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# Volatility during the Global Financial Crisis and COVID-19 pandemic through the lens of high-frequency data: a Realized GARCH approach

Denisa Banulescu-Radu, Peter Reinhard Hansen, Zhuo Huang, Marius Matei

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Denisa Banulescu-Radu, Peter Reinhard Hansen, Zhuo Huang, Marius Matei. Volatility during the Global Financial Crisis and COVID-19 pandemic through the lens of high-frequency data: a Realized GARCH approach. 2025. hal-05310604

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# Volatility during the Global Financial Crisis and COVID-19 pandemic through the lens of high-frequency data: a Realized GARCH approach*

Denisa Banulescu-Radu†

Peter Reinhard Hansen‡

Zhuo Huang§

Marius Matei¶

October 12, 2025

## Abstract

This article has two primary objectives: 1) to propose a new parametric model of volatility and 2) to identify the days with the most significant volatility shocks, while exploring the events that triggered these shocks. We analyze financial volatility during the global financial crisis and the COVID-19 pandemic, utilizing high-frequency financial data and an improved model, which incorporates asymmetry and handles outliers, to estimate volatility and determine the timing of the largest shocks within the day. For instance, for the global financial crisis, major volatility shocks coincide with events such as the bankruptcy of Lehman Brothers and the failure of the Emergency Economic Stabilization Act. However, the largest shock occurred on February 27, 2007, which, despite coinciding with market events like a Chinese stock market crash and Freddie Mac's tighter subprime loan policy, was primarily caused by a computer glitch. Our analysis underscores the value of high-frequency data in modeling financial volatility and identifying the key events driving volatility shocks.

**Keywords:** Financial crisis; COVID-19 pandemic; Volatility; High-frequency data; Realized GARCH.

**JEL Classification:** C10; C22; C80

<footnote>*Acknowledgements. We are grateful for comments received at the Bank of Italy, Rome, CERGE-EI, Prague, and the Second Workshop in Financial Econometrics, Salvador, Brazil, in particular to Eric Hillebrand for an insightful discussion. The first author acknowledges the support of "Data Science and Insurance Fraud Detection" research initiative under the aegis of the Europlace Institut of Finance, a joint initiative by Laboratoire d'Economie d'Orléans and Thélem Assurances, the "Fraud detection and Anti-Money Laundering" research project (REDFLAG APR-IA AE 2019-1850) funded by Centre-Val de Loire Region, the ANR program CaliBank (ANR-19-CE26-0002-02) and the ANR PRCE InsurFraud (ANR-23-CE26-0022). The second author would like to acknowledge the support of the ADEMU project, "A Dynamic Economic and Monetary Union", funded by the European Union's Horizon 2020 Program under grant agreement No 649396 (ADEMU)", as well as the support from CREATES - Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation. The third author acknowledges financial support by the National Natural Science Foundation of China (71201001, 71671004) and he is grateful to the Department of Economics at the University of North Carolina at Chapel Hill for their hospitality during Spring 2017. The fourth author acknowledges financial support of the EU's NextGenerationEU instrument through the National Recovery and Resilience Plan of Romania - Pillar III-C9-I8, managed by the Ministry of Research, Innovation and Digitalization, within the project: "CauseFinder: Causality in the Era of Big Data and AI and its applications to innovation management", contract no. 760049/23.05.2023, code CF 268/29.11.2023. MATLAB 2020a</footnote>

<footnote>†University of Orléans, Laboratory of Economics of Orléans (corresponding author: denisa.banulescu-radu@univ-orleans.fr)</footnote>

<footnote>‡University of North Carolina & CREATES & Copenhagen Business School</footnote>

<footnote>§Peking University, National School of Development</footnote>

<footnote>¶Bucharest University of Economic Studies, Faculty of Business Administration, in foreign languages & National Institute of Economic Research "Costin C. Kiritescu", Romanian Academy</footnote>
# 1 Introduction

Understanding financial volatility is essential both for researchers and market players, as it plays a central role in assessing market risks and pricing assets. This article addresses two main objectives: first, to propose an enhanced version of the Realized GARCH model, designed to overcome the problem of outliers in volatility modeling; second, to study financial volatility during the global financial crisis (GFC) and the COVID-19 pandemic, identifying the key events that generated the largest volatility shocks. This twofold contribution combines methodological innovation and empirical analysis, providing particularly valuable information for understanding volatility dynamics during periods of extreme market stress.

The importance of this study lies in the fact that volatility is a direct reflection of market reactions to economic and financial events. In times of crisis, it is essential for policymakers, investors and financial institutions to understand volatility trends and identify events that trigger major shocks. Moreover, high-frequency data offers a detailed view of these volatility shocks, enabling more precise identification of their timing and the nature of the underlying events. This level of granularity is often lacking in studies based solely on low-frequency data, such as daily returns. For instance, during the global financial crisis, one of the biggest volatility shocks was mainly linked to a technical problem in the trading system, which could only be discovered using high-frequency data.

<figure><img src="image_3.png" /><figcaption>Figure 1: Annualized realized volatility for the period of 1997-2009 and some of the major events.</figcaption></figure>

The relationship between significant financial/economic events during the 2008-2009 financial crisis
and the realized volatility measure is illustrated in figure 1. This figure highlights the key events that triggered clusters of increased volatility, including the collapse of Lehman Brothers and other events that disrupted the market. It is essential to understand these dynamics to comprehend how market participants react to different types of economic shocks.

This paper builds on the Realized GARCH model introduced by Hansen et al. (2012) and enhances it by addressing the issue of outliers, which has long been a concern in volatility modeling. Outliers (such as extreme returns that may result from shocks or errors) can distort volatility estimates. By incorporating leverage functions to address asymmetry and transformed innovation terms designed to reduce the impact of outliers, and by using realized measures derived from intraday data (e.g. Realized Kernel), the proposed robustified Realized GARCH model offers a more accurate representation of volatility, particularly in the presence of extreme events. This robustness is particularly relevant in applications where outliers are more frequent due to sudden shocks or unexpected events.

The growing literature on volatility modelling, driven by advances in high-frequency financial data, has explored several approaches to improving volatility estimation. Early studies, such as Andersen and Bollerslev (1998), demonstrated that realized variance, calculated from intraday returns, provides an accurate measure of daily volatility. The stochastic properties of realized variance were then analyzed in Andersen et al. (2001), Barndorff-Nielsen and Shephard (2002), Meddahi (2002), Andersen et al. (2003) and Mykland and Zhang (2009). At the same time, a number of improved volatility indicators that account for market microstructure noise have been developed, including those introduced by Zhang et al. (2005), Barndorff-Nielsen et al. (2008), and Hansen and Horel (2009), *inter alios*. Over time, sophisticated volatility models have been designed to incorporate these volatility measures. Notable examples include the MEM model proposed by Engle and Gallo (2006), the HEAVY model by Shephard and Sheppard (2010), and the Realized GARCH model by Hansen et al. (2012). The latter setting has been extended with contributions from Hansen and Huang (2016a), Huang et al. (2016); Hung et al. (2020); Cai et al. (2021), and others. Despite these advancements, volatility models remain sensitive to outliers, particularly those arising from large innovations in returns or realized measures. This paper seeks to improve on these existing approaches by introducing a robustified Realized GARCH model that dampens the influence of outliers without compromising the accuracy of volatility estimation. This feature is inspired by Harvey (2013), who proposed a score-based method to mitigate outlier effects. Our approach borrows this dampening
feature, but instead of using a score-driven model, we adapt it within the context of the Realized GARCH framework to create a robust version of the model suited for periods of financial stress. For low values of the new parameters, observations considered outliers are far less influential, ensuring the model adjusts future volatility more accurately in response to extreme events.

A key aspect of this study is the empirical investigation of volatility during the global financial crisis and the COVID-19 pandemic. Previous studies have explored the relationship between financial crises and volatility, often using daily data. However, this approach can miss the intraday dynamics of volatility shocks, which are crucial to understanding the precise impact of certain events. Using high-frequency data, this paper provides a more detailed view of the events that generated the largest volatility shocks in the two crises. Identifying the link between triggering events, shocks and volatility/returns during financial crises has already been tackled in the literature. Dungey et al. (2010) proposed an identified structural GARCH model to disentangle the dynamics of financial market crises in hypersensitivity to news from foreign markets not experiencing crisis, and the contagion effects transmitted to a tranquil domestic market from foreign crises. For other related studies, see Karanasos et al. (2014), Choudhry and Jayasekera (2014), Allen et al. (2017), and Dungey et al. (2020).

Our empirical analysis highlights different patterns of volatility shocks during the global financial crisis and the COVID-19 pandemic. During the global financial crisis, upside shocks were mainly driven by economic and financial events, including market downturns, institutional failures and policy interventions, while downside shocks were linked to positive developments such as monetary policies and economic stimuli. These events often correlate with changes in market returns. In contrast, the COVID-19 pandemic saw, as expected, health-related news, notably the increase in the number of cases and the introduction of vaccines, playing a more prominent role. Negative shocks were often linked to positive developments, while positive shocks reflected worsening health conditions or economic uncertainty. The magnitude of volatility shocks during the pandemic highlighted the higher global uncertainty compared to the global financial crisis.

The paper is organized as follows. Section 2 presents the theoretical model and the robustified Realized GARCH specification. Section 3 outlines the empirical methodology, data sources, and analysis. Section 4 identifies and discusses key events that triggered the largest volatility shocks in both crises, while Section 5 concludes. Appendices provide additional theoretical and empirical details.
# 2 Modeling Framework

## 2.1 Key Variables

In this study, we examine the volatility of asset returns, denoted $r_t$. For the empirical analysis, we use the exchange-traded fund (ETF) SPY to define daily returns, as it closely tracks the S&P 500 index and provides access to high-frequency data. The conditional variance of daily returns is expressed as:

$$h_t = \mathrm{var}(r_t | \mathcal{F}_{t-1}), \qquad (1)$$

where {$\mathcal{F}_t$} denotes the filtration to which $r_t$ is adapted.

A key variable in our analysis is the volatility shock, which is defined as:

$$v_t = \mathbb{E}(\log h_{t+1} | \mathcal{F}_t) - \mathbb{E}(\log h_{t+1} | \mathcal{F}_{t-1}), \qquad (2)$$

such that $100 \times v_t$ represents the percentage change in volatility due to new information arriving on the t^th day.

The subsequent sections detail the econometric modeling of returns and realized measures of volatility, which underlie the empirical estimation of volatility shocks. Specifically, we outline the Realized GARCH framework and present a robustified version of the model introduced in this paper. Readers primarily interested in empirical results rather than the technical econometric framework may choose to proceed directly to the empirical analysis in Section 3.

## 2.2 Realized GARCH Framework

The Realized EGARCH model of Hansen and Huang (2016b) (with a single realized measure of volatility) is specified by the following three equations:

$$r_t = \mu + \sqrt{h_t} z_t, \qquad (3)$$

$$\log h_t = \omega + \beta \log h_{t-1} + \tau(z_{t-1}) + \gamma u_{t-1}, \qquad (4)$$

$$\log x_t = \xi + \varphi \log h_t + \delta(z_t) + u_t, \qquad (5)$$
where $\tau(z) = \tau_1 z + \tau_2(z^2 - 1)$ and $\delta(z) = \delta_1 z + \delta_2(z^2 - 1)$. Here, $z_t$ and $u_t$ are typically assumed to be mutually and serially independent and modeled with the specifications: $z_t \sim \text{iid}(0, 1)$ and $u_t \sim \text{iid}(0, \sigma_u^2)$. The three equations are referred to as the *return equation*, the *GARCH equation*, and the *measurement equation*, respectively. The first two form the foundation of a GARCH-X model, similar to that estimated by Engle (2002), Barndorff-Nielsen and Shephard (2007), and Visser (2011). The measurement equation, however, is a defining feature of the Realized GARCH framework, linking the (ex-post) realized measure, $x_t$, to the latent (ex-ante) conditional variance, $h_t$. A GARCH-X model, on its own, provides an incomplete description of the data because it lacks a specification for the realized measure. The measurement equation completes the model by capturing the dynamic properties of both returns and realized measures.<fnref n="1" />

Some of the key features of this model are represented by $\beta$, which captures the persistence of volatility, and by $\tau(z_{t-1})+\gamma u_{t-1}$, which reflects the innovation in conditional volatility. Specifically, $\gamma u_{t-1}$ quantifies the impact of the realized measure on the conditional variance for the next period. The functions $\tau(z)$ and $\delta(z)$, known as leverage functions, describe the relationship between returns and volatility, commonly referred to as the *leverage effect*. Hansen et al. (2012) investigated various forms of leverage functions and concluded that a simple quadratic form is effective in practice, which we also adopt in our estimation. Moreover, the term $\tau(z)$ aligns with the *news impact curve* introduced by Engle and Ng (1993), illustrating how positive and negative returns influence expected future volatility.

## 2.3 Robustified Realized GARCH

Periods of financial crises are often characterized by unusually large shocks to returns and volatility. Such large shocks pose significant challenges to conventional GARCH models, which tend to exhibit high sensitivity to extreme returns and to overstate risk in the aftermath of shocks. This limitation motivated Harvey (2013) to propose a more robust dynamic framework based on the conditional scores of the model. These models, commonly referred to as Dynamic Conditional Score (DCS) or Generalized Autoregressive Score (GAS) models, are further elaborated upon in Harvey (2013) and Creal et al. (2012, 2013).

To address the issue of sensitivity to large shocks, we draw insights from Harvey (2013) by incorporating parameters designed to dampen the influence of outliers in returns. Specifically, we replace $z_t$ with $\tilde{z}_t = \frac{z_t}{\sqrt{1+\frac{z_t^2}{d_z^2}}}$ in the GARCH equation, where $d_z$ is a robustness parameter to be estimated. The effect of this

<footnote n="1">¹An alternative approach to extending the GARCH-X model, which incorporates additional latent variables, was proposed by Engle and Gallo (2006) and further discussed by Shephard and Sheppard (2010).</footnote>
transformation is illustrated in Figure 2, which shows the influence of extreme shocks for varying values of $d_z$. This transformation was originally derived from the score function of a standard GARCH model, focusing on univariate time series of returns (see Appendix A.1 for details). Moreover, in our context, both returns and realized measures are subject to potential outliers. Such outliers, whether they arise from extreme returns or unusual values in realized measures, translate into disproportionately large values of $z_t$ and $u_t$, respectively. To account for this, we adopt a similar adjustment for $u_t$, which quantifies shocks to volatility. Specifically, we replace $u_t$ with $\tilde{u}_t = \frac{u_t}{\sqrt{1+\frac{(u_t/\sigma_u)^2}{d_u}}}$, where $d_u$ is another robustness parameter to be estimated. Notably, the standard Realized GARCH model is recovered as $d_z, d_u \to \infty$.

<figure><img src="image_4.png" /><figcaption>Figure 2: The transformation $x \mapsto \frac{x}{\sqrt{1+x^2/d}}$ for various values of $d$.</figcaption></figure>

The robustified Realized GARCH model is specified as follows:

$$r_t = \mu + \sqrt{h_t} z_t, \qquad (6)$$

$$\log h_t = \omega + \beta \log h_{t-1} + \tau(\tilde{z}_{t-1}) + \gamma \tilde{u}_{t-1}, \qquad (7)$$

$$\log x_t = \xi + \varphi \log h_t + \delta(z_t) + u_t, \qquad (8)$$

where $\tilde{z}_t = z_t/\sqrt{1+z_t^2/d_z}$ and $\tilde{u}_t = u_t/\sqrt{1+(u_t^2/\sigma_u^2)/d_u}$. The leverage functions are defined as $\tau(\tilde{z}) = \tau_1\tilde{z} + \tau_2(\tilde{z}^2 - 1)$ and $\delta(z) = \delta_1 z + \delta_2(z^2 - 1)$. Additional variants of this robust model are estimated and discussed in Appendix B. Furthermore, to highlight the statistical superiority and improved performance of the robustified Realized GARCH model over its standard counterpart, we conducted a series of in-depth exercises. The detailed results are presented in the Appendix C. For the quasi-maximum likelihood
estimation, $z_t$ and $u_t$ are assumed to be mutually and serially independent, with $z_t \sim \text{iid}(0, 1)$ and $u_t \sim \text{iid}(0, \sigma_u^2)$. Furthermore, within the framework defined by equations (6)-(8), the volatility shock, as described in equation (1), takes the following form:

$$v_t = \tau(\tilde{z}_t) + \gamma \tilde{u}_t. \qquad (9)$$

The volatility shock $v_t$ comprises hence two components. The first is the news impact curve, a feature well-known from conventional GARCH models, which describes how return shocks affect volatility on average. The second component incorporates the additional information about future volatility contained in the realized measure of volatility. This latter term highlights one of the key advantages of employing realized measures: within the Realized GARCH framework, it enables a more precise estimation of the volatility shock.

# 3 Empirical Analysis

## 3.1 Data Description

This study uses high-frequency transaction prices for the exchange-traded fund (ETF) SPY, which closely tracks the S&P 500 index. The high-frequency data are sourced from the Trade and Quote (TAQ) database, covering the period from January 1, 1997, to March 23, 2021, which enables empirical analysis spanning significant market periods, including the 2007-2009 financial crisis and the COVID-19 pandemic.

Following standard practices in the GARCH literature, we model daily close-to-close returns. The realized measure of volatility is as an estimate of volatility during the active trading hours, which span from 9:30 a.m. to 4:00 p.m., as opposed to the full close-to-close daily volatility relevant to daily returns. However, as noted by Hansen et al. (2012), approximately 75% of daily volatility occurs during these 6.5 hours of active trading. Their findings also suggest that the realized measure is approximately proportional to the full daily volatility, as evidenced by estimates of $\phi$ close to one. For the realized measure of volatility, denoted as $x_t$, we employ the Realized Kernel (RK) estimator proposed by Barndorff-Nielsen et al. (2008).<fnref n="2" />

$$\text{RK}_t(\Delta) = \sum_{h=-H}^{H} k(h/(H+1))\gamma_{h,t},$$

<footnote n="2">²The RK estimator is defined as:</footnote>

<footnote>where $H$ represents the bandwidth, determined based on the recommendations of Barndorff-Nielsen et al. (2009). Here, $\gamma_{h,t} = \sum_{j=|h|+1}^{M} r_{t,j}r_{t,j-|h|}$, and $k(\cdot)$ is the Parzen kernel function.</footnote>
This approach ensures robustness to market microstructure noise and guarantees positive estimates, which are crucial for modeling logarithmic transformations of volatility.

Before computing intraday returns and realized measures, the high-frequency data undergo preprocess-
ing based on the cleaning procedures of Barndorff-Nielsen et al. (2009). Additionally, we exclude trading
days with shortened hours, such as those around Thanksgiving and Christmas, to avoid introducing spu-
rious outliers into the realized measures.<fnref n="3" /> To facilitate interpretation, both the conditional variance and
realized measure are reported on an annualized volatility scale. Annualized realized volatility, derived
from the RK estimates, is defined as:

$$
Rvol_t = \sqrt{250 \times \hat{c} \times RK_t}, \quad \hat{c} = \frac{\sum_t r_t^2}{\sum_t RK_t}, \qquad (10)
$$

where $\hat{c}$ adjusts for the fact that $RK_t$ measures volatility only during active trading hours. Empirical evidence indicates $\hat{c} \approx 4/3$, reflecting that approximately 75% of daily volatility occurs between 9:30 a.m. and 4:00 p.m. Similarly, annualized conditional variance (volatility) is defined as $Cvol_t = \sqrt{250 \times \bar{h}_t}$.

3.2 Estimation Results

The estimation of the robustified Realized GARCH model focuses on two distinct periods: 2006-2009, encompassing the global financial crisis, and 2017-2021, capturing the COVID-19 pandemic. This segmentation facilitates the identification of the largest volatility shocks during these periods. For illustrative purposes, we report results for the period January 1, 2006, to December 31, 2009.<fnref n="4" /> The specification imposes the constraint $\mu = 0$, as preliminary estimations revealed that $\mu$ is statistically insignificant and its inclusion does not materially affect the time series of estimated conditional variances, $\hat{h}_t$. The robustified

<footnote n="3">³The number of excluded days is minimal and does not materially affect the analysis. Following a reviewer's suggestion, we re-estimated the models after reintroducing these excluded observations for the two periods under study. The results revealed negligible differences in the estimates, and the rankings of the days with the largest volatility shocks remained consistent. These additional results are available upon request.</footnote>

<footnote n="4">⁴Estimation results for other periods, including the full sample from January 3, 1997, to December 31, 2009, are consistent and available upon request.</footnote>
Realized GARCH model is estimated as follows:

$$r_t = \sqrt{h_t} z_t,$$

$$\log h_t = \underset{(0.010)}{0.017} + \underset{(0.006)}{0.969} \log h_{t-1} + \underset{(0.061)}{0.407} \tilde{u}_{t-1} - \underset{(0.017)}{0.180} \tilde{z}_{t-1} + \underset{(0.014)}{0.056} (\tilde{z}_{t-1}^2 - 1),$$

$$\log x_t = \underset{(0.082)}{-0.532} + \underset{(0.070)}{1.019} \log h_t - \underset{(0.017)}{0.132} z_t + \underset{(0.009)}{0.036} (z_t^2 - 1) + u_t,$$

The numbers in parentheses are robust standard errors, computed using the sandwich estimator (Bollerslev and Wooldridge, 1992). The model also estimates $\hat{\sigma}_u^2 = 0.154$, $\hat{d}_z = 27.689$, and $\hat{d}u = 5.904$. While standard errors for $d_z$ and $d_u$ are not available, approximate confidence sets can be derived by inverting the likelihood ratio statistics.

The estimation results provide several key insights. The high persistence of volatility is evidenced by $\hat{\beta} = 0.969$, consistent with the extant volatility modeling literature. The parameter $\hat{\gamma} = 0.407$ underscores the informational value of the realized measure in predicting future volatility. The coefficient $\hat{\phi} = 1.019$ demonstrates that the realized measure is proportional to the conditional variance, confirming that a fixed proportion of daily volatility occurs during active trading hours. The asymmetric response of volatility to return shocks (leverage effect) is captured by $\hat{\tau}_1 = -0.180$ and $\hat{\delta}_1 = -0.132$. The news impact curve, which summarizes the response of volatility to studentized return shocks, $z_t$, is displayed in Figure 3. Both the robustified and standard Realized GARCH models exhibit pronounced asymmetry, with negative return shocks having a disproportionately larger impact on volatility than positive shocks of the same magnitude. The figure highlights that the robustified model dampens the volatility response on days with extreme negative return shocks.

The time series of the conditional variance, $h_t$, derived from the estimated model, is illustrated in Figure 4, which also highlights key events during the global financial crisis. The first significant volatility spike was recorded on February 27, 2007. Subsequent spikes aligned with major events, including the collapse of Bear Stearns, the bankruptcy of Lehman Brothers, and the unexpected rejection of the $700 billion banking-rescue package by the U.S. House of Representatives. These and other large volatility spikes will be analyzed in greater detail in later sections. Volatility shocks are formally defined as: $v_t = E(\log h_{t+1}|\mathcal{F}_t) - E(\log h_{t+1}|\mathcal{F}_{t-1}) = \tau(\tilde{z}_t) + \gamma\tilde{u}_t$, and capture the impact of news on day t on expected future volatility.<fnref n="5" /> Figure 5 displays the evolution of estimated volatility shocks, $v_t$, alongside daily returns, $r_t$, for

<footnote n="5">⁵To assess the average contributions of shocks to returns and the realized measure of volatility to volatility shocks, $v_t$,</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 3: News Impact Curves</figcaption></figure>

*Note:* The figure compares the News Impact Curves of the robustified Realized GARCH model (solid line) and the standard Realized GARCH model (dashed line).

<figure><img src="image_6.png" /><figcaption>Figure 4: Volatility and Events</figcaption></figure>

*Note:* This figure illustrates the annualized conditional variance estimated using the robustified Realized GARCH model, with markers indicating notable events during the 2007–2009 period.

the 2007-2009 period. Notably, the largest estimated volatility shock occurred on February 27, 2007. This spike was influenced by the relatively low volatility in the preceding period (approximately 9% annualized). The resulting 118% increase in expected annualized volatility, corresponding to $v_t = 1.558$, did not push volatility to record highs but represented the largest relative change during this timeframe. For the non-robust specification, the volatility shock on the same date was estimated at $v_t = 2.295$, reflecting a 215%

<footnote>we performed two regressions. The first linked $v_t$ to the news impact curve, $\tau(\tilde{z}_t)$, while the second linked it to the realized measure shock, $\gamma\tilde{u}_t$. Coefficients of determination ($R^2$) were calculated in both cases to quantify explanatory power. Analyses across multiple time periods revealed that approximately two-thirds of the contribution stems from shocks to returns, while the remaining third comes from shocks to the realized measure. These proportions remained stable over time, with only minor fluctuations. Detailed results are available upon request.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 5: Returns and Volatility Shocks</figcaption></figure>

*Note:* This figure presents the evolution of volatility shocks (green), $v_t$, as estimated by the robustified Realized GARCH model, alongside daily returns (blue), $r_t$, for the period 2007–2009.

increase in annualized volatility.⁶⁷

Figure 6 compares the non-robust Realized GARCH model with the robustified version. The upper left panel shows the two volatility series, $h_t$, alongside the realized measure of volatility (annualized). While the two $h_t$ series are generally similar, the non-robust specification occasionally exhibits higher volatility spikes than the robust version. The other three panels focus on three-week intervals surrounding the three largest volatility shocks in our sample, with notable discrepancies observed in the upper right panel after the February 27, 2007 event. Following the large realized volatility and negative return on February 27, 2007, the Realized GARCH model reacts strongly to the shock. The non-robust model significantly overestimates the subsequent volatility, while the robust model performs better, except on March 1, 2007. Approximately one week later, both models converge, yielding similar conditional variance estimates. Overall, the standard and robust models yield comparable results. This includes the periods around the second and the third largest volatility shocks, where the only noticeable difference is the day following the second largest volatility shock. The following section will focus on the largest volatility shocks during both the financial crisis and the COVID-19 pandemic.

<footnote n="6">⁶The values of $v_t$ for both the robust and non-robust models are reported in the first panel of Table B.2.</footnote>

<footnote n="7">⁷In response to reviewer feedback, we conducted a comparative analysis using alternative models, including model-free approaches, to evaluate the robustness of the shock classification. Various differences were observed between the rankings produced by the robustified Realized GARCH model and those derived from alternative approaches, such as EGARCH, HAR, SVJD, and the Realized Kernel measure. These discrepancies were evident both in the identification of specific days and their rankings, underscoring the sensitivity of volatility shock rankings to model specifications. While we acknowledge the advantages of model-free approaches, for instance (such as the absence of distributional assumptions and minimal parameter estimation) and also those of other parametric volatility models, we argue that our proposed model offers several advantages over them. These include its ability to account for leverage effects, incorporate realized measures, and effectively handle outliers. Detailed results of this comparative analysis are available upon request.</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 6: Realized Measure of Volatility and Conditional Volatility</figcaption></figure>

Note: This figure compares the realized measure of volatility and the conditional volatility for the period 2007–2009, highlighting three sub-periods around the three largest volatility shocks: February 27, 2007, September 29, 2008, and December 11, 2007. The blue line represents the realized volatility, while the red and green lines correspond to the conditional volatility from the non-robust and robustified GARCH models, respectively.
# 4 News Related to the Largest Volatility Shocks

This section provides a detailed analysis of the days associated with the largest volatility shocks, presented chronologically, as identified using the robustified Realized GARCH model. Positive (upward) shocks are consistently observed to be larger than negative (downward) shocks, both in absolute and percentage terms. To explore these phenomena, we focus on the ten largest upward shocks during the global financial crisis and the five largest upward shocks during the COVID-19 pandemic (see Tables 1 and 2). Additionally, the five largest downward shocks during both periods are detailed in Tables 3 and 4. These tables also present the percentage changes in S&P 500 expected volatility, the return level and a selection of contemporaneous news stories and their type.

Volatility is typically expressed on an annualized scale, calculated as $\sqrt{250h_t}$. The percentage volatility shock to $\sqrt{250h_t}$ is approximately: $100(e^{\frac{1}{2}v_t} - 1)$ (see Appendix A.2). In the following subsections, we analyze intraday high-frequency price data alongside realized volatility measures, calculated over 30-minute intervals. The analysis employs the Realized Kernel estimator, which is subsequently annualized using the transformation $RK \mapsto \sqrt{250 \times 13 \times \hat{c} \times RK}$, where $\hat{c} = \frac{\sum r_t^2}{\sum x_t}$ is the scaling constant defined in Eq. (10). This constant accounts for the fact that realized measures compute volatility over only a portion of the trading day. For each analyzed day, we provide a summary of the key news events and utilize high-frequency data to pinpoint the precise time of the volatility shock, along with the corresponding news, when identifiable.<fnref n="8" />

## 4.1 General Discussion

We argue that the interpretation of volatility shocks should require a close examination of the broader context in which they occur. Market dynamics often reflect the cumulative impact of several events on the same day, rather than being determined solely by an isolated trigger. For example, on February 27, 2007, a major volatility shock was observed, which the literature attributes mainly to a decline in the Chinese stock market or the subprime-related news from Freddie Mac.<fnref n="9" /> However, closer examination reveals that a simultaneous computer malfunction at the New York Stock Exchange (NYSE) may have played a more immediate role in triggering the shock.

<footnote n="8">⁸In Section 4.2, we focus on the seven (five) largest positive volatility shocks and the five (five) largest negative volatility shocks observed during the financial crisis (pandemic period).</footnote>

<footnote n="9">⁹The latter event is highlighted also by the Federal Reserve Bank of St. Louis as a key moment in the timeline of the financial crisis, https://www.stlouisfed.org/financial-crisis/full-timeline.</footnote>
Understanding the interplay of events on any given day is essential to accurately assess market behavior. Secondary events can amplify the market's reaction to a primary shock, increasing investor anxiety or panic. The combined effects of related events can intensify the volatility observed during periods of crisis. In addition, volatility, as a latent variable, reflects the overall impact of all the events that occur during a day. High-frequency data, such as minute-by-minute returns, are used to calculate measures such as the Realized Kernel, which captures cumulative changes in volatility. Consequently, most of the times, attributing a large volatility spike to a single event risks being overly simplistic, as the observed market reaction is often the result of multiple interrelated factors. Documenting the primary and secondary events associated with major volatility shocks contributes to a growing repository of knowledge on market behavior during periods of stress. By preserving a detailed account of such events, researchers can build a comprehensive database of market shocks and their causes. This resource not only facilitates future studies but also supports the development of predictive models to better understand crisis-induced market dynamics.

In addition to its academic relevance, the analysis of volatility shocks has practical implications. Policymakers can leverage this knowledge to anticipate market reactions to specific announcements, such as monetary policy adjustments or economic interventions, and to design strategies to mitigate destabilizing effects. For investors, understanding the drivers and magnitude of historical shocks enables the creation of models that predict market reactions to similar events, thus improving portfolio management strategies during periods of heightened uncertainty.

Based on the data presented in Tables 1 to 4, we observe distinct patterns in the nature and drivers of volatility shocks during the global financial crisis and the COVID-19 pandemic. During the global financial crisis, upward volatility shocks ranged from 44% to 118%, whereas the five largest downward shocks were smaller, ranging between -20% and -22%. Positive volatility shocks were primarily triggered by economic and financial sector events, including drastic market declines, bankruptcies of major institutions such as Lehman Brothers, and interventions by monetary and political authorities. Our analysis shows, based on the news collected, that the majority of news driving these shocks was economic (36%), financial sector-related (32%), or tied to economic policy (16%), with other categories accounting for the remainder. These shocks often coincided with negative market returns, such as the 8.16% drop on September 29, 2008. Conversely, downward volatility shocks were associated with positive developments like economic stimulus
measures, monetary policies, and favorable economic data, resulting in typically positive or modest market returns, as seen on October 13, 2008, when the market rose by 13.56% following government capital injections.

<table><caption>Table 1: Dates with the ten largest upwards volatility shocks and selected news - Global Financial Crisis</caption><thead><tr><th>Date</th><th>Vol. Shock</th><th>Return (r<sub>t</sub>)</th><th>News</th><th>Type</th></tr></thead><tbody><tr><td>2007-02-27</td><td>118%</td><td>-3.98%</td><td><sup>1</sup>China's stock market dropped by 8.8%.<br/><sup>2</sup>Freddie Mac announced tighter standards on subprime loans.<br/><sup>3</sup>NYSE trading was interrupted due to a computer glitch at 3:00 pm.<br/><sup>4</sup>News of a suicide bombing at the main U.S. military base in Afghanistan, coupled with negative news on U.S. economic growth.</td><td>Economic<br/>Economic Policy<br/>Technological<br/>Political/Economic</td></tr><tr><td>2008-09-29</td><td>95%</td><td>-8.16%</td><td><sup>5</sup>The U.S. House of Representatives rejected the $700 billion banking-rescue package.<br/><sup>6</sup>Wachovia announced the sale of its banking operations to Citibank.<br/><sup>7</sup>The financial crisis spread to Europe, including the Icelandic government nationalizing Glitnir bank.</td><td>Economic Policy<br/>Financial Sector<br/>Economic</td></tr><tr><td>2007-12-11</td><td>83%</td><td>-2.78%</td><td><sup>8</sup>The Federal Reserve cut the federal funds rate by 0.25% to 4.25%.<br/><sup>9</sup>Freddie Mac announced large subprime losses.</td><td>Monetary Policy<br/>Financial Sector</td></tr><tr><td>2009-02-10</td><td>54%</td><td>-4.69%</td><td><sup>10</sup>The Obama administration unveiled a new rescue package, which raised concerns about its adequacy.<br/><sup>11</sup>Large layoffs were announced by companies such as General Motors, Wal-Mart, and UBS.</td><td>Economic Policy<br/>Corporate</td></tr><tr><td>2008-06-06</td><td>51%</td><td>-3.24%</td><td><sup>12</sup>The unemployment rate for May 2008 unexpectedly increased to 5.5%, up from 5.0%.<br/><sup>13</sup>Bond guarantors MBIA and Ambac were downgraded from AAA to AA.<br/><sup>14</sup>Lehman Brothers announced plans to raise $5-6 billion in capital after revealing a large second-quarter loss.</td><td>Economic<br/>Financial Sector<br/>Corporate</td></tr><tr><td>2008-09-15</td><td>50%</td><td>-4.87%</td><td><sup>15</sup>Lehman Brothers Holdings filed for bankruptcy protection.<br/><sup>16</sup>Merrill Lynch was acquired by Bank of America.</td><td>Financial Sector<br/>Financial Sector</td></tr><tr><td>2007-07-10</td><td>48%</td><td>-1.43%</td><td><sup>17</sup>Standard and Poor's added 612 securities to the "CreditWatch negative" list due to high delinquency and foreclosure rates.<br/><sup>18</sup>Media reported growing concern about subprime lending.</td><td>Financial Sector<br/>Economic</td></tr><tr><td>2007-03-13</td><td>46%</td><td>-1.96%</td><td><sup>19</sup>The U.S. dollar tumbled against other major currencies.</td><td>Economic</td></tr><tr><td>2007-11-01</td><td>44%</td><td>-2.37%</td><td><sup>20</sup>Citigroup was downgraded.<br/><sup>21</sup>Credit Suisse reported a 31% drop in profits.<br/><sup>22</sup>Exxon Mobil reported a larger-than-expected drop in quarterly earnings.<br/><sup>23</sup>Moody's, S&P, and Fitch placed $70 billion of collateralized debt obligations on review for downgrading.<br/><sup>24</sup>Economic reports showed declines in personal income and spending, manufacturing, and increased foreclosure filings.</td><td>Financial Sector<br/>Financial Sector<br/>Corporate<br/>Financial Sector<br/>Economic</td></tr><tr><td>2007-07-26</td><td>44%</td><td>-2.39%</td><td><sup>25</sup>Wells Fargo & Co. stopped making subprime mortgages due to rising late payments and defaults.<br/><sup>26</sup>NYSE invoked trading curbs to slow trading amid large price changes.<br/><sup>27</sup>Homebuilders reported significant losses as new house sales fell by 6.6%.</td><td>Financial Sector<br/>Financial Sector<br/>Economic</td></tr></tbody></table>

Note: Volatility shocks, returns on the S&P 500 index, and key events/news. Specific events and news stories related to the numbers 1,...,27 are collected in an extensive web appendix, see Banulescu-Radu et al. (2023).
<table><caption>Table 2: Dates with the ten largest upwards volatility shocks and selected news - COVID-19 pandemic</caption><thead><tr><th>Date</th><th>Vol. Shock</th><th>Return (r<sub>t</sub>)</th><th>News</th><th>Type</th></tr></thead><tbody><tr><td>2020-09-03</td><td>157</td><td>-3.48%</td><td><sup>1</sup>Technology stocks suffer significant losses; elevated jobless claims and slowdown in the services sector.</td><td>Economic</td></tr><tr><td>2020-06-11</td><td>126</td><td>-5.92%</td><td><sup>2</sup>U.S. Federal Reserve issues cautious economic forecast.<br><sup>3</sup>New concerns over a possible resurgence of COVID-19 infections.</td><td>Economic Policy<br>Health</td></tr><tr><td>2021-01-04</td><td>105</td><td>-1.42%</td><td><sup>4</sup>New COVID-19 variant discovered, raising concerns.<br><sup>5</sup>Uncertainty surrounding U.S. Senate elections in Georgia and potential impact on Biden's agenda.</td><td>Health<br>Political</td></tr><tr><td>2021-01-27</td><td>105</td><td>-2.40%</td><td><sup>6</sup>Stocks hit worst day in three months after Fed statement signals slowing economic recovery.<br><sup>7</sup>Federal Reserve reassures markets of continued support until full economic recovery.</td><td>Economic<br>Monetary Policy</td></tr><tr><td>2020-02-24</td><td>90</td><td>-3.36%</td><td><sup>8</sup>COVID-19 spreads beyond China, heightening global concerns.<br><sup>9</sup>World Health Organization expresses alarm over rapid increase in COVID-19 cases.</td><td>Health<br>Health</td></tr><tr><td>2020-03-12</td><td>85</td><td>-10.1%</td><td><sup>10</sup>Double shock: COVID-19 acceleration and plummeting oil prices.<br><sup>11</sup>U.S. President Trump's travel ban for 26 European countries due to COVID-19.</td><td>Economic/Health<br>Political/Health</td></tr><tr><td>2021-02-25</td><td>83</td><td>-2.45%</td><td><sup>12</sup>Technology stocks under pressure due to rising U.S. bond yields.<br><sup>13</sup>Winter storms disrupt Texas infrastructure, adding uncertainty to economic outlook.<br><sup>14</sup>Fed signals continued commitment to supporting U.S. economic recovery.</td><td>Financial Sector<br>Macroeconomic<br>Monetary Policy</td></tr><tr><td>2020-03-09</td><td>66</td><td>-8.06%</td><td><sup>15</sup>Global oil-price plunge triggered by Russia-Saudi Arabia conflict.<br><sup>16</sup>COVID-19 outbreak worsens, intensifying market fears.</td><td>Commodity Market<br>Health</td></tr><tr><td>2020-06-24</td><td>65</td><td>-2.53%</td><td><sup>17</sup>COVID-19 cases surge in the U.S., raising economic concerns.<br><sup>18</sup>Concerns over potential new government lockdowns due to rising coronavirus cases.</td><td>Health/Economic<br>Health/Economic</td></tr><tr><td>2020-10-28</td><td>63</td><td>-3.50%</td><td><sup>19</sup>Global increase in coronavirus cases sparks renewed market concerns.<br><sup>20</sup>Investors worried about the possibility of a contested U.S. presidential election.</td><td>Health<br>Political</td></tr></tbody></table>

Note: Volatility shocks, returns on the S&P 500 index, and key events/news (source Refinitiv/Reuters). Specific events and news stories related to the numbers 1,...,20 are collected in an extensive web appendix, see Banulescu-Radu et al. (2023).

In contrast, the COVID-19 pandemic was characterized by a larger share of volatility shocks driven by health-related news (35%), alongside economic concerns stemming from lockdowns and global slowdowns. Key drivers included rising COVID-19 case numbers, the emergence of new variants, and uncertainty about economic recovery. Negative volatility shocks during this period were often triggered by optimistic developments, such as vaccine rollouts and government stimulus packages, while positive shocks reflected deteriorating health conditions or renewed lockdown fears. Market returns during significant volatility events were predominantly negative, such as the 10.1% drop in the S&P 500 on March 12, 2020, driven by accelerating COVID-19 cases and plummeting oil prices. Notably, the magnitude of volatility shocks during
the pandemic was generally higher than during the financial crisis, reflecting elevated global uncertainty.

<table><caption>Table 3: Dates with the five largest downward volatility shocks and selected news - Global Financial Crisis</caption><thead><tr><th>Date</th><th>Vol. Shock</th><th>Return (r<sub>t</sub>)</th><th>News</th><th>Type</th></tr></thead><tbody><tr><td>2008-10-13</td><td>-22%</td><td>13.56%</td><td><sup>1</sup>Governments announce capital injections to rescue banks.<br><sup>2</sup>European Central Bank provides unlimited euro liquidity to revive credit markets.<br><sup>3</sup>U.S. Federal Reserve to supply unlimited dollars to European banks.</td><td>Economic Policy<br>Monetary Policy<br>Monetary Policy</td></tr><tr><td>2009-11-09</td><td>-21%</td><td>2.25%</td><td><sup>4</sup>G-20 finance ministers pledge to maintain economic stimulus.</td><td>International Policy</td></tr><tr><td>2007-11-13</td><td>-20%</td><td>3%</td><td><sup>5</sup>Positive statements from Goldman Sachs and JP Morgan CEOs.<br><sup>6</sup>Wal-Mart reports higher-than-expected Q3 earnings.<br><sup>7</sup>Oil prices retreat from near-record highs.<br><sup>8</sup>Home sales index beats expectations (up 0.2% vs. expected -2.5%).</td><td>Financial Sector<br>Corporate<br>Commodity Market<br>Real Estate</td></tr><tr><td>2008-09-30</td><td>-20%</td><td>4.06%</td><td><sup>9</sup>Volatility decline due to the previous day's spike after U.S. Congress rejected the bank rescue package.</td><td>Political</td></tr><tr><td>2007-12-21</td><td>-20%</td><td>1.43%</td><td><sup>10</sup>Federal Reserve lends $20 billion to support credit markets.<br><sup>11</sup>"Super SIV" rescue fund canceled as unnecessary.<br><sup>12</sup>Encouraging economic data on personal income and spending.</td><td>Monetary Policy<br>Financial Sector<br>Macroeconomic</td></tr></tbody></table>

*Note:* The table presents volatility shocks, returns on the S&P 500 index, and key events/news associated with those shocks. Specific details of the events corresponding to the numbers 1,...,12 are available in an extensive web appendix (see Banulescu-Radu et al., 2023).

<table><caption>Table 4: Dates with the five largest downward volatility shocks and selected news - COVID-19 pandemic</caption><thead><tr><th>Date</th><th>Vol. Shock</th><th>Return (r<sub>t</sub>)</th><th>News</th><th>Type</th></tr></thead><tbody><tr><td>2020-10-07</td><td>-28%</td><td>1.72%</td><td><sup>1</sup>President Trump announces that a partial deal on U.S. fiscal stimulus may occur.<br><sup>2</sup>President Trump urges Congress to pass small bills for small businesses and airline industry relief.</td><td>Economic Policy<br>Political/Economic</td></tr><tr><td>2021-03-01</td><td>-28%</td><td>2.45%</td><td><sup>3</sup>U.S. approves another COVID-19 vaccine and fiscal stimulus expectations rise.<br><sup>4</sup>Johnson &amp; Johnson ships its single-dose COVID-19 vaccine in the U.S.<br><sup>5</sup>President Biden's $1.9 trillion coronavirus relief package passes in the House.</td><td>Health/Economic<br>Health<br>Political/Economic</td></tr><tr><td>2020-12-28</td><td>-27%</td><td>0.87%</td><td><sup>6</sup>Trump signs a $2.3 trillion stimulus package.</td><td>Economic Policy</td></tr><tr><td>2020-10-05</td><td>-27%</td><td>1.73%</td><td><sup>7</sup>Likelihood of more fiscal stimulus.<br><sup>8</sup>Trump to leave the hospital after COVID-19 treatment.<br><sup>9</sup>Trump's physician declares novel antibody treatment successful.<br><sup>10</sup>Potential agreement on additional coronavirus relief.</td><td>Economic<br>Health/Political<br>Health/Political<br>Political/Economic</td></tr><tr><td>2021-03-22</td><td>-26%</td><td>0.76%</td><td><sup>11</sup>Technology stocks rebound after a recent selloff due to surging bond yields.<br><sup>12</sup>Optimistic Federal Reserve declaration on economic recovery.</td><td>Financial Sector<br>Monetary Policy</td></tr></tbody></table>

*Note:* The table presents volatility shocks, returns on the S&P 500 index, and key events/news associated with those shocks. Specific details of the events corresponding to the numbers 1,...,12 are available in an extensive web appendix (see Banulescu-Radu et al., 2023).

These differences underscore the unique characteristics of each crisis and their respective impacts on
market volatility. The global financial crisis primarily demanded targeted financial sector bailouts and monetary interventions, while the COVID-19 pandemic necessitated widespread public health measures and broader economic stimulus initiatives. Both crises, however, highlight the critical role of government actions in shaping market responses through distinct mechanisms. Moreover, the COVID-19 crisis took advantage of the better regulated banking system following the global financial crisis, and thus, while the global financial crisis was rather a banking crisis, the COVID-19 was an investor crisis.

## 4.2 News Related to the seven (five) Largest Positive (Negative) Volatility Shocks during the Global Financial Crisis

### Tuesday, February 27, 2007 (+118%)

February 27, 2007 marks the largest volatility shock in our sample, with a volatility shock of $v_t = 1.558$, translating into an expected 118% increase in volatility. On this day, the Dow Jones Industrial Average (DJIA) fell by 416.02 points, the largest drop since 9/11 and the 19th largest daily decrease during the period 1885-2010, as noted by Schwert (2011, table 1). The S&P 500 and Nasdaq indices also experienced significant declines of approximately 3.5% and 3.9%, respectively.

By the time the U.S. markets opened, several potentially distressing news stories had emerged. The Chinese stock market had experienced a sharp decline, pessimistic reports on U.S. economic growth circulated (such as Federal Reserve Chairman Alan Greenspan's Monday announcement of a potential recession by the end of 2007) and negative data on durable goods orders and housing prices were released. Additionally, a U.S. military base in Afghanistan visited by Vice President Dick Cheney was attacked by a suicide bomber, and Freddie Mac announced tighter standards on subprime loans. While the subprime-related news from Freddie Mac is unlikely to have been a major driver of market turmoil (given that the tighter lending standards were to take effect only starting September 1, 2007) the Chinese stock market crash is a more plausible contributor. The Shanghai Composite Index fell by 8.5%, reportedly due to fears of new regulatory measures, such as potential trading taxes. However, this explanation becomes less convincing when examined alongside high-frequency data.

Figure 7 illustrates the intraday minute-by-minute price movements of SPY, along with Realized Kernel computed over 30-minute intervals. The data indicate that U.S. markets were not significantly disturbed by the aforementioned news, including the Chinese crash. Instead, the notable price fluctuations began
shortly before 3:00 PM, causing a dramatic eightfold increase in volatility over a short period. This timing coincides with a computer glitch in the trading system, which delayed the reporting of certain trades and led to stale posted prices. According to a Dow Jones spokeswoman: "*Around 2:00 PM [on that day], the market's extraordinary heavy trading volume caused a delay in the Dow Jones data systems. [...] As we identified the problem, we decided to switch to a backup system, and the result was a rapid catch-up in the published value of the Dow Jones Industrial Average.*" The backup system was activated around 3:00 PM, and at 3:02 PM, the DJIA dropped by 160 points, continuing its decline throughout the afternoon. By the end of the day, the DJIA had fallen by 546 points.

This episode provides a compelling example of the critical insights high-frequency data can offer. The precise timing and magnitude of the volatility shock underscore the necessity of high-frequency data in accurately identifying the primary sources of market turmoil, illustrating their value in disentangling complex market dynamics.

<figure><img src="image_9.png" /><figcaption>Figure 7: Intraday prices and 30-min RK – February 27, 2007</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for February 27, 2007.

## Tuesday, July 10, 2007 (+48%)

On July 10, 2007, rating agencies downgraded several subprime bonds, signaling growing concerns in the housing market. Standard and Poor's placed 612 securities backed by subprime mortgages on "CreditWatch negative". These securities represented approximately 2% of all residential mortgage-backed securities in the United States. The downgrades were prompted by delinquency rates that far exceeded historical norms
and surpassed the agency's prior expectations. This directly impacted major financial institutions such as Bear Stearns, Citigroup, JP Morgan, Merrill Lynch, and Morgan Stanley, all of which held substantial amounts of these securities in their portfolios. On the same day, Moody's downgraded 399 securities and placed an additional 32 securities under review for a possible downgrade.

These downgrades underscored the systemic risks in the housing market, with potentially far-reaching implications for financial stability, because borrowers with subprime adjustable rate mortgage (ARM) loans would have faced difficulties refinancing their loans at an increased interest rates. Stricter underwriting standards made even more difficult for borrowers to refinance out of unaffordable ARMs, and the falling prices in the housing market placed an increasing number of borrowers “under water”. Standard and Poor’s cited findings by Mortgage Asset Research Institute (MARI) as one of the reasons for the downgrades. MARI had reported a high incidence of fraud in loan applications, such as false or unsubstantiated claims about income, assets, and employment. Affected loans were known as “liar loans”.

<figure><img src="image_10.png" /><figcaption>Figure 8: Intraday prices and 30-min RK – July 10, 2007</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for July 10, 2007.

## Tuesday, November 13, 2007 (-20%)

On November 13, 2007, the Dow Jones Industrial Average rose by approximately 320 points. Goldman Sachs and JP Morgan experienced significant gains of 8.5% and 6.2%, respectively. These increases followed reassuring statements from Goldman Sachs CEO Lloyd Blankfein, who asserted that the company would not incur further substantial losses related to subprime mortgages, and JP Morgan CEO Jamie Dimon,
who downplayed the bank's exposure to subprime debt. Additional positive developments contributed to the market rally. Wal-Mart reported third-quarter earnings that exceeded expectations, accompanied by an optimistic outlook. Meanwhile, oil prices dropped significantly, with U.S. light crude oil for December delivery declining by $3.45 per barrel. Another noteworthy news event was the release of U.S. Pending Home Sales data for September 2007, which showed a 0.2% increase. This result was markedly better than the forecasted decline of -2.5% and a sharp contrast to the -6.5% decrease reported for the previous month. The release of this report coincided with the afternoon rally in the markets, further buoying investor sentiment.

<figure><img src="image_11.png" /><figcaption>Figure 9: Intraday prices and 30-min RK – November 13, 2007</figcaption></figure>

Note: The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for November 13, 2007.

## Tuesday, December 11, 2007 (+83%)

On December 11, 2007, the S&P 500 index declined by 2.5%, the Dow Jones Industrial Average fell by 294 points (a 2.1% drop), and the Nasdaq lost 2.5%. The markets remained relatively stable during the morning but entered a sharp decline at approximately 14:15, accompanied by a surge in volatility from about 10% to 70% on an annualized basis. The primary news of the day revolved around the Federal Open Market Committee (FOMC) meeting, which culminated in a 25 basis point reduction in the Federal Funds Rate (FFR) to 4.25%, announced at 14:15. This decision disappointed market participants who had anticipated a larger cut of 50 basis points. Earlier in the day, additional negative news emerged as Freddie Mac's CEO, Richard Syron, revealed that the company expected further losses ranging from $5.5 billion to $7.5 billion, in addition to the previously projected $4.5 billion in losses. As illustrated in Figure 10, the
FOMC announcement was the key trigger for the sharp decline in prices observed during the afternoon. The unexpected magnitude of the rate cut had an immediate and significant impact on market sentiment, causing heightened volatility that persisted for the remainder of the trading session, as documented by Birru and Figlewski (2010).

<figure><img src="image_12.png" /><figcaption>Figure 10: Intraday prices and 30-min RK – December 11, 2007</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for December 11, 2007.

Friday, December 21, 2007 (-20%)

<figure><img src="image_13.png" /><figcaption>Figure 11: Intraday prices and 30-min RK – December 21, 2007</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for December 21, 2007.
Stocks rose early on December 21, 2007, supported by news that Merrill Lynch, heavily impacted by the credit crisis, was in negotiations with Temasek Holdings, Singapore's state investment firm, for a potential sale of part of its assets. Additionally, the Wall Street Journal reported impressive earnings from Research in Motion, the maker of BlackBerry. As a result, the Dow Jones Industrial Average gained approximately 1.2% during the first hour of trading, while the S&P 500 and Nasdaq both climbed by 1.3%. In the afternoon, it was announced that plans for the Super SIV (structured investment vehicle), formally known as the Master Liquidity Enhancement Conduit, were abandoned. This decision was accompanied by a statement asserting that "it is not needed at this time", which markets appeared to interpret as positive news. The Super SIV had been proposed to address liquidity issues and prevent fire sales of SIV assets, which faced mounting difficulties in securing short-term financing due to concerns over exposure to subprime mortgages. The Super SIV initiative was supported by a consortium of major financial institutions, including Citigroup, JPMorgan Chase, Bank of America, Wachovia, and Fidelity, with backing from the U.S. Treasury Department. However, the project faced criticism, notably from former Federal Reserve Chairman Alan Greenspan, who argued that the Super SIV amounted to a bailout for banks and would ultimately cause more harm than good.

Friday, June 6, 2008 (+51%)

<figure><img src="image_14.png" /><figcaption>Figure 12: Intraday prices and 30-min RK – June 6, 2008</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for June 6, 2008.
Early in the morning, the Dow, Nasdaq, and S&P 500 all declined following the release of the May jobs report, which revealed the largest surge in unemployment since 1986. The unemployment rate rose sharply to 5.5% from 5.0% in April, significantly exceeding the expected increase to 5.1%. This disappointing labor market news coincided with a spike in oil prices to $134 per barrel, as the dollar weakened against the euro and the yen. Additionally, the day was marked by another significant development: Standard & Poor's downgraded the AAA ratings of the two largest bond insurers, MBIA (the world's largest bond insurer) and Ambac (the second largest). S&P cited anticipated further losses from mortgage-backed securities as the rationale for the downgrades and issued warnings of potential additional downgrades. The ratings of MBIA and Ambac were cut by two notches, from AAA to AA, triggering stricter capital requirements for the insurers. The combination of these adverse factors weighed heavily on the markets. By the end of the day, the Dow Jones Industrial Average had lost 395 points, or 3.1%, marking its largest single-day decline since the onset of the subprime mortgage crisis in February 2007.

Monday, September 15, 2008 (+50%)

<figure><img src="image_15.png" /><figcaption>Figure 13: Intraday prices and 30-min RK — September 15, 2008</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for September 15, 2008.

On September 15, 2008, the Dow Jones Industrial Average fell by 504.49 points (-4.4%), marking the largest decline since 9/11. The drop followed a weekend in which Lehman Brothers filed for bankruptcy protection, the largest bankruptcy proceeding in U.S. history. The collapse of Lehman Brothers made the severity of the crisis crystal clear and reinforced concerns that the crisis was systemic and would spread
throughout the financial sector and beyond. Merrill Lynch was also severely distressed but did not file for bankruptcy because Bank of America agreed to purchase it for $50 billion in stock. In an attempt to counter these events, the Federal Reserve doubled the size of its Term Securities Lending Facility (TSLF) program to $200 billion and widened the range of assets eligible as collateral for Treasury loans. To mitigate the spread of financial turmoil to Europe, the European Central Bank and the Bank of England injected €30 billion and £5 billion of capital, respectively.

## Monday, September 29, 2008 (+95%)

The second-largest volatility shock occurred on September 29, 2008. As shown in Figure 14, prices plunged significantly in the afternoon between 1:30 pm and 1:45 pm. At that time, the House of Representatives rejected (by a 228-205 vote) the Emergency Economic Stabilization Act of 2008, which triggered a tailspin in the stock market. The banking rescue package was designed to authorize the Treasury to spend up to $700 billion to purchase toxic assets, mainly mortgage-backed securities, and provide cash directly to banks. By the end of the day, the Dow had fallen by 777 points (the largest drop in its history) while the S&P 500 index was down by 8.8%, its largest percentage drop since the crash of '87.

<figure><img src="image_16.png" /><figcaption>Figure 14: Intraday prices and 30-min RK — September 29, 2008</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for September 29, 2008.

There was other news on September 29, 2008, that may have contributed to the market turmoil, albeit to a lesser extent. Wachovia announced it was selling its banking operations to Citigroup. While Wachovia's shares lost 81% of their value in the afternoon, Citigroup lost about 12%. The British government
nationalized the mortgage lender Bradford & Bingley PLC, and some European banks collapsed. The German commercial property lender Hypo Real Estate Group utilized a government-facilitated credit line due to difficulties in the international credit market. The government of Iceland took control of Glitnir, the country's third-largest bank, to prevent its collapse. Moreover, over the weekend, Fortis was partially nationalized, receiving a €11.2 billion capital injection from the Netherlands, Belgium, and Luxembourg.

## Tuesday, September 30, 2008 (-20%)

<figure><img src="image_17.png" /><figcaption>Figure 15: Intraday prices and 30-min RK – September 30, 2008</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for September 30, 2008.

Stock prices rebounded the day after Congress failed to pass the government's $700 billion rescue plan. The DJIA increased by 485 points, partially reversing the 777-point decline of the previous day. The Standard & Poor's 500 index and the Nasdaq Composite both gained about 5%. Most of the rebound occurred late in the day after the Federal Deposit Insurance Corporation announced an enhanced deposit insurance with increased limits, a move that was supported by both presidential candidates, Barack Obama and John McCain.

## Monday, October 13, 2008 (-22%)

Stock markets around the world rallied the day in response to several new policies introduced by the US and the European Commission. The US stock markets increased, after the European markets had increased earlier in the day: London's FTSE 100 was up 4.9%, the CAC 40 in Paris was up 6.9%, and the
DAX in Frankfurt was up 8.0%.

The leaders of 15 European nations gathered in Paris for a first formal meeting since the launch of the Euro currency in 1999. Their main goal was to adopt measures to combat the credit crisis in Europe. The meeting was organized around four panel discussions on the following themes: i) facilitating the access of banks to capital; ii) global plans for governments to rescue banks through direct capital injections (such as buying soured mortgage assets from banks and injections of capital); iii) an efficient recapitalization of distressed banks and other appropriate means to support the banking system; iv) urging regulators to ease the “mark-to-market” accounting requirements based on the evaluation of assets at their current price. There was a general agreement to act together in a comprehensive, wide-ranging plan to rescue the troubled banking system by adding capital through investment and by guaranteeing interbank lending.

<figure><img src="image_18.png" /><figcaption>Figure 16: Intraday prices and 30-min RK – October 13, 2008</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for October 13, 2008.

Shortly before stocks started trading on October 13, 2008, the British Treasury announced the investment of $63 billion in three major banks: Royal Bank of Scotland, HBOS, and Lloyds TSB. Other positive news included an unprecedented move by the Federal Reserve Bank, which announced that an unlimited amount of dollars would be available to the central banks: Bank of England, European Central Bank, and the Swiss National Bank. The French president, Nicolas Sarkozy, committed €360 billion in liquidity to French banks, the German government announced a rescue package worth $671 billion, and the prime minister of Spain, Jose Luis Rodriguez Zapatero, announced that Spain would provide up to €100 billion of guarantees for new debt issued by commercial banks in 2008. Moreover, in coordination
with other Eurozone countries, the Dutch government guaranteed interbank lending up to €200 billion.
The European Central Bank committed weekly injections of unlimited euro funds at an interest rate of
3.75%.

Tuesday, February 10, 2009 (+54%)

The day began on an optimistic note in anticipation of the new Financial Stability Plan, which was
to replace the original Troubled Asset Relief Program (TARP). The plan was detailed by US Treasury
Secretary Timothy Geithner shortly after 11:00 am and had three parts: i) the reinforcement of the stress
testing procedures within each banking institution; ii) the development of a new Public-Private Investment
Fund, which would provide government capital and government financing, helping hence to the recovery
of private markets; iii) the revival of the secondary lending markets through a commitment (together with
the Federal Reserve) of up to a trillion dollars to support a Consumer and Business Lending Initiative.
Nevertheless, the new rescue plan failed to reassure investors, who received it as "a huge disappointment",
because it lacked specific details. As a result, stocks fell during and after Geithner's speech.<fnref n="10" /> The Dow
Jones Industrial Average lost 382 points (4.6%), continuing its decline in the afternoon. The Standard &
Poor's 500 index lost 43 points, or 4.9%. The Nasdaq Composite lost 66 points, or 4.2%.

<figure><img src="image_19.png" /><figcaption>Figure 17: Intraday prices and 30-min RK – February 10, 2009</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for February 10, 2009.

<footnote n="10">¹⁰His 20-minute speech was published by CNN at 11:33 am.</footnote>
Besides Geithner's speech, there was bad news from several large companies. General Motors announced it would cut 14% of its workforce around the world and reduce salaries for the remaining employees. Wal-Mart Stores planned to lay off 800 workers, and UBS announced it would lay off 2,000 workers after reporting a $17 billion loss during the last quarter of 2008.

## Monday, November 9, 2009 (-21%)

On November 9, 2009, stock prices rose while volatility fell in response to an announcement made by the *Group of 20* that met over the weekend and confirmed they would keep economic stimulus measures in place, including the American Recovery and Reinvestment Act of 2009, also known as the Obama Stimulus Plan. This economic stimulus plan refers to the $787 billion plan approved by Congress in February 2009, which was mainly devoted to tax cuts, unemployment benefits, and job creation.

<figure><img src="image_20.png" /><figcaption>Figure 18: Intraday prices and 30-min RK – November 9, 2009</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for November 9, 2009.

## 4.3 News Related to the five Largest Positive Volatility Shocks during COVID-19 Pandemic

### Monday, February 24, 2020 (+90%)

Since the first coronavirus cases outside China were found in Italy, European stocks marked their biggest drop since mid-2016, while the oil price fell by approximately 5% on the news of a jump in coronavirus cases in Italy, South Korea, Japan, and Iran. Italy was racing to contain the largest outbreak of coronavirus
in Europe, sealing off the worst-affected towns and banning public events in the northern part of the country, as a fourth patient died of the illness. More than a dozen towns in Lombardy, with a combined population of approximately 50,000, were placed under effective quarantine, *the first such measures in Europe*. Following the pandemic news, Goldman Sachs slashed its U.S. GDP forecast for the first quarter of 2020 to 1.2% from 1.4%, foreseeing that potential output drops would impact U.S. companies' supply chains in the second quarter.

<figure><img src="image_21.png" /><figcaption>Figure 19: Intraday prices and 30-min RK – February 24, 2020</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for February 24, 2020.

Cleveland Federal Reserve Bank President envisaged that the U.S. economy should continue to perform well for the rest of the year, acknowledging that monetary policy was well positioned. However, she advised that the bank's modeling incorporated the potential impact of the coronavirus as a downside risk to that year's forecast, which included U.S. economic growth of approximately 2%, solid job market growth, and low and stable inflation. This echoed the view shared by other Fed policymakers, who were converging on the opinion that it was still too early to predict the economic impact of the virus on the U.S. economy.

Despite the calming view on the impact shared by policymakers, markets predicted that the U.S. Federal Reserve would be pressed to cut interest rates to cushion the possible hit of the coronavirus spread to economic growth. U.S. interest rate futures increased to their highest levels since the previous fall on news that the virus was spreading further outside its original epicenter in China, determining a global stock sell-off and a panicked purchase of government bonds. The benchmark U.S. 10-year Treasury yield fell to its lowest level since July 2016, U.S. stock futures slumped, while gold soared to a seven-year high.
Oil tumbled nearly 4% after the World Health Organization expressed concern about the growing number of coronavirus cases. Investors were also expecting global central banks to further ease monetary policy as the risks from the virus threatened global growth.

Thursday, June 11, 2020 (+126%)

Wall Street fell sharply on that day, and safe-haven assets rallied as investors reacted to renewed concerns of a pandemic resurgence and a sobering economic forecast from the U.S. Federal Reserve, which became uncertain on hopes for a V-shaped recovery (a sharp economic resurgence following a sharp decline) from the coronavirus pandemic, indicating that the Fed would be ultra-accommodative for a very long time. All three major U.S. equity indexes dropped more than 5%, posting their worst one-day percentage drops since March 16, 2020, when markets severely fell, triggered by the abrupt economic lockdowns put in place to contain the pandemic. The sell-off was extensive, with all 11 major sectors of the S&P 500 index dropping by between nearly 4% and over 9%.

<figure><img src="image_22.png" /><figcaption>Figure 20: Intraday prices and 30-min RK – June 11, 2020</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for June 11, 2020.

The U.S. Federal Reserve issued, following its two-day monetary policy meeting, its first pandemic-era economic outlook, after which Chair Jerome Powell warned of a “long road” to recovery. To counter the economic fallout from a pandemic that was still spreading and still had no cure or vaccine, the Fed signaled it planned years of extraordinary support for the U.S. economy, which policymakers projected would shrink by 6.5% in 2020, with unemployment at 9.3%. Economic data, including unemployment
data, supported the Fed's depressed economic projections. The VIX posted its largest one-day point gain since March 16, 2020. Market sentiment also dropped as new coronavirus infections in the United States indicated a marginal increase after five weeks of continuous decline, partly attributed to expanded testing detecting more cases. Oil prices fell due to renewed concerns about demand prospects as new COVID-19 cases increased globally.

Thursday, September 3, 2020 (+157%)

Markets closed lower on the day after investors abandoned tech-heavy stocks due to new concerns about high valuations and an uneven economic recovery. Technology stocks once again played a prominent role. At its lowest point of the day, the tech-heavy Nasdaq fell as much as 9.9% from its previous day's record high, while the S&P 500 briefly fell below its pre-crisis high reached in February. The Dow Jones Industrial Average dropped 159.42 points (-0.56%), the S&P 500 lost 28.1 points (-0.81%), and the Nasdaq Composite fell 144.97 points (-1.27%) compared to the previous day. These losses are a preview of two eventful months ahead of the November 3 U.S. presidential election. Worth mentioning is that the VIX index reached its highest level in over 11 weeks during late morning trading.

<figure><img src="image_23.png" /><figcaption>Figure 21: Intraday prices and 30-min RK – September 3, 2020</figcaption></figure>

Note: The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for September 3, 2020.

Besides the technology selloff, panic was fueled by a U.S. payrolls report that painted a weaker economic picture. According to the report, U.S. job growth slowed in August 2020 as financial aid from the government was depleted, with nonfarm payrolls increasing by 1.371 million jobs, compared to 1.734
million in the previous month. The figures did not meet expectations, which had envisaged an addition of
1.4 million jobs. Despite the fact that the unemployment rate dropped to 8.4% from 10.2%, it remained
high. Oil prices continued to decline due to demand concerns and were on track for their worst week since
mid-June. Signs that the U.S. economic recovery was faltering also raised concerns about depleting fiscal
aid.

Monday, January 4, 2021 (+105%)

<figure><img src="image_24.png" /><figcaption>Figure 22: Intraday prices and 30-min RK – January 4, 2021</figcaption></figure>

Note: The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for January 4, 2021.

Major stock indexes fell following the midday release of new data showing that coronavirus cases continued to rise at a high rate around the world, threatening to lead to further lockdown measures that could hurt the economy. Upcoming runoff elections to determine which party controls the Senate may also have contributed to the volatility. The decline came after some nations began administering the vaccines, suggesting that Wall Street may have been overly optimistic about the big economic recovery anticipated due to COVID-19 vaccines. The concerns have been mainly related to the rising number of coronavirus cases, the emergence of a mutant variant of the virus, and concerns that the rollout of the vaccine was not happening quickly enough. Volatility was also spurred by the elections in Georgia, which were seen as determining control of the Senate, with effects on the fate of President Biden's agenda, including rewriting the tax code, boosting stimulus, and infrastructure spending. Markets were also worried about the pace of economic growth as U.S. jobless claims remained persistently high, while a new round of pandemic-related
restrictions issued the previous month and the emergence of a new variant of the coronavirus casted a shadow on the economic outlook.

### Wednesday, January 27, 2021 (+105%)

Amid a slowing economic recovery, stocks experienced their largest one-day percentage drop in three months after the Federal Reserve decided to keep the overnight interest rate near zero and made no changes to its monthly bond purchases. The Fed reaffirmed its commitment to using the full range of tools to support growth until a full economic rebound occurs. The Federal Reserve opted to maintain its accommodative monetary policies, stating that business activity had moderated with the resurgence of COVID-19 cases, but expressing optimism that the rollout of vaccines would help curb the pandemic and foster economic recovery.

This decision came in a context marked by ongoing concerns related to rising coronavirus cases, the disappointingly slow pace of vaccination, and the uneven distribution of vaccine rollouts, with the U.S. economy appearing to lose momentum in the first quarter of the year. The VIX index, a proxy for investor anxiety, closed at a high of 37.21, its highest level since October 30, 2020.

<figure><img src="image_25.png" /><figcaption>Figure 23: Intraday prices and 30-min RK – January 27, 2021</figcaption></figure>

*Note:* The figure presents the evolution of 1-minute intraday prices along with the 30-minute Realized Kernel (RK) estimate for January 27, 2021.
# 5 Conclusion

This paper analyzes volatility dynamics during both the 2008 financial crisis and the COVID-19 pandemic, utilizing high-frequency data to assess volatility shocks. We employed this data in two primary ways: first, to calculate realized volatility measures and estimate the robustified version of the Realized GARCH model, identifying the days with the largest volatility shocks, and second, to pinpoint the precise timing of these shocks through intraday data analysis. By comparing the fluctuations in high-frequency data with specific events and news announcements, we were able to link volatility shocks to particular triggers while ruling out events whose timing did not align with observed shocks.

An important contribution of this study is the proposal of a modified Realized GARCH model, designed to be more robust to outliers. This modification, inspired by Harvey (2013), employs a transformation of innovations that diminishes the influence of outliers on volatility dynamics, improving the model's empirical fit, as indicated by the log-likelihood function. The robustified Realized GARCH model facilitates the extraction of volatility shocks, offering a clear measure of how expectations of future volatility shift in response to news on specific days. While alternative models exist to compute volatility shocks, the proposed model offers several grouped advantages over existing approaches. It incorporates asymmetry in the impact of positive and negative news on volatility, employs realized measures derived from intraday data such as the Realized Kernels, and effectively handles outliers in returns and realized measures. The model's ability to rapidly adjust for unusually large innovations ensures accurate forecasting of future volatility in response to extreme events.

Our comparative analysis of the global financial crisis and the COVID-19 pandemic highlights the different catalysts for market volatility during these periods. The global financial crisis was largely influenced by disruptions in the financial sector, notably the collapse of Lehman Brothers and concerns over sub-prime mortgages, as well as by key economic policy interventions. In contrast, the COVID-19 pandemic was characterized by health-related news, such as updates on the spread of the virus and the deployment of vaccines, coupled with expansive fiscal and monetary policy measures. Furthermore, analysis of significant volatility shocks revealed consistent patterns: upward shocks were generally associated with significant negative market returns, while downward shocks often coincided with positive market movements.

In summary, this study demonstrates the value of high-frequency data and a robustified Realized GARCH model in identifying and analyzing volatility shocks. The findings contribute to a better un-
derstanding of how specific events influence market volatility and provide a more precise methodology for modeling volatility dynamics, with important implications for both academic research and practical financial analysis.

# A Robustification and Volatility Shock

## A.1 Motivating the Robustified Structure

The structure of score-driven models, see Creal et al. (2012, 2013) and Harvey (2013), is motivated by the first order conditions that the true parameter values ought to satisfy. Consider the following example where $y = \sigma z$ with $z \sim t_d$, and $\sigma > 0$ being an unknown scale parameter. If we reparametrize the model with $\lambda = \log \sigma^2$, then the log-likelihood function is

$$ \ell(\lambda) = -\frac{1}{2}\lambda + c_d - \frac{d+1}{2}\log(1 + e^{-\lambda}\frac{y^2}{d}), $$

where $c_d = \log[\Gamma(\frac{d+1}{2})/\Gamma(\frac{d+1}{2})/\sqrt{d\pi}]$. The score is therefore

$$ s(\lambda) = -\frac{1}{2} + \frac{d+1}{2} \frac{e^{-\lambda} \frac{y^2}{d}}{1 + e^{-\lambda} \frac{y^2}{d}} = -\frac{1}{2} \left( 1 - \frac{\frac{d+1}{d} z^2}{1 + z^2/d} \right) \approx \frac{1}{2} (\tilde{z}^2 - 1), \quad \text{with } \tilde{z} = z/\sqrt{1 + z^2/d}. $$

A positive value of $s(\lambda)$ is a signal that the expected log-likelihood may be improved by increasing the value of $\lambda$. Similarly, $s(\lambda) < 0$ is an indication that a smaller value of $\lambda$ may improve the objective. In a time series context, with time varying parameters, $\tilde{z}_t^2 - 1 > 0$ becomes a signal to increase $\lambda_t = \log \sigma_t^2$, whereas $\tilde{z}_t^2 - 1 < 0$ is an indication that $\lambda_t$ should be lowered. Precisely how much the parameter, $\lambda_t$, ought to be changed is less obvious, but a simple starting point is to use a simple autoregressive structure such as $\lambda_t = \omega + \beta\lambda_{t-1} + \alpha s(y_{t-1})$. In the robustified Realized GARCH framework we also want to allow for leverage effects, which is the reason we adopt the specification $\tau(\tilde{z}_t) = \tau_1\tilde{z}_t + \tau_2(\tilde{z}_t^2 - 1)$. This structure, which includes a linear term, $\tau_1\tilde{z}_t$, in addition to the score-motivated term, $\tau_2(\tilde{z}_t^2 - 1)$, is identical to that in Hansen et al. (2012) with the exception that $\tilde{z}_t$ has replaced $z_t$. In our model we maintain the Gaussian distributional specification, and merely use $\tilde{z} = z/\sqrt{1 + z^2/d}$ to reduce the influence of outliers. A fully-fledged DCS/GAS structure is not needed in order to gain the robustness we seek. Adopting t-distributions for $z_t$ and $u_t$ is relatively straightforward, but would be computationally more cumbersome.
## A.2 Volatility Shock

The percentage volatility shock to $\sqrt{250h_t}$ is approximately given by $100 (\exp \frac{v_t}{2} - 1)$. This follows because

$$ \mathbb{E}(\log \sqrt{h_{t+1}} | \mathcal{F}_{t-1}) = \frac{1}{2} \mathbb{E}(\log h_{t+1} | \mathcal{F}_{t-1}) = \frac{1}{2}(\omega + \beta \log h_t) = \frac{\omega}{2} + \beta \log \sqrt{h_t} \approx \log \sqrt{h_t}, $$

where the last approximation uses that $\omega \approx 0$ and $\beta \approx 1$.

From $v_t = \log h_{t+1} - \mathbb{E}(\log h_{t+1} | \mathcal{F}_{t-1})$, it now follows that

$$ \exp \frac{1}{2} v_t = \exp[\log \sqrt{h_{t+1}} - \mathbb{E}(\log \sqrt{h_{t+1}} | \mathcal{F}_{t-1})] \approx \frac{\sqrt{h_{t+1}}}{\sqrt{h_t}}, $$

which shows that the percentage volatility shock to $\sqrt{h_{t+1}}$ (and hence $\sqrt{250h_{t+1}}$) is approximately given by $100 (\exp \frac{v_t}{2} - 1)$.

## B Comparison of Different Robust Specifications

We explored a range of model specifications, each incorporating varying levels of robustness to outliers. All the models are submodels of the following general specification:

$$ r_t = \mu + \sqrt{h_t} z_t, $$

$$ \log h_t = \omega + \beta \log h_{t-1} + \tau(\tilde{z}_{1,t-1}) + \gamma \tilde{u}_{t-1}, \quad \tau(z) = \tau_1 z + \tau_2(z^2 - 1), $$

$$ \log x_t = \xi + \varphi \log h_t + \delta(\tilde{z}_{2,t}) + u_t, \quad \delta(z) = \delta_1 z + \delta_2(z^2 - 1). $$

The structure of each model is outlined below. The baseline Realized GARCH model (M0) and the more flexible alternatives (M1-M6) are defined as follows:

$$ \text{M0: } z_t = \tilde{z}_{1,t} = \tilde{z}_{2,t} \text{ and } u_t = \tilde{u}_t $$

$$ \text{M1: } z_t = \tilde{z}_{1,t} = \tilde{z}_{2,t}, \text{ and } \tilde{u}_t = u_t / \sqrt{1 + (u_t / \sigma_u)^2 / d_u} $$

$$ \text{M2: } \tilde{z}_{1,t} = \tilde{z}_{2,t} = \tilde{z}_t \text{ with } \tilde{z}_t = z_t / \sqrt{1 + z_t^2 / d_z} \text{ and } u_t = \tilde{u}_t $$

$$ \text{M3: } \tilde{z}_{1,t} = z_t / \sqrt{1 + z_t^2 / d_{1z}}, \quad \tilde{z}_{2,t} = z_t / \sqrt{1 + z_t^2 / d_{2z}}, \quad \text{and } u_t = \tilde{u}_t $$

$$ \text{M4: } z_t = z_{2,t}, \quad \tilde{z}_{1,t} = z_t / \sqrt{1 + z_t^2 / d_{1z}} \text{ and } u_t = \tilde{u}_t $$

$$ \text{M5: } z_t = z_{2,t}, \quad \tilde{z}_{1,t} = z_t / \sqrt{1 + z_t^2 / d_{1z}} \text{ and } \tilde{u}_t = u_t / \sqrt{1 + (u_t / \sigma_u)^2 / d_u} $$
$$ \text{M6: } \tilde{z}_{1,t} = z_t/\sqrt{1+z_t^2/d_{1z}} \quad \tilde{z}_{2,t} = z_t/\sqrt{1+z_t^2/d_{2z}}, \quad \text{and} \quad \tilde{u}_t = u_t/\sqrt{1+(u_t/\sigma_u)^2/d_u} $$

<table><caption>Table B.1: Parameter estimates for each of the seven model specifications: the Realized GARCH model (M0) and the six robustified variants.</caption><thead><tr><th></th><th>M0<br>Realized<br>GARCH</th><th>M1</th><th>M2</th><th>M3</th><th>M4</th><th>M5<br>(Preferred)</th><th>M6</th></tr></thead><tbody><tr><td colspan="8" style="text-align: center; font-weight: bold;">Panel A: Financial crisis</td></tr><tr><td>d<sub>1z</sub></td><td></td><td></td><td>63.531</td><td>29.487</td><td>33.359</td><td>27.689</td><td>24.792</td></tr><tr><td>d<sub>2z</sub></td><td></td><td></td><td>63.531</td><td>290.770</td><td></td><td></td><td>290.787</td></tr><tr><td>d<sub>u</sub></td><td></td><td>12.768</td><td></td><td></td><td></td><td>5.904</td><td>5.895</td></tr><tr><td>h<sub>0</sub></td><td>0.797</td><td>0.812</td><td>0.782</td><td>0.797</td><td>0.803</td><td>0.824</td><td>0.819</td></tr><tr><td>ω</td><td>0.006</td><td>0.007</td><td>0.010</td><td>0.015</td><td>0.014</td><td>0.017</td><td>0.018</td></tr><tr><td>β</td><td>0.972</td><td>0.972</td><td>0.971</td><td>0.968</td><td>0.968</td><td>0.969</td><td>0.969</td></tr><tr><td>γ</td><td>0.368</td><td>0.402</td><td>0.364</td><td>0.354</td><td>0.351</td><td>0.407</td><td>0.411</td></tr><tr><td>τ<sub>1</sub></td><td>-0.171</td><td>-0.171</td><td>-0.177</td><td>-0.180</td><td>-0.178</td><td>-0.180</td><td>-0.182</td></tr><tr><td>τ<sub>2</sub></td><td>0.025</td><td>0.025</td><td>0.043</td><td>0.056</td><td>0.053</td><td>0.056</td><td>0.059</td></tr><tr><td>ξ</td><td>-0.518</td><td>-0.519</td><td>-0.516</td><td>-0.528</td><td>-0.531</td><td>-0.532</td><td>-0.529</td></tr><tr><td>φ</td><td>1.006</td><td>1.005</td><td>0.994</td><td>1.014</td><td>1.022</td><td>1.019</td><td>1.012</td></tr><tr><td>δ<sub>1</sub></td><td>-0.128</td><td>-0.129</td><td>-0.133</td><td>-0.130</td><td>-0.129</td><td>-0.132</td><td>-0.133</td></tr><tr><td>δ<sub>2</sub></td><td>0.037</td><td>0.036</td><td>0.052</td><td>0.042</td><td>0.038</td><td>0.036</td><td>0.040</td></tr><tr><td>σ<sup>2</sup><sub>u</sub></td><td>0.157</td><td>0.157</td><td>0.156</td><td>0.154</td><td>0.154</td><td>0.154</td><td>0.154</td></tr><tr><td>AIC</td><td>4026.1</td><td>4026.3</td><td>4024.9</td><td>4020.1</td><td>4019.2</td><td><strong>4017.4</strong></td><td>4018.3</td></tr><tr><td>BIC</td><td>4080.0</td><td>4085.1</td><td>4088.6</td><td>4083.8</td><td><strong>4078.0</strong></td><td>4081.1</td><td>4086.9</td></tr><tr><td>-logL</td><td>2002.1</td><td>2001.2</td><td>1999.5</td><td>1997.0</td><td>1997.6</td><td>1995.7</td><td><strong>1995.1</strong></td></tr><tr><td colspan="8" style="text-align: center; font-weight: bold;">Panel B: COVID-19 pandemic</td></tr><tr><td>d<sub>1z</sub></td><td></td><td></td><td>9.248</td><td>77.985</td><td>95.603</td><td>31.566</td><td>12.901</td></tr><tr><td>d<sub>2z</sub></td><td></td><td></td><td>9.248</td><td>290.417</td><td></td><td></td><td>7.500</td></tr><tr><td>d<sub>u</sub></td><td></td><td>11.612</td><td></td><td></td><td></td><td>7.498</td><td>5.770</td></tr><tr><td>h<sub>0</sub></td><td>0.455</td><td>0.429</td><td>0.465</td><td>0.454</td><td>0.455</td><td>0.423</td><td>0.419</td></tr><tr><td>ω</td><td>-0.015</td><td>-0.013</td><td>0.039</td><td>-0.009</td><td>-0.010</td><td>0.003</td><td>0.027</td></tr><tr><td>β</td><td>0.939</td><td>0.939</td><td>0.939</td><td>0.939</td><td>0.939</td><td>0.940</td><td>0.939</td></tr><tr><td>γ</td><td>0.413</td><td>0.454</td><td>0.384</td><td>0.401</td><td>0.403</td><td>0.450</td><td>0.449</td></tr><tr><td>τ<sub>1</sub></td><td>-0.214</td><td>-0.212</td><td>-0.272</td><td>-0.221</td><td>-0.220</td><td>-0.229</td><td>-0.254</td></tr><tr><td>τ<sub>2</sub></td><td>0.078</td><td>0.076</td><td>0.177</td><td>0.093</td><td>0.090</td><td>0.108</td><td>0.152</td></tr><tr><td>ξ</td><td>-0.794</td><td>-0.794</td><td>-0.743</td><td>-0.789</td><td>-0.793</td><td>-0.795</td><td>-0.729</td></tr><tr><td>φ</td><td>0.919</td><td>0.917</td><td>0.910</td><td>0.917</td><td>0.918</td><td>0.913</td><td>0.911</td></tr><tr><td>δ<sub>1</sub></td><td>-0.303</td><td>-0.301</td><td>-0.376</td><td>-0.305</td><td>-0.303</td><td>-0.301</td><td>-0.388</td></tr><tr><td>δ<sub>2</sub></td><td>0.072</td><td>0.071</td><td>0.174</td><td>0.077</td><td>0.072</td><td>0.072</td><td>0.190</td></tr><tr><td>σ<sup>2</sup><sub>u</sub></td><td>0.317</td><td>0.317</td><td>0.306</td><td>0.315</td><td>0.316</td><td>0.315</td><td>0.305</td></tr><tr><td>AIC</td><td>4239.9</td><td>4239.7</td><td>4207.4</td><td>4237.8</td><td>4239.4</td><td>4236.9</td><td><strong>4203.5</strong></td></tr><tr><td>BIC</td><td>4294.5</td><td>4299.3</td><td><strong>4271.9</strong></td><td>4302.3</td><td>4298.9</td><td>4301.4</td><td>4272.9</td></tr><tr><td>-logL</td><td>2108.9</td><td>2107.9</td><td>2090.7</td><td>2105.9</td><td>2107.7</td><td>2105.4</td><td><strong>2087.8</strong></td></tr></tbody></table>

The empirical results for the two periods under analysis are presented in Table B.1. We analyzed the robustified Realized GARCH structures, including both the general and simplified forms, and compared their performance in terms of empirical fit to the standard Realized GARCH model. For the two analyzed periods, the highest log-likelihood value was achieved by the most general specification, M6, although it was closely followed by M5. The difference between these two models was not statistically significant. However, the log-likelihood for M5 exceeded that of the classical Realized GARCH specification by six
units, indicating a significant statistical advantage of incorporating robustness into the GARCH equation. Furthermore, the new parameter associated with the transformed innovation term in the measurement equation of M6 was found to be quite large, suggesting that $\tilde{z}_{2,t} = z_t$ might be a reasonable assumption. By adopting model M5, robustness to outliers is introduced exclusively in the GARCH equation, while the measurement equation remains unchanged.<fnref n="11" />

Table B.2 presents the ten largest positive volatility shocks, along with their corresponding dates, for each of the seven estimated models. For both the global financial crisis and the pandemic periods, the models largely agree on the dates of the largest volatility shocks, although the estimated magnitudes of these shocks vary. The outlier-damping effect is evident from the estimated values of $v_t$, particularly in the robust specifications, although the reduction in magnitude is significant only for the largest volatility shocks.

<table><caption>Table B.2: Ten largest positive volatility shocks for each of the seven specifications</caption><thead><tr><th colspan="2">M0</th><th colspan="2">M1</th><th colspan="2">M2</th><th colspan="2">M3</th><th colspan="2">M4</th><th colspan="2">M5</th><th colspan="2">M6</th></tr><tr><th>Date</th><th>v<sub>t</sub></th><th>Date</th><th>v<sub>t</sub></th><th>Date</th><th>v<sub>t</sub></th><th>Date</th><th>v<sub>t</sub></th><th>Date</th><th>v<sub>t</sub></th><th>Date</th><th>v<sub>t</sub></th><th>Date</th><th>v<sub>t</sub></th></tr></thead><tbody><tr><td colspan="14" style="text-align: center; font-weight: bold;">Panel A: Financial crisis</td></tr><tr><td>20070227</td><td>2.295</td><td>20070227</td><td>2.298</td><td>20070227</td><td>2.200</td><td>20070227</td><td>1.670</td><td>20070227</td><td>1.654</td><td>20070227</td><td>1.558</td><td>20070227</td><td>1.576</td></tr><tr><td>20080929</td><td>1.314</td><td>20080929</td><td>1.304</td><td>20080929</td><td>1.391</td><td>20080929</td><td>1.373</td><td>20080929</td><td>1.375</td><td>20080929</td><td>1.334</td><td>20080929</td><td>1.331</td></tr><tr><td>20071211</td><td>1.213</td><td>20071211</td><td>1.192</td><td>20071211</td><td>1.277</td><td>20071211</td><td>1.260</td><td>20071211</td><td>1.256</td><td>20071211</td><td>1.208</td><td>20071211</td><td>1.214</td></tr><tr><td>20080606</td><td>0.791</td><td>20080606</td><td>0.785</td><td>20080606</td><td>0.845</td><td>20090210</td><td>0.871</td><td>20090210</td><td>0.868</td><td>20090210</td><td>0.866</td><td>20090210</td><td>0.869</td></tr><tr><td>20090210</td><td>0.779</td><td>20090210</td><td>0.779</td><td>20090210</td><td>0.830</td><td>20080606</td><td>0.842</td><td>20080606</td><td>0.842</td><td>20080606</td><td>0.827</td><td>20080606</td><td>0.825</td></tr><tr><td>20070726</td><td>0.731</td><td>20070726</td><td>0.712</td><td>20080915</td><td>0.763</td><td>20080915</td><td>0.802</td><td>20080915</td><td>0.800</td><td>20080915</td><td>0.808</td><td>20080915</td><td>0.808</td></tr><tr><td>20080915</td><td>0.705</td><td>20070710</td><td>0.712</td><td>20070726</td><td>0.756</td><td>20070710</td><td>0.777</td><td>20070710</td><td>0.768</td><td>20070710</td><td>0.785</td><td>20070710</td><td>0.794</td></tr><tr><td>20070710</td><td>0.701</td><td>20080915</td><td>0.707</td><td>20070710</td><td>0.750</td><td>20070726</td><td>0.767</td><td>20070726</td><td>0.761</td><td>20070313</td><td>0.760</td><td>20070313</td><td>0.765</td></tr><tr><td>20070313</td><td>0.662</td><td>20070313</td><td>0.666</td><td>20070313</td><td>0.720</td><td>20070313</td><td>0.754</td><td>20070313</td><td>0.748</td><td>20071101</td><td>0.729</td><td>20070726</td><td>0.734</td></tr><tr><td>20071101</td><td>0.638</td><td>20071101</td><td>0.647</td><td>20071101</td><td>0.685</td><td>20071101</td><td>0.710</td><td>20071101</td><td>0.706</td><td>20070726</td><td>0.728</td><td>20071101</td><td>0.733</td></tr><tr><td colspan="14" style="text-align: center; font-weight: bold;">Panel B: COVID-19 pandemic</td></tr><tr><td>20200903</td><td>2.315</td><td>20200903</td><td>2.266</td><td>20200903</td><td>1.810</td><td>20200903</td><td>2.163</td><td>20200903</td><td>2.167</td><td>20200903</td><td>1.885</td><td>20200903</td><td>1.988</td></tr><tr><td>20200611</td><td>1.976</td><td>20200611</td><td>2.068</td><td>20210104</td><td>1.440</td><td>20200611</td><td>1.789</td><td>20200611</td><td>1.790</td><td>20200611</td><td>1.633</td><td>20200611</td><td>1.594</td></tr><tr><td>20210104</td><td>1.456</td><td>20210104</td><td>1.415</td><td>20200611</td><td>1.397</td><td>20210104</td><td>1.479</td><td>20210104</td><td>1.479</td><td>20210104</td><td>1.439</td><td>20210127</td><td>1.465</td></tr><tr><td>20200224</td><td>1.381</td><td>20200224</td><td>1.414</td><td>20210127</td><td>1.286</td><td>20200224</td><td>1.350</td><td>20200224</td><td>1.350</td><td>20210127</td><td>1.433</td><td>20210104</td><td>1.419</td></tr><tr><td>20210127</td><td>1.288</td><td>20210127</td><td>1.383</td><td>20200312</td><td>1.165</td><td>20210127</td><td>1.321</td><td>20210127</td><td>1.319</td><td>20200224</td><td>1.279</td><td>20200224</td><td>1.307</td></tr><tr><td>20200312</td><td>1.224</td><td>20200312</td><td>1.232</td><td>20200224</td><td>1.156</td><td>20200312</td><td>1.223</td><td>20200312</td><td>1.226</td><td>20200312</td><td>1.228</td><td>20210225</td><td>1.218</td></tr><tr><td>20210225</td><td>1.102</td><td>20210225</td><td>1.139</td><td>20210225</td><td>1.129</td><td>20210225</td><td>1.139</td><td>20210225</td><td>1.136</td><td>20210225</td><td>1.206</td><td>20200312</td><td>1.208</td></tr><tr><td>20200309</td><td>1.024</td><td>20200309</td><td>0.995</td><td>20200309</td><td>0.985</td><td>20200309</td><td>1.042</td><td>20200309</td><td>1.041</td><td>20200309</td><td>1.008</td><td>20200309</td><td>1.002</td></tr><tr><td>20201028</td><td>0.999</td><td>20201028</td><td>0.975</td><td>20200624</td><td>0.974</td><td>20201028</td><td>1.014</td><td>20201028</td><td>1.014</td><td>20200624</td><td>0.998</td><td>20201028</td><td>0.982</td></tr><tr><td>20200220</td><td>0.925</td><td>20200624</td><td>0.928</td><td>20201028</td><td>0.953</td><td>20200624</td><td>0.955</td><td>20200624</td><td>0.955</td><td>20201028</td><td>0.980</td><td>20200624</td><td>0.968</td></tr></tbody></table>

Note: This table presents the ten largest positive volatility shocks computed as $v_t = \tau(\tilde{z}_t) + \gamma\tilde{u}_t$, along with the corresponding dates of occurrence.

<footnote n="11">¹¹For example, during the financial crisis period, the estimated parameter associated with the degrees of freedom for the transformed innovation term $\tilde{u}_t$ ($d_u = 5.904$) was lower than that for $\tilde{z}_{1,t}$ ($d_{1,z} = 27.689$). This implies that outliers related to the realized volatility series required greater dampening.</footnote>
# C Performance Evaluation

To evaluate the statistical performance and robustness of the robustified Realized GARCH model (M5) relative to the standard Realized GARCH model (M0), we conducted two experiments.

**Experiment 1: Performance Comparison During Different Periods.** We estimated the M0 and M5 models over two distinct periods: a calm period (2003-2006) and a turbulent period during the global financial crisis (2007-2009).<fnref n="12" /> The results are summarized in Table C.1, which presents key performance metrics such as log-likelihood, AIC, BIC, and robustness parameters ($d_{1z}$ and $d_u$). The results indicate that during the calm period (2003-2006), both models performed similarly, with minimal differences in AIC and BIC. However, during the global financial crisis (2007-2009), the M5 model outperformed M0 across all metrics. Specifically, the robustness parameter $d_{1z}$ decreased significantly in this period, highlighting M5 ability to downweight the influence of extreme shocks, particularly in returns.

<table><caption>Table C.1: Comparison of models M0 and M5 over calm (2003-2006) and turbulent (2007-2009) periods.</caption><thead><tr><th>Period</th><th>Model</th><th>Log-Likelihood</th><th>AIC</th><th>BIC</th><th>d<sub>1z</sub></th><th>d<sub>u</sub></th></tr></thead><tbody><tr><td rowspan="2">2003-2006</td><td>M0</td><td>1423.97</td><td>2869.93</td><td>2923.91</td><td>-</td><td>-</td></tr><tr><td>M5</td><td>1422.02</td><td>2870.04</td><td>2933.82</td><td>81.67</td><td>2.40</td></tr><tr><td rowspan="2">2007-2009</td><td>M0</td><td>1653.88</td><td>3329.76</td><td>3380.51</td><td>-</td><td>-</td></tr><tr><td>M5</td><td>1644.50</td><td>3315.00</td><td>3374.98</td><td>14.29</td><td>5.16</td></tr></tbody></table>

**Experiment 2: Time-Dependent Performance Analysis.** To assess the time-dependent performance of the models, we estimated M0 and M5 using a rolling window of 500 observations from 2003 to 2009. Figure C.1 presents the differences in AIC between M0 and M5 ($\Delta$AIC = AIC$_{M5}$ - AIC$_{M0}$). Negative values favor M5, while positive values favor M0. Thresholds at ±2 and ±10 help categorize differences as weak, moderate, or strong evidence.<fnref n="13" /> The results indicate that M5 outperforms M0 in nearly 60% of cases, with moderate to strong evidence in over 39% of cases. Conversely, M0 is favored in 40% of cases, but its advantage is weak in nearly all instances. These results highlight the superior robustness of M5 especially during periods of financial stress. Table C.2 provides a summary of the $\Delta$AIC outcomes.

<footnote n="12">¹²Data prior to 2003 were excluded to avoid contamination from significant crises, including the Asian financial crisis (1997-1998), the Russian crisis (1998), the internet bubble (2000-2002), and the Argentine crisis (1999-2002).</footnote>

<footnote n="13">¹³Since statistical tests for AIC differences are not standard, we used the following heuristic thresholds: $\Delta$AIC below -2 suggests weak evidence, between -2 and -10 indicates moderate evidence, and below -10 indicates strong evidence for M5. Similarly, positive thresholds indicate the strength of evidence for M0.</footnote>
<table><caption>Table C.2: Summary of ΔAIC = AIC<sub>M5</sub> - AIC<sub>M0</sub> outcomes between M0 and M5.</caption><thead><tr><th>Case</th><th>Description</th><th>Percentage (%)</th></tr></thead><tbody><tr><td>M5 better (ΔAIC &lt; 0)</td><td>M5 outperforms M0</td><td>59.86</td></tr><tr><td>Weak evidence (0 &gt; ΔAIC ≥ -2)</td><td>Weak evidence for M5</td><td>20.35</td></tr><tr><td>Moderate evidence (-2 &gt; ΔAIC ≥ -10)</td><td>Moderate evidence for M5</td><td>23.97</td></tr><tr><td>Strong evidence (ΔAIC &lt; -10)</td><td>Strong evidence for M5</td><td>15.53</td></tr><tr><td>M0 better (ΔAIC &gt; 0)</td><td>M0 outperforms M5</td><td>40.14</td></tr><tr><td>Weak evidence (0 &lt; ΔAIC ≤ 2)</td><td>Weak evidence for M0</td><td>29.77</td></tr><tr><td>Moderate evidence (2 &lt; ΔAIC ≤ 10)</td><td>Moderate evidence for M0</td><td>10.38</td></tr><tr><td>Strong evidence (ΔAIC &gt; 10)</td><td>Strong evidence for M0</td><td>0.00</td></tr></tbody></table>

<figure><img src="image_26.png" /><figcaption>Figure C.1: Differences in AIC between M0 and M5 across rolling windows. Horizontal lines indicate thresholds for weak, moderate, and strong evidence. ΔAIC = AIC<sub>M5</sub> - AIC<sub>M0</sub>.</figcaption></figure>

Finally, Figure C.2 presents the likelihood ratio test (LRT) statistics over time, showing the percentage of cases where M5 performance is statistically significant. The horizontal line indicates the critical value for significance. M5 is statistically superior in approximately 39.5% of the cases (and mostly during estimation span windows with high volatility).

Both experiments confirm the robustness and adaptability of the M5 model. While M0 performs adequately during calm periods, M5 demonstrates clear superiority during periods of financial stress, as evidenced by stronger performance metrics, robustness parameter estimates, and likelihood ratio tests.
<figure><img src="image_27.png" /><figcaption>Figure C.2: Likelihood Ratio Test (LRT) results comparing M0 and M5. The horizontal line indicates the critical value for significance.</figcaption></figure>

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