Forecasting volatility with time-varying leverage and volatility of volatility effects

International Journal of Forecasting xxx (xxxx) xxx

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International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast

Forecasting volatility with time-varying leverage and volatility of volatility effects ∗

Leopoldo Catania a , , Tommaso Proietti a,b a b

Aarhus University and CREATES, Denmark University of Rome Tor Vergata, Italy

article

info

Keywords: Realized volatility Leverage effect Volatility of volatility Score driven models Volatility prediction

a b s t r a c t Predicting volatility is of primary importance for business applications in risk management, asset allocation, and the pricing of derivative instruments. This paper proposes a measurement model that considers the possibly time-varying interaction of realized volatility and asset returns according to a bivariate model to capture its major characteristics: (i) the long-term memory of the volatility process, (ii) the heavy-tailedness of the distribution of returns, and (iii) the negative dependence of volatility and daily market returns. We assess the relevance of the effects of ‘‘the volatility of volatility’’ and time-varying ‘‘leverage’’ to the out-of-sample forecasting performance of the model, and evaluate the density of forecasts of market volatility. Empirical results show that our specification can outperform the benchmark HAR–GARCH model in terms of both point and density forecasts. © 2020 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction Market volatility plays an important role in applied finance. Starting from the seminal work by Andersen, Bollerslev, Diebold, and Labys (2001), the possibility of estimating market volatility using intraday returns paved the way for a myriad of applications in risk management, asset allocation, and the pricing of derivative instruments. The properties of the time series of realized volatility have since been investigated in a series of studies that have documented the long-term memory of the series. In a highly influential paper, Corsi (2009) proposed the heterogeneous autoregressive model (HAR–RV) that accounts for long-term memory via a long autoregression, parsimoniously parameterized in terms of three parameters. In addition to long-term memory, the ‘‘volatility of volatility’’ is of interest. Specifically, empirical evidence is available for the time-varying second-conditional ∗ Corresponding author. E-mail address: [email protected] (L. Catania).

moment of realized volatility measures. For instance, Bollerslev, Tauchen, and Zhou (2009) investigated this aspect with reference to the temporal variation of the expected returns. Their results suggest that accounting for the volatility of volatility is of primary importance for forecasting. Other recent applications have dealt with the related concept of ‘‘quarticity’’ and its implications for prediction (see, for example, Bollerslev, Patton, and Quaedvlieg (2016)). A number of new contributions that jointly model the realized volatility and market returns have also been proposed. The idea behind these models is to appropriately account for another important characteristic of realized volatility that concerns its negative dependence with daily market returns (i.e., the so called ‘‘leverage effect’’). The most notable contributions are by Shephard and Sheppard (2010) and Takahashi, Omori, and Watanabe (2009). While the multivariate extensions of this framework are not trivial, contributions in this direction have been made by Hansen, Lunde, and Voev (2014), Noureldin, Shephard, and Sheppard (2012), and Opschoor, Janus, Lucas, Opschoor, and Van Dijk (2016). Also of remarkable interest are the working papers by Asai, Chang,

https://doi.org/10.1016/j.ijforecast.2020.01.003 0169-2070/© 2020 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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and McAleer (2016), Hansen, Janus, and Koopman (2016), and Lucas and Opschoor (2016). In this paper, we propose a bivariate model of realized volatility and market returns to account for the well-established features mentioned above. In particular, (i) the conditional mean of the realized volatility series can show high persistence, (ii) the volatility of volatility is allowed to vary over time according to an observationdriven process, (iii) the conditional correlation of volatility and asset returns is allowed to be nonzero and timevarying, and (iv) both the returns and the realized volatility measure follow a heavy-tailed distribution. Our specification nests the model with no leverage and a constant volatility of volatility. We build our new model in a fully observation-driven framework and depart from the Realized Volatility– Stochastic Volatility (RV–SV) type of models introduced by Takahashi et al. (2009). Indeed, our proposal is rooted in recent advances in score-driven (SD) models, also known as generalized autoregressive score (GAS) and dynamic conditional score (DCS) models, see Creal, Koopman, and Lucas (2013) and Harvey (2013). SD models offer a good trade-off between the flexibility and complexity of time-series models, and can address a series of relevant empirical problems in a variety of fields while remaining in a fully observation-driven framework. For further details on the SD approach, see Koopman, Lucas, and Scharth (2016) and the papers listed on the website www.gasmodels.com. Our approach is close to the realized GARCH (RGARCH) model of Hansen, Huang, and Shek (2012) and its DCS extension by Huang, Wang, and Zhang (2014). The RGARCH is a joint model of returns and realized volatility, but our specification differs in several respects, including the way we handle the persistence of the conditional mean of the realized volatility, and the possibility that both the volatility of volatility and leverage vary over time. Moreover, the RGARCH makes no assumption on the joint conditional distribution of the series, whereas we assume a bivariate student’s t distribution. Huang et al. (2014) extended the RGARCH specification by adding a distributional assumption for the underlying disturbances. The forecasting ability of the proposed model is assessed through an extensive empirical application based on a daily time series of asset returns and realized volatility constructed from the Trades and Quotes (TAQ) database, referring to all companies listed on the Dow Jones index (DJIA) and the S&P 500 ETF (SPY) index. We investigate the usefulness of accounting for the aforementioned features for point and density prediction of market volatility. Our results show that the model can account for the main facts of the time series of asset returns and realized volatility, and can outperform the benchmark HAR–GARCH specification proposed by Corsi, Mittnik, Pigorsch, and Pigorsch (2008), which encompasses the HAR-RV model, in terms of both point and density forecasts. The remainder of this paper is organized in the following manner: Section 2 introduces the proposed model and discusses its main properties. Section 3 completes the specification of the model by providing a dynamic

model for time-varying parameters driven by the score of predictive density. Section 4 reports an analysis of the shape of the response of future volatility with respect to current shocks. Section 5 reports the empirical forecasting application. Finally, Section 6 concludes the results of this study and suggests possible developments. 2. New joint model for market returns and realized volatility Let yt ∈ R denote the financial return at time t ∈ N and let xt ∈ R be its associated realized log-volatility measure, i.e., the logarithm of the realized standard deviation. Conditional on Ft −1 = σ (yt −s , xt −s , s > 0), the bivariate vector zt = (yt , xt )′ is modeled as follows:

⏐ ⏐

zt ⏐Ft −1 ∼ T µt , Σ t , ν ,

µt =

(

[ ] 0

µt

,

)

Σt =

[ 2 σt ct

(1)

]

ct , q2t

where T denotes the bivariate student’s t distribution with ν degrees of freedom, with probability density function Γ ((ν + 2)/2) p(zt |Ft −1 ) = Γ (ν/2)(ν − 2)π|Σ t |1/2

( ×

1+

1

ν−2

)−(ν+2)/2

1 (zt − µt )′ Σ − t (zt − µt )

,

(2)

and µt = E(zt |Ft −1 ), Σ t = Var(zt |Ft −1 ). A distinctive feature of our specification is that the conditional mean of the realized log-volatility, µt ∈ R, enters the expression of the conditional standard deviation of the returns series according to the following specification:

σt = exp(α + βµt ), according to which the log-volatility of asset returns is an affine transformation of the conditional mean of xt , where α ∈ R and β ∈ R are two coefficients correcting for possible biases affecting the measurement of volatility via xt . Note that by setting the conditional mean of yt equal to zero, we assume that financial returns are a martingale difference sequence with respect to Ft −1 . The dynamic model for the evolution of the parameters µt , ct , and qt , specified in the next section, implies that they are measurable with respect to Ft −1 . For the degrees of freedom, we assume ν > 2. Our specification also implies that both yt and xt , conditional on Ft −1 , are univariate student’s t distributed, see Kotz and Nadarajah (2004, pp. 15–16). The conditional covariance between yt and xt is responsible for the leverage effect. It is parameterized as ct = σt qt ρt , where ρt ∈ (−1, 1) is the time-varying conditional correlation. In the financial econometrics literature, there is no consensus on the definition of the leverage effect. The traditional one dates back at least to Crane (1959), and relates equity returns to changes in the levels of their volatility, i.e., when stock price falls, the future volatility increases, and vice versa. The theoretical background for this reasoning has been principally developed by financial economists starting from the seminal

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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work by Black and Scholes (1973), and is grounded on the well-known Modigliani–Miller framework, see Black (1976) and Christie (1982). Statistical models have incorporated this notion in different ways. In the GARCH framework (Bollerslev, 1986; Engle, 1982), the leverage effect is the asymmetric response of the conditional volatility process to past returns, i.e., ∂ Var(yt |Ft −1 )/∂ yt −1 is larger when yt < 0 than when yt ≥ 0; see, for example, Nelson (1991), Glosten, Jagannathan, and Runkle (1993), and Zakoian (1994). As noted by Yu (2005), in the stochastic volatility (SV) framework (Taylor, 1986), two specifications coexist. Specifically, the asymmetric SV model of Harvey and Shephard (1996) (ASV1) is the following non-linear unobserved model of Gaussian components for yt : yt = ξ exp(wt /2)εt , wt +1 = φwt + κηt , where E[εt ηt ] = ρ . On the contrary, Jacquier, Polson, and Rossi (2004) assumed that E[εt ηt +1 ] = ρ (ASV2) so that if we set ut +1 = ηt , the dynamic model for log-volatility can be written in contemporaneous form, yt = ξ exp(wt /2)εt , wt = φwt −1 + κ ut , with E[εt ξt ] = ρ . This implies that in the ASV1, an ‘‘intertemporal leverage effect,’’ (i.e., the common specification), is assumed by introducing a contemporaneous correlation between shocks in the transition and in the measurement equations, E[εt ηt ] = ρ . On the contrary, in the AS2 specification, a ‘‘contemporaneous leverage effect’’ is assumed by introducing the intertemporal correlation E[εt ηt +1 ] = ρ . This difference has two important implications. First, in the ASV1, {yt } is a martingale difference sequence, E [yt |Ft −1 ] = 0, such that the model is consistent with the efficient market hypothesis of Fama (1970). The same is not true for ASV2, where E [yt |Ft −1 ] ̸ = 0 unless ρ = 0. Second, in the ASV1 model, there is no room for a ‘‘marginal leverage effect’’, that is, Cov(yt , wt ) = 0 for any choice of ρ , whereas in ASV2, Cov(yt , wt ) ̸ = 0 unless ρ = 0. As is evident from (1), our model incorporates the leverage effect by assuming a contemporaneous correlation between returns and the conditional volatility. This is still consistent with the efficient market hypothesis as the model is formulated in terms of the conditional bivariate distribution of asset returns and the log-volatility process, where the conditional mean of the former is postulated to be equal to zero. Moreover, as becomes clear in Section 4, our specification implies an intertemporal leverage effect that arises as a byproduct of the score-driven dynamics that we postulate for µt . 3. Dynamic model for time-varying parameters Elements of the conditional covariance matrix of zt , after a transformation, vary over time according to a dynamic model driven by the score of the conditional density in (2). The mechanism of updating the parameters µt , ρt , and qt is based on the recent score-driven (SD) framework introduced by Creal et al. (2013) and Harvey (2013). SD models build a dynamic update equation exploiting information contained in the score of a conditional distribution of the data. This way of introducing time variation has proven to be very flexible, and has been used in a number of different applied settings—see, e.g., Koopman et al. (2016).

3

It is convenient to formulate an SD update model for a smooth transformation of the original parameters to render the range of their values unconstrained. In particular, for ρt and qt we define:

ρt = (1 − exp(−˜ ρt ))/(1 + exp(−˜ ρt )),

(3)

qt = exp(˜ qt ),

so that, e.g., ˜ ρt is twice the Fisher transformation of the conditional correlation parameter; we formulate the following score-driven models:

µt

=

(2) κ µ + µ(1) t + µt ,

µ(i) t ˜ qt ˜ ρt

=

bi µt −1 + ai st −1 ,

= =

κ (1 − b ) + + bq˜ qt −1 , ρ ρ ρ ρ κ (1 − b ) + a st −1 + bρ˜ ρt −1 .

µ

µ µ

(i)

q

µ

i = 1, 2,

q aq st −1

q

q

(4)

ρ

The vector st = (st , st , st )′ is the score of the conditional distribution of zt , i.e., the vector of partial derivatives of the joint conditional density in (2) with respect to µt , ˜ qt , and ˜ ρt , respectively, so that:

] [ ν+2 ∂ωt ∂ωt − β, + βσt 2(1 + ωt ) ∂µt ∂σt ν + 2 ∂ωt q st = −qt − 1, 2(1 + ωt ) ∂ qt ] [ 2 exp(−˜ ρt ) ρ ν + 2 ∂ωt ρ , st = − (1 + exp(−˜ ρt ))2 1 − ρ 2 2(1 + ωt ) ∂ρt µ

st = −

(5)

where

∂ωt 2ρt σt qt − σt2 (xt − µt ) = , ∂µt (ν − 2)σt2 q2t (1 − ρt2 ) ∂ωt 2ρt σt qt yt (xt − µt ) − 2y2t q2t = , ∂σt σt3 (ν − 2)q2t (1 − ρt2 ) ∂ωt 2ρt σt qt yt (xt − µt ) − 2σt2 (xt − µt )2 , = ∂ qt (ν − 2)σt2 (1 − ρt2 )q3t [ 2 2 ] 2 ρ (yt qt + (xt − µt )2 σt2 ) − σt qt yt (xt − µt )(1 + ρ 2 ) ∂ωt . = ∂ρt (ν − 2)σt2 q2t (1 − ρt2 )2 (6) µ

In Eq. (4), the coefficients ai , aq , and aρ determine the reaction of the model parameters to the new information µ provided by the score, whereas the coefficients bi , bq , ρ and b determine the persistence of the evolution of the model parameters. To impose the weak stationarity of µt , µ ˜ qt , and ˜ ρt , we impose |bi |< 1, i = 1, 2, |bq | < 1, and |bρ | < 1, respectively. Note that under correct model specifications, st is a martingale difference sequence, implying that E(µt ) = κ µ , E(˜ qt ) = κ q , and E(˜ ρt ) = κ ρ . A key fact of log-realized volatility is its long-term memory. This is well known from the seminal contribution by Corsi (2009) and further developments, for instance Proietti (2016). Our way of introducing longterm memory in the evolution of µt follows the approach by Harvey and Sucarrat (2014), who considered a multiple-component specification for the Beta-t-EGARCH

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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model detailed in Harvey (2013).1 The idea of modeling the long-term memory in volatility by the superposition of independent AR(1) processes is rooted in the seminal paper by Granger and Joyeux (1980), and has been applied in Gallant, Hsu, and Tauchen (1999) and Engle and Lee (1999). A similar approach was considered in Barndorff-Nielsen and Shephard (2002) in the context of continuous-time volatility modeling, and by Koopman and Scharth (2012) for the case of stochastic volatility. Specifically, our long-term memory specification assumes that the process µt is the sum of two stationary AR(1) µ processes with |βi | < 1 for i = 1, 2. For identifiability, µ µ we assume that 0 ≤ b2 < b1 < 1. We refer to (1)–(6) as a score-driven realized volatility (SDRV) model for zt . Given a bivariate time-series of financial returns and realized log-volatilities {(yt , xt )′ , t = µ µ 1, . . . , T }, the model parameters ξ = (κ µ , κ ρ , κ q , a1 , a2 , µ µ ρ ρ q q ′ a , a , b1 , b2 , b , b , ν, α, β ) are estimated via numerical maximization of the likelihood function:

ˆ ξ = argmax ξ∈Ξ

T ∑

log p(zt |Ft −1 ; ξ ),

(7)

t =1

where we have highlighted the dependence of p(zt |Ft −1 ) on ξ . 4. News impact curve Understanding the reaction of the conditional volatility process to past observations is of primary importance for policymakers and market participants. In the financial econometrics literature, the News Impact Curve (NIC), introduced by Engle and Ng (1993) for GARCH models, is the key tool used to investigate this important aspect. The NIC plots volatility at time t + 1 against a shock that hits the system at time t. The leverage effect implies an asymmetric reaction of the volatility to past shocks, so that the NIC for models that incorporate this feature are asymmetric. As stated before, GARCH models incorporate the leverage effect in a simplistic way, that is, by estimating an additional parameter that increases volatility proportionally to some function of past negative shock, as for example the approaches proposed by Nelson (1991) and Glosten et al. (1993). By contrast, as we reported in the previous section, SV models incorporate leverage effect through a correlation parameter between disturbances that affect the measurement and state equations of the state space formulation. Unfortunately, for SV models, the NIC is challenging to derive because p(yt |Ft −1 ) and its moments are not available in closed form. Solutions exploiting computationally intensive simulation techniques are available (Takahashi, Omori, & Watanabe, 2013), but are rarely implemented. Our framework provides an easy way to derive the NIC and sheds light on the three different sources of 1 In a previous version of this manuscript, we employed a fractional integrated specification similar to the one used by Lucas and Opschoor (2016). However, during revisions, we found that in the context we consider, the multiple-component specification provides better forecasting results and a clearer interpretation. Thus, we follow this approach.

variation in the conditional volatility process: (i) the level of leverage, via ρt ; (ii) the fatness of the tail of the conditional distribution, measured by tail index ν ; and (iii) the volatility of volatility component, qt . The three drivers influence the NIC in different ways and offer intuitive insights into the structure of volatility, as follows: (i) The correlation between yt and xt determines the asymmetric behavior of the volatility response. This is consistent with the usual interpretation of the leverage effect. However, as is clarified, the direction of the asymmetry also depends on the realization of the log-volatility at time t, xt . (ii) The number of degrees of freedom helps make the update robust. That is, if ν is low, large shocks are tapered so that they exert little influence on tomorrow’s volatility. This mechanism is well known in the literature on score-driven techniques, and several authors have noted its benefit from an applied perspective. See, e.g., Creal et al. (2013), Harvey (2013), and Harvey and Luati (2014). The reasoning behind this behavior is that if there is statistical evidence of fat tails, extreme observations are treated as realizations from tails of the distribution and, hence, do not imply a strong signal in terms of increased volatility. (iii) The volatility of volatility influences the amplitude of the update. The signal is enhanced if the volatility of volatility is high and reduced if it is low. To illustrate the behavior of the NIC, we consider a single-component specification for µt ; as in Engle and Ng (1993), we start by setting the unrestricted parameters to their unconditional level, i.e., ˜ ρ = κρ, ˜ q = κ q , and µ µ = κ , and set ρ = (1 − exp(−˜ ρ ))/(1 + exp(−˜ ρ )), and q = exp(˜ q).2 We also assume that there is no bias: α = 0 and β = 1. In contrast to the NIC defined in Engle and Ng (1993), in our case, we observe two ‘‘shocks’’ hitting the system, and causing the conditional volatility level to be updated from σt = exp(µt ) to σt +1 = exp(µt +1 ). Formally, given the vector of new observations at time t, zt = (yt , xt )′ , the NIC is defined as: µ

NIC (yt , xt ; µ, ρ, q, ν ) = σ exp(aµ st ),

(8)

µ

where σ = exp(µ) and st , defined in (5), is evaluated at (µ, ρ, q). We now graphically investigate the different shapes of the NIC with respect to different values of the model parameters and new observations. Fig. 1 displays NIC (yt , xt ; 0, ρ, 1, ν ) as a function of yt for the following selected values of xt : (i) xt = −5.5, implying that the realized volatility decreases; (ii) xt = 0 (no changes in volatility); and (iii) xt = 5.5 (increase in volatility). The results are reported for different values of ρ : Panels (a), (b), and (c) refer to the case of a negative correlation between yt and xt (ρ = −0.7). Panels (d), (e), and (f) refer to the case of no correlation between yt and xt (ρ = 0). Panels (g), (h), and (i) are relative to the case of positive correlation between yt and xt (ρ = 0.7). We consider the sensitivity 2 Note that ρ and q are not the unconditional values of ρ and t qt , i.e., E(ρt ) ̸ = ρ and E(qt ) ̸ = q. However, this does not affect the interpretation of the NIC.

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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Fig. 1. News Impact Curve with respect to yt (horizontal axis) and selected values of xt (RVt ): (i) decrease in log-volatility xt = −5.5, (ii) no changes in volatility xt = µ, (iii) increase in volatility, xt = 5.5. The parameters of the model are fixed at their unconditional level, similar to what is usually found in empirical applications with financial returns in percentage points: µt = µ = 0, qt = q = 1. Panels (a), (b), and (c) report the case of negative correlation between yt and xt (ρ = −0.7). Panels (d), (e), and (f) report the case of the absence of a correlation between yt and xt (ρ = 0). Panels (g), (h), and (i) report the case of a positive correlation between yt and xt (ρ = 0.7). The NICs are displayed for different values of the degree of freedom parameter, ν = 3 (black, continuous), ν = 6 (red, dashed), ν = 9 (green, dotted). The horizontal axis indicates the financial return at time t, yt , while the vertical axis indicates the corresponding volatility level at time t + 1, σt +1 = exp(µt +1 ). The orange horizontal line is reported as a benchmark and indicates the volatility level observed in the case of (yt , xt )′ = (0, µ)′ and ρ = 0.

to the degree of freedom parameter ν in Fig. 1, which considers the following cases: very fat tails, ν = 3 (black, continuous), moderate fat tails, ν = 6 (red, dashed), and relatively thin tails, ν = 9 (green, dotted). The effects of ρ and ν on the NIC are evident from Fig. 1. As expected, ρ determines the levels of asymmetry of the NIC while ν its behavior when extreme observations are considered. Looking at panels (d), (e), and (f), we observe that the NIC is symmetric for ρ = 0. The value of xt determines the level of the curve: For values of

yt near zero, the model increases (decreases) in volatility if xt > 0 (xt < 0) compared with xt = µ. The cases ρt = 0.7 and ρt = −0.7 provide interesting insights into the relation between the volatility level at time t + 1 and the observations (yt , xt )′ . As, in these cases, the returns and volatility are correlated, the concordance between their signs also influences the update.3 Intuitively, if we 3 It is actually the sign of the difference between y and x from t t their averages, which we assume to be zero in both cases.

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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Fig. 2. News Impact Curve with respect to xt (horizontal axis) and selected values of yt : (i) large negative return, yt = −7, (ii) zero return, yt = 0, and (iii) large positive return, yt = 7. The parameters of the model are fixed at their unconditional levels, similar to what is usually found in empirical applications with financial returns in percentage points: µt = µ = 0, qt = q = 1. Panels (a), (b), and (c) report the case of a negative correlation between yt and xt (ρ = −0.7). Panels (d), (e), and (f) report the case of the absence of a correlation between yt and xt (ρ = 0). Panels (g), (h), and (i) report the case of a positive correlation between yt and xt (ρ = 0.7). The NICs are displayed for different values of the degree of freedom parameter, ν = 3 (black, continuous), ν = 6 (red, dashed), ν = 9 (green, dotted). The horizontal axis indicates the financial return at time t, yt , while the vertical axis indicates the corresponding volatility level at time t + 1, σt +1 = exp(µt +1 ). The orange horizontal line is reported as a benchmark and indicates the volatility level observed in the case of (yt , xt )′ = (0, µ)′ and ρ = 0.

observe that sign(yt xt ) = +1 and the correlation is negative, this is a signal of increased uncertainty because such an event is less likely than sign(yt xt ) = −1 if ρ < 0. The model translates this unlikely event into an increase in volatility as evident from panels (a) and (c). By contrast, as shown by panels (g) and (i), if ρ > 0, the opposite holds. Panels (b) and (h) are more difficult to interpret, and there is no a clear reasoning explaining the asymmetric behavior of the NIC in this circumstance. Fig. 2 displays NIC (y, xt ; 0, ρ, 1, ν ) as a function of xt for selected values of the remaining arguments. For

returns, we consider the following cases: (i) a large negative return, yt = −7; (ii) zero return, yt = 0; and (iii) a large positive return, yt = 7. As before, the results are reported for different values of ρ and ν . Again, ρ determines the asymmetry of the curve while ν its behavior with respect to extreme values of yt and xt . The symmetric case is still recovered when ρ = 0. The robustness of the filter is evident from all plots. Specifically, we note that panels (b), (e), and (h) are consistent with Figure 1 of Harvey and Luati (2014), where the authors studied the robustness of score-driven filters for location models. In our framework,

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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their results are extended to the case when the scale and location are simultaneously updated. This graphical analysis illustrates that our specification generates a flexible and interpretable NIC, and that the asymmetric response of volatility is due not only to the correlation parameter ρt , but is also dependent upon the state of the system via the levels of xt , yt , and the signs of their products. 5. Empirical application Our empirical application deals with the ability of our model to predict the realized log-volatility. We considered the financial returns and associated realized log-volatility of all companies listed on the Dow Jones Index (DJIA) and the S&P 500 ETF (SPY) from January 29, 2001 to December 31, 2013 for a total of 3250 daily observations. From the DJIA, we removed Google (GOOG), Travelers Companies (TRV), Visa (V), and Verizon Communications (VZ) owing to a limited number of available observations for them. The 29 bivariate series is identified in the tables by the ticker associated to the asset in the dataset. Data were collected from the Trades and Quotes (TAQ) database, and a preliminary cleaning of the high-frequency prices was performed following the procedure in Brownlees and Gallo (2006) and Barndorff-Nielsen, Hansen, Lunde, and Shephard (2009). To construct of the realized log volatility, we aggregate returns at a five-minute frequency and considered the standard realized volatility estimator, based on the sum of the squares of intradaily returns.4 Financial returns were demeaned to be consistent with the specification given in (1). Overnight returns were removed from the computation of the daily return and the log-realized volatility. The empirical analysis was divided in two parts. We set off by reporting the results of estimation and model selection based on the full sample. The second part dealt with the evaluation of the predictive performance of various specifications nested within the SDRV model, also in comparison with authoritative benchmarks for predicting volatility. 5.1. Full sample analysis: Model selection and estimation results The SDRV model (1)–(6) nests several specifications, that arise, e.g., by modeling the persistence in the realized log-volatility conditional mean µt by one or two (1) autoregressive components, i.e., setting µt = κ µ + µt in the former case, or imposing that ρt and/or qt are time invariant (setting aρ = bρ = 0 and/or aq = bq = 0, respectively). We labeled the unrestricted specification SDRV(2) because log-volatility features two components in the 4 A different choice of the sampling frequency or the realized volatility estimator may lead to minor changes in our results. In our analysis, we followed Liu, Patton, and Sheppard (2015), who provided strong empirical support for the use of the five-minute realized volatility, among many other estimators.

7

conditional mean. The specification with a single volatility component was indicated by SDRV(1). In both cases, nested versions recovered by imposing constraints on the coefficients of the system are indicated by the –j label, j = ρ, q. For example, the SDRV(2) model featuring a time-invariant ρ is denoted by SDRV(2)–ρ . Similarly, the specifications of a single AR component, and constant volatility of volatility and leverage are referred to as SDRV(1)–ρ q. Table 1 lists the eight model specifications considered. All specifications were estimated by maximum likelihood using the complete available sample. Model selection was carried out using the Bayesian information criterion (BIC), which is reported in Table 2. The specifications selected most often were those with two volatility components and time-invariant correlation (SDRV(2)–ρ ), and the model with two volatility components, and timeinvariant correlation and volatility of volatility (SDRV(2)– ρ q). An exception was the S&P 500 ETF (SPY), for which the more general specification with two components (SDRV(2)) delivered the minimum BIC. Selection according to the AIC (not reported) led to our mostly choosing the most general specification SDRV(2). The parameter estimates for the SDRV specifications selected by BIC are reported in Table 3. Asymptotic standard errors were generally low due to the very large sample, and were omitted. The first autoregressive component showed very high persistence, and was characµ terized by an autoregressive parameter b1 estimated to be close to one. The second component contributed significantly to the persistence of the log-volatility process, as highlighted by the sharp increase in the BIC values as we move from the specification with a single component to those with two components. The out-of-sample forecasting experiment presented in Section 5.2 confirms the importance of being able to capture the persistence in the realized log-volatility for enhancing predictive performance. The quasi non-stationarity of the process estimated for µt is the likely consequence of allowing for measurement noise in xt in the specification of the model. Evidence for non-stationarity in the systematic components of volatility, once we allow for measurement noise, was found in Proietti (2016), although in a long-term memory framework. The conditional correlation between returns and the realized log-volatility was constant in all occurrences but those of SPY, and was predominantly negative. For the time-invariant specifications, it was estimated as ρˆ = (1 − exp(κˆ ρ ))/(1 + exp(κˆ ρ )), and took values in the range (−0.09, 0.04). A more sizable leverage effect was detected for the SPY series, which was the only series for which the SDRV(2) full model found strong support, according to the BIC. Fig. 3 displays the filtered parameters for the SDRV(2) model estimated on the SPY series. The period started on 29 January, 2001 and ended on 31 December, 2013. The blue and red vertical bands indicate periods of US and European recessions (usually associated with periods of financial turmoil). Panel (a) depicts the filtered realized log-volatility level µt compared with the observed realized log-volatility. Volatility increased considerably

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L. Catania and T. Proietti / International Journal of Forecasting xxx (xxxx) xxx Table 1 List of model specifications. Starting from the most general two components score-driven Realized Volatility (SDRV(2)) model, the nested specifications are recovered by applying constraints to the coefficients of the dynamic Equation (4). With two volatility components SDRV(2) SDRV(2)–q SDRV(2)–ρ SDRV(2)–ρ q

The most general specification defined in Section 2. SDRV(2) with constant volatility of volatility, qt = q. SDRV(2) with constant correlation, ρt = ρ . SDRV(2) with constant correlation and volatility of volatility, qt = q, ρt = ρ .

With one volatility component SDRV(1) SDRV(1)–q SDRV(1)–ρ SDRV(1)–ρ q

SDRV(2) SDRV(1) SDRV(1) SDRV(1)

one volatility component. with constant volatility of volatility, qt = q. with constant correlation, ρt = ρ . with constant correlation and volatility of volatility, qt = q, ρt = ρ .

Table 2 BIC evaluated over the full sample for the model specifications defined in Table 1. The gray cells indicate models with lower BIC values and, hence, preferred from a penalized likelihood perspective. With two volatility components

With one volatility component

Ticker

SDRV(2)

SDRV(2)–q

SDRV(2)–ρ

SDRV(2)–ρ q

SDRV(1)

SDRV(1)–q

SDRV(1)–ρ

SDRV(1)–ρ q

AAPL AXP BA CAT CSCO CVX DD DIS GE GS HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT NKE PFE PG SPY UNH UTX VZ WMT XOM

13901.6 11443.2 11134.4 11292.8 11593.1 8947.1 10180.6 10488.9 10393.2 11912.3 10690.1 9000.5 11237.7 7594.3 11849.7 7791.3 9437.1 9048.7 10322.8 10127.9 10516.7 9698.5 7592.5 7741.4 11821.6 9837.4 9555.2 8633.6 9222.8

13914.9 11426.1 11121.6 11277.0 11592.9 8938.0 10166.6 10503.5 10401.2 11891.3 10689.0 8995.0 11239.4 7599.8 11840.3 7812.8 9428.7 9051.9 10318.2 10126.0 10526.3 9684.1 7568.7 7789.6 11827.1 9826.6 9556.1 8627.2 9206.6

13897.3 11433.7 11108.5 11277.0 11583.1 8935.7 10165.2 10477.8 10379.7 11877.6 10686.5 8987.7 11228.3 7588.6 11836.1 7779.7 9423.5 9032.4 10306.6 10119.0 10500.5 9684.2 7554.8 7790.0 11817.1 9828.7 9549.6 8620.9 9207.2

13909.2 11414.4 11099.7 11260.8 11583.1 8926.3 10151.1 10491.5 10387.5 11901.1 10685.1 8981.9 11228.6 7591.6 11826.8 7790.3 9417.4 9035.6 10301.9 10116.9 10512.6 9669.7 7552.5 7798.4 11811.1 9817.6 9532.7 8631.6 9191.0

14109.2 11480.1 11165.7 11350.0 11701.5 8986.9 10210.8 10536.8 10433.1 11978.3 10711.2 9032.3 11320.7 7630.3 11937.4 7842.7 9482.9 9082.7 10354.9 10167.2 10583.2 9753.7 7613.9 7766.0 11871.2 9873.8 9610.0 8691.0 9243.3

14060.6 11496.3 11157.8 11340.6 11713.5 8985.0 10200.6 10558.8 10445.1 11961.4 10710.3 9031.1 11324.8 7655.4 11933.4 7854.0 9482.1 9089.0 10353.9 10161.7 10594.3 9745.6 7640.0 7795.1 11886.8 9864.2 9608.3 8673.8 9243.3

14038.4 11445.7 11139.7 11327.0 11691.3 8974.4 10197.6 10524.9 10419.9 11942.9 10707.0 9020.0 11312.4 7623.7 11921.9 7822.2 9468.5 9066.5 10348.6 10150.5 10567.1 9732.8 7597.7 7814.3 11873.8 9863.9 9597.2 8663.5 9227.3

14057.3 11482.1 11131.7 11325.1 11703.0 8972.6 10185.9 10545.6 10434.0 11949.9 10705.5 9018.6 11315.7 7636.8 11917.8 7842.9 9470.4 9072.8 10336.7 10146.7 10582.2 9735.8 7601.6 7800.8 11871.0 9854.2 9587.8 8660.5 9227.7

during 2008–2009, which were characterized by the global financial crisis (GFC). To observe the levels of volatility comparable to those of the GFC, we needed to return to early 2001 in the middle of the DotCom bubble. Panel (b) reports the filtered volatility of the volatility parameter qt . The volatility of volatility oscillated between a longrun average of about 0.25, and moved together with the filtered realized log-volatility. Finally, panel (c) reports the correlation coefficient associated with the leverage effect, ρt = cor(yt , xt ). The estimated volatility of volatility was positively related with the unanticipated level of volatility, xt − µ ˆ t . The plot also reveals that the variation over time was driven by two episodes of sharp increases in volatility in the period 2010–2011. Moreover, the time series pattern of ρˆ t appears to be negatively

correlated to that of qˆ t . Interestingly, the estimated correlation coefficient was almost zero during the GFC and the end of the DotCom Bubble, and the contemporaneous correlation was negative and sizable during the recovery phase after the GFC. This finding suggests that negative news has a more pronounced effect on unanticipated volatility during tranquil periods. However, this result seems to be specific to the SPY series as there is no substantive evidence for a time-varying correlation for the other series in the dataset. 5.2. Out-of-sample predictive analysis We now turn our attention to the ability of our model to predict the h-step-ahead log-volatility measure, and

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Table 3 Estimated parameters for the SDRV specification for the Dow Jones constituents and the SPY index selected by BIC. Standard errors are generally small owing to the long estimation period, and are not reported. Ticker

AAPL AXP BA CAT CSCO CVX DD DIS GE GS HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT NKE PFE PG SPY UNH UTX VZ WMT XOM

µt

ρt

qt µ

µ

µ

µ

κµ

a1

b1

a2

b2

κq

aq

bq

κρ

aρ

bρ

1.519 0.639 0.401 0.381 1.222 0.180 0.448 0.875 0.417 0.851 0.504 0.170 1.160 −0.068 0.610 0.174 0.288 0.466 0.244 0.750 0.532 0.319 0.078 −0.220 0.417 0.220 0.435 0.644 0.124

0.010 0.010 0.010 0.012 0.009 0.013 0.012 0.009 0.008 0.010 0.007 0.013 0.008 0.013 0.013 0.009 0.010 0.011 0.012 0.006 0.007 0.009 0.012 0.014 0.011 0.011 0.010 0.007 0.014

0.999 0.999 0.993 0.988 0.998 0.985 0.992 0.998 0.998 0.998 0.997 0.989 0.999 0.990 0.995 0.995 0.995 0.996 0.989 0.999 0.998 0.992 0.989 0.988 0.992 0.991 0.995 0.999 0.983

0.017 0.011 0.008 0.008 0.011 0.008 0.007 0.009 0.011 0.012 0.009 0.007 0.010 0.006 0.012 0.008 0.008 0.007 0.008 0.010 0.011 0.010 0.008 0.008 0.009 0.009 0.009 0.009 0.007

0.630 0.769 0.646 0.235 0.406 0.309 0.424 0.652 0.828 0.703 0.872 0.370 0.617 0.000 0.502 0.631 0.343 0.478 0.366 0.883 0.760 0.588 0.300 0.645 0.389 0.628 0.553 0.808 0.214

−1.277 −1.414 −1.427 −1.474 −1.470 −1.484 −1.476 −1.475 −1.408 −1.444 −1.471 −1.465 −1.531 −1.410 −1.408 −1.489 −1.406 −1.461 −1.383 −1.461 −1.428 −1.455 −1.466 −1.422 −1.370 −1.452 −1.448 −1.468 −1.481

0.012 – – – – – – 0.024 0.041 0.024 – – 0.033 0.012 – 0.043 – 0.039 – – 0.005 – – 0.014 – – – 0.016 –

0.964 – – – – – – 0.868 0.479 0.699 – – 0.000 0.950 – 0.582 – 0.158 – – 0.994 – – 0.946 – – – 0.833 –

−0.116 0.034 −0.058 −0.051 −0.094 −0.153 −0.094 −0.048 0.086 −0.004 −0.036 −0.049 −0.011 0.108 −0.004 −0.026 −0.059 −0.048 0.020 −0.012 0.050 −0.048 −0.036 −0.011 −0.033 −0.009 −0.078 0.046 −0.181

– – – – – – – – – – – – – – – – – – – – – – – 0.033 – – – – –

– – – – – – – – – – – – – – – – – – – – – – – 0.984 – – – – –

α

β

ν

0.166 0.079 0.055 0.054 0.000 −0.014 −0.018 0.000 0.000 0.083 0.001 0.001 −0.001 −0.058 0.001 −0.051 0.000 −0.006 0.001 0.000 0.000 −0.048 −0.079 0.098 0.000 0.000 −0.048 −0.058 −0.024

0.868 0.984 0.931 0.975 1.040 0.966 1.012 0.985 1.020 1.006 1.025 1.026 1.015 1.043 1.056 1.028 0.956 0.970 1.009 1.070 1.014 1.084 0.974 1.014 1.056 1.061 1.054 0.987 1.004

10.646 10.848 14.699 11.400 11.948 25.191 11.159 12.430 13.519 14.197 10.799 13.085 23.502 11.121 12.614 10.607 9.692 10.085 8.055 11.953 11.371 9.991 11.222 12.965 8.826 13.919 13.368 14.524 18.796

Fig. 3. Filtered parameters for the SDRV(2) model estimated on the SPY (the S&P 500 ETF) series. The period started on the 29 January, 2001 and ended on 31 December, 2013. Panel (a) depicts the filtered log-volatility level µt (solid line) and the observed realized log-volatility (gray points). Panel (b) reports the filtered volatility of volatility, qt . Panel (c) reports the correlation coefficient ρt = cor(yt , xt ). The blue and red vertical bands indicate periods of US and European recessions according to the OECD and FRED recession indicators, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Table 4 One-step-ahead median absolute forecast error evaluated over the out-of-sample period for the models listed in Table 1 and the RGARCH model. The results are reported relative to the HAR–GARCH model of Corsi et al. (2008). Values smaller than 1 indicate that the model outperformed the benchmark. Gray boxes identify models belonging to the 75% model confidence set, see Hansen, Lunde, and Nason (2011). AAPL AXP BA CAT CSCO CVX DD DIS GE GS HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT NKE PFE PG SPY UNH UTX VZ WMT XOM

HAR-GARCH

RGARCH

SDRV(2)

SDRV(2)–q

SDRV(2)–ρ

SDRV(2)–ρ q

SDRV(1)

SDRV(1)–q

SDRV(1)–ρ

SDRV(1)–ρ q

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.77 0.68 0.75 0.75 0.81 0.65 0.69 0.77 0.74 0.75 0.67 0.76 0.78 0.75 0.68 0.76 0.77 0.72 0.75 0.74 0.72 0.73 0.75 0.64 0.78 0.75 0.72 0.78 0.70

0.73 0.66 0.73 0.72 0.79 0.68 0.69 0.74 0.70 0.76 0.66 0.74 0.76 0.74 0.66 0.73 0.74 0.71 0.73 0.74 0.72 0.70 0.75 0.64 0.72 0.73 0.70 0.77 0.69

0.71 0.65 0.74 0.71 0.78 0.68 0.69 0.74 0.70 0.75 0.66 0.74 0.76 0.74 0.66 0.73 0.75 0.71 0.73 0.73 0.72 0.70 0.76 0.63 0.71 0.72 0.70 0.77 0.69

0.73 0.66 0.74 0.72 0.78 0.68 0.69 0.74 0.71 0.76 0.67 0.74 0.76 0.75 0.66 0.73 0.75 0.71 0.73 0.73 0.72 0.70 0.75 0.63 0.72 0.72 0.71 0.77 0.69

0.72 0.66 0.74 0.71 0.78 0.68 0.69 0.74 0.70 0.76 0.66 0.74 0.76 0.75 0.66 0.73 0.75 0.71 0.73 0.74 0.72 0.70 0.75 0.63 0.71 0.72 0.70 0.77 0.68

0.75 0.66 0.75 0.73 0.81 0.67 0.69 0.75 0.73 0.77 0.67 0.75 0.76 0.75 0.68 0.74 0.75 0.74 0.75 0.74 0.74 0.72 0.76 0.65 0.72 0.73 0.72 0.78 0.70

0.74 0.67 0.75 0.73 0.79 0.67 0.69 0.76 0.73 0.76 0.67 0.74 0.76 0.75 0.68 0.75 0.75 0.73 0.74 0.74 0.73 0.72 0.76 0.64 0.72 0.73 0.72 0.78 0.70

0.74 0.66 0.74 0.73 0.82 0.67 0.69 0.75 0.72 0.77 0.67 0.74 0.77 0.75 0.67 0.75 0.75 0.74 0.75 0.74 0.74 0.72 0.76 0.64 0.72 0.73 0.72 0.78 0.71

0.74 0.67 0.75 0.73 0.80 0.67 0.69 0.76 0.73 0.76 0.67 0.74 0.76 0.74 0.67 0.76 0.75 0.73 0.75 0.74 0.74 0.72 0.76 0.64 0.72 0.73 0.72 0.77 0.70

investigate if including different features, such as (i) longterm memory, (ii) time-varying leverage effect, and (iii) time-varying volatility of volatility, is of any help. To this end, we perform a classical rolling forecasting experiment such that for each series, we consider a training set of S = 1000 consecutive observations, fit the eight SDRV specifications, compute the point and density predictions at forecasting horizons h = 1, . . . , 25, and compare them to the observed value xt +h . These steps are iterated using rolling samples obtained by adding one future observation and removing the initial one, until the end of the sample is reached, for a total of H = T − S = 2250 iterations. The model parameters were estimated using only data belonging to the training sample, and were updated each time a new rolling sample was processed until the end of the time series was reached. The point predictions were computed as:

ˆ xt +h|t = E[xt +h |Ft ] = E[µt +h |Ft ] (2) = κµ + E[µ(1) t +h |Ft ] + E[µt +h |Ft ],

where (i)

µ

µ

µ

µ

(i)

E[µt +h |Ft ] = ai (bi )h−1 st + (bi )h µt , for i = 1, 2. As for density forecasting, the one-step-ahead (h = 1) predictive distribution of xt is the student’s t with ν degrees of freedom, mean µt +1 , and variance q2t +1 . The multi-step-ahead (h > 1) distribution of xt +h |Ft is not available in closed form, and is obtained by simulation using 100,000 draws.

The conditional mean predictive performances of the specifications listed in Table 1 were compared with those of the HAR–GARCH(1,1) model by Corsi et al. (2008), and represent a valid benchmark for the realized volatility series and the Realized GARCH (RGARCH) specification of Hansen et al. (2012). The HAR–GARCH is a constrained autoregressive model of order 22, parsimoniously parameterized in terms of three coefficients, with a GARCH specification for the volatility of volatility. Specifically, the model is defined as: (w )

xt = β0 + β (d) xt −1 + β (w) xt −1 + β (m) xt −1 + ζt , (w )

∑5

(m)

(m)

(9)

∑22

1 where xt −1 = 51 s=1 xt −s and xt −1 = s=1 xt −s 22 are the weekly and monthly arithmetic moving averages √ of the log-volatility, ζt = ηt εt , where εt is an i.i.d. standardized normal inverse Gaussian random variable, and ηt = ϖ + δζt2−1 + ϕηt −1 . It is common in the HAR literature to formulate predictions via a direct forecast of the multi-step-ahead conditional mean, see Marcellino, Stock, and Watson (2006). We follow this approach with the HAR–GARCH specification. The RGARCH specification, as the SDRV, is a joint model of returns and realized measures of volatility that also incorporates an intertemporal leverage component similar to that employed in the asymmetric GARCH specifications. Specifically, the RGARCH model is defined by the following system of equations:

y t = σt ϵ t , xt = π0 + φ1 log σt2 + κ1 ϵt + κ2 (ϵt2 − 1) + ηt ,

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Fig. 4. Plot of the relative median absolute deviation for HAR–GARCH (benchmark, solid orange line at 1), RGARCH (black solid line), SDRV(2) (red dashed line with circles), SDRV(2)–q (blue dotted line with circles), SDRV(2)–ρ (purple dotdash line with circles), SDRV(2)–ρ q (green longdash line with circles), SDRV(1) (red dashed line with asterisks), SDRV(1)–q (blue dotted line with asterisks), SDRV(1)–ρ (purple dotdash line with asterisks), SDRV(1)–ρ q (green longdash line with asterisks), versus the forecast horizon h.

log σt2 = ζ0 + ζ1 xt −1 + ζ2 log σt2−1 , where {ϵt } and {ηt } are two independent i.i.d. sequences of innovations with zero mean and E[ϵt2 ] = 1, E[ηt2 ] = λ. The term κ1 ϵt on the right-hand side of the log-volatility equation accounts for the leverage effect. In contrast to HAR-GARCH, RGARCH is distribution free, and is estimated by QMLE, see Hansen et al. (2012). Predictions of the conditional mean of the log-realized volatility are available in closed form for all h, see Hansen et al. (2012).

The RGARCH model is considered only for the comparison of point predictions, as density predictions are not available due to the absence of any distributional assumption. To alleviate the effect of possible extreme observations during the out-of-sample analysis, we compared the predictions of log-realized volatility using the median absolute deviation (MAD). That is, given a sequence of predictions for the h-step-ahead log-realized volatility level, ˆ xt +h|t , t = S + 1, . . . T − h, we computed the median of the H − h absolute prediction errors |ˆ xt +h|t − xt +h |.

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Fig. 5. Continued from Fig. 4.

The calibration and accuracy of the predictive densities were assessed via the continuous ranked probability score (CRPS), introduced by Matheson and Winkler (1976). The CRPS is a proper scoring rule,5 defined as:

∫

[F (zt +h | Ft ) − I{xt +h < zt +h }]2 dzt +h , (10)

CRPSt +h|t = R

5 Given a random variable y ∈ R with continuous probability density function f , the ∫ scoring rule S(f∫ , y) is said to be proper if and only if Ef [S(f , y)] = R f (y)S(f , y)dy ≤ R f (y)S(g , y)dy = Ef [S(g , y)] for all density functions f and g.

where F (zt +h | Ft ) is the h-step-ahead cumulative density function (cdf ) evaluated in zt +h . Evidently, the CRPS measures the distance between the predicted cdf, F (xt +h | Ft ), and the empirical cdf represented as a step function in xt +h . Taking the median of the CRPS over the out-ofsample period provides the quantity at the base of our comparative analysis. Models with lower median CRPS are preferred. Analytical solutions of (10) are not available in the context of the SDRV. When h = 1, the predictive distribution of xt is available in closed form, and the integral is computed using a standard numerical integration routine. When h > 1, the predictive distribution is not

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Table 5 One-step-ahead median continuous ranked probability score evaluated over the out-of-sample period for models listed in Table 1 for the logrealized volatility. The results are reported relative to the HAR–GARCH model of Corsi et al. (2008). Values smaller than 1 indicate that the model outperformed the benchmark. Gray boxes identify models belonging to the 75% model confidence set, see Hansen et al. (2011). AAPL AXP BA CAT CSCO CVX DD DIS GE GS HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT NKE PFE PG SPY UNH UTX VZ WMT XOM

HAR-GARCH

SDRV(2)

SDRV(2)–q

SDRV(2)–ρ

SDRV(2)–ρ q

SDRV(1)

SDRV(1)–q

SDRV(1)–ρ

SDRV(1)–ρ q

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.74 0.66 0.73 0.72 0.78 0.67 0.68 0.73 0.70 0.75 0.66 0.73 0.75 0.74 0.66 0.73 0.74 0.71 0.75 0.73 0.71 0.70 0.75 0.63 0.72 0.71 0.70 0.76 0.67

0.72 0.65 0.73 0.71 0.77 0.67 0.68 0.72 0.69 0.74 0.66 0.73 0.75 0.73 0.66 0.73 0.74 0.70 0.74 0.72 0.71 0.70 0.75 0.62 0.72 0.71 0.70 0.76 0.67

0.74 0.66 0.74 0.71 0.78 0.67 0.69 0.73 0.70 0.74 0.66 0.73 0.75 0.75 0.66 0.73 0.74 0.71 0.74 0.72 0.71 0.71 0.75 0.62 0.72 0.71 0.70 0.76 0.66

0.73 0.65 0.74 0.71 0.76 0.67 0.68 0.73 0.70 0.74 0.66 0.73 0.75 0.74 0.66 0.73 0.74 0.71 0.74 0.72 0.72 0.71 0.75 0.61 0.72 0.71 0.70 0.77 0.66

0.76 0.66 0.74 0.73 0.79 0.67 0.68 0.74 0.72 0.76 0.67 0.73 0.75 0.75 0.68 0.74 0.75 0.73 0.76 0.73 0.73 0.72 0.75 0.64 0.73 0.72 0.71 0.77 0.68

0.75 0.66 0.74 0.73 0.78 0.67 0.68 0.74 0.72 0.75 0.66 0.73 0.76 0.74 0.68 0.74 0.75 0.72 0.75 0.72 0.72 0.72 0.75 0.63 0.73 0.72 0.71 0.77 0.68

0.75 0.66 0.74 0.73 0.80 0.67 0.69 0.74 0.72 0.75 0.67 0.73 0.76 0.74 0.68 0.75 0.75 0.73 0.76 0.73 0.72 0.72 0.75 0.64 0.73 0.72 0.71 0.77 0.68

0.75 0.67 0.74 0.72 0.78 0.67 0.68 0.75 0.72 0.75 0.67 0.73 0.75 0.74 0.67 0.75 0.75 0.73 0.75 0.72 0.72 0.72 0.75 0.63 0.73 0.73 0.71 0.77 0.68

available, and CRPSt +h|t is computed as CRPSt +h|t =

1 2

E |X − X ′ ||Ft − E [|X − xt ||Ft ] ,

[

]

(11)

where X and X ′ are independent copies of a random variable with distribution function equal to the h-stepahead predictive distribution of xt . In (11), the expectation is approximated using 100,000 draws from the predictive distribution. 5.2.1. Results of point forecast Table 4 reports the values of the MAD accuracy measure computed over H − h one-step-ahead volatility forecasts for all models listed in Table 1. All the results are relative to the HAR–GARCH(1,1) model by Corsi et al. (2008), which was used as a benchmark. The third column reports the relative MAD for the RGARCH model. All the SDRV specifications outperformed the benchmark. The superior predictive performance can be ascribed to its ability to track the persistent evolution of the conditional mean of the log-volatility process. In terms of MAD, the overall performance of the RGARCH model was often indistinguishable from that of the SDRV(2) specifications. To discriminate the performance of the different models through comparison, we estimated the 75% model confidence set (MCS) according to the methodology proposed by Hansen et al. (2011). The procedure is based on the Diebold and Mariano (1995) test of equal forecasting accuracy, which is based on the average absolute loss differential between rival predictors, standardized under

the null. The specifications entering the MCS are boxed in gray in Table 4. The main evidence was that SDRV(2)– ρ q was always featured in the MCS. Time variations in leverage and volatility were often supported, but did not improve forecasting ability over the more parsimonious SDRV(2)–ρ q specification, not even for the SPY series. Finally, in eight occurrences of 29, the RGARCH model entered the MCS. The case for considering the SDRV model with two components in µt is even stronger if one considers multistep predictive ability. Figs. 4 and 5 plot the forecasted MAD versus the forecasting horizon h = 1, . . . , 25, i.e., the median of the H − h absolute prediction errors |xt +h − xˆ t +h|t |, divided by those pertaining to the HAR-GARCH benchmark. While the performance of RGARCH was similar to that of the SDRV(1) specifications, and tended to converge to that of the benchmark for longer forecasting lead times, the MAD index stayed systematically below one for the SDRV(2) models. Moreover, at longer horizons, introducing time variation to qt and ρt did not lead to sizable accuracy gains. The irrelevant role of the variation over time of the volatility in volatility for point forecasting was expected, and can be ascribed to the logarithmic transformation of the realized measure, which emphasizes persistence and reduces heteroskedasticity.

5.2.2. Results of density forecast Table 5 reports the median CRPS over the test period for all models considered, with the exception of RGARCH, for the reasons provided above. The results are

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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Fig. 6. Plot of the median CRPS scores for HAR–GARCH (benchmark, constant solid orange line), SDRV(2) (red dashed line with circles), SDRV(2)–q (blue dotted line with circles), SDRV(2)–ρ (purple dotdash line with circles), SDRV(2)–ρ q (green longdash line with circles), SDRV(1) (red dashed line with asterisks), SDRV(1)–q (blue dotted line with asterisks), SDRV(1)–ρ (purple dot-dashed line with asterisks), SDRV(1)–ρ q (green long-dashed line with asterisks), versus the forecast horizon h.

relative to the CRPS of the HAR–GARCH model, and refer to the one-step-ahead density forecasts available in closed form for the specifications considered, given the maximum likelihood estimates of the parameters and information up to the time of the forecast. In accordance with the results presented for point forecasts, all the SDRV specifications outperformed the benchmark and produced density forecasts significantly more accurate according to

the CRPS metric. However, there were significant differences among the various SDRV specifications. In particular, if one constructs a 75% model confidence set, based on the Diebold and Mariano (1995) test of equal predictive accuracy using the CRPS as relevant loss function, SDRV(2)–ρ q always belongs to the model confidence set. We can equivalently allow for time variation in either qt or ρt in many occurrences, and for both in nine cases,

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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Fig. 7. Continued from Fig. 6.

while specifications with a single autoregressive component in µt never enter the MCS, except in a single case. Figs. 6 and 7 plot the relative median CRPS score versus the forecast horizon h = 1, . . . , 25. The accuracy and calibration of the multi-step-ahead predictive densities, which were obtained by a Monte Carlo simulation using the method of composition, see, e.g., Tanner (2012), again differed across the specifications with a single and two autoregressive components, with the SDRV(2) specifications leading to substantially improved

performance, regardless of the time variation in ρt and qt . All SDRV models dominated the benchmark, especially at short forecast leads, while at longer horizons, only SDRV(2) improved almost systematically over the HAR-GARCH. 6. Concluding remarks We proposed here a model for a bivariate time series consisting of financial returns and realized log-volatility measures. The model can account for several features in

Please cite this article as: L. Catania and T. Proietti, Forecasting volatility with time-varying leverage and volatility of volatility effects. International Journal of Forecasting (2020), https://doi.org/10.1016/j.ijforecast.2020.01.003.

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a coherent framework, such as: (i) long-term memory or persistence of the volatility process, (ii) the presence of time-varying leverage effect, and (iii) the time-varying volatility of volatility. We studied in detail the shape of the News Impact Curve, and showed that the proposed model can represent the leverage effect as usually intended in the literature. However, in our framework, the definition of the leverage effect also depends on the concordance between the sign of the financial return and its realized log-volatility measure as well as the level of correlation. Finally, empirical evidence provided strong support for our proposal. An extensive forecasting analysis based on a panel of 29 daily time series of asset returns and realized volatility constructed from the Trades and Quotes (TAQ) database, referring to all companies listed on the Dow Jones Index (DJIA) and the S&P 500 ETF (SPY) index, showed that from the perspectives of both point and density forecasting, our specifications can outperform a valid and solid univariate predictor of realized volatility, such as the HAR-GARCH model. Moreover, for multi-step forecasting, it outperformed the bivariate RGARCH model. Empirical analysis also revealed that the conditional correlation between returns and log-realized variance is a stable feature, and that allowing for time variation does not further improve forecasting accuracy. The main driver of the improved performance is the ability of our specification to model the persistence of the conditional mean of the realized log-volatility measure. The finding that the volatility of volatility is a stable feature can be explained by the fact that the logarithmic transformation stabilizes the variance of the realized volatility measure. In future research, we foresee extensions that can encompass additional facts, such as skewness. An interesting proposition is to consider a skewed-t distribution for our bivariate system. Moreover, as indicated by a referee, the difference in the kurtosis of returns and logvolatility might be incorporated by a different parametric assumption. For example, returns and log-volatility can be assumed to be marginally student’s t distributed with different degrees of freedom, and their joint distributions can be recovered according to a copula specification. However, note that this parameterization might obfuscate the interpretation of the leverage effect and complicate the resulting score-driven filter.

Acknowledgments This research was supported by the IIF-SAS Research Grant to Support Research on Principles of Forecasting, awarded in 2017 to the project proposal entitled Forecasting Realized Volatility and the Role of Time-varying Dependence with Market Returns. A preliminary version of this paper was presented at the 2017 International Symposium on Forecasting in Cairns, Australia, on June 25–28. We are grateful to the participants, the reviewers, and an associate editor for their comments and suggestions that led to several improvements in the paper.

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