Modelling Volatility Cycles: The MF2‐GARCH Model

Conrad, Christian; Engle, Robert F. Article — Published Version Modelling Volatility Cycles: The MF2‐GARCH Model Journal of Applied Econometrics Provided in Cooperation with: John Wiley & Sons Suggested Citation: Conrad, Christian; Engle, Robert F. (2025) : Modelling Volatility Cycles: The MF2‐ GARCH Model, Journal of Applied Econometrics, ISSN 1099-1255, Wiley, Hoboken, NJ, Vol. 40, Iss. 4, pp. 438-454, https://doi.org/10.1002/jae.3118 This Version is available at: https://hdl.handle.net/10419/323894 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Journal of Applied Econometrics RESEARCH ARTICLE OPEN ACCESS Modelling Volatility Cycles: The MF2-GARCH Model Christian Conrad1,2,3,4 | Robert F. Engle5 1Alfred-Weber-Institute, Heidelberg University, Heidelberg, Germany | 2HEiKA - Heidelberg Karlsruhe Strategic Partnership, Heidelberg University, Karlsruhe Institute of Technology, Karlsruhe, Germany | 3KOF Swiss Economic Institute, Zurich, Switzerland | 4ZEW Mannheim, Mannheim, Germany | 5Stern School of Business, New York University, New York, USA Correspondence: Christian Conrad ([email protected]) Received: 6 March 2023 | Revised: 22 October 2024 | Accepted: 15 December 2024 Funding: This study was funded by the German Federal Ministry of Education and Research (BMBF) and the Baden-Württemberg Ministry of Science as part of Germany’s Excellence Strategy (ExU 10.2.31). Keywords: longand short-term volatility | long-term forecasting | mixed frequency data | volatility component models | volatility forecasting ABSTRACT We propose a novel multiplicative factor multi-frequency GARCH (MF2-GARCH) model, which exploits the empirical fact that thedaily standardizedforecasterrorsofone-component GARCHmodels arepredictablebya movingaverageofpaststandardized forecast errors. In contrast to other multiplicative component GARCH models, the MF2-GARCH features stationary returns, and long-term volatility forecasts are mean-reverting. When applied to the S&P 500, the new component model significantly outperformstheone-componentGJR-GARCH,theGARCH-MIDAS-RV,andthelog-HARmodelinlong-termout-of-sampleforecasting. We illustrate the MF2-GARCH’s scalability by applying the new model to more than 2100 individual stocks in the Volatility Lab at NYU Stern. 1|Introduction There is strong empirical evidence that the conditional variance of stock returns consists of several components. Early evidence for volatility components was provided in, for example, Ding and Granger 1996 and Engle and Lee 1999. More recent evidence can be found in Christoffersen et al. 2008, Kim and Nelson 2013, Dorion 2016, and Conrad and Kleen 2020, among others. While the GARCH models of Ding and Granger 1996 and Engle and Lee 1999 have additive volatility components, more recent GARCH-type models decompose the conditional variance into multiplicative shortand long-term components. For example, in the Spline-GARCH model of Engle and Rangel 2008 and the multiplicative time-varying GARCH (MTV-GARCH) of Amado and Teräsvirta 2013 and Amado and Teräsvirta 2017, the long-term volatility component is a deterministic function of calendar time. In contrast, in the GARCH-MIDASofEngle,Ghysels,andSohn2013,thelong-term ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). Journal of Applied Econometrics published by John Wiley & Sons Ltd. component depends either on a rolling window realized variance (henceforth GARCH-MIDAS-RV, see also Wang and Ghysels 2015) or on low-frequency macroeconomic or financialvariables(henceforthGARCH-MIDAS-X,seealsoAsgharian, Hou, and Javed 2013, and Conrad and Loch 2015). Those multiplicative volatility models are based on the idea that returns follow a stationary GARCH process once divided by the long-term volatility component. However, there is no consensus yet on the most suitable approach for modeling the long-term component. We propose a novel specification for the long-term volatility component in multiplicative GARCH models. The specification is motivated by a new empirical fact that we document for the volatility forecast errors of one-component GARCH models: While the daily standardized forecast errors are essentially unpredictable based on past daily standardized forecast errors, a rolling window moving average of the past daily standardized forecast errors does have predictive power. This is Journal of Applied Econometrics, 2025; 40:438–454 https://doi.org/10.1002/jae.3118 438 because one-component models tend to underpredict or overpredict volatility for extended time periods. While overprediction typically happens during economic expansions, underprediction materializesduringeconomicrecessionsandothercrisesperiods. The new model combines a short-term GJR-GARCH (see Glosten, Jagannathan, and Runkle 1993) component with a long-term component specified as a multiplicative error model (MEM) for the past forecast errors of the GARCH component. That is, the long-term component exploits the predictability in the averaged standardized forecast errors of the short-term component. Intuitively, the long-term component scales the GARCH component’s volatility forecast up/down if the short-term component’s forecasts have underestimated/overestimated volatility in the recent past, that is, the model is learning from past forecast errors. Because the long-term component can either evolve at the same frequency as the short-term component or at a lower frequency, the new specification belongs to the class of mixed frequency data sampling (MIDAS) models pioneered by Ghysels, Santa-Clara,andValkanov2004.WerefertotheproposedspecificationasMultiplicativeFactorMulti-FrequencyGARCH.Asthe “MF” appears twice, we abbreviate the model as MF2-GARCH. The properties of the MF2-GARCH model clearly distinguish it from previous specifications. First, while in other multiplicative models, there is typically no feedback from the short-term to the long-term component (e.g., in the Spline-GARCH), the MF2-GARCH explicitly specifies the long-term component as a function of the short-term component’s past forecast errors. Second, because the actual economic drivers of long-term volatility are unknown and may vary over time, it is challenging to correctlyspecifythelong-termcomponentintheGARCH-MIDAS-X inrealtime.Ourspecificationavoidsthisproblemandisbasedon a simple MEM equation for the long-term component. Interestingly, the MF2-GARCH can be rewritten as a GARCH-MIDAS-X with an explanatory variable “generated within the model.” Third, because the MF2-GARCH is dynamically complete, that is, it fully specifies the dynamics of the conditional variance, it is straightforward to construct multistep ahead volatility forecasts. We obtain the following theoretical results for the MF2-GARCH: First, we derive the unconditional variance of the daily returns. While the unconditional variance is time-varying in the Spline-GARCH and infinite in the GARCH-MIDAS-RV of Wang and Ghysels 2015, returns are covariance stationary in the MF2-GARCH. Due to the feedback between the shortand long-term components, the unconditional variance depends not only on the model parameters but also on the fourth moment of the innovation. Second, we obtain the news impact curve (NIC). TheNICillustratesthatthe responsivenesstonewschanges with thelevelofvolatilitywhichisanotheressentialfeaturethatdistinguishestheMF2-GARCHfromothermodelsintheGARCHfamily. Specifically, conditional volatility is more responsive to news during low volatility periods than during high volatility periods. Third,wederiveexpressionsformultistepaheadforecastsofconditional volatility. Our results show that the forecasts are much more flexible than forecasts from the nested GARCH model. The forecastsreflectthecurrentstanceoftheconditionalvarianceand theprevailingvolatilityregime.Intheshortterm,theconditional volatility forecast will approach the forecast of the long-term component before it converges to the unconditional volatility in the long run. Forecasts from the MF2-GARCH also differ from forecasts of standard Spline-GARCH or GARCH-MIDAS models. The forecasts of the latter models are typically assumed to converge to the current level of the long-term component. Thus, the forecasts from these models do not feature mean reversion inthe long run.Forth,we discussthe quasi-maximum likelihood estimation of the MF2-GARCH and provide a Monte Carlo simulation showing that standard asymptotic results lead to valid inference. Finally, we provide an analysis of the degree of misspecification of the nested one-component GJR-GARCH and the GARCH-MIDAS-RV when the true data-generating process is an MF2-GARCH. As discussed in Patton 2020, in the presence of estimation error and depending on the employed loss function, a parsimoniously misspecified model might dominate the true but more complex model in terms of forecast performance. We show by simulations that for reasonable parameter values of the MF2-GARCH,thedegreeofmisspecificationoftheGJR-GARCH and the GARCH-MIDAS-RV is so severe that the MF2-GARCH outperforms both models when evaluated by the squared error (SE) and the QLIKE loss. Our empirical results strongly support the MF2-GARCH. First, we estimate the MF2-GARCH for the S&P 500 and 2142 US and international equities in the Volatility Laboratory (V-Lab) at NYUStern.1Ourin-sampleresultsshowthattheMF2-GARCHis clearly preferred to the nested one-component GJR-GARCH, the GARCH-MIDAS-RV and the Spline-GARCH. For the S&P 500, we show that the MF2-GARCH’s estimated long-term componentiscloselyrelatedtonewsaboutthemacroeconomyandmonetary policy, particularly news about inflation and interest rates. Thus, we provide further evidence for the close link between economic conditions and long-term volatility (see, e.g., Engle, Ghysels, and Sohn 2013, and Conrad and Loch 2015). We also illustrate that the MF2-GARCH’s volatility forecasts, which feature cyclical behavior, are much more flexible than the forecasts of the competitor models. In contrast to the (overly) smooth long-term component of the Spline-GARCH, the MF2-GARCH’s long-termcomponentadjustsinresponsetoshort-livedperiodsof market turmoil. Nevertheless, compared to the one-component GJR-GARCH, the MF2-GARCH’s forecasts avoid overestimating volatility after a short-lived surge in volatility due to the low persistence in its short-term component. While most of the literature on volatility forecasting focuses on short-term (e.g., 1-day ahead) prediction horizons, Christoffersen and Diebold 2000, Engle 2009b, and Ghysels et al. 2019, among others, have highlighted that in many areas of finance long-term risk forecasts are the relevant inputs. This leads to the question of how far ahead into the future we can forecast volatility. Hence, in evaluating the out-of-sample forecast performance of the MF2-GARCH, our focus is on mediumand long-term forecast horizons of up to 8 months. We test whether the MF2-GARCH, which is designed to capture volatility cycles, leads to better long-term predictions than the nested GJR-GARCH, the GARCH-MIDAS-RV, the Spline-GARCH, and Corsi and Reno 2012’s log-HAR with leverage. For the S&P 500, it turns out that the MF2-GARCH outperforms all competitor models when the forecast horizon is beyond 2 months. The MF2-GARCH’s forecast performance is particularly strong during periods of high volatility, where it dominates the competitor models at all forecast horizons. For a cross-section of 20 439 equities, out-of-sample results from the V-Lab confirm that the MF2-GARCH strongly outperforms the competitor models. The paper is organized as follows. In Section 2, we show that the volatility forecast errors of the GJR-GARCH are predictable. We introduce the MF2-GARCH and discuss its properties in Section 3. The empirical results are presented in Section 4and Section5concludes.Furthermodeldetails,proofs,andadditional tables and figures can be found in the Supporting Information. 2|A New Empirical Fact of Volatility Forecast Errors In this section, we provide evidence for a new empirical fact of volatility forecast errors: Rolling window moving averages of the standardized forecast errors of one-component GARCH models behave counter-cyclically and have predictive power for future standardized forecast errors. We denote the log-return on day 𝑡by 𝑟𝑡. The conditional heteroskedasticity in daily stock returns is commonly modeled as a GARCH process. Daily stock returns are written as 𝑟𝑡=√ ℎ𝑡𝜁𝑡,(1) where  ℎ𝑡denotestheconditionalvarianceandthe𝜁𝑡areassumed tobe 𝑖.𝑖.𝑑. with mean andvarianceequal to zeroand one, respectively.Forillustration,we estimate a GJR-GARCH(1,1)specification for a long time series of daily S&P 500 log-returns covering January 1971 to June 2023.2We obtain the following result:  ℎ𝑡=0.018 (0.003)+(0.022 (0.006)+0.116 (0.014)𝟏{𝑟𝑡−1<0})𝑟2 𝑡−1+0.905 (0.008) ℎ𝑡−1(2) where the numbers in parentheses are Bollerslev–Wooldridge robuststandarderrorsand𝟏{𝑟𝑡−1<0}equalsoneif𝑟𝑡−1<0,andzero else. As expected, the conditional variance is highly persistent and there is strong evidence for asymmetry. Several test statistics have been proposed to check a GARCH specification’s adequacy. For example, Engle and Ng 1993 and Halunga and Orme 2009 propose Lagrange Multiplier (LM) tests for the null hypothesis that a (GJR-)GARCH(1,1) is correctly specified. An alternative approach is to check whether Equation (1) is misspecified in the sense that 𝜁𝑡=√𝜏𝑡𝑍𝑡, where the 𝑍𝑡are 𝑖.𝑖.𝑑. and 𝜏𝑡represents an omitted multiplicative long-term volatility component.Thelong-termcomponentevolveseitheratthesame frequencyasthedailyreturnsoratalower(e.g.,monthlyorquarterly) frequency. The daily returns, 𝑟𝑡, can be either stationary or nonstationary. For example, in the Spline-GARCH model 𝜏𝑡 evolves at the daily frequency and—because the long-term component is a deterministic function of time—the daily returns have a time-varying unconditional second moment. In either case, the scaled returns, 𝑟𝑡∕√𝜏𝑡, are assumed to follow a stationary GARCH process. LM tests for an omitted 𝜏𝑡component have beenproposedinLundberghandTeräsvirta2002andAmadoand Teräsvirta 2017 for daily long-term components. The LM test of Conrad and Schienle 2020 allows for explanatory variables in the long-term component and either a daily or lower frequency 𝜏𝑡. The tests of Lundbergh and Teräsvirta 2002 and Conrad and Schienle 2020 exploit that the squared standardized errors, 𝜁2 𝑡= 𝑟2 𝑡∕ ℎ𝑡,are𝑖.𝑖.𝑑. under the null hypothesis of a constant long-term component. The Conrad and Schienle 2020 LM test checks whether 𝜁2 𝑡is predictable by 𝑥𝑡−1,𝑥 𝑡−2,…,𝑥𝑡−𝐾, where 𝑥𝑡is a predictorvariablethatcanbeexogenousor“generatedwithinthe model.”3Underthe null hypothesis,theLM test is 𝜒2distributed with 𝐾degrees of freedom. We propose to use the 𝑚-days rolling window average of the squared standardized errors,  𝑉(𝑚) 𝑡−1=1 𝑚 𝑚 ∑ 𝑗=1 𝑟2 𝑡−𝑗  ℎ𝑡−𝑗 ,(3) asthepredictorvariable.Underthenullhypothesis,𝑟2 𝑡isaconditionally unbiased proxy of the true but unobservable conditional variance and  ℎ𝑡is a one-step-ahead forecast for the same quantity. If the GARCH model is correctly specified, the standardized volatility forecast errors, 𝑟2 𝑡∕ ℎ𝑡, have an expected value of one and a variance of two if 𝜁𝑡is Gaussian. Hence, we think of  𝑉(𝑚) 𝑡 as a measure of the local bias of the GARCH conditional variance. For 𝑚=1, we obtain  𝑉(1) 𝑡−1=𝜁2 𝑡−1, which is the predictor variable in the “ARCH nested in GARCH” test of Lundbergh and Teräsvirta 2002. Figure 1shows  𝑉(𝑚) 𝑡based on the conditional variances from the GJR-GARCH in Equation (2) for 𝑚∈{1,15,25,45}. In the upper right and both lower panels, it is visible that, as expected,  𝑉(𝑚) 𝑡fluctuates around the value of one. However, as the two lower panels show, there are extended periods during which the one-component GARCH model underestimates or overestimates volatility. That is, for 𝑚=25 and 𝑚=45, the evolution of  𝑉(𝑚) 𝑡−1is inline with alocal bias ofthe GJR-GARCHconditional variance. Thelocalbiasappearstobecounter-cyclical:Theone-component GARCH model tends to overestimate volatility during expansions and to underestimate it during recessions. For 𝑚=1, 𝑉(𝑚) 𝑡 istoo noisy to revealthis bias. There arealso some spikes in  𝑉(𝑚) 𝑡. Thesespikesoccurduetoextraordinaryeventswithunexpectedly high volatility. For example, the two largest spikes are due to the stock market crashes on October 19, 1987 (“Black Monday”) and October13, 1989 (“Mini-Crash”).ThespikeinMarch 2020is due to the emergence of the Covid-19 pandemic and the spike on September 14, 2022 due to the release of higher-than-expected inflation numbers. In Panel B of Table 1, we formally test whether  𝑉(𝑚) 𝑡−1has predictive power for 𝜁2 𝑡using the Conrad and Schienle 2020 LM test with 𝐾=1. For 𝑚∈{1,5,15}, the null hypothesis of a constant long-term component is not rejected. When the true long-term component smoothly varies over time, this is to be expected because for small 𝑚,  𝑉(𝑚) 𝑡−1is too noisy to have explanatory power for 𝜁2 𝑡. In contrast, for 𝑚∈{25,35,45,55}, we strongly reject the nullhypothesis.Thus,the LM testprovides evidencefor an omitted long-term component and suggests that  𝑉(𝑚) 𝑡−1is suitable for modeling the dynamics of the long-term component when 𝑚is appropriately chosen. Mathematical Methods in the Applied Sciences, 2025 440 FIGURE 1 |AGJR-GARCH(1,1)isestimatedfordailyS&P500returndatafortheJanuary1971toJune2023period.Thefigureshows  𝑉(𝑚) 𝑡for𝑚=1 (upper left panel), 𝑚=15 (upper right panel), 𝑚=25 (lower left panel), and 𝑚=45 (lower right panel). Gray shaded areas represent NBER recession periods. TABLE 1 |Summary statistics S&P 500 and LM test. Mean SD Skewness Kurtosis Min Max AC(1) Panel A: Summary statistics 𝑟𝑡0.03 1.09 −1.00 27.11 −22.93 10.71 −0.02 𝑅𝑉𝑡0.99 2.94 18.01 486.49 0.02 101.29 0.63 Panel B: LM test: Explanatory variable  𝑉(𝑚) 𝑡 𝑚1 5 15 25 35 45 55 𝑝-value 0.930 0.960 0.450 0.030 0.001 0.001 0.010 Note: Panel A shows summary statistics for the daily returns, 𝑟𝑡, and the daily realized variances, 𝑅𝑉𝑡, of the S&P 500. The columns present the mean, the standard deviation (sd), skewness, kurtosis, the minimum (min) and maximum (max) as well as the first-order autocorrelationcoefficient (AC(1)). Daily returns for the S&P 500 cover the period January 1971 to June 2023. Realized variancesare for the period January 2010 to June 2023. Panel B shows the results of the Conrad and Schienle 2020 LM test for an omitted long-term component under the null hypothesis of a one-component GJR-GARCH. We set 𝐾=1. The table shows the 𝑝-values of the test for different choices of 𝑚. Importantly, the behavior of the standardized volatility forecast errors is not specific to the one-component GJR-GARCH. We also estimated EGARCH (Nelson 1991), FIGARCH (Baillie, Bollerslev, and Mikkelsen 1996), and Realized GARCH (Hansen, Huang, and Shek 2012) models and obtained very similar results. For example, the correlation between the  𝑉(45) 𝑡−1of the GJR-GARCH and the  𝑉(45) 𝑡−1of the EGARCH, FIGARCH, and the Realized GARCH is 0.92, 0.82, and 0.77, respectively. Figure A.1 in the Supporting Information plots  𝑉(45) 𝑡−1for all four models and confirms that there is strong co-movement. Furthermore, our findingsdo not only holdfor the S&P 500 butalso for other internationalstockindices.Forillustration,FigureA.2intheSupporting Information replicates Figure 1for the FTSE 100.4 In summary, the evidence suggests that one-component GARCH models are misspecified and that the misspecification is detectable when using suitable moving averages of past standardized forecast errors to predict the current standardized forecast error. 3|The MF2-GARCH Model This section introduces the MF2-GARCH model. In the main specification, the shortand the long-term components evolve at a daily frequency. For this specification, we derive the unconditional variance of returns, the NIC and multistep ahead forecasts. In Section 3.2, we suggest several directions in which the MF2-GARCH can be extended. In particular, we introduce a parametrization that allows for multiple frequencies, that is, the short-term component evolves at the daily frequency, while the long-termcomponentvariesatalower frequency.Further details 441 on the MF2-GARCH are provided in the Supporting Information: Section A.1 discusses quasi-maximum likelihood estimation and provides a Monte Carlo simulation. The degree of misspecification of the nested one-component GJR-GARCH and the GARCH-MIDAS-RV when the true model is an MF2-GARCH is analyzed in Section A.2 in the Supporting Information. A comparison of the MF2-GARCH with other component models is provided in Section A.3 in the Supporting Information. In general and as motivated in Section 2, daily log-returns are defined as 𝑟𝑡=𝜎𝑡𝑍𝑡=√ℎ𝑡𝜏𝑡𝑍𝑡. We denote the information set on day 𝑡by 𝑡.𝜎2 𝑡denotes the conditional variance and the shortand long-term volatility components are given by ℎ𝑡and 𝜏𝑡.We make the following assumption about the innovations 𝑍𝑡. Assumption 1. Let𝑍𝑡bei.i.d.Thedensityof𝑍𝑡issymmetric with E[𝑍𝑡]=0 and E[𝑍2 𝑡]=1. Further, 𝑍2 𝑡has a nondegenerate distribution and 𝜅=E[𝑍4 𝑡]<∞. The assumption that the density of 𝑍𝑡is symmetric is commonly made for GJR-GARCH models because it allows for a straightforwardcomputationofthemultistepaheadconditionalvariance forecast(see,e.g.,Zivot2009).Also,LingandMcAleer2002make thisassumptionwhenderivingconditionsforthestationarityand the existence of the fourth moment of the GJR-GARCH. Importantly, as shown in Alexander, Lazar, and Stanescu (2021), the symmetry of the density of 𝑍𝑡does not preclude that the 𝑠-step ahead aggregated returns exhibitskewness. The assumption that 𝜅=E[𝑍4 𝑡]<∞is a necessary condition for ensuring the finitenessofthe unconditionalvarianceof thereturnsandforthe existence of the variance of the score of the likelihood function. 3.1 |Daily Shortand Long-Term Components Wespecifytheshort-termvolatility componentasaunitvariance GJR-GARCH(1,1) ℎ𝑡=(1−𝜙)+(𝛼+𝛾𝟏{𝑟𝑡−1<0})𝑟2 𝑡−1 𝜏𝑡−1+𝛽ℎ𝑡−1,(4) where 𝜙=𝛼+𝛾∕2+𝛽. Note that the driving variable in Equation(4)is𝑟2 𝑡−1∕𝜏𝑡−1.Thisdistinguishesℎ𝑡fromthedailyconditional variance,  ℎ𝑡, in Equation (2). We make the following assumption about the parameters of the short-term component: Assumption 2. The parameters of the short-term GJR-GARCH component satisfy the conditions 𝛼>0,𝛼+𝛾> 0,𝛽>0 and 𝜙=𝛼+𝛾∕2+𝛽<1. If Assumptions 1and 2hold, then 𝑟𝑡∕√𝜏𝑡=√ℎ𝑡𝑍𝑡follows a covariance stationary GJR-GARCH(1,1) with E[ℎ𝑡𝑍2 𝑡]=E[ℎ𝑡]= 1. If the GJR-GARCH(1,1) fully captures the conditional heteroskedasticity, then 𝜏𝑡is equal to a constant and the multiplicative model reduces to a one-component GJR-GARCH for the daily returns. Following Engle 2009a, we refer to 𝑟𝑡∕√ℎ𝑡as “deGARCHed returns” and define 𝑉𝑡=𝑟2 𝑡∕ℎ𝑡=𝜏𝑡𝑍2 𝑡as the squared deGARCHed returns. If the GARCH component fully captures the conditional heteroskedasticity, then by Assumption 1 the 𝑉𝑡are 𝑖.𝑖.𝑑. However, if 𝜏𝑡is time-varying and persistent, then 𝑉𝑡is autocorrelated. Because 𝑉𝑡is a nonnegative variable, we specify the long-term component as a MEM equation for the conditional expectation of 𝑉𝑡: 𝜏𝑡=𝜆0+𝜆1𝑉(𝑚) 𝑡−1+𝜆2𝜏𝑡−1,(5) where 𝑉(𝑚) 𝑡−1=1 𝑚 𝑚 ∑ 𝑗=1𝑉𝑡−𝑗=1 𝑚 𝑚 ∑ 𝑗=1 𝑟2 𝑡−𝑗 ℎ𝑡−𝑗.(6) Recall that we can think of the squared deGARCHed returns as standardized volatility forecast errors. Hence, 𝑉(𝑚) 𝑡−1isarolling window measure of the local bias of the short-term component’s conditional variance over the previous 𝑚days. Note that Equation (5) can be written is a MEM(1,𝑚) with the restriction that the 𝑚“ARCH” coefficients are given by 𝜆1∕𝑚. The MEM(1,𝑚) is covariance stationary if the sum of the ARCH and GARCH coefficients is less than one. By construction, this is satisfied if 𝜆1+𝜆2<1. Assumption 3. Theparametersofthelong-term component satisfy the conditions 𝜆0>0,𝜆 1>0,𝜆 2>0 and 𝜆1+𝜆2<1. Under Assumptions 1and 3, it holds that 𝑉𝑡=𝜏𝑡𝑍2 𝑡is a covariance stationary MEM with E[𝑉𝑡|𝑡−1]=𝜏𝑡and E[𝑉𝑡]=𝜆0∕(1− 𝜆1−𝜆2). We refer to the parametrization given by Equations (4) and (5) as MF2-GARCH-rw-𝑚, where “rw-𝑚” stands for rolling window of length 𝑚. 3.1.1 |Unconditional Variance of Daily Returns Next, we derive the unconditional variance of the daily returns. First,notethattheunconditionalmeanandvariancearegivenby E[𝑟𝑡]=0 and Var[𝑟𝑡]=E[𝑟2 𝑡]=E[𝜏𝑡ℎ𝑡]. The following theorem provides an expression for Var[𝑟𝑡]. Theorem 1. Let Assumptions 1–3be satisfied. If the MF2-GARCH-rw-𝑚process, (𝑟𝑡)𝑡∈ℤ, is covariance stationary, then Γ𝑚=(𝜆11 𝑚𝜙𝜅+𝜆2𝜙)+𝜆1𝜙𝜅1 𝑚 𝑚 ∑ 𝑗=2𝜙𝑗−1<1(7) with 𝜙𝜅=(𝛼+𝛾∕2)𝜅+𝛽. Theunconditional variance of thedaily returns is given by Var[𝑟𝑡]=(𝜆0+𝜆0 1−𝜆1−𝜆2(1−𝜙)(𝜆1+𝜆2)+Δ 𝑚)∕(1−Γ 𝑚), (8) where Δ𝑚=(1−𝜙)𝜆1𝜙𝜆0 1−𝜆1−𝜆2(𝑚−1 𝑚+1 𝑚 𝑚 ∑ 𝑗=2 𝑗−2 ∑ 𝑘=1𝜙𝑘). In the proof of Theorem 1, we show that Γ𝑚<1 is satisfied if the returns are covariance stationary. For example, for 𝑚=1,the conditionreduces toΓ1=𝜆1[(𝛼+𝛾∕2)𝜅+𝛽]+𝜆2[𝛼+ 𝛾∕2+𝛽]<1. This illustrates that even if the conditions which ensure that 𝑟𝑡∕√𝜏𝑡and 𝑉𝑡=𝑟2 𝑡∕ℎ𝑡are individually covariance Mathematical Methods in the Applied Sciences, 2025 442 stationary (i.e., Assumptions 1–3) are satisfied, the condition in Equation (7) may be violated if 𝜅is sufficiently large. Ingeneral,thevarianceofthedailyreturnswilldependonallthe model parameters and the innovation’s fourth moment, 𝜅. The fact that the unconditional variance depends on 𝜅distinguishes the MF2-GARCH-rw-𝑚from standard GARCH models and is due to the correlation between 𝜏𝑡and ℎ𝑡. The unconditional variance of the returns increases in 𝜅and decreases as 𝑚increases.5 Theorem 1reveals that the MF2-GARCH is fundamentally different from the Spline-GARCH and the GARCH-MIDAS-RV. In the Spline-GARCH, the unconditional variance of the returns is time-varying, and Var[𝑟𝑡]=∞in the GARCH-MIDAS-RV (see Wang and Ghysels 2015). When imposing the restrictions 𝑚=1 and 𝛾=0, we obtain a model that can be considered a “multiplicative version” of the additive component model of Engle and Lee 1999.For𝑚=1, both components are symmetric, and we impose the restriction 𝛼+𝛽<𝜆 1+𝜆2<1 to ensure that 𝜏𝑡is the long-run component. In the following corollary, we derive the condition for the covariance stationarity of the daily returns when 𝑚=1 and 𝛾=0 and state their unconditional variance. Corollary 1. LetAssumptions 1–3besatisfied, 𝑚=1,and 𝛾= 0.Thenecessaryandsufficientconditionforthecovariancestationary of the MF2-GARCH-rw-1process is Γ1=𝜆1(𝛼𝜅 +𝛽)+𝜆2(𝛼+ 𝛽)<1. The unconditional variance of the returns is given by Var[𝑟𝑡]=𝜆0+𝜆0 1−𝜆1−𝜆2(1−𝛼−𝛽)(𝜆1+𝜆2) 1−[𝜆1(𝛼𝜅 +𝛽)+𝜆2(𝛼+𝛽)] .(9) If the short-term component is constant (i.e., 𝛼=𝛽=0), the expression in Equation (9) reduces to Var[𝑟𝑡]=E[𝜏𝑡]=𝜆0∕(1− 𝜆1−𝜆2). If the long-term component is constant (i.e., 𝜆1= 𝜆2=0), the expression in Equation (9) reduces to Var[𝑟𝑡]= E[𝜆0ℎ𝑡]=𝜆0. In the latter case, the MF2-GARCH-rw-1 reduces to a GARCH(1,1). Asexpected,empiricallywefindthat𝑉(1) 𝑡−1=𝑟2 𝑡−1∕ℎ𝑡−1istoonoisy to serve as a proxy of the local bias. In Section 4.2.1, we show that the optimal choice of 𝑚is around 63 for the S&P 500, which corresponds to a quarterly moving average. 3.1.2 |News Impact Curve Following Engle and Ng 1993, we use the NIC to illustrate how the conditional volatility is updated in response to new information. For a GJR-GARCH(1,1) with conditional variance  ℎ𝑡= 𝛼0+(𝛼+𝛾1{𝑟𝑡−1<0})𝑟2 𝑡−1+𝛽 ℎ𝑡−1, the NIC is defined as 𝑁𝐼𝐶𝐺𝐽𝑅 𝑡+1= ℎ𝑡+1(𝑟𝑡| ℎ𝑡)=𝐴𝐺𝐽𝑅 𝑡+(𝛼+𝛾1{𝑟𝑡<0})𝑟2 𝑡, where 𝐴𝐺𝐽𝑅 𝑡=𝛼0+𝛽 ℎ𝑡. That is, the NIC is a function of today’s return and 𝐴𝐺𝐽𝑅 𝑡is known conditional on 𝑡−1.If𝛾>0, negative news, 𝑟𝑡<0, have a stronger effect on volatility than positive news. However, the size of the effect does not depend on the current level of volatility. The NIC of the MF2-GARCH-rw-𝑚consists of three terms: 𝑁𝐼𝐶𝑀𝐹 𝑡+1=𝜎2 𝑡+1(𝑟𝑡|𝜏𝑡,ℎ𝑡)=𝐴𝑀𝐹 𝑡+𝐵𝑀𝐹 𝑡 +(𝜆1𝛽1 𝑚+𝜆2(𝛼+𝛾1{𝑟𝑡<0}))𝑟2 𝑡,(10) where 𝐴𝑀𝐹 𝑡=𝜆0(1−𝜙)+𝜆0𝛽ℎ𝑡+𝜆1(1−𝜙)1 𝑚 𝑚−1 ∑ 𝑗=1 𝑟2 𝑡−𝑗 ℎ𝑡−𝑗 +𝜆1𝛽ℎ𝑡1 𝑚 𝑚−1 ∑ 𝑗=1 𝑟2 𝑡−𝑗 ℎ𝑡−𝑗 +𝜆2(1−𝜙)𝜏𝑡+𝜆2𝛽𝜎2 𝑡 and 𝐵𝑀𝐹 𝑡=𝜆0(𝛼+𝛾1{𝑟𝑡<0})𝑟2 𝑡 𝜏𝑡 +𝜆1(1−𝜙)1 𝑚 𝑟2 𝑡 ℎ𝑡 +𝜆1(𝛼+𝛾1{𝑟𝑡<0})1 𝑚 𝑟4 𝑡 𝜎2 𝑡 +𝜆1(𝛼+𝛾1{𝑟𝑡<0})𝑟2 𝑡 𝜏𝑡 1 𝑚 𝑚−1 ∑ 𝑗=1 𝑟2 𝑡−𝑗 ℎ𝑡−𝑗. The intercept, 𝐴𝑀𝐹 𝑡, is known conditional on 𝑡−1. The last term dependson 𝑟2 𝑡andmodel parametersand, hence, issimilar tothe term(𝛼+𝛾1{𝑟𝑡<0})𝑟2 𝑡intheGJR-GARCH.However,theterm𝐵𝑀𝐹 𝑡 shows that the marginal effect of 𝑟2 𝑡also depends on the current conditional variance, 𝜎2 𝑡, as well as ℎ𝑡and 𝜏𝑡individually. Figure2showsthe(standardized)NICoftheMF2-GARCH-rw-𝑚 with𝑚=63(leftpanel)and𝑚=21(rightpanel).TheNICisplotted as a function of the return, 𝑟𝑡, and for 𝜏𝑡=1, 𝜏𝑡=1.5, and 𝜏𝑡=0.5. The parameters are chosen as in Figure A.3 in the Supporting Information. The short-term component is fixed at its unconditional expectation (i.e., ℎ𝑡=1). Note that parameters of thelong-term component arechosensuch that E[𝜏𝑡]=1.That is, thesolidblackNICrepresentsasituationinwhichboththeshortandlong-termcomponentareattheirunconditionalexpectation. Due to the asymmetry term in the short-term component, the impact of negative returns is stronger than the impact of positive returns. The other two lines show that the effect of new information becomes stronger/weaker when long-term volatility isbelow/aboveitsexpectation.Thatis,theconditionalvolatilityis moresensitivetonews during alow-volatilityperiodthan during a high-volatility period. This is reasonable because a large value of |𝑟𝑡|is to be expected during turbulent times but less so during tranquil times. In line with the observation that the variance of the returns decreases as 𝑚increases, the news impact is slightly weaker for 𝑚=63 than for 𝑚=21. 3.1.3 |Forecasting Volatility Because the MF2-GARCH-rw-𝑚model is dynamically complete, we can analytically derive volatility forecasts for any desired horizon. In the following, we assume that a researcher has observed returns up to day 𝑡. Based on the information set 𝑡, she intends to compute a volatility forecast for day 𝑡+𝑠. First, recall that the 𝑠-step ahead forecast of the short-term component can be computed as E[ℎ𝑡+𝑠|𝑡]=1+𝜙𝑠−1(ℎ𝑡+1−1),𝑠≥2 (see, e.g., Zivot 2009). The forecasts for the long-term component are slightly more involved and are presented in Appendix A.4 of the 443 FIGURE 2 |ThefigureshowstheNICforanMF2-GARCH-rw-𝑚modelwith𝑚=63(leftpanel)and𝑚=21(rightpanel)andparametersasinFigure A.3 in the Supporting Information. We fix ℎ𝑡=1 and assume that the short-term component correctly predicts volatility on days 𝑡−1to𝑡−𝑚−1, that is, we set 𝑟2 𝑡−𝑗∕ℎ𝑡−𝑗=1 for 𝑗=1,…,𝑚−1. The NICs are plotted as a function of the return, 𝑟𝑡, and for 𝜏𝑡∈{0.5,1,1.5}. The NICs are standardized such that news impact is zero for 𝑟𝑡=0 and presented as annualized volatilities, that is, we plot √252(𝜎2 𝑡+1(𝑟𝑡|𝜏𝑡,ℎ𝑡=1)−𝜎2 𝑡+1(𝑟𝑡=0|𝜏𝑡,ℎ𝑡=1)). Supporting Information. Using these results, E[𝜎2 𝑡+𝑠|𝑡]can be computed as follows: Theorem 2. Let Assumptions 1–3and the constraint in Equation (7)be satisfied. Then, in the MF2-GARCH-rw-𝑚the forecast of the conditional variance on day 𝑡+𝑠, 𝑠 ≥1, can be computed as follows: First, E[𝜎2 𝑡+1|𝑡]=ℎ𝑡+1𝜏𝑡+1. Second, for 𝑠= 2,…,𝑚the forecasts can be recursively calculated as E[𝜎2 𝑡+𝑠|𝑡]=(1−𝜙)E[𝜏𝑡+𝑠|𝑡]+𝜆0𝜙E[ℎ𝑡+𝑠−1|𝑡] +(𝜆11 𝑚𝜙𝜅+𝜆2𝜙)E[𝜎2 𝑡+𝑠−1|𝑡] +𝜆1𝜙E[ℎ𝑡+𝑠−1|𝑡]1 𝑚 𝑚 ∑ 𝑗=𝑠 𝑟2 𝑡+𝑠−𝑗 ℎ𝑡+𝑠−𝑗 +(1−𝜙)𝜆1𝜙1 𝑚 𝑠−1 ∑ 𝑗=2E[𝜏𝑡+𝑠−𝑗|𝑡](1+𝑗−2 ∑ 𝑘=1𝜙𝑘) +𝜆1𝜙𝜅𝜙1 𝑚 𝑠−1 ∑ 𝑗=2𝜙𝑗−2E[𝜎2 𝑡+𝑠−𝑗|𝑡]. (11) Third, for 𝑠>𝑚, the following recursion applies: E[𝜎2 𝑡+𝑠|𝑡]=(1−𝜙)E[𝜏𝑡+𝑠|𝑡]+𝜆0𝜙E[ℎ𝑡+𝑠−1|𝑡] +(𝜆11 𝑚𝜙𝜅+𝜆2𝜙)E[𝜎2 𝑡+𝑠−1|𝑡] +(1−𝜙)𝜆1𝜙1 𝑚 𝑚 ∑ 𝑗=2E[𝜏𝑡+𝑠−𝑗|𝑡](1+𝑗−2 ∑ 𝑘=1𝜙𝑘) +𝜆1𝜙𝜅𝜙1 𝑚 𝑚 ∑ 𝑗=2𝜙𝑗−2E[𝜎2 𝑡+𝑠−𝑗|𝑡]. (12) We will illustrate the behavior of the volatility forecasts in Section 4.2.2. 3.2 |Extensions of the MF2-GARCH-rw 3.2.1 |Modifications of the Daily Long-Term Component MF2-GARCH with beta-weights: In Equation (6), we assume that 𝑉(𝑚) 𝑡−1is based on the average of the last 𝑚standardized forecast errors. Instead of imposing equal weights, we can take a weighted average of the form 𝑉(𝑚) 𝑡−1=𝑚 ∑ 𝑗=1𝑤𝑗(𝜔)𝑉𝑡−𝑗=𝑚 ∑ 𝑗=1𝑤𝑗(𝜔)𝑟2 𝑡−𝑗 ℎ𝑡−𝑗.(13) Following a common choice in the MIDAS literature (see Ghysels, Santa-Clara, and Valkanov 2006, and Ghysels, Sinko, and Valkanov 2007), we parsimoniously model the weights 𝑤𝑗(𝜔)according to a restricted beta-weighting scheme: 𝑤𝑗(𝜔)= ((1−𝑗∕(𝑚+1))𝜔−1)∕(∑𝑚 𝑘=1(1−𝑗∕(𝑚+1))𝜔−1). By construction, theweightssumtoone.Inaddition,weimposetheconstraintthat the weights are nonincreasing (𝜔≥1). For 𝜔>1, the weights declinefromthefirstlag.For𝜔=1,theweightsaregivenby1∕𝑚, and hence, we obtain the model with rolling window 𝑉(𝑚) 𝑡.We will refer to this parametrization as MF2-GARCH-bw-𝑚, where “bw-𝑚” stands for beta-weights of length 𝑚. Realized volatility MEM: When 𝑚is small, the average standardized forecast error of the short-term component, 𝑉(𝑚) 𝑡= 1∕𝑚∑𝑚 𝑗=1𝑟2 𝑡−𝑗∕ℎ𝑡−𝑗,canbeanoisyproxyforthelocalbias.Instead, if we observe daily realized variances, 𝑅𝑉𝑡, we can base 𝜏𝑡on the realized measure: 𝑉(𝑚,𝑅𝑉 ) 𝑡=1∕𝑚∑𝑚 𝑗=1𝑅𝑉𝑡−𝑗∕ℎ𝑡−𝑗. Again, we can apply the MEM specification from Equation (5). However, without further assumptions, this specification is no longer dynamically complete. We leave this specification for future research. 3.2.2 |Low-Frequency Long-Term Component In the MF2-GARCH-rw-𝑚, 𝜏𝑡varies at the daily frequency. Instead, we can specify the MF2-GARCH so that the long-term component varies at a lower frequency. For this specification, we introduce a notation that allows for mixed frequencies. We distinguish between a low-frequency period 𝑡and a high-frequency period 𝑖. The high-frequency period 𝑖represents days while 𝑡 Mathematical Methods in the Applied Sciences, 2025 444 might represent a monthly, quarterly, or semiannual frequency. We assume that there are 𝑛days within each period 𝑡, that is, 𝑖= 1,…,𝑛, and that we observe 𝑡=1,…,𝑇 low-frequency periods. Using this notation, we denote the log-return on day 𝑖of period 𝑡by 𝑟𝑖,𝑡 (where we use the convention that 𝑟0,𝑡 =𝑟𝑛,𝑡−1). Similarly, we denote the information set on day 𝑖in period 𝑡by 𝑖,𝑡 and define 𝑡∶= 𝑛,𝑡. Note that the new notation reduces to the notation with a daily long-term component when 𝑛=1. In the following, we assume that Assumption 1holds for the innovations 𝑍𝑖,𝑡. Using the notation for multiple frequencies, we write the short-term volatility component as ℎ𝑖,𝑡 =(1−𝜙)+(𝛼+𝛾1{𝑟𝑖−1,𝑡<0})𝑟2 𝑖−1,𝑡 𝜏𝑡 +𝛽ℎ𝑖−1,𝑡 (14) for 𝑖=2,…,𝑛and with ℎ0,𝑡 =ℎ𝑛,𝑡−1. Thus, for 𝑖=1, we obtain ℎ1,𝑡 =(1−𝜙)+(𝛼+𝛾1{𝑟𝑛,𝑡−1<0})𝑟2 𝑛,𝑡−1∕𝜏𝑡−1+𝛽ℎ𝑛,𝑡−1.Asbefore,we assume that Assumption 2holds. We close the model by defininga MEM specification at the lower frequency. Byaveragingthe squared deGARCHed returns within low-frequency period 𝑡,we obtain 𝑉𝑡=1 𝑛 𝑛 ∑ 𝑖=1 𝑟2 𝑖,𝑡 ℎ𝑖,𝑡 =𝜏𝑡1 𝑛 𝑛 ∑ 𝑖=1𝑍2 𝑖,𝑡 =𝜏𝑡𝑍𝑡,(15) where 𝑍𝑡=𝑛−1∑𝑛 𝑖=1𝑍2 𝑖,𝑡 with E[𝑍𝑡]=1 and Var[𝑍𝑡]=(𝜅− 1)∕𝑛. Again, it follows from Assumption 1that the 𝑍𝑡are 𝑖.𝑖.𝑑. This suggests the specification 𝜏𝑡=𝜆0+𝜆1𝑉𝑡−1+𝜆2𝜏𝑡−1.(16) Note that the low-frequency component 𝜏𝑡still measures volatility in daily units. We will refer to this parametrization of the long-termcomponentas MF2-GARCH-lf-𝑛,where“lf”standsfor low-frequency and 𝑛refers to the choice of the low-frequency period.6For example, when setting 𝑛=21 or 𝑛=63, the long-term component varies at the monthly or quarterly frequency. If Assumptions 1and 3hold, then 𝑉𝑡=𝜏𝑡𝑍𝑡is a covariancestationaryMEM(1,1)withE[𝑉𝑡|𝑡−1]=𝜏𝑡andE[𝑉𝑡]= 𝜆0∕(1−𝜆1−𝜆2). A drawback of the low-frequency updating of thelong-termcomponentisthatitintroducesadiscontinuityinto the daily conditional variances. Thus, the daily returns are no longer covariance stationary. 4|Empirical Application 4.1 |Stock Market Data Weuse daily returndatafor the S&P500 startingin January1971 and ending in June 2023. Using the notation for 𝑛=1, daily log returns are computed as 𝑟𝑡=100(log(𝑃𝑡)−log(𝑃𝑡−1)), where 𝑃𝑡is the close price on day 𝑡. We employ realized variances based on intraday data provided by Tick Data to evaluate the forecast performance. Daily realized variances, 𝑅𝑉𝑡, are defined as the sum of the squared five-minute intraday log-returns on day 𝑡plus the squaredovernightlog-return(seeBollerslevetal.2018).Wecompute realized variances for the period January 2000 to June 2023. Table 1shows summary statistics for the daily returns and realizedvariances.Theannualized dailyreturnshaveasamplemean of 7.38%. The mean of the annualized daily realized volatility is 15.81%duringtheperiod January2010 toJune2023,which isthe period that is used for the out-of-sample forecast evaluation. In addition, in Section 4.2.3 we use return data from the V-Lab for 2142 US and international equities. 4.2 |MF2-GARCH in Action 4.2.1 |Application to S&P 500 We first apply the MF2-GARCH model to the daily log returns of the S&P 500. We mainly focus on models with a daily long-term component and estimate the MF2-GARCH-rw-𝑚and the MF2-GARCH-bw-𝑚for the entire sample period from January 1971 to June 2023. Choice of 𝒎:We first estimate both models for values of 𝑚 up to 160 and determine the optimal 𝑚as the one that minimizes the BIC.7For both models, the upper left panel of Figure 3 shows the BIC as a function of 𝑚. The lowest value of the BIC materializes for 𝑚=63 for both models. For this value of 𝑚, the MF2-GARCH-rw-𝑚is clearly preferred relative to the MF2-GARCH-bw-𝑚. To investigate whether this pattern holds moregenerally,we reestimateboth models for threesubsamples: January1971-December2009(upperrightpanel),January1980December2009(lower leftpanel),and January1980toJune 2023 (lower right panel). In all three panels, the optimal choice of 𝑚 is around 63 and the MF2-GARCH-rw-𝑚is the preferred model. Overall, the subsample analysis shows that the optimal choice of 𝑚is very stable for the S&P 500 and that equal weights are preferred to beta-weights. Parameter estimates (sample period January 1971 to June 2023): The first two rows of Table 2(labeled as “𝜏𝑡const.”) show the parameter estimates of the nested one-component GJR-GARCH. The parameter estimates of 𝛼, 𝛾, and 𝛽take typical values and indicate strong persistence in the GARCH component(𝛼+𝛾∕2+𝛽=0.982).Next,thetabledisplaystheparameter estimates of the MF2-GARCH-rw-𝑚model. First, for the optimal window length, that is, 𝑚=63, the estimates of the parameters in the long-term component, 𝜆1and 𝜆2, are both significant. As expected, the persistence in the long-term component (0.982) is much stronger than the persistence in the short-term component (0.924). Also, due to introducing a time-varying long-term component, the short-term GJR-component is much less persistent than the one-component GJR-GARCH. The BIC clearly favors the two-component MF2-GARCH-rw-63 over the one-component model. To illustrate the consequences of choosing𝑚too small and toolarge, we presentparameterestimates for a monthly (𝑚=21) and semiannual (𝑚=126) averaging of past forecasterrors.For𝑚=21,the persistence inthe long-termcomponent increases to 0.995. Presumably, this is because increasing 𝜆2smoothes the long-term component and, thereby, counteracts the effect of decreasing 𝑚.For𝑚=126, the estimates of thelong-termcomponent’sparametersarealmostthesameasfor 𝑚=63, but the standard errors increase considerably. 445 GJR-GARCH, the Spline-GARCH, the GARCH-MIDAS-RV and the log-HAR in out-of-sample forecast performance. In general, the MF2-GARCH model will benefit applications that require long-term forecasts of financial volatility, such as long-run value-at-risk predictions or measurement of systemic risk. For example, Conrad, Schoelkopf, and Tushteva 2024 show that the MF2-GARCH allows to compute “volatility news” separately for the short-term and the long-term components. Assuming a positive relation between expected returns and the conditional variance of returns in a GARCH-in-Mean type model, unexpected returns can be decomposed into cash flow and discount rate news. In this framework, discount rate news is mainly drivenbynewstotheMF2-GARCH’slong-termcomponent.This is because only news to the long-term component is persistent enoughto generatesizable variationin discount rates.Fromthis, it follows that the MF2-GARCH’s long-term volatility component is a strong predictor for the strength of the instantaneous response of the stock market to surprises in macroeconomic announcements (see Conrad, Schoelkopf, and Tushteva 2024). It will also be interesting to employ the long-term component in applications that require low-frequency estimates of volatility, for example, when analyzing the link between financial volatilityandfinancialcrises(see,forinstance,Danielsson,Valenzuela, and Zer 2018). Acknowledgments We are grateful to the co-editor, Eric Ghysels, and two anonymous referees for their comments, which greatly improved our paper. We would like to thank Jörg Breitung, Christian Brownlees, Rob Capellini, Zeno Enders, Jean-David Fermanian, Christian Francq, Christian Gourieroux, Onno Kleen, Robinson Kruse-Becher, Enno Mammen, Anne Opschoor, Lara Schadwinkel, Julius Schölkopf, Timo Teräsvirta, and Jean Michel Zakoïan as well as seminar and conference participants at CREST (January 2020), the ES World Congress (2020), the 13th Annual SoFiE Conference (2021), and the 11th ECB Conference on Forecasting Techniques (2021) for their feedback on earlier versions of the paper. Funding by the German Federal Ministry of Education and Research (BMBF) and the Baden-Württemberg Ministry of Science as part of Germany’s Excellence Strategy (ExU 10.2.31) is gratefully acknowledged. Open Access funding enabled and organized by Projekt DEAL. Data Availability Statement The authors have nothing to report. Open Research Badges This article has been awarded Open Data Badge for making publicly available the digitally-shareable data necessary to reproduce the reported results. Data is available at https://doi.org/10.15456/jae.2025013.1232487362. Endnotes 1In V-Lab, the MF2-GARCH is estimated for more than 18,000 assets from different asset classes on a weekly basis. See: https://vlab.stern. nyu.edu/docs/volatility/MF2-GARCH. 2ThedatawillbeintroducedanddiscussedinmoredetailinSection4.1. See also Panel A of Table 1. 3For details, see the discussion below Assumption 6 in Conrad and Schienle 2020, as well as Remark 7 in the Supplementary Appendix of their paper. 4We obtained similar results for the DAX and the Hang Seng Index (HSI). In addition, we found evidence for considerable co-movement instandardizedvolatilityforecasterrorsinternationally(seealsoEngle and Campos-Martins 2023). 5For a graphical illustration, Figure A.3 in the Supporting Information plots the annualized unconditional volatility as a function of 𝑚and for 𝜅∈{3,5,7}. The model parameters are chosen as 𝛼=0.02,𝛾= 0.10,𝛽=0.8,𝜆 0=0.01,𝜆 1=0.05, and 𝜆2=0.94. 6We treat 𝑛as a fixed number that is not “too large.” If 𝑛→∞,then 𝑍𝑡converges to one in probability and, hence, an identification issue arises due to the linear dependence of 𝑉𝑡and 𝜏𝑡. 7As the choice of 𝑚does not affect the number of parameters, we could also determine the optimal value based on the likelihood function. We prefer the BIC because this allows for a meaningful comparison with the nested one-component GJR-GARCH. 8Wecheckedwhetherthereisstillpredictabilityinthevolatilityforecast errorsoftheMF2-GARCH-rw-63.Wefoundnoevidenceforautocorrelation in the daily squared standardized residuals,  𝑍2 𝑡,whenaveraged atlowerfrequencies.In addition,theempiricaldensityof  𝑍𝑡iscloseto being symmetric. 9That is, the 𝑠-step ahead forecast is given by: Var[𝑟𝑡]+ (𝛼𝐺𝐴 +𝛾𝐺𝐴∕2+𝛽𝐺𝐴)𝑠−1(ℎ𝑡+1𝜏𝑡+1−Var[𝑟𝑡]),where𝛼𝐺𝐴,𝛾 𝐺𝐴 and 𝛽𝐺𝐴 are the parameters of the GJR-GARCH and Var[𝑟𝑡]is the unconditional variance of the MF2-GARCH. 10 The increase in volatility was driven by the European sovereign debt crisis and a downgrade of the U.S.’s credit rating by Standard & Poor’s. 11 The spike in volatility was associated with fears that the Federal Reserve might raise interest rates. 12 We only include stocks for which the MF2-GARCH parameter estimates satisfy Assumptions 2and 3. 13 We also considered log-HAR models with (the log of) quarterly and semiannual averages of realized variances as additional explanatory variables. 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