## 1 Introduction

^{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.

^{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.

## 2 The VAR model and impulse response functions

^{3}of the shocks \(\varvec{\epsilon }_{t}\), we have

^{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.

## 3 The joint impulse response function

^{5}For example, if \(K=4\) and \({\mathbb {J}}=\{1,3\}\), then

^{6}

## 4 Properties of the jIRF

^{7}

## 5 The joint forecast error variance decomposition

^{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

^{9}

^{10}Therefore, the jFEVD technique complements the work of Greenwood-Nimmo et al. (2021).

## 6 jIRF application: trans-Atlantic volatility transmissions

^{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.

^{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 approach

^{14}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.

^{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}

Financial institution | Ticker symbol | Location | Full sample | Pre-pandemic | Pandemic | |||
---|---|---|---|---|---|---|---|---|

1/3/2017–2/4/2021 | 1/3/2017–1/31/2020 | 2/3/2020–2/4/2021 | ||||||

Mean | Variance | Mean | Variance | Mean | Variance | |||

of volatility | of volatility | of volatility | of volatility | of volatility | of volatility | |||

Bank of America | BAC | USA | 28.0 | 576.1 | 22.2 | 99.4 | 45.7 | 1615.7 |

Bank of New York Mellon | BK | USA | 25.0 | 444.1 | 19.7 | 92.6 | 41.3 | 1167.9 |

Barclays | BCS | Europe | 29.8 | 773.6 | 22.4 | 164.5 | 52.2 | 1966.9 |

Citigroup | C | USA | 29.3 | 741.2 | 22.2 | 86.7 | 51.0 | 2113.2 |

Credit Suisse | CS | Europe | 26.1 | 539.5 | 21.4 | 127.8 | 40.2 | 1532.4 |

Deutsche Bank | DB | Europe | 33.2 | 643.3 | 28.2 | 235.0 | 48.7 | 1572.8 |

Goldman Sachs | GS | USA | 25.9 | 412.2 | 21.3 | 75.3 | 40.1 | 1174.7 |

JP Morgan Chase | JPM | USA | 24.5 | 513.8 | 18.8 | 67.2 | 42.0 | 1470.0 |

Morgan Stanley | MS | USA | 29.1 | 564.8 | 24.0 | 120.5 | 44.4 | 1607.5 |

State Street | STT | USA | 29.4 | 604.6 | 23.2 | 147.0 | 48.2 | 1533.4 |

Wells Fargo | WFC | USA | 27.1 | 583.4 | 20.0 | 79.9 | 49.0 | 1484.6 |

^{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.

^{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.

^{19}

^{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 \(h=1,2,3\ldots \).

Pre-pandemic | Pandemic | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1/3/2017–1/31/2020 | 2/3/2020–2/4/2021 | |||||||||

gFEVD: % explained by BCS | gFEVD: % explained by CS | gFEVD: % explained by DB | Sum of gFEVD elements: BCS, CS, & DB | jFEVD: % jointly explained by BCS, CS, & DB | gFEVD: % explained by BCS | gFEVD: % explained by CS | gFEVD: % explained by DB | Sum of gFEVD elements: BCS, CS, & DB | jFEVD: % jointly explained by BCS, CS, & DB | |

Bank of America | 7.2 | 11.8 | 6.0 | 25.0 | 16.0 | 45.0 | 59.5 | 48.1 | 152.5 | 63.9 |

Bank of New York Mellon | 2.6 | 3.6 | 1.7 | 7.9 | 5.2 | 44.6 | 61.6 | 46.7 | 153.0 | 64.6 |

Citigroup | 8.3 | 11.8 | 5.3 | 25.4 | 16.5 | 44.6 | 61.5 | 46.9 | 152.9 | 64.4 |

Goldman Sachs | 8.8 | 8.5 | 5.0 | 22.3 | 14.3 | 43.6 | 62.0 | 49.3 | 154.9 | 65.1 |

JP Morgan Chase | 9.3 | 11.2 | 5.9 | 26.5 | 16.9 | 47.3 | 62.5 | 47.9 | 157.7 | 66.3 |

Morgan Stanley | 5.9 | 9.1 | 5.8 | 20.9 | 13.0 | 43.4 | 63.3 | 47.7 | 154.4 | 65.4 |

State Street | 2.8 | 3.3 | 3.4 | 9.5 | 5.8 | 47.6 | 64.5 | 48.7 | 160.8 | 67.6 |

Wells Fargo | 4.4 | 6.0 | 3.6 | 14.0 | 8.7 | 44.9 | 55.2 | 43.8 | 143.9 | 60.0 |

^{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.

^{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

^{24}

^{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.

^{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.