A Smooth SEM Approach to Measure Contagion in International Markets

Transcription

1 A Smooth SEM Approach to Measure Contagion in International Markets Aneza Kadilli and Jaya Krishnakumar February 6, 2014 Preliminary Draft Abstract There is extensive empirical evidence that market linkages strengthen during crises. The aim of this paper is to suggest a flexible model for contagion between markets using a nonlinear Smooth Simultaneous Equation Model (SSEM) which allows also for the presence of market interdependence. Based on the nonlinear SEM model for contagion developed by Pesaran and Pick (2007) (which is a special case of our model), we allow the transmission of shocks to be smooth and estimate the degree of smoothness. The model also endogenizes the identification of crisis periods. It controls for global and country-specific factors and though contagion is defined as market comovement that goes beyond these links. Within an asset pricing framework, we investigate the cross-asset and cross-country contagion in the sovereign CDS spread market and the banking sector in the euro area, U.S., U.K. and Switzerland. We examine the transmission of shocks through fundamentals, investors behavior measures and pure contagion effect. Given the special market conditions during the financial crisis, and more recently the euro area sovereign debt crisis, we expect to find increasing and asymmetric contagion effects during these periods. JEL Classification: C58, G01, G12. Keywords: Contagion; Smooth SEM model; Global and country-specific variables; Investor behavior. The authors are particularly grateful to Laurent Barras, Martin Boyer, Gerda Cabe, Ines Chaieb, Alessandro Fontana, Alexandre Jeanneret, Henri Loubergé, Federico Ravenna, and to the colleagues from the Department of Economics of the University of Geneva for their valuable comments and insights. Part of the research was conducted when the first author was a visiting research scholar at HEC Montreal. Geneva School of Economics and Managament, University of Geneva, Bvd. du Pont d Arve 40, CH Geneva 4. aneza.kadilli@unige.ch Geneva School of Economics and Management, University of Geneva, Bvd.du Pont d Arve 40, CH-1211 Geneva 4. aya.krishnakumar@unige.ch

2 1 Introduction The recent events in the financial and sovereign debt market and their propagation to other interrelated countries and markets have aroused increasing interest in studying how shocks originating from one market or country are transmitted to other markets or countries. An extensive strand of literature provides evidence that such linkages strengthen during periods of crisis with important implications for policy and portfolio diversification decisions. Many studies attempt to distinguish between market interdependence and contagion (Corsetti, Pericoli and Sbarcia 2005; Dungey and Martin 2007; Forbes 2012). Interdependence is defined in the literature as a constant link between markets, potentially explained by fundamentals such as trade and financial linkages. Contagion is most frequently defined as a significant and unpredictable increase of the linkages between markets when one of them is hit by a shock (Dornbusch, Park and Claessens 2000; Forbes and Rigobon 2002; Longstaff 2010), although the literature has not come to a unique conclusion regarding this definition. The principal aim of this paper is to study the contagion phenomenon between the sovereign CDS market and the banking sector across asset classes and across countries in the euro area, U.S., U.K. and Switzerland. We employ a novel methodology which allows for the presence of contagion, interdependence and transmission of global and local shocks. Eun and Shim (1989) provide empirical evidence that asset prices are more interrelated within the same country than among countries. Nonetheless, the recent crises have shown that international linkages are far from being negligible. The channels through which the shocks are potentially transmitted are very often classified in two broad categories: (i) the fundamental-based channel and (ii) the investor behavior channel (Dornbusch, Park and Claessens 2000; Karolyi 2003; Barberis, Shleifer and Wurgler 2005). The interconnectedness once we control for these factors and for market interdependence should be interpreted as a pure contagion effect. 1 The mechanism of real linkages operates mostly through trade, financial system, exchange rates and interest rates. Allen and Gale (2000) and Longstaff (2010) discuss the crucial importance of market liquidity in transmitting negative shocks from one market to other markets, creating a downward liquidity spiral. The investor behavior mechanism, rational or irrational, is not necessarily related to the value of the fundamentals or to a global shock. Large losses in one market can trigger a general panic and loss of confidence in other markets or countries, leading to increasing risk aversion and to fire sales from investors which act in herd (Bae, Karolyi and Stulz 2003). Forbes and Rigobon (2002) underline the difficulty of directly quantifying the impact of each mechanism. Another important distinction is whether the variables through which the shocks are transmitted are of local or global nature. Given the importance of the U.S. economy in the world economy, variables stemming from this market are often considered as global variables. For instance, Longstaff, Pan, Pedersen and Singleton (2011) and Forbes (2012) use volatility measures, interest rates, credit and term spreads from the U.S. market to account for global shocks. Constructed variables from the world economy are also utilized in the literature. 1 The definition of contagion that we adopt is very close to that of Bae, Karolyi and Stulz (2003), although they use a different methodology. 2

3 Several methods are adopted to quantify contagion. 2 The most used approach is based on cross-market correlation (King and Wadhwani 1990; Forbes and Rigobon 2002; Bekaert, Hodrick and Zhang 2009). Other methods are the probability analysis (Eichengreen, Rose and Wyplosz 1996), the extreme value analysis (Bae, Karolyi and Stulz 2003), the VAR models (Eun and Shim 1989; Dees, di Mauro, Pesaran and Smith 2007; Longstaff 2010; Ehrmann, Fratzscher and Rigobon 2011), the latent factor/garch models (Bekaert, Harvey and Ng 2005; Corsetti, Pericoli and Sbracia 2005; Dungey and Martin 2007). Very recently, Aït-Sahalia, Cacho-Diaz and Laeven (2013) and Aït-Sahalia, Laeven and Pelizzon (2013) use ump-diffusion models with self-excitation and mutual excitation to examine contagion. The general outcome of many studies is supportive of highly interrelated markets / countries with a significantly larger comovement during distressed periods. 3 Despite their frequent use and some confirmed advantages, most of these methods have some important limitations (see Pesaran and Pick 2007 and Forbes 2012). First, several empirical studies provide an ex-post identification of crises, potentially introducing a selection bias (Forbes and Rigobon 2002; Bekaert, Harvey and Ng 2005; Corsetti, Pericoli and Sbracia 2005). This procedure assumes that the crisis spans over a sufficiently long period of time to allow a consistent estimation of the contagion measure. Moreover, it cannot be used for forecasting or ex-ante decisions in portfolio selection or policy responses to restrain adverse contagion effects. Contagion can be considered as a tail behavior, but a potential disadvantage of the extreme value analysis which correctly incorporates this definition, is that the number of extreme observations may be limited. The correlation-based tests - and other methods - fail to control for common or country-specific shocks, introducing an omitted variable bias. To remedy to this drawback, several papers explore for contagion effects in the model residuals. Methods such as probability analysis or cross-market correlation do not succeed in controlling for endogeneity and heteroskedasticity. Several methodologies assume a linear (and symmetric) transmission of shocks. The empirical evidence shows that the contagion effect is asymmetric and strengthens as the degree of a negative shock intensifies with positive shocks being transmitted to a much weaker extent. Based on the nonlinear Simultaneous Equation Model (SEM) of Pesaran and Pick (2007), we develop a new and more flexible methodology to measure market contagion in an asset pricing perspective, while keeping all its advantages. The novelty consists on the presence of a bounded nonlinear function as a contagion variable which allows shocks to be transmitted either gradually or abruptly from one market to the other, estimating the degree of smoothness. In Pesaran and Pick (2007) contagion is modeled by a dummy variable. This specification is a special case of our model when the smoothness parameter tends to infinity. The degree at which one market affects another, after controlling for local and global shocks, is almost zero in normal times and converges progressively to one as the magnitude of the (negative) shock amplifies. The model is flexible enough to include the transmission of positive or negative shocks, but the latter are of sensibly greater concern 2 For a recent review of these methods, their advantages and limitations see Forbes (2012). 3 Corsetti, Pericoli and Sbracia (2005) show evidence against preceding studies which find no contagion effects due to potential model misspecifications. Billio and Caporin (2010) identify the turmoil periods consistently with Forbes and Rigobon (2002), but contrary to this latter paper in which no contagion is detected, they find a strong contagion effect. 3

4 for investors and policy makers. In contrast with many empirical studies, in this methodology the identification of crises is defined endogenously through an estimated threshold which can be time-varying. The model introduces global and local factors to control for any comovement between markets stemming from such sources. The presence of the local factors is necessary for the correction of the endogeneity. The effect captured by the contagion coefficient is disentangled from interdependence which is captured by the correlation between the errors of the different equations. Using a Monte Carlo simulation, Pesaran and Pick (2007) show that ignoring endogeneity and interdependence could introduce a substantial upward bias in the contagion coefficient. Another advantage of our approach is that we are able to investigate contagion between markets within the same country and across countries. Contagion is extensively investigated in the stock market, while studies in the bond market, exchange rate, volatility, interest rates are scarce (Forbes 2012). We apply the methodology developed in this paper to investigate the interconnectedness of sovereign CDS market and banking sector in the euro area countries and some other closely linked countries such as U.S., U.K. and Switzerland. Pan and Singleton (2008) underlines the frequent trading of the CDS compared to the liquidity of the underlying bonds. This oins Ang and Longstaff (2011) who argue that the CDS spread is a more accurate measure of sovereign credit risk than the sovereign debt spread, the latter reflecting also other types of risk such as liquidity and interest rate variations. Despite this, the data on sovereign CDS spreads are available only since December 2007, consequently not allowing a comparison of the contagion effects with the period previous to the subprime crisis. To conduct a complete analysis since the creation of the euro area, we also consider the sovereign debt spread. During the subprime crisis of Fall 2007, and lately during the sovereign debt issue, banks played a crucial role in spreading out the distress to other markets. The subprime crisis was initially a banking crisis in the U.S. market, but it rapidly expanded in the financial market, causing the bankrupt of many financial institutions, and ended up in a general crisis. More recently, since late 2009, the sovereign debt crisis in the euro area not only impaired the ability of countries to reimburse or refinance government debt, but also had serious drawbacks on the banking system which faced large losses and liquidity and insolvency problems because European banks held a large part of euro-area government bonds. In addition, the banking system had feed-back effects in aggravating the crisis which degenerated on a general downturn with dramatic consequences for countries such as Greece, Spain and Portugal. In a general framework, Allen and Gale (2000) argue that in theory the banking system is highly sensible even to small shocks affecting a short segment of the market because of the tight linkages between banks through their mutual claims. Very recent studies examine contagion in the sovereign CDS market with a special focus on the euro area and the U.S., but the current research has not explored all aspects of this market. A main finding common to many studies is the dominance of the U.S. market and increasing comovoment during crisis periods (Ehrmann, Fratzscher and Rigobon 2011). 4

5 Ang and Longstaff (2011) find a tight relation within the sovereign CDS spread market and between the sovereign CDS spread and the financial system with an asymmetric effect which is stronger in the direction from the financial market versus the CDS market. These links strengthen during crises: the average correlation rises from 39% for the period to 73% for the period. Longstaff, Pan, Pedersen and Singleton (2011) examine in 26 emerging countries, the role of local and global factors in explaining sovereign CDS spreads and come up with the main finding that global factors stemming from the U.S. market are more important than local factors. Aït-Sahalia, Laeven and Pelizzon (2013) use a ump-diffusion process to study contagion in the euro area sovereign CDS market and they uncover asymmetric feed-back effects within the same country and among countries. The data analysis shows that large moves do not occur simultaneously in all countries, but they take a few days to be transmitted. Examining contagion at longer horizons as proposed in Bae, Karolyi and Stulz (2003) should avoid the lag effect. Ehrmann, Fratzscher and Rigobon (2011) study the relation between stocks, bonds, money markets and exchange rates in the U.S. and the euro area and find that there is a close interrelation within asset classes and between markets. Billio, Getmansky, Lo and Pelizzon (2011) investigate the interconnectedness between hedge funds, banks, broker/dealers and insurers and its impact on systemic risk. Their interrelations have been intensified during the last decade with the banking system having a much more important impact. In our model contagion is defined as interconnectedness between markets that is not explained by the value of the fundamentals and investor behavior. Hence, we control for variations in real growth variables, trade and financial indicators, volatility measures, credit ratings, short rates, term and credit spreads. Measures of investor sentiment are also included to control for rational or irrational behavior of individuals. Given the local or global nature of the shocks we control for both types. We expect that the following general results should come out from our analysis. We should find that the contagion coefficient is positive and statistically significant for the cross-asset and the cross-country investigations. The contagion effect should increase abruptly at the onset of the subprime and sovereign debt crises and remain at high level during the corresponding periods. Despite the finding that cross-asset linkages within the same country are stronger than cross-country linkages, we can say that given the international reach of the recent crises, between-country contagion should be almost as important as within-country contagion. We expect an accurate identification of the crises periods which are defined endogenously. In line with the current literature the contagion effects should be asymmetric with a dominant role of the U.S. market during the subprime crisis and of the most affected markets such as Greece and Spain during the euro area sovereign debt issue. This paper is organized as follows. In Section 2 we present the nonlinear SEM developed by Pesaran and Pick (2007) and further extend to render it more flexible. In Section 3 we describe the data used and in Section 4 we empirically test our model. Section 5 provides some concluding remarks and suggests further work along these lines. 5

6 2 The econometric model In this section we present the nonlinear Simultaneous Equation Model (SEM) for contagion developed in Pesaran and Pick (2007). We then extend it to a more flexible model where the 0-1 contagion variable is replaced by a smooth monotone and bounded function. The main obective is to allow the transmission of shocks, after controlling for global and local factors, to be gradual and estimate the degree of smoothness along with the location parameter which defines the threshold between the normal and the crisis regime. The model presented in this paper is general and it accommodates the Pesaran and Pick (2007) framework as a special case. In high frequency data, such as for instance daily data, it seems unlikely that a shock is simultaneously transmitted from one country to another one. The lag effect should be reduced if the frequency is lower. We start by presenting the two-country model and subsequently extend it to a multi-country setting. Next, we present the estimation procedure and discuss about the tests for contagion. 2.1 Contagion in a two-country setting We first present the two-country nonlinear SEM model for contagion of Pesaran and Pick (2007) and then extend this model to the more flexible setup of Smooth SEM. y 1t = δ 1z t + α 1x 1t + β 1 1(y 2t > c 2 σ 2,t 1 ) + u 1t (1) y 2t = δ 2z t + α 2x 2t + β 2 1(y 1t > c 1 σ 1,t 1 ) + u 2t (2) where y it for i = 1, 2 and t = 1,..., T is a performance indicator such as return, interest rate, spread, z t and x it are p- and k i -vectors of pre-determined global and country-specific indicators, respectively. z t contains a column of ones associated to the constant term and x it can contain lags of the dependent variable, leading to a dynamic system. u it is a serially uncorrelated error term with zero mean and conditional variance σ 2 u i,t. The correlation between u 1t and u 2t is given by ρ and is an indicator of market interdependence. Despite the constancy of the latter correlation, Pesaran and Pick (2007) show that the Corr(y 1t, y 2t ) can be time-varying. 1( ) is an indicator function of pure contagion effect for values of y t greater than the time-varying threshold c σ,t 1. The shock is assumed to be transmitted from country to country i if y t exceeds this threshold. c is a country-specific threshold parameter to be ointly estimated with the other parameters of the model and σ,t 1 2 is the conditional variance of the dependent variable given by Equation 3 with Ω t 1 the available information until t 1. σ,t 1 2 = Var(y t Ω t 1 ) (3) A shock that hits, let us say country 2, will be transmitted to country 1 if y 2t is above the value of the threshold level of c 2 σ 2,t 1 and its impact is measured by the contagion coefficient β 1. If y 2t remains below the threshold no shock is transmitted. The dummy variable constrains the contagion variable to be 1 whatever the strength of the shock above 6

7 the threshold from another country. In contrast, the main obective of this paper is to render the transmission of shocks more flexible. To this aim the 0-1 contagion variable is replaced by a continuous variable which is a monotone function of the strength of the shock. More specifically, we replace the indicator function by a bounded function between 0 and 1 and allow a smooth transmission of shocks going from 0 if the value of y 2t is far below the value of the threshold parameter, to 1 if y 2t is far above the threshold parameter. The model can be written as follows: y 1t = δ 1z t + α 1x 1t + β 1 G(y 2t ; γ 1, c 2 ) + u 1t (4) y 2t = δ 2z t + α 2x 2t + β 2 G(y 1t ; γ 2, c 1 ) + u 2t (5) where G(y t ; γ i, c ) is the transition function (contagion variable), γ i is the smoothness parameter designing whether the transmission of contagion from country to country i is rather abrupt or smooth, and c is the threshold level. In a very large number of studies concerned with the Smooth Transition Regression (STR) models the transition function is defined to be the logistic distribution. 4 This distribution, given in Equation 6, can be suitably used in our framework. G(y t ; γ i, c ) = {1 } + e γ 1 i(y t c σ,t 1 ) γ i > 0, i, = 1, 2, i. (6) The logistic distribution satisfies the requirement of a bounded function between 0 and 1 and is monotonically increasing in the values of y t. It possesses continuous bounded first and second derivatives of y t for γ i < + (Amemiya 1974; Eitrheim and Teräsvirta 1996). Moreover, the logistic distribution is a very close approximation of the normal distribution and it can be used in the cases where the computationally-heavy normal distribution is an appropriate alternative (Luukkonen, Saikkonen and Teräsvirta 1988). Chan and Tong (1986) point out that the normality of the transition function is not a necessary assumption. The SSEM model composed by Equations 4 and 5 nests the model of Pesaran and Pick (2007) (Equations 1 and 2) as a special case when γ i + for all i. Indeed, for large values of γ the logistic distribution converges to a heavyside function. If y t = c σ,t 1 then G( ) = 0.5 which can give a rationale to define c as a threshold between a normal regime and a crisis regime. Indeed, this is the interpretation that is very often given to c in the switching-regime literature. Figure 3 shows how the steepness of the logistic distribution changes with the values of the smoothness parameter γ. For a value of γ equal to 10 the function is already very close to a step function. 2.2 Contagion in a multi-country setting Pesaran and Pick (2007) generalize the SEM model for contagion to a multi-country setting in the following way: 4 For a review of the literature see van Dik, Teräsvirta and Franses (2002), González, Teräsvirta and van Dik (2005). 7

8 y it = δ iz t + α ix it + β i ζ it + u it, i = 1,..., N, (7) where the contagion variable which we denote by ζ it is defined in two distinct alternatives: Alternative 1: Alternative 2: ζ it = ζ it = I N =1, i N =1, i w i 1(y t > c σ,t 1 ) (8) I(y t > c σ,t 1 ). (9) In Alternative 1 the weights are defined such that w i 0 and N =1, i w i = 1. An important issue at this stage is the definition of the weights. Allen and Gale (2000) suggest three different weighting schemes, two of which assume an ordering of the countries and the third one is an equal weights scheme. Alternative 2 is a simple way to model contagion, the outer indicator function taking the value 1 if the sum is positive. The drawback is that this variable takes the value 1 when either one, or more than one country is hit by a shock (y t exceeds the threshold). To model the contemporaneous correlation at the individual dimension Pick (2007) assumes that u it follows a common factor structure of the following form: u it = λ i f t + ε it, (10) with λ i being a scalar parameter, the common unobserved factor following f t i.i.d.n(0, 1) and the remaining error term following ε it i.i.d.n(0, σε). 2 We now present the Smooth SEM model for N countries which can be written as: y it = δ iz t + α ix it + β i ψ it + u it, i = 1,..., N, (11) where ψ it is the continuous and bounded contagion variable that we suggest to define in the three following alternatives. Alternative 1: Multivariate logistic distribution. ψ it = 1 + N =1, i e γ(i) (y t c (i) σ,t 1) Alternative 2: Weighted average of univariate logistic distributions. ψ it = N =1, i 1 (12) w i G(y t ; γ (i), c (i) ) (13) Alternative 3: Univariate logistic distribution for the weighted average of y. ψ it = {1 + e γ i(yit c i σ )} 1,t 1 (14) where y it = N =1, i w iy t, σ 2,t 1 = Var(y t Ω t 1) and as previously w i 0, and 8

9 N =1, i w i = 1. The superscript (i) in Alternatives 1 and 2 indicates that the smoothness and threshold parameters are country-specific. For instance, if N = 3, a shock from country 2 is not necessarily transmitted at the same threshold value, and with the same degree of smoothness, to country 1 and country 3. Such a specification permits the country-by-country estimation of the model. Alternatives 1 and 2 require the estimation of N 1 smoothness parameters and N 1 location parameters for each equation. If N is large, the numerical procedure for the estimation of these 2(N 1) parameters can become computationally heavy and imprecise. A way to simplify the model is to assign a data-based value to every parameter c (i) and estimate the smoothness parameter γ (i). A further simplification would be to impose that γ (i) = γ (i) for all. This specification would give an estimate of an average of smoothness parameters for each country. In Alternative 3 the weights can either be defined in an ad hoc manner (e.g. (i) 1/(N-1), (ii) the neighbors have a bigger weight, (iii) the weight is proportional to a country s fundamentals such as GDP or trade), or they can be optimally estimated with the other parameters of the model. We need to do further research about the multi-country specification of the SSEM. 2.3 Consistent estimation of the contagion coefficients The contagion variable ψ it in Equation 11 is a nonlinear function of the endogenous variables of the system, leading to an endogeneity issue. The endogeneity is present even if the error terms from the different equations are uncorrelated. Our Smooth SEM model is also a nonlinear function of the parameters of the logistic distribution. Keleian (1971) and Amemiya (1974) show that the parameters of nonlinear SEM models can be consistently estimated by a Nonlinear Two-Stage Least Squares (NL2SLS) procedure. Given that the regressor ψ it is a nonlinear function of the endogenous variables, Keleian (1971) suggests that at the first stage of the estimation procedure ψ it should be regressed on a polynomial of the instrumental variables in order to improve their strength. The approximation should improve as the order d of the polynomial increases. Newey (1990) suggests nonparametric methods to provide estimates of optimal instruments. The consistent estimation of the parameters for each single equation does not require the full specification of the system. On the other hand, the use of an instrumental variable estimation method requires that each equation of the system be identified. The global factors, being present in each equation, cannot be used as instruments. Thus, the identification condition needs the presence of exogenous country-specific factors which will be used as valid instruments for the contagion variable. This requirement is satisfied in our framework. Let us denote by φ i = [δ i, α i, β i] the set of the parameters on which each equation of the system depends linearly, by h it = [z t, x it, ψ it] the set of explanatory variables for individual i and period t given in Equation 11, and by w t = [z t, x 1t, x 2t,... x Nt ] the set of instruments to be used in each equation. Below we describe the NL2SLS estimation procedure. Step 1: Take a first guess for γ (i) and c (i) for i denoted by ˆγ (i) value of the contagion variable ψ it denoted by ψ it. and ĉ (i). Estimate the 9

10 Step 2: Regress ψ it on a d-order polynomial based on all the instruments contained in w t and compute the fitted value ˆ ψit. ψ it = π i0 + π i1 w 1t + + π iq w qt + + π ik w d qt + ε it, (15) where q = p+k 1 + +k N is the total number of instruments in w t, K = d q and ε it is the remaining error term. Let us denote by π i = [π i0, π i1,..., π ik ] the (K + 1)- vector of parameters to be estimated in Equation 15, M t = [1, w 1t, w 2t,..., w d qt] the set of explanatory variables for period t. By stacking all the time periods we obtain M = [M 1,... M T ] and ψ i = [ ψ i1,... ψ it ]. The OLS estimate of π i is given by the following formula: ˆπˆπˆπ i = ( M M ) 1 M ψi (16) The fitted value of ψ i is given by ˆ ψi = Mˆπˆπˆπ i. Step 3: Regress y it on h h hit = [z t, x it, ˆ ψit ] to obtain estimates of φ i using the OLS estimation method with H H Hi = [ h h hi1,..., h h hit ] and y i = [y i1,..., y it ]. 5 ˆφˆφˆφ i = ( H H H i H H Hi ) 1 H H H i y i (17) Step 4: Compute the residual sum of squares SSR i = T t=1 û2 it which defines our obective function to be minimized. Apply a numerical procedure to find the values of γ (i) and c (i) that minimize the obective function. Step 5: Take the optimal values of γ (i) and c (i) found in Step 4 as initial values in Step 1 and repeat Steps 2 to 4 until the convergence criterion is met. Report the optimal values of ˆφˆφˆφi and ˆγ (i) and ĉ (i). Keleian (1971) and Amemiya (1974) show the consistency of the estimates for the overall set of parameters and their asymptotic normality under certain regularity conditions. van Dik, Teräsvirta and Franses (2002) point out the difficulty of a precise estimation of the smoothness parameter γ for sufficiently large values. Indeed, when this parameter is large, the logistic distribution is very close to a step function and a big variation of this parameter will not substantially affect the shape of the transition function. A precise estimation of the smoothness parameter requires a large number of observations around the location parameter and a lack of significance does not suggest that the model is linear (Teräsvirta, Tøstheim and Granger 2010). Nevertheless, these authors argue that a precise estimation of this parameter is not necessary for an accurate estimation of the other parameters of the model. We employ a grid search procedure to find appropriate initial values for γ and c and quasi-newton and two heuristic methods, namely, simulated 5 If we denote by û it the residual from Equation 11 using ˆφˆφˆφi and ψ it, the White-corrected estimate of the covariance matrix of ˆφˆφˆφi is given by ˆV ( T ) 1 ( (ˆφˆφˆφi) = t=1 h h hit h h h T T ) 1. it t=1 û2 it h h hit h h h it t=1 h h hit h h h it 10

11 annealing and Nelder-Mead algorithm to minimize the obective function in Step 4. We conduct a sensitivity analysis concerning the performance of each of them. 2.4 Testing for contagion As part of the model specification an important step is to test whether the contagion effect is significant. For simplicity, we first consider the two-equation Smooth SEM model given by Equations 4 and 5. The results for the N-equation model are developed thereafter. Let us consider the first equation of the two-equation model. The test for contagion is equivalent to testing the null hypothesis of β 1 is equal to 0 (H 0 : β 1 = 0) against the alternative that β 1 is different from 0 (H a : β 1 0). Under the null hypothesis, the smoothness parameter γ 1 and the threshold parameter c 2 are not identified and can take any value. The null hypothesis can alternatively be set as H 0 : γ 1 = 0, in which case the logistic distribution becomes a constant. 6 In this case, neither β 1, nor c 2 are identified. This issue is known under the name of nuisance parameter, the presence of which renders the distributions of the conventional tests non-standard under the null hypothesis. The analytical expressions of these tests are often not available and simulation techniques must be used (van Dik, Teräsvirta and Franses 2002). The unidentified nuisance parameter issue was first considered by Davies (1977). The strategy adopted in the Smooth Transition Regression (STR) models to sidestep the nuisance parameter issue is a Taylor expansion of an appropriate order around γ 1 = 0 first proposed by Luukkonnen, Saikkonen and Teräsvira (1988). Under this transformation, the standard tests such as LM and LR test statistics follow asymptotically their conventional chi-squared distribution. Moreover, the test is a general test of nonlinearity in the sense that it has power against other nonlinear additive models which yield the same auxiliary regression (González and Teräsvirta 2006). Applying the first-order Taylor approximation to G(y 2t ; γ 1, c 2 ) around γ 1 = 0 yields the following auxiliary regression: y 1t = β 1,0 + δ 1 z t + α 1x 1t + β 1,1y 2t + u 1t (18) where β 1,0 = β 1 ( c 2γ 1 ) + δ1,1, β 1,1 = 1 4 β 1γ 1 and u 1t = β 1 R 1 (y 2t ; γ 1, c 2 ) + u 1t. R 1 (y 2t ; γ 1, c 2 ) is the remainder of the Taylor approximation. z t is obtained by taking out from z t the first element associated with the constant term δ 1,1. In the auxiliary regression, the null hypothesis of no contagion can be set as H 0 : β 1 = 0 which is equivalent to having γ 1 = 0. This equation is free of nuisance parameters. Under the null hypothesis u 1t = u 1t since R 1 (y 2t ; γ 1, c 2 ) = 0, though not affecting the asymptotic properties of the test (González, Teräsvirta and van Dik 2005). Note that the linearization converts the testing for contagion into a testing for simultaneity within a linear SEM model. Below we give the necessary steps to perform the testing procedure. (i) Regress y 1t on [z t x 1t ] using the OLS estimation procedure. Take the residual û 1t for t = 1,..., T and compute the residual sum of squares SSE 0 = T t=1 û2 1t. 6 This constant is equal to 0.5 but the transition function can be easily rescaled such that it takes the value 0 when the smoothness parameter is equal to 0 (see Teräsvirta 1994). 11

12 (ii) Regress y 1t on [z t x 1t y 2t] using the IV estimation procedure and w it as set of instruments. Take the residual denoted by û 1t for t = 1,..., T and compute the residual sum of squares SSE 1 = T t=1 û 2 1t. (iii) Compute the test statistic as: S 1 = T (SSE 0 SSE 1 )/SSE 0. The S 1 test follows asymptotically a standard χ 2 1. High values of S 1 above the critical value show evidence against the null hypothesis of no contagion. González and Teräsvirta (2006) state that the S 1 test has good size and power properties. For small samples the F version of the test is found to have correct size compared to the LM test which is often oversized. Teräsvirta (1994) documents that the F test has power against the limiting case where the contagion variable is a dummy. The testing procedure does not require the estimation of the nonlinear smooth SEM model, which can be computationally costly. For a system with N equations the auxiliary regression for the i th general equation, and using Alternative 1 for the definition of ψ it, can be written as follows: y it = β i,0 + δ i z t + α ix it + N =1, i β i,y,t + u it, i = 1,..., N, (19) where β i,0, β i, are functions of the parameters of the Smooth SEM for equation i and u it = β 1 R i1 + u it with R i1 the remainder term of the first-order multivariate Taylor expansion. R i1 is a function of the variables y t and of the parameters γ (i), c (i) for = 1,..., N and i. The null hypothesis of no contagion can be written as H 0 : β i, = 0, i against the alternative that at least one of them is different from zero. This is equivalent to setting the null that γ (i) = 0, i against the alternative that at least one of them is different. As previously, the auxiliary regression in Equation 19 is a standard linear SEM model. For Alternative 2 the auxiliary regression is identical to Equation 19 except that the sum is weighted by w i. For Alternative 3 there is ust one parameter to be tested similar to Equation Other tests Test of no error autocorrelation See Teräsvirta, Tøstheim and Granger (2010), page 382 and the notes Test robust to heteroskedasticity See Teräsvirta, Tøstheim and Granger (2010), page 70 and the notes. 3 Data Our data includes the core euro area countries such as Germany, France, Italy and U.S., U.K and Switzerland. We dispose data on the performance of the banking sector, on sovereign debt spread and on the sovereign CDS premia of these countries. We consider 5- and 10-year maturity CDS spreads based on the evidence that they are the most frequently 12

13 traded. The CDS are derivative instruments providing insurance to bond holders in case of a default of the debt issuer - firm or sovereign (see Ang and Longstaff 2011; Aït-Sahalia, Laeven and Pelizzon 2013). The protection buyer pays a premium interchangeably called the CDS spread, CDS price or CDS rate. If the default occurs, the protection seller delivers the market value of the bond (through physical or cash settlement) as it was when the contract was signed. In the determination of the CDS spread, the rating of the underlying bond plays a crucial role. For this reason a measure of credit rating appears in the set of local factors. In the set of global and local variables which control for shocks stemming from the value of the fundamentals or from investor behavior we include variables such as short-term rates, term spread, default spread, market volatility measures, investor sentiment and market liquidity proxies. Other global and country-specific variables may also be included in the simultaneous equations system. We restrict our sample to the period from January 1999, which corresponds to the onset of the euro area, to December The period is shorter when the CDS market is considered because of data unavailability for the period previous to The data are sourced from Datastream. In Figure 1 are displayed the sovereign CDS premia for France, Germany, Greece, Italy, Portugal, Spain, U.K., and the U.S. The CDS spread increased significantly after the peak of the financial crisis in October For the euro area countries that were the most affected by the sovereign debt crisis such as Greece and Portugal the spread reach dramatically high levels and end up with the default of Greece in March Figure 2 shows the annual return of the banking sector at the aggregate level for the following countries: France, Germany, Greece, Italy, Portugal, Spain, Switzerland, U.K., U.S and the world indicator. One can easily distinguish three deep drops for all the countries represented corresponding to the burst of the dot-com bubble, the financial crisis and to the euro area sovereign debt issue. Especially, the global reach of the euro area crisis is a witness of the magnitude of the contagion phenomenon. 4 Estimation results This version is a very preliminary draft. The estimation results are not yet available. 5 Conclusion Not yet available. References [1] Aït-Sahalia, Y., Cacho-Diaz, J., Laeven, R. J. A., Modeling Financial Contagion Using Mutually Exciting Jump Processes. Working paper. [2] Aït-Sahalia, Y., Laeven, R. J. A., Pelizzon, L., Mutual Excitation in Eurozone Credit CDS. Working paper. 13

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18 Figure 1: Sovereign CDS spread for Euro area countries, U.K. and U.S. Notes: 5-year sovereign CDS spread in basis points and daily frequency. Left scale: France (FR), Germany (DE), Italy (IT), Portugal (PT), Spain (SP), U.K., U.S. Right scale: Greece (GR). Figure 2: Annual return of the banking sector index Notes: Annual return of the banking sector index for France (FR), Germany (DE), Greece (GR), Italy (IT), Portugal (PT), Spain (SP), Switzerland (SW), U.K., U.S., World (WD). 18

19 Figure 3: Logistic distribution with different values of γ Notes: In this figure is displayed the logistic distribution with different values of γ and c = 0: G(x; γ, c) = ( 1 + e γ(x c)) 1. 19