Structural Volatility Impulse Response Analysis

Abstract

We make three contributions to the volatility impulse response function (VIRF) developed by Hafner and Herwartz (2006). First, we derive its law for multivariate GARCH models of the BEKK type. Second, we present a structural embedding of the VIRF, leveraging recent advancements in the identification of multivariate generalized autoregressive conditional heteroskedasticity models. Third, we show how to endow the VIRF with a causal interpretation. We illustrate the merits of a structural VIRF analysis by investigating the impacts of historical and out-of-sample shock scenarios on the U.S. equity, government bond, and foreign exchange markets.

The impulse response function (IRF) is a primary tool for analyzing how dynamic multivariate systems respond to shocks. It illustrates the effects of an unanticipated disturbance on the modeled variables, revealing how the sign, magnitude, and overall duration of the response evolve across the forecast horizon. In vector autoregressions, IRF analysis focuses on feedback within the mean process (Lütkepohl 2010; Inoue and Kilian 2013). However, in volatility models of high-frequency speculative asset returns—where the mean equation often carries little significance—the second-order response becomes the central focus. For example, a mutual fund manager might be interested in how the variance matrix of specific asset classes responds to an unexpected policy shock. In such situations, the volatility impulse response function (VIRF) is the appropriate tool.

The VIRF, initially proposed by Hafner and Herwartz (2006), extends the concept of the generalized IRF (GIRF) due to Koop, Pesaran, and Potter (1996) to second-order moments. By conditioning on past information and an exogenous shock, it traces that shock’s nonlinear impact on volatility dynamics. Among the various VIRF specifications proposed (Gallant, Rossi, and Tauchen 1993; Liu 2018), the approach of Hafner and Herwartz (2006) is of special significance, as it permits a closed-form solution in one of the most flexible multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) models: the BEKK(p, q) model introduced by Baba, Engle, Kraft, and Kroner (Engle and Kroner 1995).

In this work, we revisit the VIRF and contribute to the literature in three key ways: First, we derive the VIRF’s asymptotic distribution in the BEKK model. Specifically, we show that, like the VIRF itself, its asymptotic variance matrix can be expressed as a function of the forecast horizon in a compact recursive form. This approach enables an efficient numerical computation of confidence intervals, which previously required time-intensive simulation techniques such as the bootstrap. A comparative study of both methods reveals that our newly available asymptotic confidence intervals provide researchers with a valuable tool for assessing the volatility impact of a given shock within an MGARCH model.

As a second contribution, we provide the VIRF with a structural interpretation by leveraging recent advances in the identification of MGARCH models (Hafner, Herwartz, and Maxand 2022; Fengler and Polivka 2024). This new interpretation materially broadens its range of applications. Traditionally, the VIRF has been used to study the impact of a realized shock on volatility—a use case we term “historical VIRF analysis.” By incorporating identified and labeled structural shocks derived from a structural MGARCH model, we can construct specific, interpretable shocks for scenario analyses, similar to those in structural vector autoregressive (VAR) models (Amisano and Giannini 2012; Kilian and Lütkepohl 2017). This approach allows for defining counterfactual scenarios and assessing the expected volatility impact of well-defined shock scenarios. We introduce the term “scenario VIRF” to describe this new application framework.

Our third contribution is to equip the VIRF with a causal interpretation. Drawing on advances in causal inference for time series by Rambachan and Shephard (2020, 2021), we examine the conditions under which the microeconometricians’ concept of causality can be applied—specifically, when the VIRF can be understood as representing dynamic causal effects of assignments on outcomes. In other words, we provide the conditions under which the VIRF, obtained for a given tail event, measures how changes in shocks (i.e., assignments) dynamically cause movements in second-order moments over a given horizon. These results align with recent efforts in macroeconometrics to provide a causal interpretation of impulse responses in structural VARs and local projections—see, for example, Cloyne, Jordà, and Taylor (2023) and Gonçalves et al. (2024).

Studying the impact of shocks on financial volatility is essential in applications across asset pricing, risk management, and portfolio optimization. The development of tools to measure volatility responses has thus been a longstanding focus in financial econometrics, beginning with the work of Gallant, Rossi, and Tauchen (1993) and Lin (1997). However, both approaches rely on correlated reduced-form errors, which limit the meaningful analysis of individual shock components. Hafner and Herwartz (2006) address this shortcoming by proposing a VIRF, which is based on mutually orthogonal shocks. Nevertheless, their identification assumption rests solely on the principal matrix square root of the variance matrix sequence. This limitation also applies to recent advances, such as the VIRF for Markov-switching MGARCH models in Cavicchioli (2019) and the asymmetric VIRF proposed by Hafner and Herwartz (2023).

Despite this history, structural advancements in volatility impulse response analysis remain in their early stages. Initial identification efforts leveraging time-varying heteroscedasticity were made by Weber (2010) and Liu (2018), both within conditional correlation models. More recently, Hafner, Herwartz, and Maxand (2022) used information from third and fourth-order moments to identify structural shocks based on independence and non-normality assumptions. While these strategies successfully identify orthogonal structural shocks, they lack economic interpretability and thus are limited to using realized model-implied shocks. To address this issue, Fengler and Polivka (2024) propose employing external instrument data for identification. A major advantage of this approach is that it yields economically labeled shocks, at least for those targeted by the instrument data. In this work, we extend that framework by conducting a structural VIRF analysis using labeled structural shocks.

Lastly, few studies address the asymptotic properties of the VIRF for statistical inference. An exception is Liu (2018), though this work relies on a different MGARCH model and an alternative VIRF definition compared to Hafner and Herwartz (2006). Additionally, we are the first to provide a discussion of causality for VIRFs.

The remainder is organized as follows. Section 1 introduces the VIRF and presents key results on its asymptotic distribution and connections to causal inference. Section 2 discusses the estimation of the structural model and provides both historical and out-of-sample scenario VIRFs for well-defined risk scenarios. Section 3 concludes.  Appendix A offers an overview of definitions, relevant matrix algebra results, and the proofs.

1 Volatility Impulse Response Analysis

1.1 Modeling Framework

The structural volatility model presented in Equation (2) was introduced by Hafner, Herwartz, and Maxand (2022), who specify $R˜$ as a rotation matrix. As in Fengler and Polivka (2024), we model $R˜$ more broadly as an orthogonal matrix. Like a structural VAR model, model (2) establishes a specific shock composition mechanism, which is represented by $R˜$ and endows the shocks ξ_t with a structural interpretation. For example, choosing $R˜$ as the identity matrix implies symmetric volatility spillovers, as $Ht1/2$ preserves the positive definiteness and symmetry of H_t. With other choices of $R˜$⁠, mixtures of the structural shocks emerge, which are then transformed into the reduced-form shocks $εt$ by $Ht1/2$⁠.

Because any choice for $R˜$ is observationally equivalent up to second order, additional information is required for identification. Hafner, Herwartz, and Maxand (2022) develop an identification scheme based on the assumption of non-Gaussianity of ξ_t. Using additional external data sources, known as instrument or proxy variables, Fengler and Polivka (2024) suggest a framework for proxy-identification. Proxy-identification, by not relying exclusively on statistical assumptions, offers the further benefit of allowing for shock labeling, that is, a meaningful economic interpretation. We sketch the ideas of proxy-identification in Section 2.1.

1.2 The Volatility Impulse Response Function

In order to assess the impact of a shock ξ_t on volatility given $Ft−1$⁠, Hafner and Herwartz (2006) define the h-step ahead VIRF as the difference between the expected h-step-ahead covariance conditioned on the shock and past information, and a natural baseline, the expected h-step-ahead covariance conditioned on past information only. Note that, although the VIRF bears the term volatility in its name, it is, in fact, a vector of impulse responses of the conditional (co)variances to the shock $ξt=ξ¯*$⁠. 

By definition, the VIRF represents the expectation of possible future scenarios, conditioned on the history and a shock. Depending on the stance a researcher takes regarding the conditioning sets, different use cases of the VIRF can be imagined.¹ In most current applications, $Ft−1$ is treated as the realized history, and the shock at time t is selected as $ξ¯*=ξ^t$⁠, where $ξ^t=R˜⊤H^t−1/2ε^t$ is derived from the estimated residuals and the filtered variance process ${H^t}t≥1$⁠, with $R˜$ chosen as the identity matrix. Then, the conditioning set in the left-hand side expectation becomes $Ft$⁠, and the VIRF captures the system’s volatility response to the actual shock, conditioned on the information available up to time t-1. We want to call this use case the “historical VIRF.”

Alternatively, one can adopt a scenario perspective, where $ξ¯*$ is not equal to $ξ^t$ but is instead specified by the researcher. In this case, the challenge lies in selecting the shock of interest, as impulse responses in multivariate models depend on the entire shock vector composition rather than on a single component. Without an identified model, however, this is impossible because $ξt=R˜⊤Ht−1/2εt$ is unobserved and, up to second order, observationally equivalent for any choice of $R˜$⁠.

Structural MGARCH models address the composition effect problem by offering a strategy to identify and estimate $R˜$⁠, and thus enable the construction of synthetic but economically meaningful structural shocks. Under the conditions provided in Section 1.5, the VIRF can be interpreted causally, allowing these shocks to be treated as counterfactuals. We introduce the term “scenario VIRF” for this application framework and demonstrate its usefulness in Section 2.3.2.

1.3 The BEKK(p, q) Model and its VIRF

To derive a closed-form expression for the VIRF, we must select a model for the dynamics of the conditional variance matrix process. BEKK GARCH models are especially popular (Bauwens, Laurent, and Rombouts 2006) and appear to be the most commonly employed for VIRFs in the literature (Jin, Lin, and Tamvakis 2012; Olson, Vivian, and Wohar 2014; Hafner and Herwartz 2023).

Denote the vectorized parameter vector of Equation (4) by $η=(vec(C)⊤,vec(Ai)⊤,vec(Bj)⊤)⊤$⁠, where $i=1,…,p;j=1,…,q$⁠.² We adopt the following assumptions: 

These assumptions ensure the consistency and asymptotic normality of the quasi-maximum likelihood estimator (Hafner and Preminger 2009). In reference to Assumption 2.1(2), identifying the BEKK(p, q) model necessitates additional sign restrictions on its parameter matrices due to the model’s quadratic structure. For example, in the BEKK(1,1) model, it is usually assumed that the diagonal elements of C and the top left matrix entries of A₁ and B₁ be positive—see Engle and Kroner (1995) for a detailed discussion. For Assumption 2.1(4), strict stationarity and ergodicity follow from mild regularity conditions detailed in Hafner and Preminger (2009). Furthermore, as discussed in Boussama, Fuchs, and Stelzer (2011, Theorem 2.4), the BEKK GARCH process is both strictly and weakly stationary if the eigenvalues of $∑i=1pAi⊗Ai+∑j=1qBj⊗Bj$ are less than one in modulus.³

The subsequent exposition requires the VMA$(∞)$ representation of the “squared” BEKK process defined by $Xt:=vech(εtεt⊤)$⁠. Additionally, define the process $Yt:=Xt−vech(Ht)$⁠, which is a weak white noise under Assumption 2.1. Denote by D_n the duplication matrix and by $Dn+$ its Moore-Penrose inverse (see Definition D.6), and recall that $n∗=n(n+1)2$⁠. 

For the BEKK(1,1) model, these expressions simplify to $Ψ0=In∗, Ψ1=A˜1$ and $Ψi=(A˜1+B˜1)Ψi−1=(A˜1+B˜1)i−1A˜1$ where $i≥2$⁠.

Proof. See  Appendix A.2. □

A closed-form expression for the h-step ahead VIRF exists for the BEKK(p, q) model. For completeness, we provide a rigorous derivation in the appendix, adapted to the case of a structural model as given in Equation (2). This derivation will be of further value in Section 1.4. 

 

Notably, the VIRF is a nonlinear but even function of the structural shock. The persistence of a shock to volatility is governed by the moving average matrices $Ψh, h≥0$⁠. 

This follows from setting the BEKK(1,1) model in its vech form—see Equation (A14) in the proof of Proposition 2.1—and inserting the VMA(⁠$∞$⁠) coefficients into Equation (6). 

where the second line follows from an application of Equations (A2) and (A3). Thus, when $ε^t$ is known or can be estimated, this expression remains unaffected by the structural mechanism encoded by $R˜$⁠. This property no longer holds in the case of the scenario VIRF, where $ξ¯*≠ξ^t$⁠.

1.4 Asymptotic Results

As central contribution, we complete the notion of the VIRF by providing its asymptotic theory. Estimation of the BEKK model, as with MGARCH models in general, typically proceeds by quasi-maximum likelihood (QML) estimation assuming multivariate normality of ξ_t. The QML estimator is defined as maximizing the log likelihood given by $LT(η)=−12T∑i=1Tℓt(η)$⁠, where $ℓt(η)=log(det(Ht))+εt⊤Ht−1εt$ and $Ht=Ht(η)$⁠. For the VIRF, we will now use the notation $Vt+h(ξ¯*;η)$ to emphasize the dependence of $Vt+h$ on the parameter vector η of the underlying MGARCH model. We treat $R˜$ as known, which, in light of Remark 2, corresponds to the historical VIRF analysis.

1.4.1 Consistency of the VIRF

   

1.4.2 Asymptotic normality of the BEKK VIRF

Given the QML estimator $η^$ of the parameters of the BEKK model, we can deduce the asymptotic distribution of the VIRF. This allows us to construct pointwise and simultaneous confidence intervals for the volatility impulse responses of asset returns to structural shocks, as we demonstrate in Section 2. 

where $Vη(ξ¯*;η0)=∂Vt+h(ξ¯*;η0)∂η⊤$ denotes the $n∗×m$ Jacobian matrix of the VIRF with respect to $η∈Rm$⁠; $ℋ(η0)=−E[∂2ℓt(η0)∂η∂η⊤]$ is the Hessian matrix of the log-likelihood contribution $ℓt$⁠, and $ℐ(η0)=E[∂ℓt(η0)∂η∂ℓt(η0)∂η⊤]$ the Fisher information matrix.

Proof. See  Appendix A.3. □ 

 

1.5 Causal Interpretation of the VIRF

Our volatility impulse response analysis aligns with recent advances in causal inference, where model outcomes are interpreted as resulting from different assignments in treatment variables. By comparing different states of the world, each linked to distinct values of the causing variable, this notion of causality contrasts with Granger-Sims causality, which focuses on predictability.⁵

A causal interpretation requires additional identifying assumptions that allow one to relate the observed data to the distribution of the potential outcome variables. Rambachan and Shephard (2020, 2021) provide conditions under which the GIRF of Koop, Pesaran, and Potter (1996) can be endowed with a causal interpretation. In contrast to the GIRF, which addresses the conditional mean, the VIRF traces the dynamic effects of a shock on the conditional second-order moments. In this section, we develop the framework that provides a causal interpretation for the VIRF.⁶

1.5.1 Definitions and assumptions

To frame our variables of interest within the context of causal inference, we interpret the structural shocks ξ_t as treatment variables, with the treatment being continuous. Additionally, we regard the vectorized outer products of the demeaned return vectors $εt$⁠, that is, $Xt:=vech(εtεt⊤)$⁠, as the observable, continuously valued, multidimensional outcomes. The outcomes are linked to the treatment variables through the potential outcome process, which specifies the outcome that would be observed for a given path of the treatment variable. We begin by adapting the definitions and assumptions of Rambachan and Shephard (2020, 2021) to our framework: 

As this notation makes clear, in general, potential outcomes may also be influenced by future treatments. This is ruled out by 

Assumption 2.2 is analogous to the non-interference assumption in classical causal inference (Rubin 1980). It restricts the potential outcomes to depend only on past and contemporaneous treatments; therefore, we can drop any dependence on future assignments.

The potential outcomes evaluated at the treatments yield the outcome process: 

Because our treatment variable is continuous, unlike in Rambachan and Shephard (2020, 2021), we formulate the probabilistic assumptions for Borel sets. Let $B(ξ¯*,ϵ)={ξ∈Wξ:d(ξ¯*,ξ)<ϵ}$ be an ϵ-neighborhood of some fixed, structural shock of interest $ξ¯*∈Wξ$⁠, where d is some metric on $Rn$ and $ϵ>0$⁠, allowing for arbitrarily small sets. The assignment of the treatments is assumed to be sequentially probabilistic, that is, at time t, any treatment vector $ξt∈B(ξ¯*,ϵ)$ can be realized with positive probability, given the history generated by both $X(ξ)$ and ξ: 

The next assumption requires that the assignment of the treatment depends solely on past outcomes and past treatments, conditioned on $Gt−1$⁠: 

Assumption 2.5 defines non-anticipating treatment paths conditional on the information available up to time t-1. From a time series perspective, one could say that future potential outcomes do not Granger-cause the current treatment ξ_t.

Lastly, we require continuity assumptions to avoid ambiguities in defining conditional distributions, when conditioning on events with measure zero—see Gill and Robins (2001) for an in-depth discussion: 

Given this framework, we define our potential outcome process as $Xt(ξ¯)=vech(Ht1/2ξ¯tξ¯t⊤Ht1/2)$⁠, where H_t follows Equation (4). It is deterministic, nonlinear in the assignment for every t, and satisfies Assumption 2.2, as X_t depends on past shocks only due to the VMA(⁠$∞$⁠) representation (Proposition 2.1).

We observe the outcome $(ξt,Xt)=(ξt,Xt(ξt))$⁠. Assumptions 2.4 and 2.5 hold because, under Assumption 2.1, the structural shocks ξ_t are independently and identically distributed, serially independent, and the MGARCH process admits a strictly stationary and non-anticipative solution.⁸ This guarantees the independence of ξ_t from future treatments and the associated potential outcomes. Lastly, it holds that $Gt⊆σ(εs:s≤t)=Ft$⁠, implying that the usual filtration for the MGARCH model nests $Gt$⁠. Assumption 2.6 regarding continuity appears natural when modeling return data.

1.5.2 Causal effect

A dynamic causal treatment effect compares potential outcomes along an assignment path to those along a counterfactual path. Given the infinite number of possible paths, we define—similarly to Rambachan and Shephard (2020)—a filtered treatment effect that averages over all possible future assignments and contrasts it with the scenario of averaging over all potential interventions, conditional on $(t−1)$-information. Thus, the assignment path we consider is given by $({ξs}1≤s≤t−1,ξ¯*,{ξs}t+1≤s≤t+h)$⁠, while $Xt+h(ξ¯*):=Xt+h({ξs}1≤s≤t−1,ξ¯*,{ξs}t+1≤s≤t+h)$ is the $(t+h)$-potential outcome along this path. Observe that $Xt+h=Xt+h(ξt)$⁠. This leads to 

The next proposition demonstrates that the VIRF, when applied to a structural shock of interest $ξ¯*$⁠, can be decomposed into the filtered treatment effect and a selection bias term, which vanishes under time series unconfoundedness: 

 

In summary, under the present assumptions, the VIRF is a filtered treatment effect and thus has a causal meaning. 

2 Empirical Application

We now present a structural VIRF analysis using a system of daily asset returns across three key asset classes: equity, fixed income, and foreign exchange markets. These asset classes are not only central to portfolio optimization but are also vital components in financial stress tests mandated by central banks (Kremer, Lo Duca, and Holló 2012).

2.1 Identification and Estimation of the Structural MGARCH Model

It follows that $E[Ztξ1t⊤]=Ψ$⁠, with the rank condition on $Ψ$ ensuring that the instruments offer non-redundant information about the structural shocks of interest. The fact that $E[Ztξ2t]=0(n−1)×1$ imposes exogeneity restrictions, which formalize the idea that the instrument is uninformative about the non-targeted shock. This framework allows multiple instruments to jointly convey information about a structural shock, so $Ψ$ is not required to be diagonal. In our empirical application, however, an overidentification test suggests that this assumption is appropriate. This identification approach leverages methods from proxy identification in structural VAR models (see, inter alia, Stock and Watson 2012; Mertens and Ravn 2013; Angelini and Fanelli 2019; Giacomini, Kitagawa, and Read 2022).

Identification requires that $m(m−1)/2$ constraints be imposed on G. This need arises because a total of $m(m+1)/2$ parameters are governed by the orthogonality of $R˜$ and the estimated parameters in $Σζ=E[ζtζt⊤]=GG⊤$⁠. From Equation (15), $(n+1)(n−1)$ zero constraints are derived from the instrument exogeneity conditions. However, this number may not suffice to meet the required $m(m−1)/2$⁠, necessitating additional conditions on $R˜, Ψ$⁠, or Σ_v. If such conditions can be established and if $Ψ$ has full column rank, it is possible to identify $R˜$ up to column signs.⁹ In Section 2.2, we discuss how we derive these additional restrictions within our application.

Note that G depends on the parameter vector $θ=(vec(R˜)⊤,ψ⊤,vec(Σv1/2)⊤)⊤$⁠, where ψ represents a vector collecting the free (non-zero) parameters of $Ψ$⁠. Consequently, the structural parameter matrix $R˜$ will be estimated, up to sign, as part of the parameters that maximize the log-likelihood. The challenge of maximizing the log-likelihood arises from the non-convexity of the orthogonality constraints. As highlighted in Fengler and Polivka (2024), this issue can be efficiently addressed using methods from Riemannian optimization.

2.2 Data and Estimation

We borrow elements of the empirical analysis from Fengler and Polivka (2024) and examine a system of daily returns computed from the Standard and Poor’s 500 Composite Index, the yield of the U.S. constant maturity 10-year Treasury note, and the USD Index.¹⁰ The data cover the period from January 1, 1998 to December 31, 2014—see Figure 1 for the log returns and Table 1 for descriptive statistics.

We select the proxy variables in accordance with a core tenet of financial econometrics: fundamental news drives stock returns—see Jeon, McCurdy, and Zhao (2021) for evidence on the link between news and significant intraday stock returns. Our goal is to identify two types of shocks—an equity price shock that is informative about the economic fundamentals of equities, and a bond price shock that reflects shifts, for example, in real interest rates, inflation expectations, or monetary policy. To achieve this, we use two series of news analytics data from the Thomson Reuters MarketPsych Indices (TRMI) as proxies for the underlying structural shocks, specifically the U.S. stock index news sentiment and U.S. bond news sentiment. The TRMIs are constructed by means of a proprietary supervised natural language processing scheme applied to news items from a broad range of media outlets. Each item is scored for relevance, novelty, and sentiment to construct the indices.¹¹ The TRMI data are available at a daily frequency, and we apply flexible autoregressive moving average (ARMA) models to extract the unexpected innovations, which we use as proxies.

As discussed above, to achieve identification, we require $m(m−1)/2=10$ constraints, because $m=2n−1=5$ with n = 3 returns and $n−1=2$ instruments. The zero constraints provide $(n+1)(n−1)=8$ of these—see Equation (15). One additional constraint is obtained by imposing symmetry on $Σv1/2$⁠. Consequently, the system is just-identified by applying a single zero restriction on $Ψ$⁠. However, we impose diagonality on $Ψ$⁠, which introduces two zero constraints and results in an overidentified system.

For the MGARCH dynamics, we estimate Equation (1), where we set $μt=0$ and use the BEKK(1, 1) specification given in Equation (4); the results are reported in Table 2 and Figure 2. Table 3 presents the structural model estimates. The orthogonal matrix diverges from the identity matrix, indicating a departure from the symmetric volatility spillover pattern that would result if $R˜=In$⁠. This shift reallocates mass asymmetrically from the unit diagonal entries of the identity matrix to the off-diagonal elements of the orthogonal matrix. Additionally, the significant estimates of the diagonal elements in $Ψ^$ confirm that the chosen TRMI series Z₁ and Z₂ are indeed relevant instruments (Table 3). A likelihood ratio test assessing the overidentifying restriction supports the diagonal structure of $Ψ$⁠, as the test statistic $LRT=0.44$ corresponds to a p-value under the $χ2(1)$ distribution.

Due to the identification strategy of using stock market news sentiment for ξ₁ and bond price sentiment for ξ₂, we refer to these shocks as the equity price shock and bond price shock, respectively. The remaining shock is identified too, although it is not directly targeted by our identification approach. As visible in the orthogonal matrix in Table 3, this shock has the largest impact on the USD Index equation, suggesting it may represent a currency shock. However, we refer to it simply as the ‘third’ shock.

2.3 Structural Volatility Impulse Response Analysis

One of the key applications of MGARCH models is analyzing how volatility responds to shocks (Bauwens, Laurent, and Rombouts 2006). By utilizing identified labeled shocks, we can offer a more refined analysis of the VIRF patterns than has been possible to date. We begin with historical VIRFs, followed by scenario-based VIRFs.

2.3.1 Historical VIRFs

We consider three historical events from our sample: First, we analyze the NASDAQ crash on April 14, 2000, which marked the bursting of the Dotcom bubble, as an example of a 1% marginal equity price shock. Second, we examine the semiannual monetary policy report delivered by Federal Reserve Chairman Alan Greenspan to the US Senate on March 7, 2002, illustrating a 1% tail event in both the bond price shock and the third shock. Lastly, we investigate the concerns in the U.S. equity markets over the European debt crisis on August 4, 2011, as another example of an equity price shock. The corresponding shock vectors are documented in Table 4. In Figure 2, the selected case studies are indicated by red vertical lines, placing these days in the context of the filtered variance and covariance evolutions.

Figures 3–5 present the resulting VIRFs (solid black lines). Additionally, each plot includes the following 95% confidence intervals: the individual asymptotic pointwise 95% confidence intervals (black dashed line), based on the QML variance matrix; the individual pointwise 95% confidence intervals (light grey area) obtained from the residual bootstrap procedure, as described in Hafner and Herwartz (2023); and, for completeness, the simultaneous (i.e., across all VIRFs) asymptotic confidence intervals (dot-dashed blue line), derived from the $χ2(6)$ approximation of the Wald statistic, also based on the QML variance matrix.¹²

The structural shock of the NASDAQ crash on April 14, 2000 initially leads to a strong positive response in the predicted variance of the S&P 500, which is statistically significant at the 5% level, regardless of the confidence interval used (see Figure 3). While the significance diminishes after 50 days when judged by the joint confidence interval, the impulse to the level of volatility in the S&P 500 remains significant for 90–100 days based on the individual pointwise intervals. The strong reaction in the covariance of the S&P 500 and the 10-year Treasury yield returns is marginally significant for about 10–15 days according to the simultaneous intervals, and between 50 and 75 days according to the pointwise asymptotic confidence intervals.

The remaining shocks to variance are ambiguous, as the confidence intervals point in contradictory directions. Notably, although the third shock is relatively large at 1.6 standard deviations (see Table 4), its impact on the USD Index and its VIRF is dampened due to the nonlinear nature of shock propagation within the BEKK model.

For our second case study, we focus on March 7, 2002, when the chair of the Federal Reserve Board, Alan Greenspan, made an appearance in Congress that caught markets off guard. In his speech to the Senate, he presented a much more optimistic view of the economic outlook than he had in his testimony to the House of Representatives just seven days earlier. The structural shock associated with this event was moderately positive in the equity coordinate, but very large and negative in the coordinates of the bond price shock and the third shock (see Table 4). This resulted in a minor shock to the S&P 500, a strong increase in yields (as a negative shock to bond sentiment implies higher yields), and a depreciation of the U.S. Dollar relative to the currencies of major trading partners.

This shock vector triggered remarkably complex patterns of shock propagation in the system (see Figure 4). Despite the relatively small equity component of the shock, we observe a slight but statistically significant decrease in the predicted variance of the S&P 500. Based on the pointwise intervals, the effect persisted for about 10–40 days before becoming insignificant and converging to zero. Thus, this serves as a rare example of a shock having a calming effect on equity market variance, likely driven by the outlook for an accelerated recovery in the US economy.

At the same time, the sharp negative shock to the bond market triggered a substantial short-term rise in predicted yield volatility, which remained significant for at least 50 days, likely reflecting market uncertainty regarding future rate hikes. Increases in yield (return) volatility following Federal Open Market Committee (FOMC) announcements and monetary policy reports have, of course, been well-documented in the literature—see, for example, Jones, Lamont, and Lumsdaine (1998) and Rosa (2013), along with the references therein.

The predicted covariance impulse responses of the S&P 500 returns with yield returns and with USD Index returns are both significant but relatively short-lived. In contrast, the structural shock had a more lasting impact, with a positive effect on the predicted variance of the USD Index return and a negative effect on the covariance between the USD Index and the yield returns. The latter decline may appear counterintuitive, as standard exchange rate models tend to suggest a positive relationship between yields and the domestic currency—see the lower right panel in Figure 2; however, the VIRF does not provide insights into the levels of covariance, but merely indicates a negative impulse to them.

We now turn to our third case study, the structural shock observed during the European debt crisis (see Figure 5). With the exception of the equity price shock, the other components of the structural shock on August 4, 2011 remain well within one standard deviation around zero (see Table 4). Nevertheless, the shock had strong, statistically significant effects on the predicted (co-)variances of all asset return components.

In Figure 5, we observe a positive impact on the predicted variances of the S&P 500 returns, the Treasury yield returns, and their covariance. This pattern is consistent with heightened market concerns over potential European debt defaults, as well as flight-to-safety investments driven by declining equity markets amid the European debt crisis.

The impact on the USD Index return variance is only significant when judged by the pointwise confidence intervals. In contrast, we find a significant and pronounced decrease in the predicted covariance between the S&P 500 and the USD Index returns. This aligns with the literature documenting that volatility spillover effects between equity and foreign exchange markets are typically small during normal market conditions but become stronger in periods leading up to crises (see, e.g., Grobys, 2015; Cenedese and Mallucci, 2016). The even sharper drop in the predicted covariance between the USD Index and the 10-year Treasury yield returns reflects two factors: a surge in the U.S. Dollar relative to the Euro, as evidenced by the positive USD Index return on the shock date, and a decline in yields, potentially driven by increased demand for safe-haven investments (see Table 4).

In summary, our historical VIRF analysis underscores the importance of considering confidence intervals to fully appreciate volatility impulse responses. The bootstrapped and asymptotic confidence intervals (based on the QML variance matrix) align well, particularly given the highly nonlinear transformation of the model’s parameters into the VIRF. The bootstrap intervals, however, tend to be slightly tighter. The largest discrepancies between the two methods arise when, after the initial impact, the bootstrap method samples parameter constellations that result in VIRFs exhibiting always the same sign. In contrast, the asymptotic intervals, being symmetric by construction, tend to eventually cover zero.

Moreover, the discussion underscores the value of having an economic interpretation of the shocks—a feature lacking in the existing VIRF literature. Our structural model provides this by identifying which specific shocks impacted which returns, and thereby offers a broader understanding of portfolio risks.

2.3.2 Scenario VIRFs

Historical VIRFs are valuable for understanding past volatility events, but their utility is limited since events like those highlighted in Section 2.3 are unlikely to recur. However, as discussed in Section 1.5, the structural model enables the analysis of causal effects from meaningful shock scenarios, whether in an out-of-sample context or as counterfactuals.

For illustration, we adopt a risk manager’s perspective and investigate the VIRFs under two scenarios: (a) a 1% marginal equity price shock, and (b) a 1% marginal bond price shock, both on the out-of-sample date January 2, 2015. Assuming independence, we estimate the multivariate density of the structural shocks by fitting the marginal distributions with a Gaussian kernel density estimator. We use these estimates to draw independent 10,000 observations from the 1% quantile of the shock component of interest, while sampling from the full distribution for the remaining components. This approach avoids setting other structural shocks to zero. To illustrate the responses, we compute the pointwise median VIRFs, along with the 25% and 75% quantile VIRFs for each forecast horizon h. Additionally, we include the analytical asymptotic simultaneous 95% confidence intervals for the median target VIRF, calculated following Fry and Pagan (2011). It is important to note that these confidence intervals reflect the parameter estimation uncertainty of a single VIRF—the median target VIRF—and are therefore unrelated to the interquartile range, which is computed across the VIRFs based on different shock vectors.

Figure 6 displays the VIRFs resulting from shock scenario (a). According to our analysis in Section 1.5, these responses represent the causal effects associated with the 1% quantile of the equity price shock. We observe a pronounced positive median impact on the forecasted conditional (co-)variances of S&P 500 and Treasury yield returns, with a more muted effect on the (co-)variances involving the USD Index. Given the prevailing conditional covariance patterns, such a shock scenario may be expected to increase the variance levels of both equity and fixed income markets, as well as their correlation, over the medium term. Consequently, the risk manager might consider implementing further diversification strategies to reduce exposure to equity price shocks.

In contrast, Figure 7 shows that the bond price shocks sampled from the 1% quantile in scenario (b) have a negligible impact on the predicted conditional covariance between the Treasury yield and the S&P 500 returns, as well as the forecasted conditional variance of the S&P 500 returns. However, there is a strong positive effect on the predicted variance of yield returns and the predicted conditional covariance with the USD Index returns. In this scenario, the shock-induced increase in covariance between the fixed income and FX markets could be a primary concern for risk assessment.

In summary, Figures 6 and 7 demonstrate that different shock scenarios lead to distinct predictions for (co-)variance levels in the asset return system, providing valuable insights for risk managers to take scenario-specific precautions.

3 Conclusion

In this article, we revisited the VIRF introduced by Hafner and Herwartz (2006), a useful tool for analyzing the impact of shocks on conditional variance matrices in MGARCH models. By deriving the asymptotic distribution of the VIRF within the BEKK model, we enhance its potential by offering asymptotic confidence intervals. We demonstrate that the asymptotic variance matrix, similar to the VIRF itself, can be expressed as a function of the forecast horizon in a compact recursive form, enabling efficient numerical evaluation.

Building on recent advances in identifying MGARCH models, we extend the VIRF to take advantage of structural volatility models. With interpretable, labeled shocks and clearly defined structural propagation channels, we broaden the VIRF’s application to include counterfactual and out-of-sample scenario analyses. Beyond the structural interpretation, we demonstrate how to provide the VIRF with a causal interpretation. This approach enables the application of the microeconometricians’ concept of causality for assessing the impact of well-defined shock scenarios. In an empirical application to an identified system of equity, government bond, and foreign exchange returns, we illustrate two key use cases: the historical VIRF and the scenario VIRF. Our applications highlight the importance of assessing the statistical significance of volatility impulse responses.

Funding

This work was supported by the Swiss National Science Foundation, grant number 176684, Project “Structural Models of Volatility,” and by the Mobi.Doc Grant of the University of St.Gallen 1031603, Project “Time-varying shock transmission mechanisms and structural analysis of financial volatility”.

The views expressed in this publication are solely those of the author(s) and do not necessarily reflect the opinions or policies of the Zürcher Kantonalbank or its staff.

Footnotes

Appendix A

A.1 Notation and Results from Matrix Algebra

         

A.2 Prerequisites for the VIRF

  

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A.3 Asymptotic Theory for VIRFs

 

For the analytical expressions of $∂vec(Ht)∂η⊤$ within the BEKK model, see Hafner and Herwartz (2008).

In summary, based on Equation (A23), the recursion for the derivative of the BEKK(p, q) VIRF is given by Equation (A24). It can be implemented utilizing Equations (A25), (A26), (A27), and (A28).

In this case, we need to calculate $∂vec(A˜1+B˜11{h>1})η⊤$ only once to establish the recursion for $∂Vt+h(ξ¯*;η)∂η⊤$⁠. This completes the proof. □

A.4 Causality of the VIRF

 

References