Contagion and Global Financial Crises: Lessons from Nine Crisis Episodes

Abstract

Episodes of extraordinary turbulence in global financial markets are examined during nine crises ranging from the Asian crisis in 1997–98 to the recent European debt crisis of 2010–13. After dating each crisis using a regime switching model, the analysis focuses on changes in the dependence structures of equity markets through correlation, coskewness and covolatility to address a range of hypotheses regarding contagion transmission. The results show that the great recession is a true global financial crisis. Finance linkages are more likely to result in crisis transmission than trade and emerging market crises transmit unexpectedly, particularly to developed markets.

Appendices

Appendix A: Estimation of the Regime Switching Model

The regime switching model is used to estimate the crisis dates using knowledge of trigger events. Prior distributions are combined with the likelihood function to obtain the joint posterior distribution via the Bayes rule in a Bayesian estimation framework. The Gibbs sampler is then used to obtain draws from the joint posterior distribution required for the analysis.

The (complete-data) likelihood function of the model in Eqs. 1 to 2 is

$$f(y|\Theta ,s)=(2\pi)^{-\frac{T}{2}}\prod^{T}_{t=1}\left\vert \sigma_{s_{t}}^{2}\right\vert^{-\frac{1}{2}}\exp\left\{-\frac{1}{2}\sum\limits_{t=1}^{T}\left[y_{t}-\mu_{s_{t}}\right]^{\prime} \left(\sigma_{s_{t}}^{2}\right)^{-1}\left[y_{t}-\mu_{s_{t}}\right]\right\},$$

(14)

where $\Theta =\left ( \mu _{0},\mu _{1},\sigma _{0}^{2},\sigma _{1}^{2}\right )$ and s _t ∈ {0, 1}. Here, y = (y ₁, … , y _T )′ and s = (s ₁, … , s _T )′.

The prior for the model parameters are specified as

$$\mu_{s_{t}} \sim N\left( \underline{\mu },\underline{V}_{\mu }\right),$$

(15)

$$\sigma_{s_{t}}^{2} \sim IW\left( \underline{\tau }_{\sigma^{2}},\underline{S}_{\sigma^{2}}\right),$$

(16)

$$\Pr \left( s_{t}=1\right) = p_{t},\Pr \left( s_{t}=0\right) =1-p_{t},$$

(17)

where $IW\left ( \underline {\tau }_{\sigma ^{2}},\underline {S}_{\sigma ^{2}}\right )$ denotes the inverse-Wishart distribution with degree of freedom $\underline {\tau }_{\sigma ^{2}}$ and scale matrix $\underline {S}_{\sigma ^{2}}$. The prior mean and variance for $\mu _{s_{t}}$ are set to $\underline {\mu }$ and $\underline {V}_{\mu }=\phi _{\mu }$.

The joint posterior distribution is calculated by multiplying the prior distributions with the likelihood function via Bayes rule. Then, posterior draws from the joint posterior distribution are obtained via the Gibbs sampler:

• Step1: Specify starting values for $\Theta ^{(0)}=\left (\mu _{l}^{(0)},\left (\sigma _{l}^{2}\right )^{(0)}\right )$ with l = 0, 1. Set counter loop = 1, … , n.

• Steps2: Generate s ^(loop)from π (s| y, Θ^{(loop − 1))}.

• Step3: Generate $\mu _{l}^{(loop)}$ from $\pi \left (\mu _{l}|y,\left (\sigma _{l}^{2}\right )^{(loop-1)},s^{\left (loop\right )}\right )$.

• Step4: Generate $\left (\sigma _{l}^{2}\right )^{(loop)}$ from $\pi \left (\sigma _{l}^{2}|y,\mu _{l}^{\left (loop-1\right )},s^{(loop)}\right )$.

• Step5: Set loop = loop + 1 and go to Step 2.

The number of iterations set for Steps 2 to 4 is n. The first n ₀ of these are discarded as burn-in draws, and the remaining n ₁draws are retained to compute the parameter estimates. The full conditional distributions are given below and their derivations are available on request.

The posterior distribution for μ _l , l = 0, 1, conditional on y, $\sigma _{0}^{2}$, $\sigma _{1}^{2}$ and s is an univariate normal distribution given by

$$\left( \mu_{l}|y,\sigma_{l}^{2},s\right) \sim N\left( \widehat{\mu}_{l},D_{\mu_{l}}\right) ,l=0,1,$$

where $D_{\mu _{l}}=\left ( \underline {V}_{\mu }^{-1}+\left ( \sigma _{l}^{2}\right )^{-1}\right )^{-1}$ and $\widehat {\mu }_{l}\,=\,D_{\mu _{l}}\left [\underline {V}_{\mu }^{-1}\underline {\mu }\,+\,\sum \limits ^{T}_{t=1}1\left (s_{t}\,=\,l\right )\left (\sigma _{s_{t}}^{2}\right )^{-1}y_{t}\right ]$.

The posterior distribution for $\sigma _{l}^{2}$, l = 0, 1, conditional on y, μ ₀, μ ₁ and s is an inverse-Wishart distribution

$$\left( \sigma_{l}^{2}|y,\mu_{l},s\right) \sim IW\left( \tau_{\sigma_{l}^{2}},S_{\sigma_{l}^{2}}\right),$$

where $\tau _{\sigma _{l}^{2}}\,=\,\underline {\tau }_{\sigma ^{2}}\,+\,\sum \limits ^{T}_{t=1}1\left (s_{t}\,=\,l\right )$ and $S_{\sigma _{l}^{2}}=\underline {S}_{\sigma ^{2}}+\sum \limits ^{T}_{t=1}1\left (s_{t}=l\right ) \left ( y_{t}\,-\,\mu _{s_{t}}\right ) \left ( y_{t}\,-\,\mu _{s_{t}}\right )^{\prime }$.

To generate regime variable s _t , the multi-move Gibbs sampling method is used. Since the regime variable s _t evolves independently of its own past values, the regimes s ₁, … , s _T are conditionally independent of each other given the data and other parameters:

$$\pi \left( s|y,\Theta \right) =\prod\limits_{t=1}^{T}\pi\left( s_{t}|y,\Theta \right),$$

where the success probability can be calculated as

$$\Pr \left( s_{t}=1|y,\Theta \right) =\frac{\pi \left( s_{t}=1|y,\Theta\right)}{\pi\left(s_{t}=0|y,\Theta \right) +\pi \left( s_{t}=1|y,\Theta\right)}.$$

Once the above probability is calculated, a random number from a uniform distribution between 0 and 1 is generated to compare with the calculated value of Prs _t = 1|y, Θ. If the probability Prs _t = 1|y, Θ is greater than the generated number, the regime variable s _t = 1; otherwise, s _t = 0.

Appendix B: Derivation of the Covolatility Test Statistic for Contagion

This appendix summarizes the key points in the derivation of the covolatility statistic for contagion. Further details are presented in Hsiao (2012). Consider the following generalized exponential distribution which has as its base the bivariate normal distribution, but with the addition of fourth order comoments labelled covolatility:

$$\begin{array}{@{}rcl@{}} f(r_{1,t},r_{2,t}) &\,=\,&\exp \left[ -\frac{1}{2}\left( \frac{1}{1\,-\,\rho^{2}}\right) \left( \left( \frac{r_{1,t}\,-\,\mu_{1}}{\sigma_{1}}\right)^{2} \,+\, \left( \frac{r_{2,t}\,-\,\mu_{2}}{\sigma_{2}}\right)^{2} \right. \right.\notag\\ &&\left.\left.-2\rho\left(\frac{r_{1,t}\,-\,\mu_{1}}{\sigma_{1}}\right) \left(\frac{r_{2,t}\,-\,\mu_{2}}{\sigma_{2}}\right)\right)\,+\,\theta\left(\frac{r_{1,t}\,-\,\mu_{1}}{\sigma_{1}}\right)^{2} \left(\frac{r_{2,t}\,-\,\mu_{2}}{\sigma_{2}}\right)^{2}\,-\,\eta \right],\\ \end{array}$$

(18)

where $\eta =\ln \iint \exp [ h] dr_{1}dr_{2}$, and

$$\begin{array}{@{}rcl@{}} h &=&-\frac{1}{2}\left( \frac{1}{1-\rho^{2}}\right) \left( \left( \frac{ r_{1,t}-\mu_{1}}{\sigma_{1}}\right)^{2}+\left( \frac{r_{2,t}-\mu_{2}}{\sigma_{2}}\right)^{2} \right. \notag \\ &&\left. -2\rho \left( \frac{r_{1,t}-\mu_{1}}{\sigma_{1}}\right) \left( \frac{r_{2,t}-\mu_{2}}{\sigma_{2}}\right) \right)+\theta \left( \frac{r_{1,t}-\mu_{1}}{\sigma_{1}}\right)^{2}\left(\frac{r_{2,t}-\mu_{2}}{\sigma_{2}}\right)^{2}. \end{array}$$

(19)

A test for bivariate normality in this distribution is a test of the parameter

$$H_{0}:\theta =0.$$

(20)

Let the parameters of Eq. 18 be $\Theta =\left \{ \mu _{1},\mu _{2},\sigma _{1}^{2},\sigma _{2}^{2},\rho ,\theta \right \}$. Under the null hypothesis, the maximum likelihood estimators of the unknown parameters are

$$\begin{array}{@{}rcl@{}} \widehat{\mu }_{i}&=&\frac{1}{T}\overset{T}{\underset{t=1}{\sum }}r_{i,t}; \widehat{\sigma }_{i}^{2}=\frac{1}{T}\overset{T}{\underset{t=1}{\sum }} \left( r_{i,t}-\widehat{\mu }_{i}\right)^{2};\widehat{\rho }\notag\\ &=&\frac{1}{T} \overset{T}{\underset{t=1}{\sum }}\left( \frac{r_{1,t}-\widehat{\mu }_{1}}{ \widehat{\sigma }_{1}}\right) \left( \frac{r_{2,t}-\widehat{\mu }_{2}}{ \widehat{\sigma }_{2}}\right) ;i=1,2. \end{array}$$

(21)

The log likelihood function of the expression in Eq. 18 at time t is

$$\begin{array}{@{}rcl@{}} \ln L_{t}(\Theta ) &\,=\,&-\frac{1}{2}\left( \frac{1}{1-\rho^{2}}\right) \left( \left( \frac{r_{1,t}-\mu_{1}}{\sigma_{1}}\right)^{2}+\left( \frac{ r_{2,t}-\mu_{2}}{\sigma_{2}}\right)^{2} \notag \right. \\ && \left. -2\rho \left( \frac{r_{1,t}\,-\,\mu_{1}}{\sigma_{1}}\right) \left( \frac{r_{2,t}\,-\,\mu_{2}}{\sigma_{2}}\right) \right)\,+\,\theta \left( \frac{r_{1,t}\,-\,\mu_{1}}{\sigma_{1}}\right)^{2}\left( \frac{r_{2,t}\,-\,\mu_{2}}{\sigma_{2}}\right)^{2}\,-\,\eta \notag\\ &=&h-\eta , \end{array}$$

(22)

where h is given by equations (19) and $\eta =\ln \iint \exp [ h] dr_{1}dr_{2}$.

The asymptotic information matrix, derived in Fry et al. (2010), is

$$\begin{array}{@{}rcl@{}} I_{t}( \Theta ) &=&-E\left[ \frac{\partial^{2}\ln L_{t}(\Theta ) }{\partial \Theta \partial \Theta^{\prime }}\right] \notag \\ &=&E\left[ \frac{\partial h}{\partial \Theta }\frac{\partial h}{\partial \Theta^{\prime }}\right] -E\left[ \frac{\partial h}{\partial \Theta }\right] E\left[ \frac{\partial h}{\partial \Theta^{\prime }}\right] . \end{array}$$

(23)

Using Eq. 23 and the properties of the bivariate normal distribution, the information matrix under H ₀ is

$$\begin{array}{@{}rcl@{}} I\left( \widehat{\Theta }\right) &=& T\left( E\left[ \frac{\partial h}{ \partial \Theta }\frac{\partial h}{\partial \Theta^{\prime }}\right]_{\theta =0}-E\left[ \frac{\partial h}{\partial \Theta }\right]_{_{\theta =0}}E\left[ \frac{\partial h}{\partial \Theta^{\prime }}\right]_{_{\theta =0}}\right) \\ &=& \left( \frac{T}{1-\widehat{\rho }^{2}}\right) \times \notag \\ && \left[ \begin{aligned} \frac{1}{\widehat{\sigma }_{1}^{2}} & \frac{-\widehat{\rho }}{\widehat{ \sigma }_{1}\widehat{\sigma }_{2}} & 0 & 0 & 0 & 0 \\ \frac{-\widehat{\rho }}{\widehat{\sigma }_{1}\widehat{\sigma }_{2}} & \frac{1 }{\widehat{\sigma }_{2}^{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{-\widehat{\rho }^{2}+2}{4\widehat{\sigma }_{1}^{4}} & \frac{- \widehat{\rho }^{2}}{4\widehat{\sigma }_{1}^{2}\widehat{\sigma }_{2}^{2}} & \frac{-\widehat{\rho }}{2\widehat{\sigma }_{1}^{2}} & \frac{\left(2\widehat{\rho } ^{2}+1\right)\left(1-\widehat{\rho }^{2}\right)}{\widehat{\sigma }_{1}^{2}} \\ 0 & 0 & \frac{-\widehat{\rho }^{2}}{4\widehat{\sigma }_{1}^{2}\widehat{ \sigma }_{2}^{2}} & \frac{-\widehat{\rho }^{2}+2}{4\widehat{\sigma }_{2}^{4}} & \frac{-\widehat{\rho }}{2\widehat{\sigma }_{2}^{2}} & \frac{\left(2\widehat{ \rho }^{2}+1\right)\left(1-\widehat{\rho }^{2}\right)}{\widehat{\sigma }_{2}^{2}} \\ 0 & 0 & \frac{-\widehat{\rho }}{2\widehat{\sigma }_{1}^{2}} & \frac{- \widehat{\rho }}{2\widehat{\sigma }_{2}^{2}} & \frac{\widehat{\rho }^{2}+1}{ 1-\widehat{\rho }^{2}} & 4\widehat{\rho }\left(1-\widehat{\rho }^{2}\right) \\ 0 & 0 & \frac{\left(2\widehat{\rho }^{2}+1\right)\left(1-\widehat{\rho }^{2}\right)}{\widehat{ \sigma }_{1}^{2}} & \frac{\left(2\widehat{\rho }^{2}+1\right)\left(1-\widehat{\rho }^{2}\right)}{ \widehat{\sigma }_{2}^{2}} & 4\widehat{\rho }\left(1-\widehat{\rho }^{2}\right) & \left(8+68 \widehat{\rho }^{2}+20\widehat{\rho }^{4}\right)\left(1-\widehat{\rho }^{2}\right) \end{aligned} \right] . \notag \end{array}$$

(24)

Evaluating the gradient of 𝜃 under the null gives

$$\begin{array}{@{}rcl@{}} \frac{\partial \ln L_{t}}{\partial \theta } &=&\sum\limits_{t=1}^{T}\left( \frac{\partial h}{\partial \theta }\right) -T\left( \frac{\partial \eta }{\partial \theta }\right) \\ &=&\sum\limits_{t=1}^{T}\left( \frac{r_{1,t}-\mu_{1}}{\sigma_{1}}\right)^{2}\left( \frac{r_{2,t}-\mu_{2}}{\sigma_{2}}\right)^{2}-T\left( 1+2\rho^{2}\right) . \end{array}$$

(25)

The score function under the null is

$$\begin{array}{@{}rcl@{}} q\left( \widehat{\Theta }\right) &=&\frac{\partial \ln L_{t}(\Theta )}{ \partial \Theta }|_{\theta =0} \\ &=&\left[\begin{aligned} 0 & 0 & 0 & 0 & 0 & \sum\limits_{t=1}^{T}\left( \frac{r_{1,t}- \widehat{\mu }_{1}}{\widehat{\sigma }_{1}}\right)^{2}\left( \frac{r_{2,t}- \widehat{\mu }_{2}}{\widehat{\sigma }_{2}}\right)^{2}-T\left( 1+2\widehat{ \rho }^{2}\right) \end{aligned} \right]^{\prime}. \end{array}$$

(26)

The Lagrange multiplier statistic is obtained by substituting the expressions in Eqs. 24 and 26 into

$$LM=q\left( \widehat{\Theta }\right)^{\prime }I^{-1}\left( \widehat{\Theta } \right) q\left( \widehat{\Theta }\right) .$$

(27)

The test statistic for contagion through the covolatility channel is

$$LM=\left( \frac{\frac{1}{T}\sum\limits_{t=1}^{T}\left( \frac{ r_{1,t}-\widehat{\mu }_{1}}{\widehat{\sigma }_{1}}\right)^{2}\left( \frac{ r_{2,t}-\widehat{\mu }_{2}}{\widehat{\sigma }_{2}}\right)^{2}-\left( 1+2 \widehat{\rho }^{2}\right) }{\sqrt{\frac{\left(4\widehat{\rho }^{4}+16 \widehat{\rho }^{2}+4\right) }{T}}}\right)^{2},$$

(28)

which is denoted by CV in the paper.

Appendix C: Summary of Crisis Dating Assumptions in Selected Papers for the Period of 1997 to 2013

Table 9 Summary of crisis dating in papers written on the Asian financial crisis

Table 10 Summary of crisis dating in papers written on the Russian and LTCM crises

Table 11 Summary of crisis dating in papers written on the Brazilian, dot-com and Argentinian crises

Table 12 Summary of crisis dating in papers written on the sub-prime crisis and the great recession

Table 13 Summary of crisis dating in papers written on the European debt crisis