Is there life in the old dogs yet? Making break-tests work on financial contagion

Abstract

Many tests of financial contagion require a definition of the dates separating calm from crisis periods. We propose to use a battery of break search procedures for individual time series to objectively identify potential break dates in relationships between countries. Applied to the biggest European stock markets and combined with two well established tests for financial contagion, this approach results in break dates which correctly identify the timing of changes in cross-country transmission mechanisms. Application of break search procedures breathes new life into the established contagion tests, allowing for an objective, data-driven timing of crisis periods.

Appendix

An overview of the employed methods of testing for breaks in the time series.

The I&T test is based on the statistic:

$$ {\text{I}}\& {\text{T}} = { \sup }_{\text{k}} \left| {\sqrt {{\text{T}}(2{\text{D}}_{\text{k}} )^{ - 1} } } \right|,\;{\text{D}}_{\text{k}} = \frac{{{\text{C}}_{\text{k}} }}{{{\text{C}}_{\text{T}} }} - \frac{\text{k}}{\text{T}} $$

where C_k is the cumulative sum of squares of ε _t. If ε _t are identically and independently normally distributed random variables with zero mean and σ ² variance, the asymptotic distribution of the test converges to the sup_r|W^*(r)| where W^*(r) ≡ W(r) − r W^*(1) is a Brownian bridge and W(r) is a standard Brownian motion.

The SAC₁ test is based on the statistic:

$$ {\text{SAC}}_{1} = { \sup }_{\text{k}} \left| {\sqrt {\text{T}}^{ - 1} {\text{B}}_{\text{k}} } \right|,\;{\text{B}}_{\text{k}} = \left( {{\text{C}}_{\text{k}} - \frac{\text{k}}{\text{T}} \cdot {\text{C}}_{\text{T}} } \right)\left( {\sqrt {\hat{\alpha }_{4} - \hat{\sigma }^{4} } } \right)^{ - 1} $$

where the estimate of α ₄ is given by the mean cumulative sum of squared squares of ε _t and C_k as before. The limit distribution is as again the supremum of a Brownian bridge.

The SAC₂ test is based on the statistic:

$$ {\text{SAC}}_{2} = { \sup }_{\text{k}} \left| {\sqrt {\text{T}} {\text{G}}_{\text{k}} } \right|,\;{\text{G}}_{\text{k}} = \sqrt {\hat{\omega }_{4} } \left( {{\text{C}}_{\text{k}} - \frac{\text{k}}{\text{T}} \cdot {\text{C}}_{\text{T}} } \right) $$

and in our case the consistent estimate of the kurtosis is the non-parametric:

$$ \frac{1}{\text{T}}\sum\limits_{{{\text{t}} = 1}}^{\text{T}} {\left( {\varepsilon_{\text{t}}^{2} - \hat{\sigma }^{2} } \right)^{2} } + \frac{2}{\text{T}}\sum\limits_{{{\text{l}} = 1}}^{\text{m}} {{\text{w}}(1,{\text{m}})} \sum\limits_{{{\text{t}} = {\text{l}} + 1}}^{\text{T}} {\left( {\varepsilon_{\text{t}}^{2} - \hat{\sigma }^{2} } \right)\left( {\varepsilon_{{{\text{t}} - 1}}^{2} - \hat{\sigma }^{2} } \right)} $$

where w(l,m) is the Bartlett (BT) or Quadratic Spectral (QS) lag window with bandwidth m selected by the automatic procedure of Newey–West (1994). The VARHAC kernel (VH) of Den Haan and Levin (1998) is also used as a method of bypassing the problem of selecting the bandwidth. The limit distribution of this test is as before.

The K&L test is based on the statistic:

$$ {\hat{\text{k}}} = \min \left\{ {{\text{k}}:\left| {{\text{U}}_{\text{T}} ({\text{k}})} \right| = { \sup }_{\text{k}} \left| {{\text{U}}_{\text{T}} ({\text{j}})} \right|} \right\},\;{\text{U}}_{\text{T}} ({\text{k}}) = \left( {\frac{1}{{\sqrt {\text{T}} }} \cdot {\text{C}}_{\text{k}} - \frac{\text{k}}{{{\text{T}}\sqrt {\text{T}} }} \cdot {\text{C}}_{\text{T}} } \right) $$

and the rest as before.