Determinants of government bond spreads in the euro area: in good times as in bad

Abstract

Government bond spreads increased rapidly during the financial turmoil in the euro area. In general, government bond spreads in the euro area are attributed to solvency and liquidity risks and determinants thereof. This paper proposes the use of latent processes to model the time variation present in the evaluation of these determinants. In contrast to approaches using global measures like the US corporate bond spreads or short-term interest rates to approximate time variation, our model is also flexible enough to deal with the unfolding of the financial crisis. The findings suggest that the expected debt-to-GDP ratio explains a major part of the differences in bond yields in the euro area between 2003 and the unfolding of the financial crises. Coefficients for many determinants increased rapidly during the financial crises. Especially market capitalization gained relative importance in winter 2008/2009.

Appendix

1.1 Modified Kalman filter

The model in Eqs. 1 through 4 can be directly interpreted as a state-space model. The modified Kalman filter needed to calculate the likelihood and to estimate β _t is given as follows.

The predicted β _t equals the filtered one due to the random walk assumption

$$ \beta_{t|t-1}=\beta_{t-1|t-1}. $$

Thus, the corresponding variance in the prediction is given as the filtered one plus the variance in the innovations:

$$ P_{t|t-1}= P_{t-1|t-1}+\Upsigma_{t|t-1}. $$

If one assumed time invariant variances, $\Upsigma_{t|t-1}$ would be constant over all t. However, the GARCH process has to be taken into account. Therefore the diagonal elements of $\Upsigma_{t|t-1}$ follow

$$ \sigma^2_{k,t|t-1}=\gamma_{k,0}+\gamma_{k,1}\sigma^2_{k,t-1|t-2} +\gamma_{k,2}u^2_{k,t-1|t-1}, $$

where u ² _{k,t−1|t−1} is calculated via the filtered expectation and variance as

$$ u^2_{k,t-1|t-1}=(u_{k,t-1|t-1})^2+P_{k,t-1|t-1}, $$

where filtered values are calculated as follows:

$$ \begin{aligned} P_{t|t}&=P_{t|t-1}-P_{t|t-1}'*X_t'\left(s^2_{t|t-1}{\mathcal{I}}\right)^{-1} X_tP_{t|t-1},\\ u_{t|t}&=P_{t|t-1}'*X_t'\left(s^2_{t|t-1}{\mathcal{I}}\right)^{-1} \varepsilon_{t|t-1}. \end{aligned} $$

Note that the filtered β _t applies simply by adding the predicted β _t|t-1 and u _t|t . The GARCH process of the ideosyncratic errors is modeled as follows¹²:

$$ s^2_{t|t-1}=\alpha_{0}+\alpha_{1}s^2_{t-1|t-1}+\alpha_{2}\frac{1}{I}\sum_{i=1}^I \varepsilon^2_{i,t-1|t-1}, $$

where

$$ \varepsilon^2_{t-1|t-1}=Q_{t-|t-1}+(\varepsilon_{t-1|t-1})^2 $$

and

$$ Q_{t|t}=X_t'P_{t|t}X_t+s^2_{t|t-1} $$

denotes the filtered variance of the filtered errors and

$$ \varepsilon_{t|t}=y_t-X_t\beta_{t|t} $$

represents the corresponding expectations.

1.2 Approximate partial coefficient of determination

The time-varying coefficient of determination is defined as follows:

$$ R^2_t=\frac{Var(\widehat Y_t)}{Var(Y_t)}, $$

whereby $\widehat Y_t$ denotes the estimated vector of spreads based on the smoothed values of β _t . Thus, $\widehat y_{i,t}=X_{i,t}\beta_{t|T}.$

The approximate partial coefficient of determination of different variables is calculated as the squared coefficient of correlation of some auxiliary variables. In all cases, the cross-sectionally demeaned spreads are employed, $y_{i,t}^*=y_{i,t}-\frac{1}{n}\sum_i y_{i,t}.$ Further, a cross sectionally demeaned variable is constructed for each regressor variable, too, $x_{k,i,t}^*=x_{k,i,t}-\frac{1}{n}\sum_i x_{k,i,t}.$ Note that index k runs from 2 to K as k = 1 represents the constant in Model 1. Correspondingly, a cross sectionally demeaned variable is constructed representing the “rest” of the model: $z_{k,i,t}^*=z_{k,i,t}-\frac{1}{n}\sum_i z_{k,i,t},$ whereby z _k,i,t is given as X ^k _i,t β ^k _t|T and X ^k _i,t represents all regressors of X _i,t without the constant and the variable k.

Afterwards, for each $k:2\rightarrow K,$ the demeaned spreads $y_{i,t}^*$ are regressed on $z_{k,i,t}^*$, and $x_{k,i,t}^*$ are also regressed on $x_{k,i,t}^*$. The squared coefficient of correlation between the resulting residual series is used as an approximate partial coefficient of determination.