Overview

Global shocks can affect almost immediately the financial markets all over the world, making them move at the same time. The global COVOL model (Engle and CamposMartins, 2023) provides a very broad measure of risk. It includes risks from geopolitical (see also Caldara and Iacoviello (2022) or Baker, Bloom, and Davis (2016)) to climate driven (Campos-Martins and Hendry, 2023). Global COVOL can be defined as the exposure to common shocks to the volatility of a wide range of financial assets. Such events are assumed to affect all countries, all asset classes, and all sectors. These shocks can originate from natural disasters to political or terrorist actions. The key feature of these shocks is that they move financial prices globally at the same time.

It is well known that volatilities of asset returns co-move. It is natural to observe common variation when assets are all exposed to the same factors. If financial returns are linear combinations of common factors, time-varying factors will imply a volatility factor structure. However, even when these factors are taken out of the data, the idiosyncratic returns still have correlated volatilities (Herskovic et al., 2016). Since volatility is also well known to be predictable, the co-movements of volatilities are most likely caused by correlation between the shocks to volatility. The fundamental observation underlying the global COVOL model is that even though the volatility of standardized residuals are orthogonal in both times series and cross section with unit variances, their squares can be correlated. This observation in time series was the key motivation for the original ARCH model of Engle (1982) and is now the key motivation for the global COVOL model of Engle and Campos-Martins (2023) in cross section. Results extend the literature on idiosyncratic volatility including Connor et al. (2006) and Ang et al. (2006).

Modeling multiplicative volatility factors using numerical methods is easy to implement and to replicate. Another appealing feature of the multiplicative decomposition proposed is that it implies a one-factor structure of the squared innovation covariance matrix to which factor (or principal component analysis) can be applied.

Despite volatility shocks that affect all portfolios, some assets are more sensitive to the volatility shocks than others. Hence, there is a role for risk diversification. The global COVOL model allows for a new criterion for portfolio optimality which complements mean-variance efficiency by reducing the exposure to geopolitical risk.

The Global COVOL Model

To measure global common financial risk, we use market prices which are assumed to incorporate all available information. In what follows, we show how the global COVOL model provides a novel explanation for why idiosyncratic volatilities co-move. It also introduces a new way to formulate multiplicative factors for volatility rather than the more traditional additive decomposition.

An extended multivariate GARCH

Multivariate volatility models are typically formulated in terms of the conditional mean and covariance matrix of a vector of random variables, here denoted by $r_t$, and the jointly standardized residuals $e_t$. If the conditional mean and covariance matrix are correctly specified, then$$\begin{array}{}\text{(1)}& {𝔼}_{t-1}({𝒆}_t{𝒆}_t^{\prime })=𝕀\end{array}$$

These residuals are therefore assumed to be uncorrelated but not independent. In this paper we focus on the covariance matrix of the square of these residuals. Letting $e_t^2$ be the vector of the squares of the jointly standardized residuals and $\iota$ a vector of ones, we consider$$\begin{array}{}\text{(2)}& {𝔼}_{t-1}[({𝒆}_t^2-𝜾)({𝒆}_t^2-𝜾{)}^{\prime }]\equiv {\mathrm{\Psi }}_t\end{array}$$If the $e$’s are independent then $\Psi$ will be diagonal. If they are Gaussian, then $\mathrm{\Psi }=2𝕀$.

This extension of the multivariate GARCH specification is important in order to explain why volatilities of many assets peak at the same time. If $\mathrm{\Psi }$ has off diagonal elements, then multiple assets will have high volatility shocks at the same time. Without this contemporaneous response to shocks, the model must find that the volatility of one asset will rise first and this propagates to other assets.

Pricing factors and correlated squared idiosyncrasies

For financial applications, the standard asset pricing model can be formulated for a vector of returns ${𝒓}_t\equiv (r_{1t},\dots ,r_{Nt}{)}^{\prime }$ as$$\begin{array}{lrlr}\text{(3)}& & {𝒇}_t& ={𝒘}_{t-1}^{\prime }{𝒓}_t& & {𝒓}_t& =r^f+𝜷{𝒇}_t+𝖽𝗂𝖺𝗀\left\{\sqrt{{𝒉}_t^i}\right\}{𝒆}_t^i,& \end{array}$$where $𝜷$ is a matrix of risk exposures, ${𝒇}_t$ is a vector of factors, ${𝒆}_t^i\equiv (e_{1t}^i,\dots ,e_{Nt}^i{)}^{\prime }$ is the vector of residuals from factors, and ${𝒉}_t^i\equiv (h_{1t}^i,\dots ,h_{Nt}^i{)}^{\prime }$ contains the conditional variances.

If the factor model above is correctly specified and factors fully explain the cross sectional correlation, then ${𝒆}_t^i$ contains idiosyncratic returns and ${𝒉}_t^i$ the idiosyncratic conditional variances. The standard assumptions on ${𝒆}_t^i$ state that the standardized residuals are uncorrelated in both time series and cross section with unit variances. Hence, if factors are sufficient to reduce the contemporaneous correlations to zero,$$\begin{array}{}\text{(4)}& & {𝔼}_{t-1}\left({𝒆}_t^i{𝒆}_t^{i\prime }\right)=𝕀& \end{array}$$The factors are linear combinations of returns and will themselves have conditional volatilities and residuals. They satisfy:$$\begin{array}{}\text{(5)}& & {𝔼}_{t-1}({𝒇}_t)\equiv {𝝁}^f,{𝔼}_{t-1}[({𝒇}_t-{𝝁}^f)({𝒇}_t-{𝝁}^f{)}^{\prime }]\equiv H_t^f,{𝒆}_t^f=(H_t^f{)}^{1⁄2}({𝒇}_t-{𝝁}^f)& \end{array}$$Assuming that the factors can be rotated to be orthogonal then $H^f$ will be diagonal and the residuals will be uncorrelated. Furthermore, the factor residuals will be uncorrelated with the idiosyncratic residuals. Hence$$\begin{array}{}\text{(6)}& & {𝔼}_{t-1}\left({𝒆}_t^f{𝒆}_t^{f\prime }\right)=𝕀\text{ and }{𝔼}_{t-1}\left({𝒆}_t^f{𝒆}_t^{i\prime }\right)=0& \end{array}$$Therefore letting ${𝒆}_t=({𝒆}_t^{i\prime },{𝒆}_t^{f\prime }{)}^{\prime }$, we get equations $\text{(1)}$ and $\text{(2)}$ above. As is well known in the finance literature, if there are too few factors in the model, then the idiosyncrasies will be correlated.

The statistical specification

There is very strong evidence that the squared jointly standardized residuals of returns are positively correlated. This observation in the time series was the motivation for the original ARCH model of Engle (1982) and the same observation in the cross section is the motivation for global COVOL of Engle and Campos-Martins (2023) as a measure of global common risk. Global COVOL will be high when squared standardized residuals are high for a wide range of assets. Thus it is a measure of the magnitude of shocks to volatility that are common to a collection of assets.

To estimate global COVOL we must introduce parametric assumptions on the form of this relation. Let global COVOL be represented by $\sqrt{x}$, where $x$ is a vector of latent variables, and let $𝒔$ be a vector of parameters interpreted as factor loadings satisfying the assumptions:$$\begin{array}{rl}{𝔼}_{t-1}(x_t)& =1,{𝔼}_{t-1}(x_t-1{)}^2=v_t,x_t>0,t=1,\dots ,T,\\ {𝜺}_t& \sim \text{IIN}(\mathrm{0,1}),\\ s_i& \in [\mathrm{0,1}],i=1,\dots ,N.\end{array}$$We specify a function $g(s_i,x_t)$ that is a data generating process for the random variables, $e_{it}$ from $x_t,s_i$ and ${\epsilon }_{it}$ and assume$$\begin{array}{}\text{(7)}& & e_{it}=\sqrt{g(s_i,x_t)}{\epsilon }_{it},g(s_i,x_t)\equiv s_i{x}_t+1-s_i,& \end{array}$$but other specifications are certainly possible. Specification above implies that $g(s_i,x_t)$ is non-negative with expected value $1$ and therefore satisfies $\text{(4)}$. Given ${𝔼}_{t-1}[(e_{i,t}^2-1)(e_{j,t}^2-1)]\equiv {\mathrm{\Psi }}_{ij,t}=s_i{s}_j{v}_t$ and ${𝔼}_{t-1}[(e_{i,t}^2-1{)}^2]\equiv {\mathrm{\Psi }}_{ii,t}=3s_i^2{v}_t+2$, the sample covariance matrix can be constructed by averaging over $t$, which gives$$\begin{array}{}\text{(8)}& & \mathrm{\Psi }=(1⁄T)𝔼\left[\sum _{t=1}^T{𝒆}_t^2{𝒆}_t^{2\prime }\right]=(1⁄T)\sum _{t=1}^T{\mathrm{\Psi }}_t=𝒔{𝒔}^{\prime }\bar{v}+D,& \end{array}$$where $D=𝖽𝗂𝖺𝗀\{3s_i^2\bar{v}+2\}$, $\bar{v}=(1⁄T)\sum _{t=1}^T{v}_t$. It is clear that $\mathrm{\Psi }$ is a factor matrix with $x$ as the factor and $𝒔$ as the vector of factor loadings. Thus principal component analysis of the empirical version of $\mathrm{\Psi }$ will give preliminary estimates of both $𝒔$ and $x$.

Estimation

The log-likelihood function can be written for this model as follows.$$\begin{array}{}\text{(9)}& & L(𝒔,x;𝒆)=-(1⁄2)\sum _{i=1,t=1}^{N,T}\{\log [g(s_i,x_t)]+{\displaystyle \frac{e_{it}^2}{g(s_i,x_t)}}\}.& \end{array}$$The iteration solves the first order conditions$$\begin{array}{}\text{(10)}& & {\displaystyle \frac{\partial L(𝒔,x;𝒆)}{\partial s_i}}=0,{\displaystyle \frac{\partial L(𝒔,x;𝒆)}{\partial x_t}}=0& \end{array}$$sequentially until parameters are found that solve both jointly.

In practice, we estimate each set of unknowns conditional on the other by maximum likelihood. The first order conditions above give the two heteroscedasticity relationships:$$\begin{array}{lrlr}\text{(11)}& & \text{Cross section: }{e}_{it}& ={\epsilon }_{it}\sqrt{{\hat{s}}_i(x_t-1)+1}\text{ for }t=1,\dots ,T,& & \text{Time series: }{e}_{it}& ={\epsilon }_{it}\sqrt{s_i({\hat{x}}_t-1)+1}\text{ for }i=1,\dots ,N.& \end{array}$$The estimation algorithm can be interpreted as a $\text{𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛-𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛}$ procedure; See Hastie, Tibshirani, and Friedman (2009) for more details. The initial estimates are given by principal components of the empirical $\mathrm{\Psi }$ matrix; See Trzcinka (1986), Connor and Korajczyk (1993) or Jones (2001) for one-factor models of return variances. Convergence occurs typically after something like 15 to 30 iterations. In each iteration, $s_i,i=1,\dots ,N$, and $x_t,t=1,\dots ,T$, are constrained to be, respectively, in the interval $[\mathrm{0,1}]$ and positive. Scaling is also imposed to guarantee ${\bar{x}}_t=(1⁄T)\sum _{t=1}^T{x}_t=1$ and ${𝒔}^{\prime }𝒔=1$. The global COVOL model can be estimated with the R package $\text{𝑔𝑒𝑜𝑣𝑜𝑙}$ (Campos-Martins, 2021).

Testing

Empirical evidence for global COVOL is easy to find using the sample covariance matrix of ${𝒆}_t^2$. The null hypothesis of no correlation in ${𝒆}_t$ is given by $\text{(4)}$ and should be satisfied by the choice of factors. Similarly, the null hypothesis for ${𝒆}_t^2$ is simply $\text{(8)}$ with $\bar{v}=0$, which implies$$\begin{array}{}\text{(12)}& & {ℍ}_0:\mathrm{\Psi }=2𝕀& \end{array}$$under normality.

When the equal factor loading model is the alternative, all pairs of assets are affected by the same shock, and they will have the same correlation under the alternative. This is an equicorrelated panel and testing the null that the equicorrelation, denoted by ${\rho }_{{𝒆}^2}$, is zero against the alternative that it is greater than zero is asymptotically optimal for detecting global COVOL. Under this alternative, the global volatility factor varies over time inducing comovements and positive correlations between the squared standardized residuals as observed. Hence, ${ℍ}_1:\bar{v}>0⟹{\rho }_{{𝒆}^2}>0$.

For $N(N-1)⁄2$ correlations, the test statistic$$\begin{array}{}\text{(13)}& & \xi ={\displaystyle \frac{\sqrt{{\displaystyle \frac{NT}{(N-1)⁄2}}}{\sum }_{i>j,j=1}^N{\sum }_{t=1}^T(e_{it}^2-1)(e_{jt}^2-1)}{{\sum }_{i=1}^N{\sum }_{t=1}^T(e_{it}^2-1{)}^2}}\stackrel{d}{\to }N(\mathrm{0,1})\text{ under }{ℍ}_0.& \end{array}$$

The properties of the test in finite samples have been studied by means of Monte Carlo simulations; For more details, we refer to Engle and Campos-Martins (2023). The test for global COVOL can be run with the R package $\text{𝑔𝑒𝑜𝑣𝑜𝑙}$ (Campos-Martins, 2021).