A joint impulse response function for vector autoregressive models

Abstract

Many applications call for measuring the response due to shocks from several variables at once. We introduce a joint impulse response function (jIRF) that is independent of the order of the variables and allows for simultaneous shocks from multiple variables in the VAR, rather than one at a time as in the generalized IRF. The proposed jIRF controls for the cross-correlations of the several simultaneous shocks. As an application of the jIRF, we study the effect of the COVID-19 pandemic on trans-Atlantic volatility transmissions across large financial institutions and show that simply summing the generalized IRFs overestimates volatility transmissions.

1 Introduction

In financial and macroeconomic modeling, it is often of interest to study the dynamic impact on a system of variables due to simultaneous correlated shocks from some subset of variables. Ideally, a detailed structural model would be used to estimate and simulate the degree of shock transmissions in question. However, structural models are complex and often include subjective or controversial assumptions. In response, following Sims (1980), empirical time series econometrics has been dominated by vector autoregressive (VAR) models. Although easy to construct and estimate, the parameters of VAR models are difficult to interpret directly. Consequently, impulse response functions (IRFs) are constructed to reveal the response through time for each variable in the model due to a shock from another variable. In this paper, we focus on those cases where the researcher has selected to use a VAR model. We introduce a new method, the joint impulse response function (jIRF), to compute and interpret impulse responses that are independent of the order of the variables and allow for simultaneous shocks from multiple variables at a time in order to quantify the total effect.

A structural interpretation of the IRFs requires that the impulses be consistent with the underlying economic theory. In practice, many issues may arise when making the assumptions required to construct meaningful and unique structural IRFs. For example, using the traditional Cholesky decomposition method proposed by Sims (1980) requires assumptions concerning the specific recursive ordering of the K variables in the VAR. However, there are K! possible orderings with, in many applications, little guidance as to which might be the correct ordering, leaving open the possibility of a wide range of potential results to interpret. Another approach is to choose some specific structural VAR (SVAR) specification (e.g., Christiano et al. 1996; Blanchard and Quah 1989), but this approach is also unlikely to have unequivocal results as different researchers may disagree as to what identifying restrictions are reasonable. Baumeister and Hamilton (2019) observe that, in order to uniquely identify the structure, one must also assume that the true structural shocks are uncorrelated, which may be a strong assumption.¹ They argue that the usual VAR assumptions are an “all-or-nothing” approach where we know with certainty the structural identification for some parameters but nothing about others. One approach to identification for SVAR models is to impose a set of structural restrictions on the variance-covariance matrix of the shocks. Inoue and Rossi (2021), for instance, discuss the very interesting notion of “functional shocks” which use structural relationships among the shocks as an alternative to restrictions on the covariances. In their term structure example, Inoue and Rossi (2021) also demonstrate that multiple shocks may be considered with the functional shocks approach. Additionally, Lanne et al. (2017) show that an SVAR can also be identified if the structural shocks are composed of mutually independent non-Gaussian distributions. In this paper, we will be assuming Gaussian shocks because that admits a convenient closed-form solution for the jIRF that aids in understanding the method. More generally, the jIRF method does not require Gaussian shocks but, in this case, numerical methods would be required to compute the response functions.

As an alternative to focusing on identification strategies, Baumeister and Hamilton (2019) propose that researchers should acknowledge the uncertainty about the identifying assumptions themselves and use a middle ground between the two extremes of dogmatic and uninformative priors by instead utilizing Bayesian priors. Indeed, often the structural restrictions are unknown or controversial, and the uncertainty about the restrictions should be acknowledged. We concur with this sentiment so, in this paper, we focus on cases where a specific structural identification is ambiguous and nonstructural IRFs are sufficient—or are the only IRFs available—to the analyst. We propose the jIRF as an alternative approach to deal with the uncertainty about identifying restrictions and which also has the advantage of allowing us to compute the impact of multiple simultaneous and correlated shocks from several variables.

Because the explicit set of structural restrictions are often unknown, the order-independent and unique generalized impulse response function (gIRF), introduced by Koop et al. (1996) and Pesaran and Shin (1998), has become a popular tool in many diverse applications across numerous financial and macroeconomic subdisciplines. We argue below that the gIRF deals with uncertainty about the identifying restrictions of the VAR by leaving the specific identifying restrictions as ambiguous while constructing the IRFs to be consistent with the observed data. In other words, the gIRF allows the data to speak for itself regarding the nature of the data generating process because no “all-or-nothing” restrictions are imposed. Some early financial applications of the gIRF include Ewing (2002) who measure how shocks are transmitted between S &P sector-specific stock market indices and Ewing et al. (2003) who show how these same indices respond to macroeconomic shocks. More recent financial applications that use the gIRF include Smith and Yamagata (2011) who analyze the return-volatility relationship across industry sectors; Roll et al. (2014) who investigate trading volume dynamics between the S &P 500 index, options, futures, and exchange-traded funds and their relationship with the macroeconomy; and Barigozzi and Conti (2018) who show that a monetary overhang variable, constructed as the residual of real money balances from a time-varying VECM, is a leading indicator of financial stress and inflation in the Euro area. Applications that use the gIRF in international economics include Yang et al. (2006) who estimate the effect of the 1998 Russian financial crisis on the linkages between the American, German, and four Eastern European stock markets; Dees et al. (2007) who assess how the Euro area economy responds to macroeconomic shocks from foreign variables such as oil prices, the US stock market, and US interest rates; Caggiano et al. (2020) who investigate how US economic policy uncertainty shocks impact the Canadian unemployment rate during booms and busts; Phylaktis and Chen (2009) who investigate the dynamic interaction between indicative and transaction prices in foreign exchange markets; and Calice et al. (2015) who examine how the stationary and random walk components of European credit default swaps respond due to shocks from the sovereign bond yield slope and stock returns. Recent applications of the gIRF in forecasting include Loungani et al. (2013) who examine information rigidity in real GDP forecasts across 46 countries and Lahiri and Zhao (2019) who measure how quickly forecasters from one country assimilate the relevant news of shocks from other countries. The gIRF has also been used to add new insights to previous research such as Esfahani et al. (2014) who reexamine how real output in major oil exporting countries responds due to oil export shocks; Lanne and Nyberg (2016) who reexamine the interaction of US output growth and the interest rate spread; and Kim et al. (2011) who research what causes the time-varying degree of equity market efficiency and return predictability.

In reduced-form dynamic models, it is relatively straightforward to compute the impact on the variables in the system resulting from a shock of a single variable in that system. The gIRF, which only allows for one shock at a time, can be used to compute exactly that. But in many circumstances, the response due to a single shock is not what the researcher needs to know. For example, a crisis may involve simultaneous shocks from several of the model’s variables at once and these variables may themselves be interrelated. Additionally, policymakers may use multiple policy tools simultaneously, which may again be interrelated themselves. We are unaware of any method to compute the total impact resulting from contemporaneously correlated shocks of several variables in a reduced-form model like ours.² Adding up the impacts from a set of individually shocked variables is not correct due to the correlations among the shocks. Our methodological contribution in this paper extends the joint, multivariate conditioning sets of Lastrapes and Wiesen (2021) to compute a unique joint impulse response function (jIRF) and joint forecast error variance decomposition (jFEVD) from simultaneous correlated impulses from several variables in a reduced-form model. Importantly, our jIRF allows us to quantify the total effect due to several shocks in the system, and the jFEVD allows us to quantify the total explanatory power of several variables in the system. Like the gIRF, the jIRF is agnostic to structural identification.

To make our discussion more concrete, we consider a specific question: How did the emergence of COVID-19 affect the transmission mechanism of volatility shocks between European and US financial institutions? After its first detection in China, the novel coronavirus emerged substantially throughout Europe before emerging substantially in the USA. In Europe, many countries followed a fairly uniform, simultaneous, and coordinated approach to combat the virus and its economic impacts. Consequently, the COVID-19 related shocks to major financial institutions in Europe were essentially simultaneous shocks. Section 6 will discuss this example in detail, but here we focus on the motivation for the question and the need for a new methodology. Specifically, let $\varvec{Y}_{t}$ be a vector of the daily volatilities of stock returns for K banks in the USA and Europe. We wish to determine the system-wide dynamic impact resulting from simultaneous and contemporaneously correlated shocks from a subset of the K banks, specifically those in Europe. We know that financial firms, including major European and US banks, are highly interconnected (Diebold and Yilmaz 2016; Demirer et al. 2018), and so it is valuable to investigate the volatility transmission between European and American banks as that has implications for portfolio allocations, financial monitoring by regulatory bodies, contagion analysis, quantifying systemic risks, etcetera. Moreover, it is critical to control for the cross-relationships between banks when measuring the total spillover from a group of financial institutions. Thus, our jIRF allows us to measure the total effect on an American bank due to joint, simultaneous shocks from the multiple European banks in the model, while controlling for the cross-relationships between the several European banks.

Existing approaches compute the IRFs with respect to shocks from only one variable at a time. However, in many circumstances, such as our example, this is not what the analyst or policymaker needs to know. If we shock the volatility of European bank A and compute the response of US bank C, and then separately shock the volatility of European bank B and compute the response of US bank C, and add those two responses together, the result is not the same as shocking both banks A and B simultaneously and computing the joint response of bank C. Ignoring the correlation between the contemporaneous shocks from banks A and B will lead to biased estimates in the responses of bank C even if we use a gIRF approach. While, upon reflection, this may seem obvious, we believe the jIRF may be the first method to explicitly address this issue among reduced-form models to capture the total joint effect. Further, we will show that the gIRF is a special case of the jIRF such that the statistical construction of and incorporation of shock information by the jIRF are more general than the gIRF. Consequently, we can reproduce the gIRF from the jIRF, but it is not possible to construct the jIRF results from any set of gIRF results unless the shocks are uncorrelated. In that sense, the jIRF is a multi-shock generalization of the gIRF.

Our concept of a joint impulse response function is distinct from the literature on calculating joint confidence intervals of impulse response functions (Inoue and Kilian 2013, 2016) that computes joint confidence sets for multiple impulse response functions—that is, across the responses of different variables in the VAR due to a single impulse from another variable. Another separate literature (Sims and Zha 1999; Staszewska 2007; Jordà 2009; Bruder and Wolf 2018; Lütkepohl et al. 2015, 2018, 2020) computes joint confidence bands for the shape of particular impulse response functions across time. The concepts proposed in our paper are distinct from these groups of literature in that our jIRF method focuses on the responses due to joint, simultaneous shocks from multiple variables.

2 The VAR model and impulse response functions

To establish notation and definitions, we begin with a brief overview of VAR models and impulse response functions. Consider the stationary, structural simultaneous equation model in K-variables

$$\begin{aligned} \varvec{B}_{0}\varvec{Y}_{t} =\tilde{\varvec{\alpha }}+ \varvec{B}_{1} \varvec{Y}_{t-1} +\varvec{B}_{2} \varvec{Y}_{t-2} + \cdots + \varvec{B}_{P} \varvec{Y}_{t-P} +\varvec{u}_{t}, \end{aligned}$$

(1)

where matrices $\varvec{B}_{i}, \ i = 0,\ldots ,P$ are the structural parameters. Here, $\varvec{u}_{t} \sim IID (\varvec{0}, \varvec{\Sigma }_{u})$ is the vector of structural shocks which is typically assumed to be orthogonal in the literature, but as Pesaran (2015, p. 591) argues, the structural shocks need not be orthogonal to derive the gIRF. And so we allow $\varvec{\Sigma }_{u}$ to be unrestricted. Assuming that $\varvec{B}_{0}$ is invertible, we can rewrite the structural model as a dynamic reduced-form VAR(P) model,

$$\begin{aligned} \varvec{Y}_{t} =\varvec{\alpha }+ \varvec{\Phi }_{1} \varvec{Y}_{t-1} + \varvec{\Phi }_{2} \varvec{Y}_{t-2} + \cdots + \varvec{\Phi }_{P} \varvec{Y}_{t-P} + \varvec{\epsilon }_{t}, \end{aligned}$$

(2)

where $\varvec{\alpha }=\varvec{B}_{0}^{-1}\tilde{\varvec{\alpha }}$ is the VAR constant vector, $\varvec{\Phi }_1 = \varvec{B}_{0}^{-1}\varvec{B}_{1},\ldots , \varvec{\Phi }_P=\varvec{B}_{0}^{-1}\varvec{B}_{P}$ are the VAR coefficient matrices, and $\varvec{\epsilon }_{t} = \varvec{B}_{0}^{-1}\varvec{u}_{t}$ is the K-dimensional reduced-form shock vector with mean zero, covariance matrix $E(\varvec{\epsilon }_{t}\varvec{\epsilon }'_{t})= \varvec{B}_{0}^{-1} \varvec{\Sigma }_{u} \left( \varvec{B}_{0}^\prime \right) ^{-1} =\varvec{\Sigma }_\epsilon$, and $E(\varvec{\epsilon }_{t} \varvec{\epsilon }'_{t+h})=\varvec{0}_{K \times K}$ $\forall h\ne 0$. Although there are two types of shock vectors, the structural $\varvec{u}_{t}$ and the reduced-form $\varvec{\epsilon }_{t}$, we will broadly refer to $\varvec{\epsilon }_{t}$ as “the shocks” in most of this paper. Denote the vector moving average (VMA) representation of the VAR(P) in Eq. (2) as

$$\begin{aligned} \varvec{Y}_{t} =\varvec{c}+ \sum _{h=0}^{\infty } \varvec{A}_{h} \varvec{\epsilon }_{t-h}, \end{aligned}$$

(3)

where $\varvec{c}$ is the VMA constant vector, and $\varvec{A}_0,\varvec{A}_1,\varvec{A}_2,\ldots$ are the VMA coefficient matrices with $\varvec{A}_0=\varvec{I}_{K}$.

The IRF measures the response of $\varvec{Y}_{t+h}$, $h=0,1,2,\ldots$ due to an impulse $\varvec{\delta }$ at time t, where $\varvec{\delta }$ is the vector impulse to the system. The issue is how to choose a meaningful impulse $\varvec{\delta }$. The process is complicated by the fact that the elements of the shock vector $\varvec{\epsilon }_{t}$ are not independent, so it is meaningless to construct an impulse $\varvec{\delta }$ by setting all elements equal to zero except for some nonzero element j.

One approach is to impose identifying restrictions on the VAR using an order dependent Cholesky decomposition, referred to as an orthogonal impulse response function (oIRF). Effectively, this imposes a specific triangular structure upon the $\varvec{B}_{0}$ matrix which, along with the assumption that $\varvec{\Sigma }_{u}$ is diagonal, is sufficient to identify the oIRF with a specific recursive dynamic structural model. Another approach is to choose some set of structural restrictions on the parameters $\varvec{B}_0,\ldots ,\varvec{B}_P$ and $\varvec{\Sigma }_{u}$, leading to a structural impulse response function (sIRF). Since the restrictions used to identify the oIRF or sIRF are not unique, analysts often disagree over the appropriate set of restrictions. The functional shocks of Inoue and Rossi (2021) and non-Gaussian shocks of Lanne et al. (2017) mentioned above are interesting examples of ways to identify unique VMA coefficient matrices and impulse response functions with a structural interpretation.

Another increasingly popular approach to compute the IRF is the generalized impulse response function (gIRF) of Koop et al. (1996) and Pesaran and Shin (1998). The essential difference between the oIRF and the gIRF is that the latter does not transform the shocks in the VAR via orthogonalization but instead uses a conditional expectation of the correlated shocks to produce a unique, reduced-form solution. The gIRF at time $t+h$ is defined as

$$\begin{aligned} \varvec{gIRF}(h, \delta _j, \varvec{\Omega }_{t-1})= E(\varvec{Y}_{t+h} \mid \epsilon _{j,t}=\delta _j,\varvec{\Omega }_{t-1})-E(\varvec{Y}_{t+h} \mid \varvec{\Omega }_{t-1}), \end{aligned}$$

(4)

where $\delta _j$ is the size of the shock from variable j and $\varvec{\Omega }_{t-1}$ is the total information set available at time $t-1$. Typically, for a linear VAR, $\varvec{\Omega }_{t-1} = (\varvec{Y}_{t-1},\varvec{Y}_{t-2},\ldots )$. Assuming multivariate normality³ of the shocks $\varvec{\epsilon }_{t}$, we have

$$\begin{aligned} E(\varvec{\epsilon }_t \mid \epsilon _{j,t}=\delta _j) =\varvec{\Sigma }_\epsilon \varvec{e}_j \left( \sigma _{jj} \right) ^{-1} \delta _j, \end{aligned}$$

(5)

where $\varvec{e}_j$ is the selection vector (a K-dimensional column vector of all zeros and one as the jth element) and $\sigma _{jj}$ is the variance of shock j.

The unscaled gIRF is computed as

$$\begin{aligned} \varvec{gIRF}(h, \delta _j, \varvec{\Omega }_{t-1}) =\varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_j \left( \sigma _{jj} \right) ^{-1} \delta _j, \end{aligned}$$

(6)

for some scalar impulse $\delta _{j}$ producing a $K \times 1$ response vector at time $t+h$. Koop et al. (1996) and Pesaran and Shin (1998) set $\delta _j=\sqrt{\sigma _{jj}}$ such that the size of the shock is one standard deviation. Thus, the scaled gIRF is

$$\begin{aligned} \varvec{gIRF}(h, \delta _j=\sqrt{\sigma _{jj}}, \varvec{\Omega }_{t-1}) =\varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_j \left( \sigma _{jj} \right) ^{-1/2}, \end{aligned}$$

(7)

which measures the response of $\varvec{Y}$ at h periods ahead due to a one standard deviation shock from $\epsilon _{j,t}$. Note that, in terms of the structural parameters, $\varvec{A}_{h}$ is a function of $\varvec{B}_{i}, i = 0,1,\ldots ,P$, and $\varvec{\Sigma }_\epsilon = \varvec{B}_{0}^{-1} \varvec{\Sigma }_{u} \left( \varvec{B}_{0}^\prime \right) ^{-1}$ so that the gIRF is consistent with any set of structural parameters satisfying these expressions. Unlike the oIRF approach, the gIRF approach does not take a particular stand on what is the correct structural interpretation. Rather, the gIRF empirically selects an identification that is consistent with the observed data but does not explicitly identify the row of $\varvec{B}_{0}$ associated with the shocked variable (Pesaran 2015, p. 591). In this sense, we can interpret the oIRF as taking a strong, but possibly incorrect, stand on the structural model, while the gIRF takes an ambiguous stand on identification but one that is consistent with the reduced-form estimated residuals.

Although the implied structural model is ambiguous, the gIRF is unique.⁴ Note that, given a shock from variable j, the gIRF and oIRF are identical only if that variable is put first in the ordering. However, for no other variable $k\ne j=1$ does the gIRF match the oIRF. Thus, to reproduce the gIRF from the oIRFs would require us to compute K separate oIRFs with each variable in turn being put first in the order. Each of these oIRFs imply a specific $\varvec{B}_{0}$ matrix that is inconsistent with the implied identification of the other oIRFs. Thus, the implied identification of the gIRF cannot be constructed from any specific ordering and identification of the oIRFs.

To this end, the gIRF is dissimilar to the oIRF and the broad set of sIRFs. The oIRF and sIRF make explicit identifying restrictions on the structural parameters $\varvec{B}_0,\varvec{B}_1,\ldots ,\varvec{B}_P$, and $\varvec{\Sigma }_{u}$. In contrast, the gIRF is agnostic to identifying restrictions while incorporating shock covariance information that is consistent with the data generating process. The gIRF itself is unique, but the structural parameters that are consistent with the gIRF need not be unique. In that regard, the gIRF can be thought of as “reduced-form.” Importantly, the jIRF presented in the next section will have the same agnostic approach to the structural parameters as the gIRF. In one category are the oIRF and sIRF, which make explicit identifying restrictions on the structural parameters. In another category are the gIRF and jIRF, which are unique and “reduced-form,” but do not require unique identification of the structural parameters. Therefore, the most appropriate comparison of the jIRF is to the gIRF (or more specifically, the sum of the gIRFs). Ergo, for the remainder of the paper, we will focus on comparing the jIRF to the gIRF.

3 The joint impulse response function

In contrast to the gIRF which measures the response due to one shock at a time, the jIRF allows for joint, simultaneous shocks from a set of variables. Let ${\mathbb {J}}$ be the subset of the K variables in our system that the simultaneous shocks originate from. Thus, our impulse vector $\varvec{\delta }_{{\mathbb {J}}}$ would be a vector whose elements are indicated by the elements of ${\mathbb {J}}$. For example, if ${\mathbb {J}}=\{j,k,\ell \}$, then our impulse vector would be $\varvec{\delta }_{{\mathbb {J}}}=[\delta _{j} \;\;\; \delta _{k} \;\;\; \delta _{\ell }]'$.

We define the jIRF at time $t+h$ to be

$$\begin{aligned} \varvec{jIRF}(h, \varvec{\delta }_{{\mathbb {J}}}, \varvec{\Omega }_{t-1})=E(\varvec{Y}_{t+h} \mid \varvec{\epsilon }_{{\mathbb {J}},t}=\varvec{\delta }_{{\mathbb {J}}}, \varvec{\Omega }_{t-1})-E(\varvec{Y}_{t+h} \mid \varvec{\Omega }_{t-1}). \end{aligned}$$

(8)

Note the use of a $\#{\mathbb {J}}$-dimensional vector $\varvec{\delta }_{{\mathbb {J}}}$ in Eq. (8), while in Eq. (4) a scalar $\delta _{j}$ was used. Again assuming multivariate normality of the shocks, we have

$$\begin{aligned} E(\varvec{\epsilon }_t \mid \varvec{\epsilon }_{{\mathbb {J}},t} =\varvec{\delta }_{{\mathbb {J}}}) = \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \left( \varvec{e}'_{{\mathbb {J}}} \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \right) ^{-1} \varvec{\delta }_{{\mathbb {J}}}, \end{aligned}$$

(9)

where $\varvec{e}_{{\mathbb {J}}}$ is the $K \times \#{\mathbb {J}}$ selection matrix constructed using the set ${\mathbb {J}}$ columns of the identity matrix.⁵ For example, if $K=4$ and ${\mathbb {J}}=\{1,3\}$, then

$$\begin{aligned} \varvec{e}_{{\mathbb {J}}}=\begin{bmatrix} 1 &{} 0 \\ 0 &{} 0 \\ 0 &{} 1 \\ 0 &{} 0 \end{bmatrix}. \end{aligned}$$

(10)

The unscaled jIRF vector at time $t+h$ is

$$\begin{aligned} \varvec{jIRF}(h, \varvec{\delta }_{{\mathbb {J}}}, \varvec{\Omega }_{t-1}) =\varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \left( \varvec{e}'_{{\mathbb {J}}} \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \right) ^{-1} \varvec{\delta }_{{\mathbb {J}}}. \end{aligned}$$

(11)

Again, following the scaling convention for shocks in Koop et al. (1996) and Pesaran and Shin (1998), we set $\varvec{\delta }_{{\mathbb {J}}}=\varvec{s}_{{\mathbb {J}}}$ such that each of the shocks in set ${\mathbb {J}}$ are one standard deviation in size. For instance, if ${\mathbb {J}}=\{j,k,\ell \}$, then $\varvec{s}_{{\mathbb {J}}}=[\sqrt{\sigma _{jj}} \;\;\; \sqrt{\sigma _{kk}} \;\;\; \sqrt{\sigma _{\ell \ell }}]'$. Then, the scaled jIRF is

$$\begin{aligned} \varvec{jIRF}(h, \varvec{\delta }_{{\mathbb {J}}}=\varvec{s}_{{\mathbb {J}}}, \varvec{\Omega }_{t-1})= \varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \left( \varvec{e}'_{{\mathbb {J}}} \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \right) ^{-1} \varvec{s}_{{\mathbb {J}}}, \end{aligned}$$

(12)

producing a $K \times 1$ vector measuring the response of $\varvec{Y}$ at h periods ahead due to joint, simultaneous one standard deviation shocks from the variables in set ${\mathbb {J}}$.⁶

We observe that if there is only a single shock from one variable, instead of multiple variables, then the jIRF in Eq. (12) collapses to the gIRF in Eq. (7). Thus, by utilizing multivariate conditioning sets—similar to those used by Lastrapes and Wiesen (2021)—the jIRF can be thought of as a multi-shock generalization of the gIRF.

One feature of the jIRF is that it can identify the impact of some latent factor that affects some, but not all, variables in a system directly and affects the remaining variables indirectly. In this sense, the jIRF is related to—although methodologically distinct from—dynamic factor models (see Stock and Watson (2011)), as the jIRF captures the effect of some latent common shock across the variables in set ${\mathbb {J}}$. Since the jIRF does not explicitly model the dynamics of the latent factor, it does not require the direct estimation of factors nor the use of nonlinear methods often employed in dynamic factor models. Namely, the jIRF is simple and robust.

4 Properties of the jIRF

An important feature of the jIRF is that it controls for the shock correlations among the set of variables ${\mathbb {J}}$ where the shocks originate. To the extent that the variables in set ${\mathbb {J}}$ are caused by a common latent factor, our jIRF appropriately measures the contribution of the underlying factor by accounting for the cross-correlations among the directly impacted variables. Therefore, the sum of the gIRFs over the set ${\mathbb {J}}$ is not equivalent to the jIRF over that same set of variables. To illustrate this, consider the K-variable VAR from Eq. (2) and let ${\mathbb {J}}$ again denote a subset of size $\#{\mathbb {J}}$ of those K variables where the simultaneous shocks originate. The h-step ahead $K\times 1$ jIRF vector, given by Eq. (11), may be rewritten as

$$\begin{aligned} \varvec{jIRF}(h, \varvec{\delta }_{{\mathbb {J}}}, \varvec{\Omega }_{t-1}) = \varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \left( \varvec{\Sigma }_{{\mathbb {J}}} \right) ^{-1} \varvec{\delta }_{{\mathbb {J}}} \end{aligned}$$

(13)

where $\varvec{\Sigma }_{{\mathbb {J}}}=\varvec{e}'_{{\mathbb {J}}} \varvec{\Sigma }_{\epsilon }\varvec{e}_{{\mathbb {J}}}$ is the appropriate $\#{\mathbb {J}} \times \#{\mathbb {J}}$ subset of the $K\times K$ covariance matrix $\varvec{\Sigma }_\epsilon$. The sum of the $\#{\mathbb {J}}$ h-step ahead $K\times 1$ gIRF vectors is

$$\begin{aligned} \sum _{j \in {\mathbb {J}}} \varvec{gIRF}(h, \delta _j, \varvec{\Omega }_{t-1})&= \sum _{j \in {\mathbb {J}}} \varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_j \left( \sigma _{jj} \right) ^{-1} \delta _j \end{aligned}$$

(14)

$$\begin{aligned}&= \varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \left( \text {diag}\varvec{\Sigma }_{{\mathbb {J}}} \right) ^{-1} \varvec{\delta }_{{\mathbb {J}}} \end{aligned}$$

(15)

where $\left( \text {diag}\varvec{\Sigma }_{{\mathbb {J}}} \right) ^{-1}$ is the inverse of the diagonal elements of $\varvec{\Sigma }_{{\mathbb {J}}}$.

Finally, the difference between the sum of the gIRFs and the jIRF vectors at step h is

$$\begin{aligned} \sum _{j \in {\mathbb {J}}} \varvec{gIRF}(h, \delta _j, \varvec{\Omega }_{t-1}) - \varvec{jIRF}(h, \varvec{\delta }_{{\mathbb {J}}}, \varvec{\Omega }_{t-1}) =\varvec{A}_h \varvec{\Sigma }_\epsilon \varvec{e}_{{\mathbb {J}}} \left( \left( \text {diag}\varvec{\Sigma }_{{\mathbb {J}}}\right) ^{-1} -\varvec{\Sigma }_{{\mathbb {J}}}^{-1} \right) \varvec{\delta }_{{\mathbb {J}}}. \end{aligned}$$

(16)

Equation (16) shows that the jIRF and the sum of the gIRFs are not equal except in the special case where the shocks in set ${\mathbb {J}}$ are mutually uncorrelated, in which case, $\text {diag}\varvec{\Sigma }_{{\mathbb {J}}} =\varvec{\Sigma }_{{\mathbb {J}}}$.

To more concretely demonstrate how the jIRF controls for the cross-correlations among the shocks in set ${\mathbb {J}}$, consider a 4-variable VAR and let the symmetric and positive-definite shock covariance matrix be

$$\begin{aligned} \varvec{\Sigma }_{\epsilon }=\begin{bmatrix} 1 &{}\quad \rho _{12} &{}\quad \rho _{13} &{}\quad \rho _{14} \\ \rho _{12} &{}\quad 1 &{}\quad \rho _{23} &{}\quad \rho _{24} \\ \rho _{13} &{}\quad \rho _{23} &{}\quad 1 &{}\quad \rho _{34} \\ \rho _{14} &{}\quad \rho _{24} &{}\quad \rho _{34} &{}\quad 1 \end{bmatrix}. \end{aligned}$$

(17)

In this simple illustration, we restrict our example such that all shock variances are one, so that the shock covariance matrix is equal to the shock correlation matrix. Say we wish to measure how variable 1 responds due to joint, simultaneous shocks from variables 2 and 3 of size $\delta _2=\sqrt{\sigma _{22}} =1$ and $\delta _3= \sqrt{\sigma _{33}} = 1$, so that ${\mathbb {J}}=\left\{ 2,3 \right\}$ and $\varvec{\delta }_{{\mathbb {J}}} = \left[ 1 \;\;\; 1 \right] '$. First, consider the immediate ($h=0$) response so that $\varvec{A}_0=\varvec{I}_K$ and the VMA coefficient matrix falls out of the expression. In this example, we find that the jIRF for variable 1 is

$$\begin{aligned} \varvec{e}_{1}' \varvec{jIRF}(h=0, \varvec{\delta }_{{\mathbb {J}}} =\left[ 1 \;\;\; 1 \right] ', \varvec{\Omega }_{t-1}) = \frac{\rho _{12} + \rho _{13}}{1 + \rho _{23}}, \end{aligned}$$

(18)

and the sum of the two corresponding gIRFs is

$$\begin{aligned} \sum _{j=2}^3 \varvec{e}_{1}' \varvec{gIRF}(h=0, \delta _j = 1, \varvec{\Omega }_{t-1}) = \rho _{12} + \rho _{13}, \end{aligned}$$

(19)

so that the difference between them is

$$\begin{aligned}&\sum _{j=2}^3 \varvec{e}_1^\prime \varvec{gIRF}(h=0, \delta _j = 1, \varvec{\Omega }_{t-1}) - \varvec{e}_{1}' \varvec{jIRF}(h=0, \varvec{\delta }_{{\mathbb {J}}} =\left[ 1 \;\;\; 1 \right] ', \varvec{\Omega }_{t-1}) \nonumber \\&\quad = \rho _{23} \left( \frac{\rho _{12} + \rho _{13}}{1 + \rho _{23}} \right) = \Delta . \end{aligned}$$

(20)

The critical point here is that the gIRFs do not contain the cross-correlation $\rho _{23}$ because both gIRFs condition on a single variable’s shock individually. In contrast, the jIRF conditions on a set of shocks jointly and thus controls for the cross-correlation in a multiple regression sense.

For the IRFs at time steps $h \ge 1$, $\varvec{A}_h$ does not drop out of the expressions, and thus the way $\rho _{23}$ appears in the impulse responses is more complex and less transparent. To illustrate the differences in dynamics between the gIRF and jIRF when $h \ge 1$, we continue the same example and use a VAR(1) coefficient matrix given by the symmetric Toeplitz matrix

$$\begin{aligned} \varvec{\Phi }_{1}=\begin{bmatrix} 0.55 &{}\quad 0.1 &{}\quad 0.1 &{}\quad 0.1 \\ 0.1 &{}\quad 0.55 &{}\quad 0.1 &{}\quad 0.1 \\ 0.1 &{}\quad 0.1 &{}\quad 0.55 &{}\quad 0.1 \\ 0.1 &{}\quad 0.1 &{} \quad 0.1 &{}\quad 0.55 \end{bmatrix}, \end{aligned}$$

(21)

along with a selection of representative covariance matrices $\varvec{\Sigma }_\epsilon$. As before, our interest is in measuring how variable 1 responds due to simultaneous one standard deviation shocks from variables 2 and 3.

Figure 1 shows a collection of example impulse response plots. The black solid line with circle markers is the jIRF showing how variable 1 responds due to joint, simultaneous one standard deviation shocks from variables 2 and 3. The blue dotted line is the gIRF showing how variable 1 responds due to a one standard deviation shock from variable 2 alone. The red dashed line is the gIRF showing how variable 1 responds due to a one standard deviation shock from variable 3 alone. The brown solid line is the sum of the two gIRFs. Each of the eight graphs in Fig. 1 corresponds to a different shock correlation (covariance) matrix. The shock correlation between variables 2 and 3 (namely, $\rho _{23}$) is 0.8, 0.3, 0, and $-0.25$ in the first, second, third, and fourth rows, respectively. Thus, when comparing row to row, we can see how the jIRF evolves as the correlation between the shocks in set ${\mathbb {J}}$ changes. In the first column of plots, the shock correlation between variables 1 and 2 is $\rho _{12}=0.5$, and the shock correlation between variables 1 and 3 is $\rho _{13}=-0.1$, so that the correlations $\rho _{12}$ and $\rho _{13}$ have different signs. In the second column of plots, the shock correlation between variables 1 and 2 is still $\rho _{12}=0.5$, but the shock correlation between variables 1 and 3 is $\rho _{13}=0.2$, so that the correlations $\rho _{12}$ and $\rho _{13}$ have the same sign. All the other shock correlations are constant across the plots. The specific shock correlation (covariance) matrix used in each plot is shown below each graph.

Fig. 1

Example joint and generalized impulse response plots using different data generating processes

Recalling that we defined $\Delta = \rho _{23} \left( \frac{\rho _{12} + \rho _{13}}{1 + \rho _{23}} \right)$, there are four key features that the example plots in Fig. 1 illustrate. First, in rows one and two we have $\Delta > 0$ and the gIRF sum overestimates the joint response relative to the jIRF at $h=0$. In this example, that overestimation persists for $h>0$. Second, when $\rho _{23}=0$, as in row three, we have $\Delta = 0$ so the jIRF is equal to the sum of the two gIRFs for all time horizons $h=0,1,2,3,\ldots$. Third, when we have $\Delta < 0$, as in row four, the gIRF sum underestimates the joint response relative to the jIRF at $h=0$. In this example, that underestimation persists for $h>0$. Lastly, when $\rho _{23}$ is quite large, as in row one, then the immediate ($h=0$) jIRF is close to the average of the two immediate ($h=0$) gIRFs. Indeed, as $\rho _{23} \rightarrow 1$, the jIRF at $h=0$ approaches the average of the two gIRFs, and as $\rho _{23} \rightarrow -1$ the jIRF diverges away from the sum of the two gIRFs.⁷

As we have argued in Sect. 2, the jIRF and gIRFs are conceptually distinct from the oIRF. Nevertheless, given the historical popularity of the oIRF dating back to Sims (1980), it could be informative to compare the jIRF to the oIRF. Thus, we repeat the same experiment as above, but we compare the jIRFs to the sum of the oIRFs of variable 1 due to a shock from variable 2 plus the response of variable 1 due to a shock from variable 3. Because the oIRFs are not unique, we compute all $K!=4!=24$ possible permutations of the oIRFs to compare the jIRFs to the full range of possible oIRF sums. The equivalent of Fig. 1 using the oIRF sum can be found in the Appendix of Supplemental Results. The results show that the range of possible oIRF sums is very wide and the jIRF need not be bounded by the range of the oIRF sums. This is not surprising given that the jIRF and oIRF are quite different in construction and motivation. It is worth stressing that the gIRF, not the oIRF, is the tool most comparable to our proposed jIRF. The jIRF is a multi-shock “generalization” of the gIRF and both use the same ambiguous structural identification approach which is distinct from any of the order-specific structural identifications implied in the oIRFs. Importantly, not all of the possible oIRF structurally identified models can be “correct,” whereas the ambiguous identification implied by the jIRF, although not explicitly defined, is the one suggested by the data.

5 The joint forecast error variance decomposition

Forecast error variance decompositions are closely related to impulse response functions. In this section, we derive the joint forecast error variance decomposition (jFEVD) and provide intuitive interpretations to highlight the differences between our proposed jFEVD and the extant generalized forecast error variance decomposition (gFEVD). Given the relationship between forecast error variance decompositions and impulse response functions, by analyzing the differences between the jFEVD and gFEVD we are more fully able to understand the differences between the jIRF and gIRF.

From the VMA representation in model (3), the H-step ahead forecast error is

$$\begin{aligned} \varvec{\xi }_t(H)=\sum _{h=0}^{H-1}\varvec{A}_h \varvec{\epsilon }_{t+H-h}, \end{aligned}$$

(22)

and the H-step $K \times K$ forecast error covariance matrix is

$$\begin{aligned} cov(\varvec{\xi }_t(H))=\sum _{h=0}^{H-1} \varvec{A}_h\varvec{\Sigma }_{\epsilon } \varvec{A}_h'. \end{aligned}$$

(23)

The forecast error variance decomposition decomposes the diagonal elements of the forecast error covariance matrix in Eq. (23). Wiesen et al. (2018) provide an intuitive interpretation of the elements of the forecast error variance decomposition matrix in terms of coefficients of determination from hypothetical regressions of the H-step ahead forecast errors on various sets of future variable shocks. From Koop et al. (1996) and Pesaran and Shin (1998), the (i, j)th element of the $K\times K$ generalized forecast error variance decomposition (gFEVD) matrix $\varvec{\zeta }^{gen}(H)$ is

$$\begin{aligned} \zeta _{ij}^{gen}(H) = \frac{\sum _{h=0}^{H-1}(\varvec{e}_{i}'\varvec{A}_{h} \varvec{\Sigma }_{\epsilon }\varvec{e}_{j})^{2}}{ \sigma _{jj} \sum _{h=0}^{H-1}( \varvec{e}_{i}' \varvec{A}_{h}\varvec{\Sigma }_{\epsilon } \varvec{A}_{h}' \varvec{e}_{i})}, \end{aligned}$$

(24)

which measures the proportion of the H-step forecast error variance of variable i that can be explained by the shocks of variable j, and where $\sigma _{jj}$ is the variance of shock j.⁸ The (i, j)th element of the gFEVD matrix may be thought of as the coefficient of determination from regressing the H-step forecast error of variable i on the future shocks of variable j (Wiesen et al. 2018). Specifically, consider the hypothetical regression

$$\begin{aligned} \xi _{i,t}(H) = \sum _{h=1}^{H}\beta _{ij,h} \epsilon _{j,t+h} + \textrm{error}_{i,t}. \end{aligned}$$

(25)

The coefficient of determination from regression (25), denoted as $R_{i \mid j}^2$, is equivalent to $\zeta _{ij}^{gen}(H)$ from Eq. (24) and can be interpreted as the proportion of variance in the forecast error of variable i that is explained by controlling for the future shocks of variable j alone. This result is independent of the order of the variables in the VAR.

Next, consider the joint forecast error variance decomposition (jFEVD) associated with some set of variables ${\mathbb {J}}$ as defined in Sect. 3. The jFEVD of variable i, controlling for the shocks of the variables in set ${\mathbb {J}}$, is computed as

$$\begin{aligned} \zeta _{i \mid {\mathbb {J}}}^{jnt}(H) =\frac{\sum _{h=0}^{H-1} \varvec{e}'_{i}\varvec{A}_h \varvec{\Sigma }_{\epsilon }\varvec{e}_{{\mathbb {J}}} \left( \varvec{e}'_{{\mathbb {J}}} \varvec{\Sigma }_{\epsilon } \varvec{e}_{{\mathbb {J}}} \right) ^{-1} \varvec{e}'_{{\mathbb {J}}} \varvec{\Sigma }_{\epsilon } \varvec{A}'_{h} \varvec{e}_{i} }{\sum _{h=0}^{H-1} \varvec{e}'_{i}\varvec{A}_h \varvec{\Sigma }_{\epsilon } \varvec{A}_{h}' \varvec{e}_{i}}, \end{aligned}$$

(26)

which measures the proportion of the H-step forecast error variance of variable i that can be explained by the shocks of the variables in set ${\mathbb {J}}$.

Now extend hypothetical regression (25) to include the set ${\mathbb {J}}$ shocks. Specifically, consider the regression

$$\begin{aligned} \xi _{i,t}(H) = \sum _{h=1}^{H} \sum _{j \in {\mathbb {J}}} \beta _{ij,h} \epsilon _{j,t+h} + \textrm{error}_{i,t}, \end{aligned}$$

(27)

and denote the coefficient of determination from this regression as $R_{i \mid {\mathbb {J}}}^2$. Once again, we can show that $R_{i \mid {\mathbb {J}}}^2 = \zeta _{i \mid {\mathbb {J}}}^{jnt}(H)$ so that $\zeta _{i \mid {\mathbb {J}}}^{jnt}(H)$ from Eq. (26) can be interpreted as the proportion of variance in the forecast error of variable i that is explained by controlling for the future shocks of the variables in set ${\mathbb {J}}$. This result is also independent of the order of the variables in the VAR. Simply summing the elements of the gFEVD matrix risks double counting or undercounting the total explanatory power of the shocks in set ${\mathbb {J}}$. Since the shocks in set ${\mathbb {J}}$ are contemporaneously correlated, we generally observe that

$$\begin{aligned} R_{i \mid {\mathbb {J}}}^2 = \zeta _{i \mid {\mathbb {J}}}^{jnt}(H) \ne \sum _{j \in {\mathbb {J}}} \zeta _{ij}^{gen}(H) =\sum _{j \in {\mathbb {J}}} R_{i \mid j}^2. \end{aligned}$$

(28)

Note that the jFEVD need not be defined as a $K \times K$ matrix. For a specific set ${\mathbb {J}}$, the jFEVD in Eq. (26) can be computed $\forall i=1,\ldots ,K$ to form a $K \times 1$ vector, but different ${\mathbb {J}}$ sets can be used for different decompositions.⁹

The jFEVD can also be used to construct partially disaggregated measures of spillovers. As noted in Greenwood-Nimmo et al. (2021), the connectedness measures of Diebold and Yilmaz (2012, 2014, 2016) operate at two extremes: (1) measuring pair-wise spillovers of the K variables in a fully disaggregated manner and (2) measuring the overall system-wide connectedness in a fully aggregated manner. The connectedness measures of Diebold and Yilmaz (2012, 2014, 2016) are not well-equipped for measuring partially disaggregated connections (e.g., the spillovers from a set of variables in the VAR to a particular variable). However, our proposed jFEVD can indeed capture exactly that, because it measures the fraction a variable’s forecast error variance jointly explained by shocks from a set of $\# {\mathbb {J}}$ variables where $1 \le \# {\mathbb {J}} \le K$.¹⁰ Therefore, the jFEVD technique complements the work of Greenwood-Nimmo et al. (2021).

6 jIRF application: trans-Atlantic volatility transmissions

To illustrate the practical usefulness of the jIRF, we employ it to measure how the transmission of equity market volatility shocks from European financial institutions to American financial institutions has been affected by the COVID-19 crisis.¹¹ Volatility transmissions associated with the COVID-19 pandemic are important because the pandemic induced widespread economic crises throughout the world, and volatility is an indicator of investor fear and uncertainty. Our application is motivated by the highly cited paper of Diebold and Yilmaz (2014) which measures stock market volatility spillovers during the 2007–2008 global financial crises between thirteen major American financial institutions using a VAR framework. As was first proposed in Diebold and Yilmaz (2012), Diebold and Yilmaz (2014) construct a generalized spillover table measuring bilateral directional spillovers and other aggregate and net spillover measures that are based upon the gFEVD which, as noted in Sect. 5, is closely related to the gIRF.¹² Building on their earlier work, Diebold and Yilmaz (2016) use their generalized spillover metrics to measure trans-Atlantic volatility spillovers between seventeen American major financial institutions and eighteen European financial institutions during the 2007–2008 financial crisis. The central focus of both of these papers (Diebold and Yilmaz 2014, 2016) is to examine how volatility connectedness changed during that critical financial crisis period; they conclude that understanding connectedness between trans-Atlantic financial institutions is key for understanding the financial crisis.

Our application measuring how trans-Atlantic volatility transmission changed during a time of crisis differs from Diebold and Yilmaz (2016) in three significant ways. First, we observe that the COVID-19 pandemic—which was first detected in China in late 2019—spread substantially in Europe prior to appearing substantially in the USA. On March 1, 2020, Europe had 2261 cumulative confirmed COVID-19 cases, while North America had only 62 cumulative confirmed cases, only 32 of those being in the USA. On March 15, 2020, Europe had 56,894 cumulative confirmed COVID-19 cases, while North America had only 3622 cumulative confirmed cases, only 3212 of those being in the USA. A cross-correlation analysis of the daily percentage change in total cases during February and March 2020 shows that cases in Europe led those in the US by about eight days.¹³ Consequently, the volatility impacts to the financial sector induced by COVID-19 would have been felt by European banks prior to American banks. Second, instead of utilizing bilateral, net, and aggregate generalized spillover metrics, we measure volatility transmissions more directly via a joint impulse response framework that captures the common shock to all banks in the region. Third, rather than use a rolling-window approach¹⁴ through the crisis period, we segregate the model parameters into pre-pandemic and during pandemic periods by first examining the transmission mechanism in the pre-pandemic period and then re-estimating the process in the pandemic period to determine how the transmission structure has changed over the two periods. Our primary focus is to illustrate that the correct measurement of trans-Atlantic impulse responses requires an accurate accounting of the correlations among the contemporaneous shocks.

For our analysis, we use daily stock market volatility of the eleven systemically important financial institutions as listed by the Federal Reserve’s Large Institution Supervision Coordinating Committee (LISCC) as of August 2020.¹⁵ These eleven financial institutions consist of eight American institutions (Bank of America, Bank of New York Mellon, Citigroup, Goldman Sachs, JP Morgan Chase, Morgan Stanley, State Street, and Wells Fargo) and three European institutions (Barclays, Credit Suisse, and Deutsche Bank). For each bank, we construct daily volatility using high, low, open, and close (HLOC) price data from Yahoo Finance. An early and commonly used estimator of daily volatility using HLOC price data is the range-based method of Parkinson (1980) who assumes the underlying price follows a geometric Brownian motion process. The Parkinson (1980) volatility estimator uses only the daily high and low prices and is unbiased and efficient only if: (1) there are no jumps between the previous day’s closing price and the current day’s opening price, and (2) there is no drift in the underlying geometric Brownian motion price process. There is strong evidence that neither of these assumptions is true. Thus, in our analysis we will use the method of Yang and Zhang (2000) who extend the method of Rogers and Satchell (1991) to derive a minimum-variance unbiased estimator of volatility which is independent of both drift and opening jumps.¹⁶

Our full data sample of interest ranges from January 3, 2017 to February 4, 2021, for a total of 1027 usable daily volatility observations. We split the data into two periods: a pre-pandemic period from January 3, 2017 to January 31, 2020, and a pandemic period from February 3, 2020 to February 4, 2021. The World Health Organization (WHO) declared COVID-19 to be a global health emergency on January 30, 2020 and declared it to be a global pandemic on March 11, 2020. We chose to break our sample at the earlier date because, by March 11, a substantial number of positive cases had already emerged in Europe. Financial markets, being forward-looking, were likely already reacting to the increasing COVID-19 case counts before the WHO global pandemic declaration on March 11. Consequently, we chose to begin the pandemic period on February 3, 2020 to avoid potential contamination of our pre-pandemic data. Figure 2 shows a plot of the daily volatilities in annualized percentages over the full sample for all eleven banks. The dramatic spike in daily volatility from a normal range typically less than 100% and reaching 300–400% in mid-March 2020 reflects the turbulence in financial markets when the scope of the crisis was first becoming clear. This disruption likely caused a structural shift in the VAR’s parameters and is the motivation for splitting the sample into pre-pandemic and pandemic periods.

Fig. 2

Daily equity volatility in annualized percentages for the eleven systemically important financial institutions calculated using the Yang and Zhang (2000) estimator. The vertical dotted line denotes the January 31, 2020 separation between our pre-pandemic and pandemic periods

Table 1 shows summary statistics for the eleven systemically important financial institutions over the full-sample, pre-pandemic, and pandemic periods. For both American and European financial institutions, the mean annualized daily volatility in the pandemic period is about twice as large on average than in the pre-pandemic period. Furthermore, the variance of the computed daily volatility levels in the pandemic period is substantially higher by a factor ranging from 6.7 times to 24.4 times the pre-pandemic period. The mean volatility of Barclays in Europe and Citigroup and Wells Fargo in the USA seem to have been particularly affected by the pandemic. As expected, the average volatility over the full sample is approximately a weighted average of the two periods.

Table 1 Mean and variance of daily equity market volatility in annualized percentages for the eleven systemically important financial institutions

To investigate the impact of COVID-19 on trans-Atlantic volatility transmissions from Europe to the US, we utilized the jIRF to measure how each of the American financial institutions responds due to joint, simultaneous one standard deviation shocks from the three European financial institutions in both pre- and pandemic periods. Specifically, using the stock market volatility data for all eleven systemically important financial firms, we first estimate a VAR over the pre-pandemic period using daily data from January 3, 2017 through January 31, 2020 and then estimate another VAR using daily data from the pandemic period of February 3, 2020 through February 4, 2021.¹⁷ We acknowledge that our approach will produce two VAR models neither of which would be appropriate for post-pandemic forecasts. If forecasting is the goal, a better approach would be to compensate for the parameter instability induced by the COVID-19 shock. Lenza and Primiceri (2022), for example, scale the variance-covariance matrices over time to capture the overall changes in volatility as the COVID-19 event transitions through its cycle. Alternatively, Bobeica and Hartwig (2023) use a fat-tailed distribution for the shocks to absorb the increased volatility. They also recommend using additional information in the model (e.g., COVID-19 case counts) to help capture the shocks. Both approaches attempt to dampen the impact of the COVID-19 shock on the VAR parameter estimates to stabilize forecasts and IRFs in the pre-pandemic, pandemic, and post-pandemic periods. However, in our application, we wish to examine changes in the variance-covariance matrices across the pre-pandemic and pandemic periods by allowing the parameters and covariances to change without restrictions. While this approach allows us to compare the pre-pandemic and pandemic period IRFs, we should note that neither the pre-pandemic nor the pandemic estimated VARs would produce suitable post-pandemic forecasting models.

In Fig. 3, we show the $h=0,1,\ldots ,8$ day ahead impulse responses of various American financial institutions due to one standard deviation volatility shocks from Barclays, Credit Suisse, and Deutsche Bank. The left column of panels is for the pre-pandemic period, and the right column of panels is for the pandemic period. Row one of the panels shows the responses of Bank of America (BAC), row two shows the responses of Citigroup (C), and row three shows the responses of JP Morgan Chase (JPM).¹⁸ In each of the six panels of Fig. 3, there are five lines depicting the responses of the American financial institutions to various shocks from the European financial institutions. In each panel, the black line with circle markers is the jIRF measuring how the American financial institution’s volatility responds due to joint, simultaneous shocks from the three European financial institutions, and the gray shaded region shows the 80% confidence interval for the jIRF. The blue dotted line is the gIRF measuring how the American financial institution’s volatility responds due to a shock from Barclays (BCS) alone. The red dashed line is the gIRF measuring how the American financial institution’s volatility responds due to a shock from Credit Suisse (CS) alone. The green dot-dash line is the gIRF measuring how the American financial institution’s volatility responds due to a shock from Deutsche Bank (DB) alone. Finally, the solid brown line is the sum of the three gIRFs, and the shaded brown region shows the 80% confidence interval for the sum of the gIRFs. To aid comparison, the same vertical scale is used within a column of the different banks’ response functions. However, notice the vertical scale in the pre-pandemic period column of plots is different from the vertical scale in the pandemic period column of plots.

Fig. 3

Joint and generalized impulse response functions showing how the equity market volatility of Bank of America (first row), Citigroup (second row), and JP Morgan Chase (third row) respond over the next eight days due to one standard deviation shocks from Barclays, Credit Suisse, and Deutsche Bank

To obtain the confidence intervals shown in Fig. 3 we use the “residual” bootstrap method developed by Efron (1979) and as described for VAR models by Lütkepohl (2000). The procedure is: (i) estimate the VAR model and extract the vectors of residuals, $\hat{\varvec{\epsilon }_t}$; (ii) randomly sample the vectors of residuals with replacement and re-center each new sample of residuals so its mean is zero; (iii) taking the first data observation as given, use the original VAR estimates to recursively generate new data from the bootstrapped residuals; (iv) use the new data to re-estimate the VAR model and compute new impulse response functions; (v) repeat this process many times (in our case 12,000 times) to obtain empirical distributions of the jIRF and sum of the gIRFs; (vi) extract, in our case, the tenth and ninetieth percentiles of the IRFs’ empirical distributions to form the 80% confidence intervals.

Several implications emerge from the jIRF results shown in Fig. 3. Considering the full set of plots overall, we first observe that the impact on a particular American financial institution due to a volatility shock from a European financial institution is quite similar for each of the three European financial institutions. In fact, the individual gIRFs are hard to distinguish on the plots, especially during the pandemic period. Second, we observe that the jIRF is typically slightly larger than any of the individual financial institution’s gIRFs. In this application, all the shock correlations between the eleven institutions are positive, so we are in a situation similar to the top two example plots in the right column of panels in Fig. 1.

In viewing the pre-pandemic impulse responses in the left column of panels of Fig. 3, we observe, as seems reasonable in an environment where options trading is ubiquitous, the effects of a shock from the volatility of one financial institution on the volatility of another financial institution are not particularly large and die out in about a week. However, during the pandemic, the size of the responses due to volatility shocks are dramatically larger and are more persistent. Again notice that the vertical scale in the pandemic period plots is significantly larger.

In the pre-pandemic period, we observe that the sum of the three immediate gIRFs is larger than the immediate jIRF by a factor of about 1.65. Thus, before the pandemic, the initial impact on an American financial institution due to joint, simultaneous shocks from all three systemically important European financial institutions would be overestimated by about $65 \%$ if measured by ignoring the cross-correlations and using the sum of the gIRFs rather than the more appropriate jIRF. In comparison, the right column of panels shows that the transmission of shocks from European financial institutions to American financial institutions is vastly higher during the pandemic period than before. Note that the shocks during the pandemic period are still one standard deviation, so the “scale” difference in the volatilities between the two periods is being accounted for. Clearly, the pandemic has had the effect of increasing the volatility transmissions between major European and American financial institutions.

In the pandemic period, we similarly observe that the sum of the three immediate gIRFs is larger than the immediate jIRF, but by a larger factor of about 2.46. Thus, during the pandemic, the initial impact on an American financial institution due to joint, simultaneous shocks from all three systemically important European financial institutions would be overestimated by about $146 \%$ if measured using the sum of the gIRFs rather than the jIRF. While the pandemic clearly increased the volatility transmissions from European to American financial institutions, how much the transmissions increased is dependent on which impulse response functions are used to measure it. For example, Bank of America’s immediate jIRF increased by a factor of 5.91 during the pandemic relative to its immediate jIRF for the pre-pandemic period. However, Bank of America’s immediate gIRF sum increased by a much larger factor of 8.71 during the pandemic relative to its immediate gIRF sum for the pre-pandemic period. Similarly, Citigroup’s immediate jIRF increased by a factor of 6.85 during the pandemic relative to its immediate jIRF for the pre-pandemic period. However, Citigroup’s immediate gIRF sum increased by a much larger factor of 10.26 during the pandemic relative to its immediate gIRF sum for the pre-pandemic period. Critically, the increase in initial volatility transmission from European to American financial institutions is overestimated by the sum of the gIRFs. The jIRF still demonstrates an increase in volatility spillovers during the pandemic, but by accounting for the cross-correlations among the European shocks, the jIRF shows a more modest increase in volatility spillovers from Europe to the USA. Our results indicate that the increase in trans-Atlantic volatility transmissions is overestimated by naively summing the gIRFs, thus demonstrating that properly controlling for the cross-correlations, as the jIRF does, matters for measuring the propagation of joint shocks in a VAR model.

To further confirm that the sum of the gIRFs overestimates the impact of the simultaneous shocks from European banks, we compute confidence intervals for the difference between the sum of the gIRFs and the jIRF. Figure 4 plots the difference between the sum of the gIRFs (i.e., the gIRF for an American bank’s volatility response due to a volatility shock from Barclays alone, plus the gIRF due to a shock from Credit Suisse alone, plus the gIRF due to a shock from Deutsche Bank alone) minus the jIRF due to joint, simultaneous shocks from the three European banks. Essentially, the purple lines in Fig. 4 show the difference between the brown and black lines in Fig. 3. The shaded purple regions show the 80% and 99% confidence intervals, obtained with the same bootstrap procedure described above. As with earlier figures, the left column of panels is for the pre-pandemic period, and the right column of panels is for the pandemic period. The first row of panels shows the responses of Bank of America (BAC), the second row shows the responses of Citigroup (C), and the third row shows the responses of JP Morgan Chase (JPM).¹⁹

Fig. 4

Difference between the sum of the generalized impulse response functions and the joint impulse response function for Bank of America (first row), Citigroup (second row), and JP Morgan Chase (third row)

Figure 4 illustrates that the sum of the immediate ($h=0$) gIRFs is statistically different from the immediate ($h=0$) jIRF for all American banks at the 99% confidence level. This statistically significant overestimation by the gIRF sum persists for $h=1,2,\ldots ,8$ for most of the American banks at the 99% confidence level.²⁰ This empirical result agrees with what was found in the simulation study of Sect. 4, which computed the jIRF and sum of the gIRFs for various known data generating processes. In that simulation study, we similarly observed that when all the shock correlations are positive (as is the case for financial institution volatility) the sum of the gIRFs is greater than the jIRF for $h=0$, and this tends to persist for $h=1,2,3\ldots$.

Table 2 summarizes the jFEVD and gFEVD results. As with the jIRF and gIRF plots in Fig. 3, the goal of Table 2 is to measure volatility transmissions from European financial institutions to American financial institutions before and during the pandemic. The left panel of the table shows the decomposition of the forecast errors for the pre-pandemic period and the right panel shows the decomposition for the pandemic period. In both periods, a forecast horizon of $H=30$ days was used to capture any effects of the higher persistence during the pandemic period. Consider, for example, the Bank of America (BAC) row of Table 2. During the pre-pandemic period, the gFEVD method attributes $7.2\%$ of Bank of America’s forecast error variance to Barclays (BCS), $11.8\%$ to Credit Suisse (CS), and $6.0\%$ to Deutsche Bank (DB), for a total of $25.0\%$. During the pandemic period, the gFEVD shares for Bank of America from these three European banks sum to $152.5\%$. The fact that these sums can exceed $100\%$ is why Diebold and Yilmaz (2012, 2014, 2016) chose to normalize the gFEVD by the row sums when constructing their generalized spillover tables. However, as pointed out by Caloia et al. (2019) and Lastrapes and Wiesen (2021), this row-sum normalization can cause issues of its own. As discussed earlier, the reason that these sums can exceed $100\%$ is that the explanatory power of the European banks is being double counted because the cross-correlations among Barclays, Credit Suisse, and Deutsche Bank are not accounted for by the gFEVD. Thus, summing the elements of the gFEVD overestimates the total volatility transmissions from European to American financial institutions. The jFEVD, by contrast, attributes $16.0\%$ of Bank of America’s forecast error variance to joint shocks from Barclays, Credit Suisse, and Deutsche Bank during the pre-pandemic period and $63.9\%$ during the pandemic period. Focusing only on the two jFEVD columns, we see that volatility transmission increased in the pandemic period relative to the pre-pandemic period by a factor ranging from 3.91 for Citigroup to 12.36 for Bank of New York Mellon with an average increase factor of 6.54 across all eight American financial institutions.

Table 2 Joint and generalized forecast error variance decompositions corresponding to volatility shocks from European financial institutions to American financial institutions

Although the sum of the gFEVD elements produce overstated magnitudes, the pandemic period gFEVD percentages are on average ten times as large as the pre-pandemic period.²¹ In contrast, the jFEVD percentages in the pandemic period are on average 6.5 times larger than the jFEVD percentages in the pre-pandemic period. This qualitatively agrees with an earlier result using the impulse response functions. Namely, both the jFEVD and sum of gFEVD elements demonstrate an increase in total volatility spillover from Europe to the USA. However, by accounting for the cross-correlations among the European shocks, the jFEVD shows a more modest increase in total volatility spillovers during the pandemic. The increase in trans-Atlantic volatility transmissions during the pandemic compared to the pre-pandemic period is overestimated by the sum of the gFEVD elements.

Lastly, we check how robust the impulse responses are to the sample of European banks included in the set of joint shocks. Some changes in the impulse responses should be anticipated when the sources of the shocks are altered, but the degree to which the impulse responses change will depend on how the total effects of the joint shocks are measured. To investigate this, we added UBS Group to the sample of banks, which, as mentioned in footnote 15, was previously a member of the Federal Reserve’s LISCC portfolio.²² When the shock correlations are all positive and relatively strong, as is the case with equity market volatility, adding a new variable to the set of joint shocks will have a relatively minor effect on the jIRF, but it will have a substantial effect on the sum of the gIRFs. Adding UBS Group to the sample which already includes another Swiss bank (Credit Suisse) will result in nontrivial information overlap and cross-correlation. This cross-correlation will be correctly accounted for by the jIRF, but it will cause a double counting problem with the sum of the gIRFs. For example, when UBS Group was added, the pre-pandemic jIRF showing how Bank of America’s volatility responds due to joint, simultaneous shocks from Barclays, Credit Suisse, Deutsche Bank, and UBS Group increased by a factor of 1.08 (an $8 \%$ increase) compared to the jIRF in the top left panel in Fig. 3 which did not include UBS Group. However, the sum of the four individual gIRFs increased by a factor of 1.39 (a $39 \%$ increase) compared to the gIRF sum in the top left panel in Fig. 3.²³

7 Comparing the jIRF to single variable alternatives

In Sect. 6, we employ the jIRF to measure how the equity market volatility of an American bank responds due to joint, simultaneous volatility shocks from three European banks. The goal of the jIRF in that application is to quantify the total effect on an American bank due to multiple shocks from European banks. However, it is reasonable to ask: Can the same results of the jIRF be obtained by a simpler single variable alternative? In other words, what if we summarize the multiple European banks’ volatility into a single variable, estimate a VAR containing this European summary variable, and then compute the gIRF measuring the response of an American bank due to a shock from the European summary variable? Will this gIRF be identical to the jIRFs computed in Sect. 6? This section is meant to address this question. In short, the answer to the question is “no” because a summary variable underestimates the total variance and is thus unable to capture the full effect of the multiple shocks.

We consider several ways of summarizing the daily volatilities of the three European banks—Barclays, Credit Suisse, and Deutsche Bank—into a single variable which can be entered into the VAR model, thus replacing the three separate European variables. First, we consider principal component analysis. After initially centering and standardizing the European banks’ volatility data so the means are all equal to zero and the standard deviations are all equal to one, we perform principal component analysis and extract the first European principal component, which explains the largest variance share of the components, in our case about $84 \%$. We then estimate a VAR model for the pre-pandemic period (January 3, 2017 to January 31, 2020) and a VAR model for the pandemic period (February 3, 2020 to February 4, 2021) using the first European principal component and the volatilities of the eight American banks. These VARs exclude the three individual European banks’ volatilities. This analysis is conceptually similar to the factor augmented vector autoregressions (FAVAR) of Bernanke et al. (2005) and the large class of dynamic factor models such as those summarized in Stock and Watson (2011). As mentioned earlier in this paper, the jIRF has a similar motivation to these models in that it can identify the impact of some common latent factor across the European banks. However, as we will show, the jIRF does this without losing any of the total variance of the multiple European banks. Thus, the jIRF is a more accurate measure of the total impact due to the several, simultaneous shocks.

The second way we summarize the daily volatilities of the three European banks into a single variable is through a market capitalization weighted index of the volatilities. In particular, we obtain the market capitalization for each day and for each of the three European banks and then construct a market capitalization weighted average of the three banks’ volatilities to form this European index.²⁴

In Fig. 5, we show the impulse responses of various American financial institutions due to volatility shocks from Europe. The left column of panels is for the pre-pandemic period, and the right column of panels is for the pandemic period. The first, second, and third rows of panels show the responses of Bank of America (BAC), Citigroup (C), and JP Morgan Chase (JPM), respectively.²⁵ In each panel of Fig. 5, the orange line with upside down triangle markers is the gIRF measuring how the American bank’s volatility responds due to a shock from the first principal component of European volatilities. To be clear, this gIRF was obtained from a VAR that included the eight American banks’ volatilities and the first European principal component, while excluding the three separate volatilities for the European banks. In each panel of Fig. 5, the pink line with right side up triangle markers is the gIRF measuring how the American bank’s volatility responds due to a shock from the European index of market capitalization weighted volatilities. To be clear, this gIRF was obtained from a VAR that included the eight American banks’ volatilities and the market capitalization weighted index of the European bank volatilities, while excluding the three separate volatilities for the European banks. For comparison, the black lines with circle markers in Fig. 5 are the jIRFs measuring how the American bank’s volatility responds due to joint, simultaneous volatility shocks from the three European banks. These jIRFs are identical to the jIRFs shown in Fig. 3, which were obtained from a VAR that included the eight American banks and three European banks. Notice the vertical scale in the three pre-pandemic period panels is different from the scale in the three pandemic period panels.

Fig. 5

Impulse response functions for the equity market volatility of Bank of America (first row), Citigroup (second row), and JP Morgan Chase (third row). The orange line with upside down triangle markers is the gIRF showing how an American bank responds due to a shock from the first European principal component. The pink line with right side up triangle markers is the gIRF showing how an American bank responds due to a shock from the European index of weighted volatilities. And the black line with circle markers is the jIRF showing how an American bank responds due to joint, simultaneous shocks from the three European banks

There are two important features in the plots of Fig. 5. First, the gIRF due to a shock from the first European principal component and the gIRF due to a shock from the European index of weighted volatilities are strikingly similar. In fact, for the pandemic period, those two gIRFs are practically indistinguishable in the graphs.²⁶ In this application, it does not appear to matter how you choose to summarize the three European banks into a single variable; the results are very similar.

Second, and more importantly, the gIRFs in Fig. 5 are consistently smaller than the jIRFs. We assert that there is an intuitive explanation behind the aggregated single variable gIRF underestimation: the European summary variables used to estimate the gIRFs are averages, and any average loses some of the total variance in the original variables. The first European principal component is a linear combination of the original European banks’ volatilities, and it only explains a fraction of the total variance in the three European banks. The European index is directly a weighted average where the weights are based upon the time-varying market capitalization. Both European summary variables capture only a proportion of the total variance in the three European banks. This intuitively explains why the gIRFs in Fig. 5 underestimate the effect of the shocks. In contrast, the jIRF allows us to measure the total impact of simultaneous shocks from the three European banks—there is no loss of variance with the jIRF. Critically, the jIRF produces more appropriately sized responses compared to the gIRFs of single variable alternatives.

Furthermore, there is another benefit to utilizing the jIRF over the gIRF and single variable alternatives of some summary variable. In some ways, the jIRF is a more flexible econometric tool than employing the gIRF with single variable alternatives. In our analysis, we chose to set the three volatility shocks from Europe to be their expected size (in absolute value). Namely, we set the three joint, simultaneous shocks from Europe to be one standard deviation in size. In Eq. (12), we chose to set $\varvec{\delta }_{{\mathbb {J}}}=\varvec{s}_{{\mathbb {J}}}=[\sqrt{\sigma _{jj}} \;\;\; \sqrt{\sigma _{kk}} \;\;\; \sqrt{\sigma _{\ell \ell }}]'$ assuming the variables in set ${\mathbb {J}}$ were ${\mathbb {J}}= \left\{ j, k, \ell \right\}$. However, when utilizing the jIRF, researchers have the ability to select other appropriately sized shocks, which can be different for each of the variables in set ${\mathbb {J}}$; the shocks do not necessarily all have to equal the standard deviation. The shock sizes can be different in both magnitude and sign, which can be particularly important in policy analysis where several shocks occur simultaneously but have different signs. Critically, this flexibility to choose the size and sign of the multiple shocks is not afforded to researchers who choose to summarize the multiple shocks into a single summary variable.

8 Concluding remarks

Dynamic reduced-form VAR models are often used by analysts to explore the transmission of shocks from one variable to another. However, the existing generalized IRF approach only allows for shocks from one variable at a time, which may not be appropriate when events affect several variables simultaneously. In such cases when there is correlation among the several shocks, it is not appropriate to simply sum the gIRFs due to separate shocks because that introduces double counting or undercounting. To deal with these types of issues, we introduced a joint impulse response function (jIRF) that allows for simultaneous shocks from a set of multiple variables in a VAR rather than a shock from one variable at a time. Like the generalized IRF, our joint IRF is unique, independent of the ordering of the variables, and is agnostic to structure while remaining consistent with the data. In this sense, both the gIRF and our jIRF can be thought of as “reduced-form” tools. The jIRF is interpreted as the response of $\varvec{Y}$ at h periods ahead due to joint, simultaneous one standard deviation shocks from a subset of the variables while controlling for the cross-correlations among the shocks. Critically, the jIRF cannot be constructed from a sequence of oIRFs or gIRFs except in the very special case when all the variables in the subset have mutually uncorrelated shocks.

To illustrate the jIRF, we examined the transmission of volatility shocks of equity market returns from systemically important financial institutions in Europe to their counterparts in the US during the COVID-19 pandemic. The pandemic substantially emerged in Europe before it emerged substantially in the US, and most of Europe responded to the crisis quickly and nearly simultaneously. In such a scenario, as our empirical results confirm, it is inappropriate to compile a collection of shocks from the European financial institutions and simply sum across those to get the total impact on an American financial institution. The cross-correlations of the European financial institutions’ shocks should be accounted for, which is precisely what the jIRF does. Consistent with a common latent factor in Europe, our jIRF and jFEVD allow us to measure the total volatility transmission from all three European banks to an American bank. We find that volatility transmission from Europe to the USA substantially increased during the pandemic and became more persistent. However, how much the impulse response increased is a matter of choice of metric. Simply summing the gIRFs due to individual shocks from the European banks leads to an overestimation of increased trans-Atlantic volatility transmissions. In contrast, properly accounting for the cross-correlations of the European financial institutions, as the jIRF does, leads to a more modest sized increase in trans-Atlantic volatility transmissions.

Additional applications for the jIRF are numerous. For example, many US states have incentive programs to attract firms from specific industries to their states. These incentives, typically in the form of tax rebates, are often applied across several key industry groups simultaneously. Since the growth of employment in many of these industries is correlated, judging the impact of such incentive programs would require the use of the jIRF approach rather than evaluating the program industry-by-industry. In addition, resources for these incentive programs are generally quite limited. By examining the jIRFs of many subsets of qualified industry groups, state development agencies would be able to maximize their return on investment by identifying an optimal subset of targeted industries that produce the maximum employment growth spillover into other industries. In a related example, the jIRF could be used to determine how shocks from the returns or volatilities of a group of industry sectors who often experience common shocks—such as energy and utilities—affect other industry sectors. Similarly, one may want to examine the effects of currency shocks from a region of countries—such as East Asia—on European or North American countries. To properly investigate any of these questions, a researcher would likely need to account for the cross-correlations of the targeted industries, equity market sectors, or set of countries in a region, as the jIRF does. Indeed, any situation where a researcher wishes to measure how a variable responds due to multiple simultaneous shocks is a potential application of the jIRF.