that $\{{\xi}_{t}\}$ is loglinear. As we have already shown that an FI-log-GARCH process can be formalized as a $\mathrm{FARIMA}({q}^{*},p)$ process, as given by (2.4), $\{{\xi}_{t}\}$ is indeed a loglinear process. Subsequently, this parametrization allows (2.4) or (3.4) to be estimated by means of any standard R package for FARIMA models and that well-developed SEMIFAR algorithms are applicable for estimating the deterministic component $g({\tau}_{t,n})$ as well.

### 4.1 Local polynomial smoothing

In this paper a local polynomial estimator for ${g}^{(\nu )}({\tau}_{t,n})$, the $\nu $th derivative, is considered (see, for example, Beran and Feng 2002a,b,c; Beran et al 2013). Assume that $g$ is an at least ($l+1$)-times differentiable function on $[0,1]$. Then $g({\tau}_{t,n})$ can be approximated by a local polynomial of order $l$ for ${\tau}_{t,n}$ in a neighborhood of ${\tau}_{0,n}$. The approximation is given by

$$g({\tau}_{t,n})=g({\tau}_{0,n})+{g}^{(1)}({\tau}_{0,n})({\tau}_{t,n}-{\tau}_{0,n})+\mathrm{\cdots}+\frac{{g}^{(l)}({\tau}_{0,n}){({\tau}_{t,n}-{\tau}_{0,n})}^{l}}{l!}+{R}_{l},$$ | (4.1) |

where ${R}_{l}$ denotes a remainder term. Following Gasser and Müller (1979), we define the weight function to be a kernel of order $2$ with compact support $[-1,1]$ having the polynomial form $K(x)={\sum}_{i=0}^{r}{a}_{i}{x}^{2i}$ for $|x|\le 1$, where $K(x)=0$ if $|x|>1$, $r\in (0,1,2,\mathrm{\dots})$ and ${a}_{i}$ are such that ${\int}_{-1}^{1}K(x)dx=1$ holds. Thus, ${\widehat{g}}^{(\nu )}$ ($\nu \le l$) can now be obtained by solving the locally weighted least squares problem

$$Q=\sum _{t=1}^{n}{\left[{Y}_{t}-\sum _{j=0}^{l}{\beta}_{j}{({\tau}_{t,n}-{\tau}_{0,n})}^{j}\right]}^{2}K\left(\frac{{\tau}_{t,n}-{\tau}_{0,n}}{b}\right),$$ | (4.2) |

where $b$ denotes the bandwidth and $K[({\tau}_{t,n}-{\tau}_{0,n})/b]$ are the weights ensuring that only observations in the neighborhood of ${\tau}_{0,n}$ are used. We can see from (4.1) that ${\widehat{g}}^{(\nu )}({\tau}_{0,n})=\nu !{\widehat{\beta}}_{\nu}$, where $\widehat{\beta}=({\widehat{\beta}}_{0},{\widehat{\beta}}_{1},\mathrm{\dots},{\widehat{\beta}}_{l},)$ and $\nu =0,1,\mathrm{\dots},l$. Consider the case where $l-\nu $ is odd and let $m=l+1$. Then we have $m\ge \nu +2$ and $m-\nu $ is even. A point ${\tau}_{t,n}$ is said to be in the interior for each ${\tau}_{t,n}\in [b,1-b]$, at the left boundary if ${\tau}_{t,n}\in [0,b)$, and at the right boundary if ${\tau}_{t,n}\in (1-b,1]$. Following Beran and Feng (2002b), a common definition for a left boundary point is ${\tau}_{t,n}=cb$ with $c\in [0,1)$. For each interior point ${\tau}_{t,n}\in [b,1-b]$ we have $c=1$.

Beran and Feng (2002a,b) obtained asymptotic expressions for the bias, variance and mean integrated squared error (MISE) of $\widehat{g}$. According to Beran and Feng (2002b, Theorem 1) the bias and variance are respectively given by

$$E({\widehat{g}}^{(\nu )}-{g}^{(\nu )})={b}^{m-\nu}\frac{{g}^{(m)}(\tau ){\beta}_{(\nu ,m,c)}}{m!}o({b}^{(m-\nu )})$$ | (4.3) |

and

$${(nb)}^{1-2d}{b}^{2\nu}\mathrm{var}[{\widehat{g}}^{(\nu )}]=V(c)+o(1).$$ | (4.4) |

For an interior point, Beran and Feng (2002b) presented a simple explicit expression for $V(1)$:

$$V(1)=\{\begin{array}{cc}2\pi {c}_{f}{\int}_{-1}^{1}{K}_{(\nu ,m)}^{2}(x)dx,\hfill & d=0,\hfill \\ \hfill & \\ 2{c}_{f}\mathrm{\Gamma}(1-2d)\mathrm{sin}(\pi d){\int}_{-1}^{1}{\int}_{-1}^{1}{K}_{(\nu ,m)}(x){K}_{(\nu ,m)}(y){|x-y|}^{2d-1}dxdy,\hfill & d>0,\hfill \end{array}$$ |

where ${c}_{f}$ denotes the spectral density of the ARMA part of (3.4) at zero frequency. In the case of antipersistence (ie, for $$) the formula for $V(1)$ is quite complex and is omitted. Moreover, Beran and Feng (2002b) derived an explicit expression for the asymptotic MISE in order to determine the asymptotically optimal bandwidth, which can be obtained by

$${b}_{\mathrm{opt}}={C}_{\mathrm{opt}}{n}^{(2d-1)/(2m+1-2d)},$$ | (4.5) |

with

$${C}_{\mathrm{opt}}={\left(\frac{{[m!]}^{2}}{2(m-\nu )}\frac{(2\nu +1-2d)}{{\beta}^{2}}\frac{({d}_{b}-{c}_{b})V(1)}{I[{g}^{(m)}]}\right)}^{1/(2m+1-2d)},$$ | (4.6) |

where

$$I({g}^{(m)})={\int}_{{c}_{b}}^{{d}_{b}}[{g}^{m}(\tau )]d\tau $$ |

with $$ being small positive constants controlling for the boundary effect and $\beta ={\int}_{-1}^{1}{x}^{m}K(x)dx$. Based on these results Beran and Feng (2002a) proposed two iterative plug-in algorithms. In this paper we only consider a strongly adapted version of their Algorithm B, which is presented below.

### 4.2 A plug-in algorithm for SEMIFARIMA models

Based on the iterative plug-in (IPI) algorithms for SEMIFAR models introduced by Beran and Feng (2002a), Letmathe et al (2021) developed an IPI-procedure for SEMIFARIMA models by translating and adapting the main features of the IPI-algorithm for SEMIFAR models from the S programming language to R, in order to enhance its overall accessibility and applicability. The algorithm processes as follows.

- (i)
In the first iteration, start with an initial bandwidth ${h}_{0}$ set beforehand and select $q$ and $p$, denoting the AR-order and MA-order, respectively.

- (ii)
Estimate $g$ from ${Y}_{t}$ by minimizing (4.2) with ${h}_{j-1}$. Calculate the residuals ${\stackrel{~}{Z}}_{t}={Y}_{t}-\widehat{g}({\tau}_{t,n})$. Obtain $d$ and $V$ by fitting a FARIMA (with predefined AR- and MA-orders in step (i)) to ${\stackrel{~}{Z}}_{t}$.

- (iii)
Set ${b}_{j}={({b}_{j-1})}^{\alpha}$, where $\alpha $ denotes an inflation factor. Estimate ${g}^{(m)}$ with ${b}_{j}$ and a local polynomial of order ${l}^{*}=l+2$. Now, we obtain

$${b}_{j}={\left(\frac{{[m!]}^{2}}{2m}\frac{(1-2\widehat{d})}{{\beta}^{2}}\frac{({d}_{b}-{c}_{b})\widehat{V}(1)}{I[{\widehat{g}}^{(m)}]}\right)}^{1/(2m+1-2\widehat{d})}{n}^{(2\widehat{d}-1)/(2m+1-2\widehat{d})}.$$ (4.7) - (iv)
Repeat steps (ii) and (iii) until convergence or a given number of iterations has been reached, and set ${\widehat{b}}_{\mathrm{opt}}={b}_{j}$.

After estimating $g$ with ${\widehat{b}}_{\mathrm{opt}}$ the residuals ${\stackrel{~}{Z}}_{t}$ or ${\widehat{\xi}}_{t}$ can be further analyzed by means of the FI-log-GARCH or any other LM-GARCH model. Note that the results presented in Beran and Feng (2002a,b,c), Beran and Ocker (1999, 2001) and Beran et al (2013) remain valid for the IPI for SEMIFARIMA models. For a more detailed documentation of the procedure, changes and adaptations of this IPI-algorithm we refer the reader to Feng et al (2021) and Letmathe et al (2021).

### 4.3 Rolling one-step-ahead forecasts

Volatility estimation for the conventional semi-parametric LM-GARCH models is carried out by means of the statistical software OxMetrics and the related G@RCH 8.0 package. For the semi-FI-log-GARCH we use the free R software (R Core Team 2021). There are various possibilities for estimating a FARIMA with R. In this paper we use the fracdiff() function from the package having the same name. The volatility is then derived from the estimates of the conditional means. The latter can be obtained via a truncated $\mathrm{AR}(\mathrm{\infty})$ representation of the fitted model, which is given by

$$ | (4.8) |

where $L$ denotes the number of lags, ${\widehat{\lambda}}_{i}$ are the coefficients of

$$\widehat{\lambda}(B)={(1-B)}^{\widehat{d}}\widehat{\varphi}(B){\widehat{\beta}}^{-1}(B)=1-\sum _{i=1}^{L}{\widehat{\lambda}}_{i}{B}^{i}$$ |

with $i=1,\mathrm{\dots},L$ and ${\stackrel{~}{Z}}_{t}={Y}_{t}-\widehat{g}({\tau}_{t,n})$. To obtain the total volatility, we follow the three-step estimation procedure as proposed by Sucarrat (2019) and described in Section 3.3, with minor adjustments. To begin with, in our paper the deterministic component $g({\tau}_{t,n})$ in (3.3) is estimated nonparametrically via a data-driven local polynomial smoother.

- (i)
Estimate $g({\tau}_{t,n})$ in (3.3) by means of the IPI for the SEMIFARIMA models and calculate the residuals ${\stackrel{~}{Z}}_{t}={Y}_{t}-\widehat{g}({\tau}_{t,n})$.

- (ii)
- (iii)
The total volatilities are then obtained by

${\widehat{\zeta}}_{t}$ $=\sqrt{{\widehat{C}}_{\sigma}}\mathrm{exp}[\widehat{g}(\frac{1}{2}{\tau}_{t,n})]\sqrt{{\widehat{C}}_{h}}\mathrm{exp}[\frac{1}{2}{\widehat{Z}}_{t}]$ $=\widehat{\sigma}({\tau}_{t,n})\sqrt{{\widehat{h}}_{t}}.$ (4.9)

The conditional volatility can be calculated analogously by replacing (4.9) with $\sqrt{{\widehat{h}}_{t}}=\sqrt{{\widehat{C}}_{h}}\mathrm{exp}(\frac{1}{2}{\widehat{Z}}_{t})$. Note in this context that the parameters ${C}_{\sigma}$ and ${C}_{h}$ can be estimated equivalently, as in Sucarrat (2019) (see also Sucarrat et al 2016; Escribano and Sucarrat 2018). This yields

$${\widehat{C}}_{\sigma}=\mathrm{var}[\mathrm{exp}(\frac{1}{2}{\stackrel{~}{Z}}_{t})]=\mathrm{exp}(-{\widehat{\mu}}_{l{\xi}^{2}})$$ |

and

$${\widehat{C}}_{h}=\mathrm{var}\left[\frac{{\widehat{\xi}}_{t}}{\mathrm{exp}(\frac{1}{2}{\widehat{Z}}_{t})}\right]=\mathrm{exp}({\widehat{\mu}}_{l{\xi}^{2}}-{\widehat{\mu}}_{l{\epsilon}^{2}}),$$ |

where

${\widehat{\mu}}_{l{\xi}^{2}}=-\mathrm{ln}\left[{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}}\mathrm{exp}({\stackrel{~}{Z}}_{t})\right]\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\widehat{\mu}}_{l{\epsilon}^{2}}=-\mathrm{ln}\left[{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}}\mathrm{exp}({\widehat{\eta}}_{t})\right]$ |

are smearing estimators (see Duan 1983) of $E(\mathrm{ln}{\xi}_{t}^{2})$ and $E(\mathrm{ln}{\epsilon}_{t}^{2})$, respectively. Further, note that calculating the total volatilities in step (iii) only requires an estimate of ${\mu}_{l{\epsilon}^{2}}$, since ${\widehat{C}}_{\sigma}{\widehat{C}}_{h}=\mathrm{exp}(-{\widehat{\mu}}_{l{\epsilon}^{2}})$. Moreover, the rolling one-day forecasts are given by

$${\widehat{Z}}_{n+k}=\sum _{i=1}^{L}{\widehat{\lambda}}_{i}{\stackrel{~}{Z}}_{n+k-i},$$ | (4.10) |

where $k=1,\mathrm{\dots},K$, with $K$ being the number of out-of-sample observations. We propose to extrapolate the last estimate of the deterministic component $g({\tau}_{t,n})$ for the in-sample period as a forecast for the unconditional standard deviations for the out-of-sample period. That means ${\stackrel{~}{Z}}_{n+k-i}={Y}_{t}-\widehat{g}({\tau}_{t,n})$ if $n+k-i\le n$ and ${\stackrel{~}{Z}}_{n+k-i}={Y}_{n+k-i}-\widehat{g}({\tau}_{n})$ otherwise. Rolling one-day forecasts for the total volatility can then be obtained by plugging ${\widehat{Z}}_{n+k}$ and $\widehat{\sigma}({\tau}_{n})$ into (4.9).

## 5 Application to value-at-risk and expected shortfall

It has already been shown that eliminating the deterministic component from the data can improve the estimation of quantitative risk measures (see, for example, Peitz 2016). However, GARCH and semi-GARCH models lack the possibility of an underlying long-memory structure in the conditional dynamics of daily returns. Consequently, the employment of semi-LM-GARCH models can further improve the estimation quality of quantitative risk measures. Our approach proceeds as follows. A semi-LM-GARCH model is fitted to an in-sample return series with ${n}_{\mathrm{in}}=n-K$ trading days, where $K=250$ denotes the number of trading days in one year. Then, the out-of-sample one-step-ahead forecasts of the VaR and ES with a forecast horizon of $K$ are calculated with confidence levels of ${\alpha}_{V}=99\%$ for VaR and ${\alpha}_{E}=97.5\%$ for ES, as proposed by the BCBS (Basel Committee on Banking Supervision 2016, 2017).

### 5.1 One-day ahead forecasts of VaR and ES

Note that we only consider the conditional $t$-distribution with degree of freedom $\nu >2$. For simplicity we propose to use the last estimate of $\sigma ({\tau}_{t,n})$ as the forecast for the unconditional standard deviations for the out-of-sample period such that ${\widehat{\zeta}}_{{n}_{\mathrm{in}}+k}=\widehat{\sigma}({\tau}_{{n}_{\mathrm{in}}})\sqrt{{\widehat{h}}_{{n}_{\mathrm{in}}+k}}$. Then, for the one-day rolling forecasts of VaR we have

${\widehat{\mathrm{VaR}}}_{{n}_{\mathrm{in}}+k}(\alpha )$ | $=-\overline{r}+{\widehat{\zeta}}_{{n}_{\mathrm{in}}+k}{F}_{\widehat{\nu}}^{-1}(\alpha )\sqrt{(\widehat{\nu}-2)/\widehat{\nu}},$ | (5.1) |

where $k=1,\mathrm{\dots},K$ and ${F}_{\nu}$ denotes the cumulative distribution function of a $t$-distribution with variance $\nu /(\nu -2)$, $\widehat{\nu}$ is an estimate of the degrees of freedom $\nu $, and $\overline{r}$ is the sample mean of the in-sample returns. The one-day rolling forecasts of the ES are given by

${\mathrm{ES}}_{{n}_{\mathrm{in}}+k}(\alpha )=-\overline{r}+{\widehat{\zeta}}_{{n}_{\mathrm{in}}+k}{\mathrm{ES}}_{\epsilon ,\alpha},$ | (5.2) |

where ${\widehat{\mathrm{ES}}}_{\epsilon}(\alpha )$ is the ES of a standardized $t$-distribution with unit variance. According to McNeil et al (2015, (2.25)) we have, for $\nu >2$,

$${\mathrm{ES}}_{\epsilon}(\alpha )=\frac{{f}_{\nu}[{F}_{\nu}^{-1}(\alpha )]}{1-\alpha}\frac{\nu +{[{F}_{\nu}^{-1}(\alpha )]}^{2}}{\nu -1}\sqrt{\frac{\nu -2}{\nu}},$$ | (5.3) |

where ${f}_{\nu}$ is the density function of a $t$-distribution. It can be shown that under the conditional $t$-distribution we have ${\mathrm{ES}}_{\epsilon}(\alpha )\widehat{=}\mathrm{VaR}({\alpha}^{*})$, where

$${\alpha}^{*}(\nu )={F}_{\nu}[{\mathrm{ES}}_{\epsilon}(\alpha )\sqrt{\nu /(\nu -2)}].$$ |

For $\alpha =0.975$ we have ${\alpha}^{*}$ marginally larger than but almost equal to $0.99$. Consequently, $\widehat{\mathrm{ES}}(0.975)$ is slightly larger than $\widehat{\mathrm{VaR}}(0.99)$. For conventional models ${\widehat{h}}_{{n}_{\mathrm{in}}+k}$ can be obtained by means of OxMetrics, whereas for the semi-FI-log-GARCH ${\widehat{h}}_{{n}_{\mathrm{in}}+k}$ can be calculated directly with (4.10), where the coefficients can be obtained by means of any statistical programming language capable of estimating FARIMA models.

### 5.2 Backtesting VaR and ES

We propose to carry out a traffic-light test for the VaR, as stipulated in Basel Committee on Banking Supervision (2016). This is based on the number of violations (ie, where the losses exceed VaR estimates). Let

${I}_{{n}_{\mathrm{in}}+k}=\{\begin{array}{cc}1\hfill & \text{if}-{r}_{{n}_{\mathrm{in}}+k}{\widehat{\mathrm{VaR}}}_{{n}_{\mathrm{in}}+k}(\alpha ),\hfill \\ \hfill & \\ 0\hfill & \text{otherwise},\hfill \end{array}$ | (5.4) |

be an empirical hit sequence. Then the number of violations $\{{I}_{{n}_{\mathrm{in}}+k}(\alpha )=1\}$ will be denoted by ${N}_{1}$ at $\alpha =97.5\%$ and ${N}_{2}$ at $\alpha =99\%$. In line with Basel Committee on Banking Supervision (2016), the green zone for VaR at $\alpha =97.5\%$ is set to $0\le {N}_{1}\le 10$ and for VaR at $\alpha =99\%$ it is stipulated as $0\le {N}_{2}\le 4$. Then we have ${\mu}_{1}=E({N}_{1})=6.25$ and ${\mu}_{2}=E({N}_{2})=2.5$. Further, we adapt the idea of Costanzino and Curran (2018) for backtesting ES. Define

$${\widehat{\epsilon}}_{{n}_{\mathrm{in}}+k}^{*}=\frac{-({r}_{{n}_{\mathrm{in}}+k}-\overline{r})}{{\widehat{\zeta}}_{{n}_{\mathrm{in}}+k}}\sqrt{\frac{\nu}{\nu -2}}.$$ | (5.5) |

In order to satisfy the conditions required by Costanzino and Curran (2018, (14)), it is assumed that ${\epsilon}_{n+k}^{*}$ are iid $t$-distributed random variables. Moreover, we define

$${w}_{{n}_{\mathrm{in}}+k}=\{\begin{array}{cc}1-\frac{1-{F}_{\nu}({\epsilon}_{{n}_{\mathrm{in}}+k}^{*})}{1-\alpha},\hfill & {I}_{{n}_{\mathrm{in}}+k}=1,\hfill \\ \hfill & \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$ | (5.6) |

Then the test statistic for the ES is a weighted sum of the negative returns that exceed ${\widehat{\mathrm{VaR}}}_{{n}_{\mathrm{in}}+k}(\alpha )$ and is given by

$${T}_{\mathrm{ES}}=\sum _{k=1}^{K}{w}_{{n}_{\mathrm{in}}+k}.$$ | (5.7) |

Note that, according to Costanzino and Curran (2015), $({T}_{\mathrm{ES}}-{\mu}_{T})/\sqrt{K}\sim N({\mu}_{T},{\sigma}_{t}^{2})$ (ie, it is asymptotically normally distributed) with ${\mu}_{T}=(1-\alpha )K/2$ and ${\sigma}_{T}^{2}=(1-\alpha )(1+3\alpha )/12$. For $\alpha =0.975$ and $K=250$, Costanzino and Curran (2018) approximated the asymptotic boundary of the green zone for ${T}_{\mathrm{ES}}$ (with a cumulative probability up to 95%) with 5.48. Moreover, they derived a finite sample distribution for ${T}_{\mathrm{ES}}$ and obtained a finite sample boundary of 5.70 for the green zone. In the following we show that this new traffic light approach works very well in practice, and subsequently we will focus solely on backtesting ES based on ${T}_{\mathrm{ES}}$. A model is considered to pass our backtest if ${N}_{1}$, ${N}_{2}$ and ${T}_{\mathrm{ES}}$ are all situated in the green zone. The best model is then identified by means of a novel selection criterion (the weighted absolute deviation (WAD) criterion) developed by Feng et al (2020), which is defined as

$$\mathrm{WAD}=\frac{|{N}_{1}-{\mu}_{1}|}{{\mu}_{1}}+\frac{|{N}_{2}-{\mu}_{2}|}{{\mu}_{2}}+\frac{|{T}_{\mathrm{ES}}-{\mu}_{T}|}{{\mu}_{T}}.$$ | (5.8) |

An advantage of using the WAD criterion instead of conventional loss functions (see, for example, Sarma et al 2003) is that the WAD score of a model usually does not contradict its corresponding backtest result. The larger the quantities ${N}_{1}$, ${N}_{2}$ and ${T}_{\mathrm{ES}}$, the greater the WAD. By contrast, the firm’s so-called loss function becomes smaller with increasing ${N}_{1}$, ${N}_{2}$ and ${T}_{\mathrm{ES}}$.

## 6 Empirical results

We fit the semi-FI-log-GARCH (SFIL), semi-FIGARCH (SFIG), semi-FIEGARCH (SFIE), semi-FIAPARCH (SFIA) and their parametric equivalents (FIL, FIG, FIE and FIA) with respect to 22 return series of major stock indexes over the period from January 1999 to December 2019. A time span of 20 years is sufficiently large and includes all relevant financial crises (in particular, the 2007–9 global financial crisis) with possible long-term volatility dynamics. Consequently, the implementation of long-memory models, particularly for forecasting VaR and ES, is justified and consistent. Historical return series of major stock indexes from Asia, Europe and the United States are considered in order to evaluate the performance of our proposals in different world markets. We employ the following stock indexes: AEX Index (AEX), ATHEX Composite (ATH), Austrian Trades Index (ATX), CAC 40 (CAC), S&P/TSX Composite Index (CAD), DAX 30 (DAX), Dow Jones Industrial (DJI), EURO STOXX 50 (EST), Financial Times Stock Exchange 100 (FTS), Ireland Stock Exchange Overall Index (ISQ), Hang Seng (HSI), Korea Stock Exchange Composite (KOR), Madrid Stock Exchange General (MAD), Mexico IPC (MEX), Nasdaq Composite (NSQ), Nikkei 225 (NIK), New York Stock Exchange (NYS), OMX Stockholm (OMX), Portugal PSI General (PSI), Russell 2000 (RUS), Standard and Poor’s 500 (S&P) and Swiss Market (SWI). We split the data into a training sample (in-sample) and a test sample (out-of-sample). The training set contains $N-K$ observations, with $N$ being the total number of observations and $K=250$ days for the test-sample.

Initially, local polynomial trend estimation by means of the IPI for SEMIFARIMA models, which is introduced in Section 4.2, is applied to the training data with regard to the semiparametric models. Subsequently, the parametric parts of the models are fitted to the trend-adjusted return series, in order to calculate the one-day rolling forecasts for the test-samples. The semi-FI-log-GARCH is fitted by using the R package fracdiff, and the rolling forecasts are calculated manually according to (4.10). The parametric components of the other models and the rolling forecasts are fitted and obtained by means of the G@RCH 8.0 package, which is implemented in OxMetrics. All models are fitted with order $(1,d,1)$ and are tested at coverage probabilities 97.5% and 99% under conditional normal and $t$-distributions, as required by Basel Committee on Banking Supervision (2016).^{2}^{2} 2 We do not present the results from tests assuming a conditional normal distribution, but they are available from the corresponding author on request.

Semi-FIGARCH | Semi-FIEGARCH | |||||||||

$\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | $\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | ${\widehat{\bm{\theta}}}_{\text{\U0001d7cf}}$ | ${\widehat{\bm{\theta}}}_{\text{\U0001d7d0}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | |

S&P | 0.039 | $-$0.533 | 0.543 | 6.152 | 0.487 | $-$0.982 | $-$0.166 | 0.120 | $-$0.222 | 7.631 |

DAX | 0.079 | $-$0.457 | 0.405 | 8.625 | 0.314 | $-$0.972 | $-$0.141 | 0.122 | $-$0.178 | 9.307 |

EST | 0.087 | $-$0.420 | 0.365 | 8.549 | 0.293 | $-$0.972 | $-$0.162 | 0.105 | $-$0.184 | 10.128 |

NIK | $-$0.018 | $-$0.267 | 0.325 | 7.550 | 0.874 | $-$0.976 | $-$0.109 | 0.135 | $-$0.311 | 8.574 |

FTS | 0.132 | $-$0.453 | 0.404 | 8.978 | $-$0.336 | $-$0.839 | $-$0.163 | 0.141 | 0.361 | 10.919 |

RUS | 0.059 | $-$0.389 | 0.366 | 16.237 | 0.547 | $-$0.978 | $-$0.108 | 0.101 | $-$0.199 | 19.880 |

DJI | 0.062 | $-$0.573 | 0.567 | 6.393 | 0.464 | $-$0.975 | $-$0.136 | 0.126 | $-$0.175 | 8.066 |

NSQ | 0.084 | $-$0.449 | 0.404 | 8.348 | $-$0.167 | $-$0.844 | $-$0.166 | 0.107 | 0.270 | 9.978 |

AEX | 0.078 | $-$0.443 | 0.427 | 9.026 | 0.353 | $-$0.982 | $-$0.156 | 0.120 | $-$0.214 | 11.752 |

ATH | $-$0.029 | $-$0.186 | 0.280 | 5.494 | 0.359 | $-$0.853 | $-$0.050 | 0.125 | 0.160 | 5.755 |

ATX | 0.118 | $-$0.315 | 0.294 | 8.103 | 0.419 | $-$0.982 | $-$0.107 | 0.167 | $-$0.309 | 9.723 |

CAC | 0.065 | $-$0.382 | 0.353 | 8.857 | 0.175 | $-$0.967 | $-$0.170 | 0.107 | $-$0.129 | 10.167 |

CAD | 0.168 | $-$0.482 | 0.373 | 9.014 | 0.245 | $-$0.978 | $-$0.114 | 0.103 | $-$0.169 | 10.974 |

HSI | 0.196 | $-$0.550 | 0.350 | 7.773 | 1.934 | $-$0.970 | $-$0.027 | 0.043 | $-$0.095 | 8.491 |

ISQ | $-$0.133 | $-$0.006 | 0.221 | 7.574 | 0.326 | $-$0.952 | $-$0.090 | 0.132 | $-$0.154 | 8.578 |

KOR | 0.034 | $-$0.309 | 0.293 | 6.800 | 0.668 | $-$0.975 | $-$0.104 | 0.099 | $-$0.254 | 7.363 |

MAD | 0.110 | $-$0.402 | 0.342 | 8.792 | 0.305 | $-$0.978 | $-$0.132 | 0.120 | $-$0.220 | 9.810 |

MEX | 0.189 | $-$0.472 | 0.362 | 7.279 | 0.357 | $-$0.986 | $-$0.102 | 0.134 | $-$0.215 | 8.808 |

NYS | 0.049 | $-$0.487 | 0.483 | 6.877 | 0.537 | $-$0.977 | $-$0.126 | 0.115 | $-$0.186 | 8.578 |

OMX | 0.108 | $-$0.366 | 0.316 | 10.166 | $-$0.349 | $-$0.871 | $-$0.142 | 0.153 | 0.284 | 11.445 |

PSI | 0.155 | $-$0.357 | 0.352 | 6.450 | 0.053 | $-$0.974 | $-$0.128 | 0.241 | $-$0.248 | 7.187 |

SWI | 0.145 | $-$0.551 | 0.506 | 7.332 | $-$0.044 | $-$0.966 | $-$0.173 | 0.157 | $-$0.072 | 9.053 |

Semi-FIAPARCH | Semi-FI-log-GARCH | |||||||||

$\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | $\widehat{\bm{\gamma}}$ | $\widehat{\bm{\delta}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | $\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | |

S&P | — | — | — | — | — | — | 0.137 | $-$0.394 | 0.250 | 5.276 |

DAX | — | — | — | — | — | — | 0.249 | $-$0.487 | 0.252 | 7.545 |

EST | — | — | — | — | — | — | 0.189 | $-$0.381 | 0.196 | 7.216 |

NIK | 0.020 | $-$0.236 | 0.744 | 1.054 | 0.294 | 8.238 | 0.151 | $-$0.356 | 0.204 | 6.602 |

FTS | — | — | — | — | — | — | 0.417 | $-$0.513 | 0.180 | 5.922 |

RUS | — | — | — | — | — | — | 0.253 | $-$0.462 | 0.219 | 12.052 |

DJI | — | — | — | — | — | — | 0.204 | $-$0.467 | 0.278 | 5.789 |

NSQ | — | — | — | — | — | — | 0.152 | $-$0.397 | 0.237 | 6.811 |

AEX | — | — | — | — | — | — | 0.217 | $-$0.411 | 0.215 | 7.598 |

ATH | 0.115 | $-$0.383 | 0.359 | 1.319 | 0.341 | 5.861 | 0.187 | $-$0.277 | 0.157 | 4.378 |

ATX | 0.153 | $-$0.321 | 0.600 | 1.242 | 0.261 | 9.546 | 0.303 | $-$0.442 | 0.189 | 7.035 |

CAC | — | — | — | — | — | — | 0.136 | $-$0.353 | 0.214 | 7.564 |

CAD | 0.291 | $-$0.520 | 1.000 | 1.067 | 0.288 | 10.854 | 0.398 | $-$0.595 | 0.229 | 7.661 |

HSI | 0.217 | $-$0.479 | 0.616 | 1.258 | 0.285 | 8.363 | 0.286 | $-$0.535 | 0.227 | 7.216 |

ISQ | 0.166 | $-$0.334 | 0.634 | 1.160 | 0.242 | 8.341 | 0.056 | $-$0.181 | 0.138 | 6.978 |

KOR | — | — | — | — | — | — | 0.219 | $-$0.391 | 0.167 | 6.101 |

MAD | — | — | — | — | — | — | 0.237 | $-$0.468 | 0.232 | 7.471 |

MEX | 0.280 | $-$0.532 | 0.721 | 1.275 | 0.329 | 8.784 | 0.337 | $-$0.557 | 0.254 | 6.586 |

NYS | — | — | — | — | — | — | 0.128 | $-$0.363 | 0.239 | 6.135 |

OMX | — | — | — | — | — | — | 0.215 | $-$0.300 | 0.138 | 7.104 |

PSI | 0.212 | $-$0.366 | 0.503 | 1.200 | 0.300 | 6.967 | 0.733 | $-$0.741 | 0.117 | 4.821 |

SWI | — | — | — | — | — | — | 0.395 | $-$0.525 | 0.232 | 5.465 |

### 6.1 Fitted model parameters

FIGARCH | FIEGARCH | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | $\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | ${\widehat{\bm{\theta}}}_{\text{\U0001d7cf}}$ | ${\widehat{\bm{\theta}}}_{\text{\U0001d7d0}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | |

S&P | 0.054 | $-$0.590 | 0.581 | 6.677 | 0.370 | $-$0.334 | $-$0.180 | 0.132 | 0.597 | 7.276 |

DAX | 0.074 | $-$0.621 | 0.580 | 8.800 | 0.307 | $-$0.348 | $-$0.130 | 0.145 | 0.621 | 9.126 |

EST | 0.087 | $-$0.599 | 0.549 | 8.506 | 0.238 | $-$0.407 | $-$0.144 | 0.136 | 0.615 | 9.248 |

NIK | 0.060 | $-$0.488 | 0.467 | 7.665 | 1.064 | $-$0.204 | $-$0.080 | 0.150 | 0.577 | 7.911 |

FTS | 0.128 | $-$0.559 | 0.522 | 9.022 | $-$0.039 | $-$0.543 | $-$0.156 | 0.143 | 0.591 | 11.072 |

RUS | 0.076 | $-$0.533 | 0.495 | 15.438 | 0.349 | $-$0.467 | $-$0.104 | 0.112 | 0.590 | 16.725 |

DJI | 0.074 | $-$0.646 | 0.628 | 6.843 | 0.291 | $-$0.431 | $-$0.137 | 0.150 | 0.602 | 7.445 |

NSQ | 0.090 | $-$0.593 | 0.542 | 8.549 | 0.447 | $-$0.232 | $-$0.144 | 0.143 | 0.626 | 9.407 |

AEX | 0.079 | $-$0.581 | 0.569 | 9.120 | $-$0.963 | $-$0.988 | $-$0.166 | 0.378 | 0.765 | 7.786 |

ATH | 0.074 | $-$0.425 | 0.421 | 5.619 | 0.444 | $-$0.467 | $-$0.046 | 0.147 | 0.550 | 5.633 |

ATX | 0.156 | $-$0.442 | 0.388 | 8.205 | 0.617 | 0.058 | $-$0.104 | 0.196 | 0.620 | 9.353 |

CAC | 0.094 | $-$0.599 | 0.548 | 8.850 | 0.228 | $-$0.371 | $-$0.151 | 0.143 | 0.622 | 9.769 |

CAD | 0.155 | $-$0.667 | 0.582 | 8.626 | $-$0.068 | $-$0.632 | $-$0.086 | 0.136 | 0.612 | 9.489 |

HSI | 0.155 | $-$0.707 | 0.554 | 7.693 | 3.818 | $-$0.469 | $-$0.011 | 0.030 | 0.680 | 7.741 |

ISQ | 0.136 | $-$0.427 | 0.386 | 7.382 | 0.117 | $-$0.377 | $-$0.090 | 0.171 | 0.613 | 8.125 |

KOR | 0.120 | $-$0.542 | 0.444 | 7.006 | 0.631 | $-$0.387 | $-$0.064 | 0.119 | 0.646 | 7.025 |

MAD | 0.118 | $-$0.551 | 0.486 | 8.814 | $-$0.330 | $-$0.764 | $-$0.130 | 0.141 | 0.506 | 9.712 |

MEX | 0.205 | $-$0.604 | 0.486 | 7.240 | 0.603 | $-$0.014 | $-$0.098 | 0.174 | 0.684 | 8.369 |

NYS | 0.060 | $-$0.576 | 0.561 | 7.037 | 0.462 | $-$0.304 | $-$0.141 | 0.132 | 0.610 | 7.940 |

OMX | 0.164 | $-$0.613 | 0.518 | 9.993 | 0.176 | $-$0.262 | $-$0.125 | 0.183 | 0.651 | 10.614 |

PSI | 0.195 | $-$0.465 | 0.413 | 7.036 | $-$0.180 | $-$0.534 | $-$0.116 | 0.273 | 0.524 | 6.793 |

SWI | 0.138 | $-$0.607 | 0.573 | 7.449 | $-$0.435 | $-$0.843 | $-$0.182 | 0.170 | 0.413 | 9.073 |

FIAPARCH | FI-log-GARCH | |||||||||

$\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | $\widehat{\bm{\gamma}}$ | $\widehat{\bm{\delta}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | $\widehat{\mathit{\varphi}}$ | $\widehat{\bm{\psi}}$ | $\widehat{\bm{d}}$ | $\bm{\nu}$ | |

S&P | — | — | — | — | — | — | 0.184 | $-$0.586 | 0.406 | 5.085 |

DAX | — | — | — | — | — | — | 0.246 | $-$0.642 | 0.421 | 7.208 |

EST | — | — | — | — | — | — | 0.235 | $-$0.590 | 0.372 | 6.388 |

NIK | 0.107 | $-$0.424 | 0.636 | 1.198 | 0.391 | 8.596 | 0.198 | $-$0.500 | 0.310 | 6.548 |

FTS | — | — | — | — | — | — | 0.397 | $-$0.650 | 0.349 | 5.664 |

RUS | — | — | — | — | — | — | 0.266 | $-$0.617 | 0.371 | 10.720 |

DJI | — | — | — | — | — | — | 0.210 | $-$0.595 | 0.408 | 5.355 |

NSQ | — | — | — | — | — | — | 0.184 | $-$0.611 | 0.432 | 6.556 |

AEX | — | — | — | — | — | — | 0.253 | $-$0.610 | 0.392 | 6.926 |

ATH | 0.128 | $-$0.463 | 0.273 | 1.700 | 0.412 | 5.913 | 0.274 | $-$0.482 | 0.282 | 3.867 |

ATX | 0.219 | $-$0.437 | 0.595 | 1.380 | 0.312 | 10.223 | 0.333 | $-$0.647 | 0.378 | 6.614 |

CAC | — | — | — | — | — | — | 0.191 | $-$0.554 | 0.370 | 7.044 |

CAD | 0.155 | $-$0.667 | 0.582 | 8.626 | $-$0.068 | $-$0.632 | $-$0.086 | 0.136 | 0.612 | 9.489 |

HSI | 0.187 | $-$0.649 | 0.305 | 1.681 | 0.484 | 8.183 | 0.248 | $-$0.694 | 0.436 | 6.985 |

ISQ | 0.241 | $-$0.509 | 0.556 | 1.411 | 0.348 | 8.174 | 0.254 | $-$0.567 | 0.343 | 6.158 |

KOR | 0.143 | $-$0.417 | 0.479 | 1.768 | 0.318 | 6.961 | 0.260 | $-$0.664 | 0.417 | 5.397 |

MAD | — | — | — | — | — | — | 0.245 | $-$0.632 | 0.400 | 6.923 |

MEX | 0.252 | $-$0.571 | 0.465 | 1.699 | 0.404 | 8.310 | 0.315 | $-$0.671 | 0.398 | 5.938 |

NYS | — | — | — | — | — | — | 0.190 | $-$0.566 | 0.391 | 5.570 |

OMX | $-$0.085 | $-$0.978 | 0.697 | 1.292 | 1.155 | 10.845 | 0.334 | $-$0.595 | 0.328 | 6.295 |

PSI | 0.241 | $-$0.422 | 0.412 | 1.695 | 0.325 | 7.119 | 0.547 | $-$0.720 | 0.311 | 4.196 |

SWI | — | — | — | — | — | — | 0.382 | $-$0.613 | 0.341 | 5.126 |

FIG | SFIG | FIE | SFIE | FIA | SFIA | FIL | SFIL | |
---|---|---|---|---|---|---|---|---|

S&P | 0.581 | 0.543 | 0.597 | $-$0.222 | — | — | 0.406 | 0.250 |

DAX | 0.580 | 0.405 | 0.621 | $-$0.178 | — | — | 0.421 | 0.252 |

EST | 0.549 | 0.365 | 0.615 | $-$0.184 | — | — | 0.372 | 0.196 |

NIK | 0.467 | 0.325 | 0.577 | $-$0.311 | 0.391 | 0.294 | 0.310 | 0.204 |

FTS | 0.522 | 0.404 | 0.591 | 0.361 | — | — | 0.349 | 0.180 |

RUS | 0.495 | 0.366 | 0.590 | $-$0.199 | — | — | 0.371 | 0.219 |

DJI | 0.628 | 0.567 | 0.602 | $-$0.175 | — | — | 0.408 | 0.278 |

NSQ | 0.542 | 0.404 | 0.626 | 0.270 | — | — | 0.432 | 0.237 |

AEX | 0.569 | 0.427 | 0.765 | $-$0.214 | — | — | 0.392 | 0.215 |

ATH | 0.421 | 0.280 | 0.550 | 0.160 | 0.412 | 0.341 | 0.282 | 0.157 |

ATX | 0.388 | 0.294 | 0.620 | $-$0.309 | 0.312 | 0.261 | 0.378 | 0.189 |

CAC | 0.548 | 0.353 | 0.622 | $-$0.129 | — | — | 0.370 | 0.214 |

CAD | 0.582 | 0.373 | 0.612 | $-$0.169 | 0.440 | 0.288 | 0.434 | 0.229 |

HSI | 0.554 | 0.350 | 0.680 | $-$0.095 | 0.484 | 0.285 | 0.436 | 0.227 |

ISQ | 0.386 | 0.221 | 0.613 | $-$0.154 | 0.348 | 0.242 | 0.343 | 0.138 |

KOR | 0.444 | 0.293 | 0.646 | $-$0.254 | 0.318 | — | 0.417 | 0.167 |

MAD | 0.486 | 0.342 | 0.506 | $-$0.220 | — | — | 0.400 | 0.232 |

MEX | 0.486 | 0.362 | 0.684 | $-$0.215 | 0.404 | 0.329 | 0.398 | 0.254 |

NYS | 0.561 | 0.483 | 0.610 | $-$0.186 | — | — | 0.391 | 0.239 |

OMX | 0.518 | 0.316 | 0.651 | 0.284 | 1.155 | — | 0.328 | 0.138 |

PSI | 0.413 | 0.352 | 0.524 | $-$0.248 | 0.325 | 0.300 | 0.311 | 0.117 |

SWI | 0.573 | 0.506 | 0.413 | $-$0.072 | — | — | 0.341 | 0.232 |

The estimated parameters $\widehat{\varphi}$, $\widehat{\psi}$, the long-memory parameter $\widehat{d}$ and the degrees of freedom $\widehat{\nu}$ for all semi-parametric and parametric models are shown in Tables 1 and 2.^{3}^{3} 3 Standard errors are omitted to save space but are available from the corresponding author upon request. For the SFIE and FIE, sign and magnitude effects, denoted by ${\widehat{\theta}}_{1}$ and ${\widehat{\theta}}_{2}$ are additionally shown. Finally, $\widehat{\gamma}$ and $\widehat{\delta}$ represent the leverage-effect parameter and the transformation parameter of SFIA and FIA, respectively. It can be seen that the range of values of $\widehat{\varphi}$ and $\widehat{\psi}$ for SFIG, SFIA and SFIL are comparable. However, $\widehat{d}$ of SFIG is usually larger and shows values larger than 0.5. The estimated parameters for SFIE deviate substantially. For this model, $\widehat{d}$ is negative and the absolute values of $\widehat{\varphi}$ and $\widehat{\psi}$ are clearly larger than the corresponding estimates for the other semi-parametric models in most cases. Moreover, $\widehat{d}$ exhibits smaller values in the semi-parametric models compared with the parametric counterparts (see Table 3).^{4}^{4} 4 All estimates are statistically significant at conventional confidence levels. This is due to the fact that the deterministic component is removed from the original process, before the stochastic part is further analyzed by means of a parametric model. As a consequence, if an underlying deterministic scale function is ignored, it could be falsely captured as persistence or short-term dependence in the data. This can lead to unstable volatility predictions, which in turn translate to risk measures. A significantly smaller $\widehat{d}$ can be observed for all SFIL models, and surprisingly, the long-memory parameter is negative for all SFIE models, although it is above 0.5 in most cases for all FIE models (see Tables 1–3).

The fitted FIE models deliver surprising results in some cases. In particular with regard to the AEX (respectively, HSI) series, it seems that the FIE extremely underestimates (respectively, overestimates) the AR component, while these obviously biased estimates are somewhat corrected by the SFIE. In most cases, it is not possible to estimate the FIA or SFIA models, as the estimation algorithm is not converging, which might be caused by the in-sample size (although it is relatively large) still being too small.^{5}^{5} 5 We found during our research that increasing the in-sample period up to 30 years or more resulted in a more stable behavior of the FIAPARCH model. Estimation results for the FIAPARCH of major stock indexes (namely S&P, DJI, NSQ, RUS, DAX, EST, NIK and FTS) are available upon request. Nevertheless, the estimated model parameters of the SFIL, SFIG, FIL and FIG are robust throughout all the stock indexes.

### 6.2 Backtesting results

Semi-FIGARCH | Semi-FIEGARCH | Semi-FIAPARCH | Semi-FI-log-GARCH | |||||||||||||||||

${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | ${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | ${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | ${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | |

S&P | 7 | 2 | 2 | 3.31 | 0.38 | 6 | 3 | 3 | 3.67 | 0.42 | — | — | — | — | — | 6 | 2 | 2 | 2.65 | 0.39 |

DAX | 7 | 5 | 4 | 4.78 | 1.65 | 7 | 5 | 5 | 5.33 | 1.83 | — | — | — | — | — | 7 | 5 | 4 | 4.79 | 1.65 |

EST | 11 | 6 | 6 | 6.37 | 3.20 | 10 | 7 | 7 | 6.85 | 3.59 | — | — | — | — | — | 11 | 7 | 6 | 6.87 | 3.76 |

NIK | 6 | 2 | 2 | 2.35 | 0.49 | 3 | 1 | 1 | 1.41 | 1.67 | 3 | 1 | 1 | 1.64 | 1.60 | 3 | 2 | 2 | 1.38 | 1.28 |

FTS | 7 | 5 | 4 | 4.97 | 1.71 | 7 | 4 | 4 | 4.58 | 1.18 | — | — | — | — | — | 6 | 3 | 3 | 3.51 | 0.36 |

RUS | 5 | 3 | 3 | 3.15 | 0.41 | 4 | 2 | 2 | 2.11 | 0.89 | — | — | — | — | — | 3 | 2 | 2 | 1.76 | 1.16 |

DJI | 6 | 1 | 1 | 2.64 | 0.80 | 5 | 1 | 1 | 1.54 | 1.31 | — | — | — | — | — | 5 | 3 | 2 | 2.44 | 0.62 |

NSQ | 5 | 2 | 2 | 2.39 | 0.64 | 4 | 2 | 2 | 2.11 | 0.88 | — | — | — | — | — | 3 | 2 | 2 | 2.01 | 1.08 |

AEX | 6 | 3 | 3 | 3.55 | 0.37 | 4 | 3 | 3 | 2.91 | 0.63 | — | — | — | — | — | 4 | 3 | 3 | 3.24 | 0.60 |

ATH | 4 | 4 | 4 | 3.14 | 0.97 | 4 | 4 | 3 | 3.27 | 1.01 | 4 | 4 | 4 | 3.30 | 1.02 | 5 | 2 | 2 | 2.26 | 0.68 |

ATX | 2 | 0 | 0 | 0.78 | 2.43 | 1 | 0 | 0 | 0.19 | 2.78 | 1 | 0 | 0 | 0.28 | 2.75 | 1 | 0 | 0 | 0.32 | 2.74 |

CAC | 5 | 3 | 3 | 3.26 | 0.44 | 5 | 4 | 4 | 3.64 | 0.96 | — | — | — | — | — | 5 | 3 | 3 | 2.80 | 0.51 |

CAD | 2 | 1 | 1 | 1.00 | 1.96 | 1 | 0 | 0 | 0.43 | 2.70 | 1 | 0 | 0 | 0.54 | 2.67 | 1 | 0 | 0 | 0.28 | 2.75 |

HSI | 8 | 4 | 4 | 4.21 | 1.23 | 9 | 2 | 2 | 3.83 | 0.87 | 8 | 2 | 2 | 3.56 | 0.62 | 9 | 3 | 3 | 3.90 | 0.89 |

ISQ | 2 | 0 | 0 | 0.51 | 2.52 | 0 | 0 | 0 | 0.00 | 3.00 | 0 | 0 | 0 | 0.00 | 3.00 | 0 | 0 | 0 | 0.00 | 3.00 |

KOR | 8 | 2 | 2 | 3.76 | 0.68 | 6 | 2 | 2 | 2.51 | 0.44 | — | — | — | — | — | 7 | 2 | 2 | 3.03 | 0.35 |

MAD | 4 | 3 | 3 | 2.77 | 0.67 | 4 | 3 | 3 | 2.47 | 0.77 | — | — | — | — | — | 5 | 2 | 1 | 2.38 | 0.64 |

MEX | 2 | 0 | 0 | 0.22 | 2.61 | 1 | 0 | 0 | 0.35 | 2.73 | 1 | 0 | 0 | 0.21 | 2.77 | 1 | 0 | 0 | 0.12 | 2.80 |

NYS | 5 | 1 | 1 | 2.94 | 0.86 | 4 | 2 | 1 | 2.04 | 0.91 | — | — | — | — | — | 5 | 1 | 1 | 1.98 | 1.17 |

OMX | 8 | 3 | 3 | 3.48 | 0.60 | 8 | 2 | 2 | 2.33 | 0.73 | — | — | — | — | — | 4 | 0 | 0 | 1.26 | 1.96 |

PSI | 6 | 2 | 2 | 2.39 | 0.47 | 5 | 2 | 1 | 2.01 | 0.76 | 5 | 2 | 2 | 2.00 | 0.76 | 3 | 1 | 0 | 1.11 | 1.76 |

SWI | 5 | 2 | 2 | 2.61 | 0.57 | 5 | 1 | 1 | 1.67 | 1.27 | — | — | — | — | — | 2 | 0 | 0 | 0.45 | 2.54 |

FIGARCH | FIEGARCH | FIAPARCH | FI-log-GARCH | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | ${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | ${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | ${\bm{N}}_{\text{\U0001d7cf}}$ | ${\bm{N}}_{\text{\U0001d7d0}}$ | ES | ${\bm{T}}_{\text{\mathbf{E}\mathbf{S}}}$ | WAD | |

S&P | 7 | 4 | 3 | 4.53 | 1.17 | 7 | 6 | 6 | 4.79 | 2.05 | — | — | — | — | — | 6 | 5 | 4 | 4.17 | 1.37 |

DAX | 8 | 6 | 5 | 5.18 | 2.34 | 9 | 5 | 5 | 5.41 | 2.17 | — | — | — | — | — | 7 | 6 | 5 | 5.15 | 2.17 |

EST | 10 | 7 | 7 | 6.80 | 3.58 | 11 | 6 | 6 | 7.35 | 3.51 | — | — | — | — | — | 10 | 6 | 6 | 7.08 | 3.27 |

NIK | 9 | 4 | 4 | 4.75 | 1.56 | 7 | 4 | 3 | 4.04 | 1.01 | 7 | 2 | 2 | 3.73 | 0.51 | 7 | 2 | 2 | 2.69 | 0.46 |

FTS | 7 | 4 | 4 | 5.04 | 1.33 | 7 | 4 | 4 | 4.93 | 1.30 | — | — | — | — | — | 5 | 3 | 3 | 3.24 | 0.44 |

RUS | 6 | 4 | 3 | 4.07 | 0.94 | 7 | 4 | 3 | 4.44 | 1.14 | — | — | — | — | — | 6 | 3 | 2 | 2.69 | 0.38 |

DJI | 9 | 6 | 5 | 5.33 | 2.54 | 9 | 6 | 6 | 5.28 | 2.53 | — | — | — | — | — | 7 | 5 | 4 | 4.91 | 1.69 |

NSQ | 6 | 3 | 3 | 3.74 | 0.44 | 6 | 5 | 5 | 4.49 | 1.48 | — | — | — | — | — | 4 | 3 | 3 | 3.14 | 0.56 |

AEX | 7 | 3 | 3 | 3.70 | 0.50 | 8 | 3 | 3 | 4.41 | 0.89 | — | — | — | — | — | 5 | 3 | 3 | 3.62 | 0.56 |

ATH | 4 | 4 | 4 | 3.41 | 1.05 | 4 | 4 | 3 | 3.41 | 1.05 | 4 | 4 | 4 | 3.46 | 1.07 | 3 | 1 | 1 | 1.36 | 1.68 |

ATX | 4 | 1 | 1 | 1.77 | 1.39 | 1 | 1 | 1 | 0.73 | 2.21 | 2 | 1 | 1 | 0.79 | 2.03 | 2 | 1 | 1 | 0.84 | 2.01 |

CAC | 9 | 5 | 5 | 5.42 | 2.17 | 9 | 5 | 5 | 5.54 | 2.21 | — | — | — | — | — | 8 | 4 | 3 | 4.59 | 1.35 |

CAD | 4 | 2 | 2 | 1.79 | 0.99 | 6 | 2 | 2 | 2.10 | 0.57 | 4 | 2 | 2 | 1.93 | 0.94 | 1 | 1 | 0 | 0.63 | 2.24 |

HSI | 9 | 5 | 5 | 5.25 | 2.12 | 8 | 3 | 3 | 4.23 | 0.83 | 8 | 3 | 3 | 4.95 | 1.07 | 9 | 3 | 3 | 5.58 | 1.43 |

ISQ | 6 | 2 | 2 | 1.99 | 0.60 | 5 | 1 | 0 | 2.02 | 1.15 | 6 | 1 | 1 | 2.27 | 0.91 | 5 | 0 | 0 | 1.32 | 1.78 |

KOR | 12 | 7 | 5 | 7.75 | 4.20 | 10 | 3 | 3 | 5.35 | 1.51 | 13 | 5 | 5 | 6.99 | 3.32 | 8 | 3 | 3 | 4.88 | 1.04 |

MAD | 7 | 4 | 3 | 3.72 | 0.91 | 6 | 3 | 3 | 3.63 | 0.40 | — | — | — | — | — | 6 | 4 | 4 | 3.65 | 0.81 |

MEX | 7 | 3 | 3 | 3.73 | 0.52 | 6 | 2 | 2 | 2.85 | 0.33 | 6 | 3 | 2 | 3.23 | 0.27 | 6 | 2 | 2 | 2.49 | 0.44 |

NYS | 9 | 5 | 4 | 4.95 | 2.02 | 8 | 5 | 4 | 4.79 | 1.81 | — | — | — | — | — | 7 | 4 | 3 | 3.85 | 0.95 |

OMX | 11 | 5 | 4 | 5.86 | 2.64 | 11 | 5 | 5 | 5.83 | 2.63 | 10 | 2 | 2 | 4.67 | 1.29 | 5 | 0 | 0 | 1.87 | 1.60 |

PSI | 7 | 2 | 2 | 3.47 | 0.43 | 5 | 2 | 2 | 2.54 | 0.59 | 7 | 2 | 2 | 2.95 | 0.38 | 4 | 1 | 0 | 1.85 | 1.37 |

SWI | 5 | 4 | 4 | 3.37 | 0.88 | 7 | 2 | 2 | 3.63 | 0.48 | — | — | — | — | — | 4 | 0 | 0 | 1.12 | 2.00 |

The test statistics, ${N}_{1}$, ${N}_{2}$, ${T}_{\mathrm{ES}}$ and the WAD values for all semi-parametric and parametric models are listed in Tables 4 and 5, respectively. A model is considered as having passed the traffic-light test if its quantities ${N}_{1}$, ${N}_{2}$ and ${T}_{\mathrm{ES}}$ are all within their green zones. Among such models, the one with the smallest WAD value is determined to be the most suitable or accurate in terms of predicting VaR and ES. Here, ${N}_{\mathrm{ES}}$ denotes the number of times the out-of-sample losses exceeded the ES. However, this statistic is only provided for further information and is not considered in the traffic-light test. We have ${N}_{\mathrm{ES}}\le {N}_{2}$. Most models pass the traffic-light test. In fact, there is no suitable model for only two return series (DAX and EST). Overall, our results indicate that taking into account a potential underlying deterministic scale function could be crucial for passing the traffic-light test. This becomes particularly obvious for the FIG and SFIG. As illustrated in Tables 4 and 5, the FIG models for DJI, CAC, HSI, KOR, NYS and OMX do not pass the traffic-light test, whereas the SFIG models for the same series do. The same is true for the FIE and SFIE for DJI, NSQ, CAC, NYS and OMX as well as for the FIL and SFIL for S&P and DJI.

Figure 1 illustrates the one-step rolling forecasts of the VaR at 97.5% and 99% for the DJI series. The blue and red dashed lines indicate the VaR at 97.5% and 99%, respectively. The corresponding breaches are exemplified by the colored circles and triangles. Apparently, the parametric models underestimate VaR at 99%, indicated by the fact that all three models are located in the yellow zone, with ${N}_{2}=6$ for FIG and FIE and ${N}_{2}=5$ for the FIL (see Table 5). In contrast, when considering a nonparametric trend in the data, the number of breaches is reduced to ${N}_{2}=1$ for SFIG and SFIE and ${N}_{2}=3$ for the SFIL (see Figure 1 and Table 4). Moreover, we observe that $$ is empirically implied if both ${N}_{1}$ and ${N}_{2}$ are located within their green zones. Nonetheless, we observed some cases where ${N}_{2}$ is in its yellow zones but $$ is still satisfied. However, if both quantities are clearly settled in their yellow zones, ${T}_{\mathrm{ES}}$ will also be in the same zone, which indicates that ${T}_{\mathrm{ES}}$ is a useful statistic for backtesting the ES.

FIG | SFIG | FIE | SFIE | FIA | SFIA | FIL | SFIL | |

S&P | 1.17 | 0.38 | 2.05 | 0.42 | — | — | 1.37 | 0.39 |

DAX | 2.34 | (1.65) | 2.17 | 1.83 | — | — | 2.17 | (1.65) |

EST | 3.58 | (3.20) | 3.51 | 3.59 | — | — | 3.27 | 3.76 |

NIK | 1.56 | 0.49 | 1.01 | 1.67 | 0.51 | 1.60 | 0.46 | 1.28 |

FTS | 1.33 | 1.71 | 1.30 | 1.18 | — | — | 0.44 | 0.36 |

RUS | 0.94 | 0.41 | 1.14 | 0.89 | — | — | 0.38 | 1.16 |

DJI | 2.54 | 0.80 | 2.53 | 1.31 | — | — | 1.69 | 0.62 |

NSQ | 0.44 | 0.64 | 1.48 | 0.88 | — | — | 0.56 | 1.08 |

AEX | 0.50 | 0.37 | 0.89 | 0.63 | — | — | 0.56 | 0.60 |

ATH | 1.05 | 0.97 | 1.05 | 1.01 | 1.07 | 1.02 | 1.68 | 0.68 |

ATX | 1.39 | 2.43 | 2.21 | 2.78 | 2.03 | 2.75 | 2.01 | 2.74 |

CAC | 2.17 | 0.44 | 2.21 | 0.96 | — | — | 1.35 | 0.51 |

CAD | 0.99 | 1.96 | 0.57 | 2.70 | 0.94 | 2.67 | 2.24 | 2.75 |

HSI | 2.12 | 1.23 | 0.83 | 0.87 | 1.07 | 0.62 | 1.43 | 0.89 |

ISQ | 0.60 | 2.52 | 1.15 | 3.00 | 0.91 | 3.00 | 1.78 | 3.00 |

KOR | 4.20 | 0.68 | 1.51 | 0.44 | 3.32 | — | 1.04 | 0.35 |

MAD | 0.91 | 0.67 | 0.40 | 0.77 | — | — | 0.81 | 0.64 |

MEX | 0.52 | 2.61 | 0.33 | 2.73 | 0.27 | 2.77 | 0.44 | 2.80 |

NYS | 2.02 | 0.86 | 1.81 | 0.91 | — | — | 0.95 | 1.17 |

OMX | 2.64 | 0.60 | 2.63 | 0.73 | 1.29 | — | 1.60 | 1.96 |

PSI | 0.43 | 0.47 | 0.59 | 0.76 | 0.38 | 0.76 | 1.37 | 1.76 |

SWI | 0.88 | 0.57 | 0.48 | 1.27 | — | — | 2.00 | 2.54 |

$\text{min}(\text{WAD})$ | 3 | 5+(2) | 3 | 0 | 2 | 1 | 2 | 4+(1) |

Further, the proposed WAD model selection criterion measures the overall forecasting quality of a fitted model. An overview of the WAD values for all models across all series is given in Table 6. As shown, the FIG has the lowest WAD for NSQ, ATX and ISQ, whereas the SFIG performs best for S&P, DAX, EST, AEX, CAC, NYS and OMX. The FIE is superior for CAD, MAD as well as SWI, while the SFIE could not outperform the other models. The FIA performs best for MEX as well as PSI, while the SFIA performs best for HSI. The FIL exhibits minimum WAD values for NIK as well as RUS, while the SFIL shows minimum WAD values for DAX, FTS, DJI, ATH and KOR. In total, 12 semi-parametric models and 10 parametric models have a minimum WAD. However, the models for DAX and EST did not pass the traffic-light test (see Tables 4 and 5).

Overall, our results show that semi-LM-GARCH models, in particular the SFIG and SFIL, provide a convincing alternative to conventional parametric models when measuring market risk. If a parametric model does not perform well, it is worthwhile checking whether a semi-parametric model might deliver better results.

## 7 Conclusion

This paper introduced different classes of semi-LM-GARCH models that are able to improve the simultaneous modeling of conditional heteroscedasticity and a slowly changing unconditional variance. We employed a SEMIFARIMA model with a local polynomial smoother in order to estimate the deterministic component. Bandwidth selection was carried out by means of the SEMIFARIMA-algorithm, which was translated from the S programming language to R by Letmathe et al (2021).

We employed our semi-LM-GARCH models for an out-of-sample forecasting of VaR and ES. The performance of our proposals was assessed via traffic-light tests for both risk measures. In addition, a recently introduced model selection criterion, the WAD, was applied. Our results indicate that semi-LM-GARCH approaches are a meaningful substitute for parametric LM-GARCH models. Against this background, our models and results may help both banking supervisors and banks to further improve the ES measure and the processes for backtesting it. Further, the performance of each model is dependent on the market. Therefore, it is advisable for both risk managers and regulators to constantly monitor and benchmark a variety of models.

A comprehensive comparative study of our proposals and conventional methods (eg, historical simulation) will be conducted in future research. Moreover, the accuracy of VaR and ES forecasting could be improved further still by extending our proposals with conditional distributions that allow for modeling skewness (see Iqbal et al 2020). However, to the best of our knowledge, closed-form expressions of asymmetric, fat-tailed distributions for calculating the ES have only been partially established so far. Alternatively, the empirical distribution function of the observed return series or estimated VaR and ES could be bootstrapped based on the empirical quantiles of the residuals (see El Ghourabi et al (2016) and Francq and Zakoïan (2015) for ideas along these lines).

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## Acknowledgements

This work was supported by the German DFG Project FE 1500/2-1. We thank the two anonymous referees as well as Shujie Li and Dominik Schulz for helpful comments and discussions.

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