1. Introduction
Several governments of the European Monetary Union (EMU) have experienced fiscal crises as a consequence of the 2008 subprime mortgage crisis of the United States (US) and the subsequent global economic and financial meltdown. This motivated a large number of studies about EMU member states’ public debt during the crisis period and afterwards (e.g., [1-4]). As the crisis impacted the Eurozone, investors updated their judgments and preference s about the riskiness of sovereign bonds issued by Eurozone governments and many creditors moved from risky to safe investments, exhibiting flight to quality. In the financial markets, this process is considered as a channel of contagion (see [5-8]). In the context of credit instruments, contagion caused by updated risk judgment of investors is documented in the literature. Several studies capture revised risk assessment of investors by analyzing credit default swap prices ([8-10]). Several authors report contagion effects in international debt markets ([11,12]). Furthermore, [13-16] also provide evidence of contagion in the context of debt instruments.
In this study, we test for contagion and interdependence of debt to gross domestic product (GDP) among the Eurozone member states over the period 2000 Q4 to 2012 Q1. We apply the econometric framework of [17] to measure contagion and interdependence of government debt in the Eurozone. We proceed in two steps.
In the first step, we employ unit root tests involving structural breaks (by [18,19]) and break trend regression models in order to define an indicator variable. This variable indicates if the debt to GDP time series of a Eurozone member state has had an increasing trend during some part of the period 2007 Q4 to 2012 Q1 (i.e., after the start of the US financial crisis). We also perform an extended unit root test with three structural breaks to validate the robustness of the unit root tests applied.
In the second step, we include the indicator variable in the contagion model of [17]. In this model, indicator variables capture contagion of debt to GDP. Furthermore, interdependence is driven by a dynamic latent common factor. We consider a dynamic common factor in the EMU public debt, since the operational framework (e.g., key interest rate decisions; open market operations) of the European Central Bank (ECB) influences government bond yields and, therefore, the level of debt to GDP in the Eurozone.
We extend the latent-factor specification of [17], since the latent variables representing the common factor can be serially correlated. We estimate the extended model by the maximum simulated likelihood method, using the efficient importance sampling (EIS) technique of [20]. Furthermore, parameter estimates are validated by residual diagnostic tests.
The remaining part of this work is organized as follows. Structural breaks of public finances in the Eurozone are estimated and indicator variables are defined in Section 2. Contagion and interdependence of debt to GDP are analyzed in Section 3. We conclude in Section 4. The EIS based maximum simulated likelihood method and computation of the common factor are presented in two appendices.
2. Structural Breaks of Public Finances in the Eurozone
We study quarterly government debt to GDP data for 17 member states of the EMU for the time period 2000 Q4 to 2012 Q1.(Data source: Eurostat Statistics Database of the European Commission.)We denote the debt to GDP ratio of country 
in quarter 
by
. Table 1 presents the list of the EMU member states in 2012, shows in which year each country has adopted the euro currency, and exhibits some descriptive statistics of
.
2.1. Estimation of Break Point Dates
We perform three alternative unit root tests for the debt to GDP time series of each country to identify the number and date(s) of the structural break(s) over the period 2000 Q4 to 2012 Q1. Each test uses a particular number of break(s) and suggests the corresponding breakpoint date(s). The best unit root test is selected according to the adjusted R-squared 
metric of the unit root test equation.
The first test applied for the 
time series is the augmented Dickey-Fuller (ADF) unit root test introduced by [21]. The specification estimated includes a constant term and a linear time trend. This test does not consider structural changes in the time series. Table 2 presents the ADF test statistic, its significance and the 
of the ADF test equation for each country.
The second unit root test applied is by [19], which involves one structural break in the constant as well as in the linear time trend. The breakpoint date is estimated by the model. [19] extend previous unit root tests by considering a unit root with break under the null hypothesis,
. There are several unit root tests that consider the possibility of one structural break in a time series ([22-25]). A common feature of these tests is that they omit the possibility of a unit root process with break un
Table 1 .Descriptive statistics of government debt to GDP.
Names of the EMU founder states are written by bold letters. Start denotes the year of euro adoption for each country. Min, Max, SD and ADF denote maximum, minimum, standard deviation and augmented Dickey-Fuller, respectively. The last column shows the p-value of the ADF test statistic.
Table 2. Structural breaks in government debt to GDP.
Augmented Dickey-Fuller (ADF [21]). Lee and Strazicich (LS [18,19]). Test statistic (TS). Adjusted R-squared
. Not available (NA). 
and 
denote the quarter of the structural break, where the first two digits show the year and the last digit indicates the quarter of year. The ADF and LS tests are with constant(s) and with linear time trend(s). *, ** and *** denote significance at the 10, 5 and 1 percent levels, respectively. 
corresponds to the test equation of each unit root test. 
and break dates with bold font indicate the best unit root test for each country. The 
values are parameter estimates of breaking trend regressions. The 
coefficients underlined indicate that the debt to GDP has an increasing trend from 2007 Q4.
der the null hypothesis. Therefore, spurious rejections of 
may occur. In the [19] test, the following test equation is estimated:
, (1)
where 
denotes the first difference of the corresponding time series, 
and
. 
denotes the vector of coefficients estimated by a regression of 
on
. The 
terms are included to correct for serial correlation. See [19] for the selection of the number of augmentation terms, k.
and
where
and 
denotes the time of the structural break for country
. Table 2 presents the [19] test statistic, its significance and the 
of the [19] test equation for each country.
The third unit root test applied is the [18] test, which involves two structural breaks in the constant as well as in the linear time trend. The two breakpoint dates are estimated by the model. [18] extended previous models by including two structural breaks in
. Unit root tests with a single structural break do not consider the possibility that the variable has several structural breaks. This fact motivated [26] to include two structural breaks in their unit root test. These authors do not consider structural breaks under the null hypothesis, which may yield spurious rejection of
. The test equation of [18] is Equation (1) with
and
where
and 
is the time of the structural break 
for country
. Table 2 presents the [18] test statistic, its significance and the 
of the [18] test equation for each country.
For each country, we use the results of the unit root test with the highest 
value. Table 2 shows that the unit root test equation with the highest explanatory power is always a unit root test involving one or two structural breaks. Furthermore, Table 2 shows that the unit root null hypothesis is always rejected for the best model, i.e. a breaking trend process is suggested for all countries.
2.2. Robustness Analysis of Structural Breaks
The unit root test results reported in Table 2 show that, in several cases, the unit root test with two structural breaks has the highest 
value. However, unit root tests with more than two structural breaks may explain better the evolution of debt to GDP. Therefore, we perform a unit root test with three structural breaks for the countries where the [18] model has the highest 
metric. In this test, we use the 
and 
dates estimated by the [18] model, while 
is determined endogenously using the methodology of [18,19]. Conditioning on two previously estimated breaks, reduces the computation time substantially. This approach is similar to the idea of [27], who test for 
versus 
breaks conditioning on the locations of 
breaks and the additional breakpoint is determined endogenously; see also [28,29]. The test results, presented in Table 3, show that the 
model selection metric is always lower than that of the [18] model, i.e. two breakpoints are preferred to
Table 3 . Robustness analysis of unit root tests.
Test statistic (TS). Adjusted R-squared
. 
denotes the quarter of the structural break, where the first two digits show the year and the last digit indicates the quarter of year. ** and *** denote significance at the 5 and 1 percent levels, respectively.
three breakpoints according to the
. This finding supports the use of the [18] unit root test.
2.3. Increasing Debt to GDP during the Crisis
In this section, we define an indicator variable, 
that represents if debt to GDP has an increasing tendency during some part of the period 2007 Q4 to 2012 Q1 (i.e., from the start of the US financial crisis).We use the results for the highest 
unit root test for each country. The best unit root test estimates the date(s) of structural break(s) in debt to GDP and suggests a breaking trend process for the evolution of debt to GDP for all countries (see Table 2). We estimate the tendency of 
by two alternative breaking trend regression models. When the [19] test provides the breakpoint dates, we estimate the following regression model:
, (2)
where 
is a possibly serially correlated error term.
and zero otherwise,
and zero otherwise,
and zero otherwise, and
and zero otherwise. When the [18] test provides the breakpoint dates, we estimate the following regression model:
, (3)
where 
is a possibly serially correlated error term.
and zero otherwise,
and zero otherwise,
and zero otherwise,
and zero otherwise,
and zero otherwise, and
and zero otherwise.
Table 2 shows the 
parameter estimates for the breaking trend regressions. The 
coefficients underlined indicate that the debt to GDP has an increasing trend during some part of the period 2007 Q4 to 2012 Q1. The estimated breakpoint dates around the 
underlined indicate the sub-period when the debt to GDP increases for country
. The indicator variable, 
for an EMU member country 
in period 
takes the value one when the 
underlined is significantly positive. 
will be used in Section 3 to construct a measure of contagion of public debt within the Eurozone.
3. Contagion of Public Debt in the Eurozone
3.1. Contagion Model
We apply the econometric model of contagion of [17] to measure contagion of public finances in the Eurozone. These authors distinguish contagion from interdependence and identify the parameters measuring contagion effects. The ADF results of debt to GDP presented in Table 1 evidence that the unit root null hypothesis cannot be rejected for any country in the Eurozone. This motivates us to formulate the econometric model of contagion for first differences, 
, as follows:
(4)
(5)
(6)
For 
and
. The model includes a dummy variable, 
defined as follows: 
if country 
is EMU member in period 
and zero otherwise (see Table 1). Moreover, the model also includes the indicator variable, 
, defined in Section 2. The definition of the indicator variable in a first step (i.e., Section 2) is similar to the approach of [17]. They also choose the threshold parameter to define the crisis indicator dummy in a first step. 
is a first-order autoregressive (AR) latent factor with dynamic parameter
. In Equation (6), the constant term in the latent process is restricted to zero and the variance of 
is restricted to one due to identification reasons.
In this model, interdependence is captured by 
and contagion effects are measured by 
(see [17]). We multiply by the EMU-membership dummy, 
both the contagion and the latent factor terms of the model, since interdependence and contagion is measured only among the EMU member states. The model includes the following parameters. Initial conditions of the dynamic model,
. 
and 
measure the dynamic effects of 
and
, respectively. The quadratic term is included to capture possible non-linear dynamics. We find the 
parameter statistically significant for most countries; see Table4 
measures the impact of the average of other EMU states’ indicator variables, i.e., contagion of public debt. 
captures the impact of the latent factor,
. 
measures the standard deviation of the country-specific error term,
.We assume that the country-specific standard deviation is constant. Motivated by the discussion in [30], we also estimated the model with a GARCH 
error term ([31]). However, the results did not show significant GARCH dynamic in the error term.
This model is an extension of the latent-factor specification of [17], since the common factor, 
is serially correlated. For this reason, we use a simulation-based estimation method designed for the estimation of models with dynamic latent variables. The parameters of the model, i.e.,
with
, are estimated by the maximum simulated likelihood method (see [32]), using the EIS technique of [20]. The EIS technique has been applied successfully for the precise evaluation of likelihood functions involving high-dimensional integrals (e.g., [33-35]). We present the computation of the likelihood function in Appendix A.
Table 4 presents the parameter estimates of the contagion model. The last column of this table shows the relative importance of the contagion component of debt to GDP. For each country, the table presents the average of
, (7)
computed over the period 2007 Q4 to 2012 Q1 (i.e., from the start of the US financial crisis). Results show that the average contribution of debt contagion of other EMU member states is the highest for Luxembourg (19%), followed by, Ireland (10%), the Netherlands (7%), Slovenia (7%), Greece (6%), Spain (6%) and Slovakia (5%).
Table 5 presents the estimates of interdependence, 
, among the Eurozone states’ debt to GDP. Results suggest significant and, in most cases, positive interdependence of debt to GDP within the EMU.
3.2. Residual Diagnostic Tests
The correct specification of the error term, 
is analyzed by residual diagnostic tests. Residuals are computed as follows:
. (8)
Table 4. Parameter estimates of the contagion model.
*, ** and *** denote significance at the 10, 5 and 1 percent levels, respectively.
Table 5. Interdependence of debt to GDP among the EMU states.
Interdependence is defined as
. Correlation coefficients significant at the 5 percent level are presented by bold numbers. Austria (AUS); Belgium (BEL); Cyprus (CYP); Estonia (EST); Finland (FIN); France (FRA); Germany (GER); Greece (GRE); Ireland (IRE); Italy (ITA); Luxembourg (LUX); Malta (MAL); Netherlands (NED); Portugal (POR); Slovakia (SVK); Slovenia (SLO); Spain (SPA).
We present the estimation of the latent factor, 
in Appendix B. Table 6 shows the results of two residual diagnostic tests for each country. First, the Z test is performed with the following null hypothesis: 
. Second, the [36] test is performed with the following null hypothesis: 
are not serially correlated. For both tests, p-values presented in Table 6 suggest that the null hypotheses are never rejected at the 10 percent level of significance. These results support the specification of the 
error term in the contagion model and the consistency of the EISbased maximum simulated likelihood method.
4. Conclusion
In this study, we report contagion and interdependence of debt to GDP among the member states of the Eurozone over the period 2000 Q4 to 2012 Q1. We apply the framework of [17] to measure contagion and interdependence in two steps. First, we estimate the breakpoint dates in debt to GDP by using several unit root tests. For all countries, the unit root null hypotheses are rejected, i.e., a breaking trend process is suggested. We perform an extended unit root test with three structural breaks to validate the number of structural breaks estimated. We define an indicator variable which takes the value one if a Eurozone government has had an increasing tendency in debt to GDP during some parts of the period 2007 Q4 to 2012 Q1 or zero otherwise. Second, the indicator variable is included in the contagion model of [17], where contagion and interdependence components of debt to GDP are estimated. Contagion is driven by the mean of the indicator variables of other Eurozone governments. Interdependence is driven by a dynamic common factor, extending the latent-factor specification of [17]. We estimate the contagion model by the maximum simulated likelihood method, using the EIS technique of [20]. Furthermore, we perform residual diagnostic tests to validate the EIS parameter estimates. Results show significant and country-dependent contagion effects and interdependence of debt to GDP within the Eurozone.
Acknowledgements
Previous versions of this work were presented at the Macro-Finance Workshop, Universidad de Navarra, Pamplona, November 2011; MKE Annual Conference, Budapest, December 2011. The authors thank Gábor Kőrösi for helpful discussions and comments. The authors also thank the research financing of the School of Business of
Table 6 . Residual diagnostics for the contagion model.
Mean and SD present the mean and standard deviation, respectively, of
.
. Null hypothesis of the Z test:
. Null hypothesis of the Ljung and Box (LB 1978) test: 
are not serially correlated. LB (16) is computed for 16th-order serial correlation.
Universidad Francisco Marroquín.
Appendix A
We write the specification to be estimated as follows:
(1)
(2)
(3)
For 
and
. We introduce the following notation related to the observable and latent variables:
,
,
,
,
,
and
. The density of 
conditional on 
is
, (4)
where the conditional mean of 
is
. (5)
The density of the latent factor, 
conditional on 
is
. (6)
Since the 
is not observed, we integrate out all latent variables from the likelihood function with respect to the assumed normal distribution, to get the marginal density of
. The integrated likelihood function is the following 
-dimensional integral:
. (7)
We can rewrite the likelihood of 
by using a more compact notation as follows:
. (8)
where 
is the joint density of debt to GDP and the latent factor. For given parameter values, we evaluate the 
-dimensional integral numerically in 
by Monte Carlo (MC) simulation method, using the EIS technique ([19]). The EIS procedure is nested into a typical likelihood function maximization procedure. To maintain the stability of this procedure, the same set of i.i.d. 
random numbers (i.e., common random numbers) is used for every set of parameters to estimate the value of the integrated likelihood function (see [19]). The EIS methodology consists of the following elements. First, an auxiliary sampler, 
is introduced and it is included in the likelihood function as follows:
, (9)
where 
denotes the parameters of the auxiliary sampler in period
. Then, the importance MC estimate of 
for given 
is computed as
(10)
where 
is the vector of parameters of the auxiliary sampler including all 
with
, and 
denotes the 
-th trajectory of i.i.d.
draws from 
and
. Regarding the auxiliary sampler, the following two questions are to be answered:
1) How to simulate from
? [19] suggest defining the auxiliary sampler, 
with its associated density kernel, 
, defined as
, (11)
where
(12)
denotes the integrating constant associated to 
in period
. [19] suggest choosing 
as a kernel of the normal distribution, i.e. the exponential of a secondorder polynomial. Following [34], 
is included into the auxiliary sampler,
. Therefore, the normal density kernel in period 
has the following form:
, (13)
where the vector of parameters 
determines the conditional mean and the conditional variance of the auxiliary sampler, 
in period
. From Equation (13), we can derive that the conditional mean, 
and conditional variance, 
of 
in period 
are
(14)
and
, (15)
respectively; see also [34]. Therefore, for given parameters of the auxiliary sampler, a trajectory of 
can be generated from the following AR (1) process:
, (16)
where 
are i.i.d. common random numbers.
2) How to choose the 
parameters of
? The EIS methodology relies on the optimal choice of parameters of the auxiliary samplers in the sense that for a given
, the variance of 
is minimized, i.e.,
. (17)
From Equation (10), one can guess that this variance is “small” when 
provides a “good fit” to 
in each period
. Equation (11) shows that m 
may provide a “good fit” to 
if
(18)
for each period
. Using the fact that 
is a second-order polynomial of 
for a normal distribution, [19] show that the MC variance minimization problem stated in Equation (17) can be reduced to the following recursive sequence of 
ordinary least squares (OLS) problems:
(19)
For 
and
. 
and 
is the OLS estimate of
. Thus, for each period
, one has to compute the OLS estimate of the parameters of the auxiliary sampler,
. The regressions have a recursive structure because the 
estimates are used to compute the integrating constant for the next, 
- th OLS regression. Thus, the regressions are run backwards, i.e. from 
to 1. The sample size of each regression is equal to the number of trajectories drawn,
. One of the advantages of the EIS algorithm is that these auxiliary regressions are typically run with relatively low sample sizes. In this article, the number of trajectories simulated for the latent variables is
.
In summary, the EIS technique consists of the following steps:
a) Draw 
trajectories 
from the natural sampler,
.
b) For each 
(from 
to 1), estimate the regression in Equation (19).
c) Given the OLS estimates of 
obtained in 2), draw 
trajectories 
from the auxiliary samplers,
. Iterate 2) and 3) five times ([19]).
d) From Equation (11), express 
for each period as:
. (20)
It can be deduced from Equations (12) and (13) that
. (21)
Therefore, we can compute the importance MC estimate of 
of (11) by Equations (13), (20) and (21).
Appendix B
The value of the latent factor is approximated by estimating the expectation of 
conditional on the past observable information set, i.e.,
; see [34]. This conditional expectation can be computed as
, (22)
where 
denotes the conditional density of
. We can compute 
as follows:
. (23)
Substituting Equation (23) into (22), and using the fact that the denominator in (23) is not a function of
, we obtain that
, (24)
where the joint density, 
is given by
. (25)
The high-dimensional integrals in Equation (24) are estimated by the EIS technique.