Causality-in-variance and causality-in-mean among European government bond markets

Abstract

This article examines causality in volatility spillover (causality-in-variance) for the six major European government bond markets. Using tests of temporal causality and directed acyclic graphs, we find evidence of contemporaneous causality-in-variance, indicating that volatility spillover in the government bond markets is a short-lived phenomenon. However, we find no evidence of contemporaneous causality-in-mean for bond index returns. The tests reveal that the markets are bidirectionally linked, and reasonably well integrated.

Acknowledgements

We would like to thank seminar participants at the 2006 Midwest Finance Association for helpful comments.

Notes

¹

² Hong (2001) subtracts unity from squared standardized errors when computing sample cross correlations, finding no difference in statistic size or power.

³ Empirical studies have found that financial asset (or market) cross correlations decay to zero with lag length.

⁴ Another advantage of Hong's test is that by construction, it filters out any causilty-in-mean effects and is therefore not subject to criticism of Pantelidis and Pittis (2004), which applies the Cheung–Ng test. See Hong (2001).

⁵ The corresponding finite sample statistic is:

⁶ Hong (2001) found that several commonly used nonuniform kernels have similar size and power characteristics. We use the Daniell kernel favoured by the paper.

⁷ For example, the Z-score for the 10, 5 and 1% confidence levels are 1.282, 1.645 and 2.326, respectively.

⁸ A noted drawback of Hong's test is the tendency to ‘over-reject the null a little’ at the 5% significance level; we address this weakness by indicating significance for both the 5 and 10% levels. A weakness of both S- and Q-tests (that does not apply to our data) is the inability to detect causation patterns resulting in zero cross correlation.

⁹ More formally, where Pr denotes probability, v_i , the current realization for a vector of variables and pv_i , the current realization of some sub-set of those variables that ‘precede’ v_i in a causal sense, the DAG depicts the conditional independence relation:

¹⁰ First order refers to the correlation between two variables conditional on a third; second order is conditional on two variables, and so on. The first-order matrix can be computed by inverting and scaling the unconditional correlation matrix for triplets of variables (Swanson and Granger, 1997).

¹¹ Fisher's z-statistic is used to test conditional correlation significance.

¹² For example, AC ≠ 0, but AC|B = 0 (not statistically different from zero).

¹³ These studies, in addition to DeGennaro et al. (1994) and Sutton (2000), examine monthly return data.

¹⁴ The null hypothesis of KPSS is that the data are level or trend stationary. ADF and PP have a null hypothesis of a unit root, versus the alternative of stationarity.

¹⁵ Restrictions are as follows: GARCH(1, 1) [α ₁ ψ ₁ = ψ ₂ = 0], GARCH(1, 1)-M [α ₁ = ψ ₂ = 0], MA(1) GARCH(1, 1) [α ₁ = ψ ₁ = 0] and AR(1)GARCH(1, 1) [ψ ₁ = ψ ₂ = 0]. I_t represents all available information at time t − 1, μ_t is conditional on I_t .

¹⁶ Only the nine lag Q-test for France is significant at the 10% level.

¹⁷ Our test mirrors that of Daigler and Wiley (1999), which tests for day-of-the-week volatility effects.

¹⁸ GM→FR, NL→FR, NL→BL, lags 8–12, 9–12 and 10–12, respectively, at the reported lag-12 significance levels. Test results for all 12 lags are available upon request.

¹⁹ BL→FR in lags 3 and 4. A potential criticism of our approach to using results of bivariate tests in deriving a causal map is that it may ignore the effects of conditional correlation between variables. Analysis using DAG, which derives a causal map using that information, re-dresses this issue.

²⁰ Swanson and Granger (1997) and Breitung and Swanson (2002) point out that if data are temporally aggregated, and the time intervals are too large (or data frequency to low), true unidirectional Granger causality can result in evidence contemporaneous correlation that does not in-fact exist. The problem is the failure to differentiate data with sufficient temporal precision. As a partial remedy, Swanson and Granger (1997) argue testing for contemporaneous causality.

²¹ TETRAD II software (Scheines et al., 1994) is used to compute the directed graphs for this study. The data generated by this software indicate at what order of conditioning a link is removed, the corresponding correlation between variables, and the orientation (if any) of the surviving links.

²² Graphs generated at the 5% significance level contain fewer links than those determined at the 10% level.

²³ As of 2003, government debt in billions of Euro and as a percentage of GDP are as follows: Italy (1383, 105.8%), Germany (1366, 64.2%), France (966, 63.9%), UK (623, 39.7%), Belgium (269, 100.0%) and the Netherlands (247, 54.3%). Finfacts Business News (2005), UK is converted from the British pound, at the end of year at a rate of 1.425.