Coexceedances in Financial Markets - A Quantile Regression Analysis of Contagion

Coexceedances in Financial Markets - A Quantile Regression Analysis of Contagion Dirk Baur European Commission, Joint Research Center T.E.R.M, Ispra (VA), Italy Niels Schulze Department of Economics, University of Tuebingen, Germany September 3 Abstract This paper analyzes simultaneous exceedances (coexceedances) of several stock index returns for different thresholds with a focus on the Asian crisis in 1997. We introduce a new concept of computing and estimating time-varying coexceedances and use the quantile regression model to analyze contagion. We find contagion (i.e. more and larger coexceedances) during the Asian crisis among countries and also across regions. The results are based on a comparison of coexceedances of the full period to a benchmark period conditional on certain regimes of extreme market movements. An analysis of the evolution of coexceedances also reveals the time-varying character of contagion. Conditional density estimates additionally point to the existence of multiple equilibria in crisis periods. JEL Classification: C1, C5, G1 Keywords: Financial Crises, Contagion, Coexceedances, Quantile Regression Email: dirk.baur@jrc.it Email: niels.schulze@uni-tuebingen.de The authors thank Gerd Ronning as well as participants of the 15th Banking and Finance Conference in Sydney () and of the Econometric Society North American Summer Meeting in Chicago (3) for helpful comments on an earlier version. 1

1 Introduction The analysis of the interdependence of financial markets is an important topic and of special interest in times of market turmoil or financial crisis. One way to analyze the interdependence of financial markets in normal and crisis times is to investigate the correlation of financial markets with Pearson s correlation coefficient (see King and Wadhwani, 199, Lee and Kim, 1993, Calvo and Reinhart, 1996, Baig and Goldfajn, 1999 and Forbes and Rigobon, among others). Other approaches analyze interdependence within a regression model (e.g. Hamao et al., 199, Edwards, 1998, Corsetti et al., 1) or by estimating co-integrating relationships (e.g. Cashin et al., 1995). When discussing the issue of changing interdependencies of financial markets during certain periods, the measures used to estimate these changes and the terms to describe these changes play a crucial role. However, the expressions commonly used are rarely defined explicitly. For example, Forbes and Rigobon () use the terms cross-market linkage, correlation and comovement without defining them nor distinguishing between them. Since measures of contagion are usually based on such terms, the clarity of the underlying expressions is fundamental for the evaluation of the adequateness of the approach to measure contagion. One common concept to detect contagion in financial markets 1 is defined as the increase of the correlation between index returns in a period of market turbulence compared to a normal situation (see Baig and Goldfajn, 1999 and Forbes and Rigobon, ). In this case, the common relation changes in a direction that questions the whole concept of portfolio diversification. Decreasing comovement or interdependencies would be more desirable as downward movements would be less common among all financial markets. We argue that this concept of contagion based on the correlation coefficient is inadequate for the following reasons: First, the correlation coefficient is not an adequate measure of comovement or interdependence and difficult to interpret due to its sensitivity to heteroscedasticity (see Boyer, Gibson and Loretan, 1999, Forbes and Rigobon, and Longin and Solnik, 1). Second, the correlation coefficient is a linear measure which is inappropriate if contagion is not a linear phenomenon but rather an event that is characterized by nonlinear changes of market association (see Bae et al., 3). Thus, we base our analysis of contagion on another measure to assess the joint movement of financial markets. We use coexceedances introduced by Bae et al. (3) to evaluate any changes in the comovement of financial markets and thus test the hypothesis whether there is increased comovement (i.e. contagion) among financial markets in par- 1 A list of different definitions is provided by the World Bank (http://www1.worldbank.org/ contagion/definitions.html) Embrechts et al. () summarize the shortcomings of the correlation coefficient and discuss other types of dependence measures.

ticular periods of market turbulence or not. Coexceedances are joint exceedances of two financial market returns below or above a certain threshold. This measure circumvents the problems associated with the correlation coefficient: coexceedances (i) are not biased in periods of high volatility 3, (ii) can easily be computed (for every point of time t) and (iii) are not restricted to model linear phenomena. We contribute to the literature in three ways: first, we introduce coexceedances for every point of time t without arbitrarily categorizing the coexceedances, second, we use the Quantile Regression (QR subsequently) model of Koenker and Bassett (1978) to analyze the behavior of extreme coexceedances for different regimes of coexceedances. Third, we use the model to detect contagion among financial markets by additionally analyzing certain crisis periods. Finally, conditional quantile estimates show the evolution of coexceedances through time and conditional density estimates point to the existence of multiple equilibria. The remainder of the paper is organized as follows: section two describes the computation of time-varying coexceedances and analyzes their behavior, section three introduces the econometric framework for analyzing coexceedances, section four introduces the data set used and section five presents the empirical results. Section six concludes. Coexceedances The term coexceedance has been introduced by Bae et al. (3) and is defined as the joint occurrence of two exceedances (i.e. large absolute returns above a certain threshold) of two financial market returns at a certain point of time t. Bae et al. (3) use the five-percent tails of the overall (unconditional) return distribution as a threshold that defines an exceedance. Positive and negative returns are treated separately. Counting the number of joint occurrences for the return pairs available reveals the number of coexceedances and also shows whether coexceedances are symmetric or asymmetric in the positive and negative tail. We propose another approach that does not arbitrarily choose a certain threshold but computes coexceedances with a varying threshold for every point of time t. The potential coexceedance at time t of two return pairs r 1 and r is computed as follows: min(r 1t,r t ) if r 1t > r t > Φ t (r 1,r ) = max(r 1t,r t ) if r 1t < r t < (1) otherwise 3 This does not mean that coexceedances are not sensitive to heteroscedasticity. 3

where Φ t is the coexceedance at time t. 4 We use the term potential since the computed value is not necessarily above a prespecified threshold; it is just the data-related coexceedance at time t. It can be shown that the coexceedance Φ t is related to lower and upper tail dependence (see for example Joe, 1997) in the following way 5 Prob(Φ t a) =Prob( i : r i a) () which is equal to lower tail dependence if the scalar a is sufficiently small or if u for a = F 1 Φ (u). The measure is also related to copulas as given by Prob(Φ t a) =H( i : r i a) =C( i : F ri (a)) (3) where H denotes the joint distribution function of the returns r i, C is the copula function which is equal to the Frechet upper bound (see Nelson, 1999, for an introduction to copulas) and F ri is the marginal distribution function of return r i. 5 5 % quantile of r1(x axis >= 5) and r(y axis >= 4) 4 3 1 45 Line coexceedance> 1 coexceedance< coexceedance= 3 4 8 6 4 4 6 8 1 5 % quantile of r1(x axis <= 4.5) and r(y axis <=.5) Figure 1: Construction of coexceedances Figure 1 aims to clarify our approach. Coexceedances are always located on the 45 degree line by construction: a return pair (-1,-1) leads to a coexceedance Φ t = 1 since both returns have the same value. A return pair (-,-3) leads to a coexceedance Φ t = since the maximum of both shocks is. Two positive shocks (,3) lead to a coexceedance 4 The coexceedance Φ t of two random variables u and v could also be formulated as follows: Φ t(u, v) = min(max(u, v), ) + max(min(u, v), ) 5 Φ t a is a subset of the space Ω that contains all possible realizations of Φ. 4

Φ t =since the minimum of these shocks is. Return pairs with different signs as e.g. (3,-1) lead to a coexceedance Φ t =since there is no joint exceedance for any threshold. Figure 1 also contains an example of the coexceedances proposed by Bae et al. (3). Their measure is computed for a given threshold, e.g. the five percent smallest and the five percent largest returns. Assumed that the lower five percent of two return series r 1 and r are given by r 1 4.5 and r.5, respectively, and the upper five percent of r 1 and r are characterized by r 1 5 and r 4, the coexceedances are defined by the two areas that are located in the left bottom part of the figure and the right upper part of the figure, respectively. To further illustrate our approach, we compute a simple example: r 1 and r are two simulated GARCH-processes generated from two normally distributed returns with mean zero, variance one and a constant correlation of.6 6. The coexceedance Φ t (r 1,r ) of r 1 and r and its characteristics are given in table 1. Figure exhibits the relationship between the returns and the coexceedances. The distribution (with and without values equal to zero) of the computed coexceedance is shown in figure 3 which also contains a plot of the unconditional quantiles of the coexceedances. Table 1: Example variable obs mean std dev min max skewness kurtosis r1 1.16.9714-4.6433 4.135.88 3.499 r 1.3.9898-4.599 5.464 -.33 3.755 coex 1 -.7.6368 -.7997 3.411.468 5.454 coex 71 -.9.766 -.7997 3.411.43 3.5369 r 4 4 4 4 r1 Coexceedance of r1 and r 4 4 4 4 r1 Figure : Scatter diagrams of r versus r 1, and of Φ(r 1,r ) versus r 1 The main advantage of this method is that we do not need to arbitrarily define a threshold to categorize the coexceedances. More generally, coexceedances are superior to the correlation coefficient if non-linearities in market behavior exist since coexceedances as com- 6 The conditional volatility was generated by h t =.1 +.5ε t 1 +.94h t 1. 5

Fraction.1..3.4 4 4 Coexceedance Fraction..4.6 4 4 Coexceedance (values unequal to zero) Coexceedance of r1 and r 1 1..4.6.8 1 Quantile Figure 3: Histogram and (unconditional) quantiles of Φ(r 1,r ) puted above are not restricted to describe linear market behavior. This can be explained by the fact that the coexceedances differentiate among the quadrants of a joint distribution which is not true for the correlation coefficient (see for example Pindyck and Rubinfeld, 1998, page 4). We also argue that coexceedances describe interdependence and comovement more adequately than the standard correlation coefficient. For example, the coexceedance of two positive shocks of 5 percent is smaller than the coexceedance of two positive shocks of 1 percent. Given that the probability of larger shocks decreases for most empirical distributions, we assume that higher coexceedances also indicate higher comovement. Furthermore, two positive shocks of 5 and 1 percent (1 and percent) lead to the same coexceedance as two positive shocks of 5 percent each (1 percent each). This means that increased inequality of shocks does not lead to increased comovement since markets do not move more coordinated. This is not the case for the correlation coefficient which always increases if standardized shocks increase, even if they become less similar. 3 Estimation Framework Bae et al. (3) analyze coexceedances within a multinomial logistic regression framework with different categories depending on the number of coexceedances in order to investigate the influence of exogenous variables like exchange rates, interest rates and volatilities on the evolution of these coexceedances. Due to the use of a multinomial logistic regression, only the number of coexceedances is modelled by categories, but the actual values of these coexceedances are not used in the estimation process. In contrast, we propose to estimate coexceedances (as defined in this paper) with the Quantile Regression model (Koenker and Bassett, 1978) 7 in order to account for the ordinal nature of the measure. An additional advantage of the QR model is that no distributional assumptions have to be made as for example in Bae et al. (3) or in applications of Extreme Value Theory (see Longin and Solnik, 1). 7 see the appendix for details 6

The use of the Quantile Regression (QR) model to analyze extreme coexceedances, i.e. extreme negative and positive coexceedances, enables us to analyze any of the q-percent lowest or highest coexceedances without prespecifying any distribution or threshold. On the contrary, we can estimate the threshold for the q-percent lowest or highest coexceedances and analyze any factors that explain these coexceedances like e.g. the return and the volatility of a regional or global factor, exchange and interest rates, a dummy variable capturing the (Asian) crisis period or lagged values of the coexceedances. Analyzing extreme joint market movements with the QR model is an adequate alternative to other approaches like Extreme Value Theory (see Longin and Solnik, 1, among others), copulas (e.g. see Hu, 3) and related dependence measures (e.g. Embrechts et al.,, for an overview). A simple quantile regression equation is given as follows: Φ = Xβ(q) (4) where Φ denotes the (n 1) vector of the coexceedances, X is a (n k) matrix of k exogenous variables and β(q) represents a (k 1) parameter vector. Note that different coefficients are obtained for every desired quantile q. 3.1 Analysis of Contagion As outlined in the introduction, we identify contagion as a change of the interdependence of financial markets. Any increase of the interdependence in a period of market turbulence compared to a normal situation is referred to contagion (see Baig and Goldfajn, 1999 and Forbes and Rigobon, ). Using the coexceedances as a measure of interdependence, we can answer this question without facing the shortcomings of the correlation coefficient. Since we want to focus on the question whether the number or magnitude of coexceedances increases in a crisis period compared to a normal situation (in a given quantile), we estimate the following equation: Φ t = c(q)+β 1 (q)dt crisis (5) where Φ t is the coexceedance at time t, c(q) is a constant, β 1 (q) is the parameter estimating the effect of the dummy Dt crisis which is one if t is in the crisis period and zero otherwise. q again refers to the (conditional) quantile under investigation. For small quantiles such as the 1-, 5- or 1-percent quantile, a parameter β(q) significantly smaller than zero implies that there are larger negative coexceedances during the crisis period compared to the benchmark period. For large quantiles such as the 9-, 95- or 99-percent quantile, a β(q) significantly greater than zero implies that there are larger positive coexceedances in the crisis period compared to the benchmark period. The benchmark period is usually the full sample space. Our approach compares the crisis period and the benchmark period conditional on certain regimes of coexceedances. Since the quantile 7

regression model accounts for different regimes of coexceedances, our approach is more conservative than existing definitions and measures of contagion. Hence, contagion is not detected just because values are usually larger in the tails. In contrast, the approach can detect different degrees of dependence in a crisis period conditional on the structure of the dependence. 8 The simple specification given by equation (5) neglects any other variables that can potentially affect the number of coexceedances like for example the return and the volatility of a regional or global factor, interest rates or exchange rates. Bae et al. (3) use such variables and define contagion as an increase of the number of coexceedances not explained by these covariates. We also use a similar definition and view the model in equation (5) as a benchmark model that provides fundamental information about the behavior of the coexceedance in the crisis period. A model that does not neglect the influence of a regional or global market is given by the following equation: Φ t = c(q)+β 1 (q)d crisis t + β (q)r Mt + β 3 (q)ĥmt (6) where r Mt is the return of a global or regional market and ĥmt is its estimated conditional variance. If β 1 (q) is significantly smaller than zero even after controlling for the effect of a global or regional stock index, then there is evidence of contagion. Eventually, we include the lagged coexceedance as an additional regressor to account for any potential persistence of coexceedances. Thus, the final model that is estimated is given as follows: Φ t = c(q)+β 1 (q)d crisis t + β (q)r Mt + β 3 (q)ĥmt + β 4 (q)φ t 1 (7) where the parameter β 4 (q) captures the effects of the lagged coexceedance Φ t 1. The next section presents the data that is used to estimate the models given by equations (5) and (7). 4 Data We use daily (close-to-close) 9 continuously compounded stock index returns of eleven Asian stock markets calculated in U.S. dollars 1 : China, Hong Kong, India, Indonesia, Japan, South Korea, Malaysia, Philippines, Singapore, Taiwan and Thailand. Furthermore, four regional stock indices are analyzed: Emerging Markets Free Asia 11, Emerging Markets 8 Hu (3) introduced this differentiation between degree of dependence and structure of dependence. 9 We are aware of the potential bias that is introduced by using this type of returns since trading hours are not synchronous. 1 The data is provided by Morgan Stanley Capital International Inc. (MSCI) and can be retrieved under www.mscidata.com. 11 The MSCI Free indices reflect investable opportunities for global investors by taking into account local market restrictions on share ownership by foreigners. 8

Free Latin America, Europe and the USA. The indices span a time-period of four and a half years from April, 3th 1997 until October, 31th 1. The number of observations is T = 1176. Table presents several descriptive statistics for the fifteen time series. It can be seen that the mean is negative for Asian countries and Latin America, but positive for Europe and the U.S. The skewness exhibits a different behavior among the analyzed markets, whereas all returns are (in some cases considerably) leptokurtic. Except for the USA, all autocorrelations are positive. Table : Markets Country median mean std dev min max skewness kurtosis autocorr China -.13 -.14.67 -.1444.174.1333 5.758.1563 Hongkong. -.4. -.1377.161.1767 9.638.34 India. -.3.189 -.73.78 -.1556 4.745.933 Indonesia -.11 -.19.448 -.436.381 -.7614 16.393.137 Japan -.9 -.3.165 -.716.17.4743 6.5383.3 Korea -. -..359 -.167.688.3418 9.5961.156 Malaysia -.8 -.8.38 -.3695.568 -.5733 3.534.868 Philippines -.8 -.13.16 -.136.119 1.76 16.811.181 Singapore -.5 -.7.6 -.13.155.55 8.7563.1483 Taiwan -.9 -.8.4 -.1113.739 -.56 5.58.391 Thailand -.13 -.13.94 -.1489.1644.6388 7.177.1683 EMF Asia -.9 -.9.161 -.753.761.41 5.56.178 EMF La.Am..4 -.3.188 -.1448.137 -.49 11.7116.146 Europe.1.4.143 -.68.619 -.359 4.784.589 USA...17 -.697.488 -.55 5.89 -.3 Table 3 shows the unconditional correlation structure between all variables for the whole time span. One can see that all values are positive, thereby reflecting regional and economic relationships. Table 3: Correlations CHN HON INA IND JAP KOR MAL PHI SIN TAI THA ASI LAT EUR USA China 1. Hongkong.6 1. India.18.1 1. Indonesia.6.35.1 1. Japan.8.36.15.1 1. Korea.5.9.19.17.6 1. Malaysia.7.31.11.33.3. 1. Philippines.31.37.14.38..1.6 1. Singapore.44.61.17.46.38.6.39.44 1. Taiwan.3.6.1.18.19..18.18.8 1. Thailand.31.38.17.38.5.31.37.4.48.3 1. EMF Asia.49.54.45.47.38.63.56.45.58.6.57 1. EMF LA.13.1.1.8.14.18.11.14..7.17.1 1. Europe.19.37.15.1.18.1.13.13.3.14..8.47 1. USA.4.13.5.1.6.1..7.13.3.5.8.58.43 1. Table 4 lists the corresponding values during the crisis period under the assumption that (i) the Hongkong market is the origin of the crisis (October, 17th until November, 17th 9

1997) and, alternatively, (ii) the financial market of Thailand is the crisis-breeding element (July, nd until November, 17th 1997). In most cases, the correlation rises during the crisis in the Hongkong case, whereas the Thailand values decrease, even leading to two negative outcomes. This result already shows that the definition of the crisis period can be crucial. We will show below that conditional quantile estimates help to detect real crisis periods. Note that the time period for the Hongkong crisis is equal to the one used by Forbes and Rigobon (). Table 4: Crisis correlations CHN HON INA IND JAP KOR MAL PHI SIN TAI THA ASI LAT EUR USA HON.6 1..1.35.36.9.31.37.61.6.38.54.1.37.13 HON Crisis.81 1..1.63.47.18.58.67.79.11.1.5.1.83.5 THA.31.38.17.38.5.31.37.4.48.3 1..57.17..5 THA Crisis -.3..19.4.3.5.8.4.14.16 1..48..7 -.8 In order to clarify our approach, table 5 shows the Hongkong and Malaysian returns as well as the calculated coexceedances in exemplary fashion. The upper part of the table contains unstandardized values and the lower part the standardized analogues. We advocate the use of the standardized values in order to ease the interpretation of the results. The last three columns show that the characteristics of the time series are not changed by the standardization. It can be seen that the coexceedances are somewhat less leptokurtic than the original returns, especially if only the values different from zero are considered. The autocorrelations are slightly higher than those of the original returns. Table 5: Example Variable obs median mean std dev min max skewness kurtosis autocorr Hongkong 1175. -.4. -.1377.161.1767 9.638.34 Malaysia 1175 -.8 -.8.38 -.3695.568 -.5733 3.534.868 Coexceedance 1175. -.7.13 -.83.888 -.388 13.65.977 Coexceedance 668 -.5 -.1.163 -.83.888.686 7.5787.96 stdrhong 1175.176. 1. -6.518 7.33.1767 9.638.34 stdrmal 1175.1. 1. -11.968 8.3595 -.5733 3.534.868 coex 1175. -.14.478 -.5785 4.573.4899 15.351.116 coex 76.16 -.173.6169 -.5785 4.573.4135 9.316.143 Figure 4 plots the histogram of the computed coexceedance. Even the middle graph omitting all values equal to zero reveals that the distribution is not normal. The figure also presents the values of the unconditional quantiles of the constructed coexceedances between Hongkong and Malaysia. The question whether joint negative shocks are more common or more pronounced than joint positive shocks is often analyzed with the correlation coefficient (see Ang and Chen,, and Longin and Solnik, 1). Such an analysis can also be performed with the coexceedance measure by analyzing the percentages of positive and negative coexceedances 1

Fraction.1..3.4.5 4 coexstdrhongrmal Fraction.5.1.15 4 coexstdrhongrmal (values unequal to zero) coexstdrhongrmal 1 1..4.6.8 1 Quantile Figure 4: Histogram and (unconditional) quantiles of Φ(r HON,r MAL ) as well as the skewness of the coexceedances. Results are shown in tables 6 and 7, respectively. Table 6 indicates that joint negative shocks are less frequent than joint positive shocks (the difference seems to be bigger for Hongkong than for Thailand). Table 7 shows that joint negative shocks are not generally more pronounced than joint positive shocks. Results are mixed within Asia, whereas the skewness tends to be negative for coexceedances across regions. These findings are evidently counter to the findings in the literature and may partly be explained by the different properties of the coexceedance measure compared to the correlation coefficient. Table 6: Percentages of Coexceedances CHN HON INA IND JAP KOR MAL PHI SIN TAI THA LAT EUR USA negative (HON) 34.1 xxxx 4.3 5.9 8. 8. 8.1 6.6 3.4 7. 8.3 4.8 8.3 5.8 positive (HON) 38.3 xxxx 31. 31.3 3.7 3.3 3.1 31.9 35. 3.9 3.8 31.8 34.4 8.4 negative (THA) 9.9 8.3 5. 8. 9.3 3.3 9.9 9.5 31. 7.8 xxxx 6. 8. 5.7 positive (THA) 3.7 3.8 8.6 3.7 8.4 31. 3.5 31.5 3.1 8.3 xxxx 9.9 3.7 5. Table 7: Skewness of Coexceedances CHN HON INA IND JAP KOR MAL PHI SIN TAI THA LAT EUR USA Hongkong -.7 xxxx -.51 1.9 -.9 -..49.96.53.16.78 -.65 -.54 -.97 HON (coexstd ) -.4 xxxx -.36.81 -.18 -..41.74.45.11.6 -.47 -.41 -.7 Thailand.45.78.51 1.4..7. 1.18.99.8 xxxx -.13 -.56 -.7 THA (coexstd ).38.6.43 1.1..59..95.81.9 xxxx -.5 -.38.4 5 Empirical Results In this section, we present the empirical results of the benchmark model and the full model as introduced in section three. First, an analysis of coexceedances of market indices belonging to the same region (in our case the Asian market) is conducted and second, effects across regions are analyzed. For the coexceedances across regions, we also analyze the evolution of the conditional quantile estimates and present a (to our knowledge) new concept 11

of conditional density estimates. 5.1 Coexceedances within Regions Table 8 lists some selected estimation results for the coexceedance between Hongkong and Malaysia for the crisis period assuming that the Hongkong market is the origin of the crisis. The benchmark model indicates the regression of the coexceedance on a crisis dummy (one during October 17th until November 17th 1997) and a constant. Looking at the first, it can be stated that the coefficient is highly significant in the negative tail implying more and larger coexceedances in the crisis period. This result can be viewed as evidence of contagion since the QR model already accounts for relatively large joint negative shocks. Table 8: Estimation Results for Hongkong and Malaysia Regression q q4 q6 q8 q1 LS q9 q9 q94 q96 q98 Benchmark Model Pseudo-R.53.39.3.3..4.5.8.1.13.1 Dummy -1.374*** -1.787*** -1.341** -1.433** -1.454** -.**.66.613.485 1.83** 1.51* [3.13] [3.7] [.8] [.6] [.3] [.16] [.8] [.74] [.55] [.9] [1.8] Constant -1.5*** -.79*** -.61*** -.59*** -.396*** -.6.374***.449***.576***.71*** 1.3*** [14.33] [9.74] [1.5] [11.94] [9.89] [.44] [11.78] [11.77] [11.] [1.83] [11.67] Full Model Pseudo-R.369.344.31.83.61.363.34.59.86.39.41 Dummy -.776* -.99** -.61 -.677 -.71 -.114 -.15.31.6.35 -.4 [1.9] [.53] [1.47] [1.57] [1.5] [1.35] [.4] [.81] [.67] [1.3] [.1] Emfasia.365***.319***.31***.34***.78***.85***.69***.97***.334***.366***.376*** [9.33] [1.13] [1.4] [11.44] [9.71] [5.15] [1.6] [11.99] [9.38] [8.33] [8.58] Egarch -.143* -.189*** -.146*** -.134*** -.118***.13.114***.19***.13***.16***.1*** [1.93] [4.3] [4.47] [4.6] [4.46] [1.16] [4.39] [3.7] [.87] [.9] [4.9] Coex t 1 -.189 -.35 -.8 -.1 -.37.14 -.1.1.8.81.76 [1.14] [.8] [.3] [.5] [.51] [.59] [.3] [.1] [1.] [.86] [.6] Constant -.865*** -.618*** -.541*** -.475*** -.414*** -.8.374***.46***.51***.648***.8*** [1.93] [15.] [17.96] [15.5] [13.3] [.73] [15.7] [1.58] [9.9] [1.17] [16.96] * indicates that, statistically, the coefficient is significantly different from zero at the 9%-level (** at the 95% level, *** at the 99% level); The t-values are calculated by bootstrapping with replications. The estimates of the full model mainly show that the significance of the crisis dummy coefficient vanishes by including the regional market return EMFAsia, its volatility and the lagged coexceedance. The regional market return and its volatility have a significant influence on the coexceedance of all reported quantiles and seem to capture most of the shocks in the lower quantiles rendering the crisis dummy insignificant for almost all analyzed quantiles. Figures 5 and 6 plot the regression results (coefficients) of the full model for all quantiles. 1 The shaded areas represent the 95% confidence intervals calculated by bootstrapping with replications. The figures show the considerable influence of the regional market return and its volatility and the rather negligible influence of the other covariates. 1 It has to be noticed that due to the construction of the coexceedances (allocation of value zero for opposite returns) no relevant outcomes are to be expected for the middle quantiles (roughly between 3% and 7%). 1

Since all figures also include the least-square estimates (represented by a solid horizontal line), the additional information that is provided by the Quantile Regression in general and also in this application is evident. It is important to stress that the pseudo-r is not comparable with its least-square analogue as it is a local rather than a global measure of goodness of fit (see Koenker and Machado, 1999). Pseudo R (model81rhongrmalfourbs).1..3.4.5..4.6.8 1 Quantile Constant (model81rhongrmalfourbs) 1.5.5 1..4.6.8 1 Quantile crisis (model81rhongrmalfourbs) 1 1..4.6.8 1 Quantile Figure 5: Pseudo-R, Constant and Crisis Dummy (Φ(r HON,r MAL )) stdremfasia (model81rhongrmalfourbs).1..3.4.5..4.6.8 1 Quantile stdegarchremfasia (model81rhongrmalfourbs).4. 5.55e 17..4..4.6.8 1 Quantile lagcoexstd (model81rhongrmalfourbs).6.4. 5.55e 17..4..4.6.8 1 Quantile Figure 6: Market Return, Volatility and Lagged Coexceedance (Φ(r HON,r MAL )) Having analyzed the outcomes for Hongkong and Malaysia, now the other Asian countries are included into the consideration. Furthermore, Thailand is taken as an alternative source of potential contagion. Since we are mainly interested in the effect of the crisis dummy and to simplify and clarify the analysis, tables 9 and 1 provide a summary of the crisis dummy coefficients for all analyzed cases. The tables clearly show that contagion is mainly a phenomenon that can be found in lower quantiles with significant negative coefficients as already seen in the example presented above. An exception for the benchmark model of the Hongkong crisis (upper part of table 9) are India and Thailand that also exhibit significantly negative estimates in the upper quantiles. This implies that the crisis period is clearly characterized by jointly negative shocks. Significantly positive estimates in the upper quantiles can be found for China, Indonesia, Malaysia, the Philippines and Singapore and indicate that crisis periods can also be characterized by extreme positive shocks. Estimation results are less pronounced for the full model but the central conclusions can remain untouched. It is striking, though, that the 4-percent quantile exhibits most of the significant crisis dummy estimates while the 13

Table 9: Hongkong results (crisis dummy) Benchmark Model q q4 q6 q8 q1 q9 q9 q94 q96 q98 China -1.567*** -1.913*** -.1*** -.1*** -,994,399 1,97,95 1.511** 1.15* India -.77 -.431 -.58* -.645* -.564 -.486*** -.55*** -.65*** -.638*** -.86*** Indonesia -1.998* -.378** -.44 -.39 -.41 -.97 1.388 1.65 1.77* 1.49* Japan -.794*** -1.9*** -1.1*** -1.111*** -.938*** -.53.89.778 1.56.748 Korea -1.1*** -1.596*** -1.474** -1.643*** -1.11 -.387.15 -.17 1.36.84 Malaysia -1.374*** -1.787*** -1.341** -1.433** -1.454**.66.613.485 1.83** 1.51* Philippines -.389*** -.748*** -1.685* -1.799* -1.477.53.449.335.475**.46* Singapore -.355*** -.734*** -1.78* -1.97* -1.91.47 1.866 1.74.45**.87** Taiwan -.5*** -.57*** -1.81* -1.97* -1.19 -.46.45.35.897.578 Thailand -.115** -.446** -1.195-1.33 -.689 -.48*** -.398** -.55*** -.577*** -.956*** Full Model q q4 q6 q8 q1 q9 q9 q94 q96 q98 China -1.541** -1.844** -,478 -,516 -,616,8,771,751 1.58* 1.368* India -. -.185 -.4 -.41* -.35.61.54.43.651*.459 Indonesia -.46 -.666 -.177 -.5 -.17.17.84*.619.48 -.13 Japan -.713** -.985*** -.63 -.84** -.37 -.48.9.145 1.17*.87 Korea -.96-1.13* -.44 -.36 -.477.9.86.36 -.75 -.5* Malaysia -.776* -.99** -.61 -.677 -.71 -.15.31.6.35 -.4 Philippines -1.8*** -1.39*** -1.37** -1.46** -1.77.9.377.61 1.876** 1.951** Singapore -.966-1.447** -.969-1.1* -.467.18.13 -.11.938.5 Taiwan -.946** -.966** -.891 -.91 -.47.76 -.1 -.5.38.147 Thailand -.44 -.911** -.5 -.418 -.36 -.86 -.15 -.16 -.93 -.188 * indicates that, statistically, the coefficient is significantly different from zero at the 9%-level (** at the 95% level, *** at the 99% level); The t-values were calculated by bootstrapping with repetitions. Table 1: Thailand results (crisis dummy) Benchmark Model q q4 q6 q8 q1 q9 q9 q94 q96 q98 China -,557 -,4 -,11 -,4 -,114 -,53,1 -,115 -,148,65 Hongkong -.646 -.66 -.3 -.134 -.163 -.199 -.17 -.87 -.5 -.96 India -.38 -.38 -.197 -.18 -.15 -.113 -.3 -.79.177 -.16 Indonesia -.341 -.47 -.3 -.54 -.71* -.14 -.19 -.51 -.98 -.45 Japan -.16 -.49** -.419* -.31 -.3 -.148 -.179 -.131 -.81 -.497 Korea -.34 -.36.15.76.1 -.31*** -.347*** -.49** -.47 -.633 Malaysia -1.7* -.319 -.79 -.35* -.338** -.15 -.144 -.69 -.111.867 Philippines -.669 -.474 -.474** -.55*** -.555***.16.35.5 -.71 -.73 Singapore -.369.3.47 -.53 -.83 -.19 -.14 -.64 -.76 -.116 Taiwan -.757 -.31 -.343 -.168 -.47.157.11.4 -.11 -.9 Full Model q q4 q6 q8 q1 q9 q9 q94 q96 q98 China -.464** -,9 -,98 -,87 -,1,85,151.48*,56.737* Hongkong -.333 -.6 -.11 -.31 -.17 -.48.3 -.5.139.67 India -.177 -.197 -.131 -.14 -.184*.113.16.91*.435**.478* Indonesia -.459 -.13 -.9 -.58 -.57.8 -.13 -.59. -.84 Japan -.111 -.17 -.151 -.11 -.13 -.33 -.4 -.58.339.31 Korea -.11 -.149 -.7* -.13 -.15.38.9.65 -.7 -.11 Malaysia.6 -.18 -.1 -.13 -.116 -.35 -.57 -.5 -.1.349 Philippines -.59 -.34 -.33** -.6* -.19.44.18.1*.34 -.5 Singapore.144.. -.78 -.66 -.17 -.5 -.14 -.5 -.168 Taiwan -.47 -.3 -.74** -.1* -.143.9.11**.157*.147.9 * indicates that, statistically, the coefficient is significantly different from zero at the 9%-level (** at the 95% level, *** at the 99% level); The t-values were calculated by bootstrapping with repetitions. 14

-percent quantile is now mainly explained by the other covariates. The different conclusions depending on the considered quantile is further analyzed below in a subsection that presents the conditional quantile estimates. The results for the crisis assumed to be associated with occurrences in Thailand are in general less pronounced. The insignificance of the coefficient estimates can be partly explained by the longer crisis period compared to the one assumed in the Hongkong case. 5. Coexceedances across regions In this section, we want to answer the question whether contagion can also be found across different regions. Therefore, we calculate coexceedances between the Hongkong market as the assumed crisis origin and several indices, namely the United States, Europe and the emerging markets of Latin America. In this analysis, we use the return and the volatility of the MSCI World index to account for common shocks. Table 11 lists the effect of the crisis dummy in the full model for both crisis periods. Table 11: Coexceedances across regions (crisis dummy, full model) q q4 q6 q8 q1 q9 q9 q94 q96 q98 Hongkong Latin-A. -1.98** -1.446**.4 -.55 -.6.114.668.574 1.146**.738 Europe -.347** -.563** -,97-1,9 -,4,54,45,61.79* 1,374 USA.43 -.31 -.37 -.78 -.346*.315.41.373.638**.375 Thailand Latin-A..16 -.179 -.196** -.9** -.187*.1.1.188.169.16 Europe -.8 -.31 -.38 -.5 -.9.4.49.188.164.51 USA -.135 -. -.65 -.16* -.6.115.8.13*.195.148 The results show that there is a strong impact of Hongkong market shocks on Latin America and Europe at the lowest quantiles. In contrast, the U.S. market seems to be rather used to shocks originating in Hongkong. It is worth stressing that this finding does not imply that the U.S. market is insulated from the Hongkong shocks. It only means that the chosen crisis period does not represent a unique event of extreme market movements. 13 The outcome shows a strong asymmetry since negative coexceedances are larger in the crisis period than positive coexceedances. For the Thailand crisis, there is negligible evidence of any increased coexceedances. Comparing our results with other studies such as Forbes and Rigobon (), Longin and Solnik (1) and Bae et al. (3) reveals the following differences: Forbes and Rigobon () find contagion for the Hong Kong crisis in Indonesia, Korea and the Philippines based on a correlation coefficient not corrected for heteroscedasticity. Using the adjusted (corrected for heteroscedasticity) correlation coefficient, no contagion is found in any 13 This statement can be viewed to be in contract to the conclusions drawn by Bae et al. (3, page 7). 15

analyzed Asian market. Longin and Solnik (1) find that correlations increase in bear markets but not in bull markets. We can state that our findings bear some resemblance to these results since coexceedances tend to be larger in crisis times than in non-crisis times. The study of Bae et al. (3) differs to our analysis since they use a multinomial logistic regression model and focus on the question whether coexceedances are predictable by certain variables and whether the probability of a coexceedance in the negative (or positive) tail is increased by certain variables. However, they do not focus on particular periods of time (i.e. crisis periods) but use the whole sample period for their analysis. The different model and its focus on probability changes restrict us from a direct comparison of the results. 5..1 Evolution of coexceedances In this subsection, we analyze the evolution of the estimated conditional coexceedances between the Hong Kong market and three regional indices. The analysis of the coexceedances over time as recently performed by Chan-Lau et al. () in a similar manner provides information on the questions whether extreme joint market movements have increased in recent years, whether negative movements are more pronounced than positive movements and whether the volatility of such movements has increased or stayed rather constant. We present the quantile estimate of the (1)-percent quantile and the 98(9)-percent quantile of the three coexceedances. Since we use the Quantile Regression model, our approach is clearly different to the Chan-Lau et al. method that computes moving averages of coexceedances with a prespecified threshold. The plots of the estimated conditional -,5- and 98-percent quantiles along with the realized values are given in figure 7. Figure 8 presents the 1- and the 9-percent quantiles of the time-varying coexceedances. In each case, the assumed crisis interval is marked by the shaded area. The figures show that the chosen crisis period can be justified since extreme market movements are clearly pronounced. However, the plots also show that there are other clusters (periods) of extremes, e.g. at the end of the sample. Furthermore, there seems to be a slight positive trend of more frequent extreme market movements. The estimated coexceedance of Hongkong and the U.S. further show that the U.S. cannot be assumed to be insulated from any Asian contagion as concluded by Bae et al. (3, page 7). The plots of the conditional quantile estimates for the Thailand crisis period are given in figures 9 and 1 and show that the crisis period could also be detected endogenously since the figures clearly show that the largest negative and positive coexceedances occurred during the period defined as the Hongkong crisis period which is a subset (the last part) of the Thailand crisis period. This is an important finding and shows an alternative to the crisis detection through outliers proposed by Favero and Giavazzi (). 16

Oct1 coexstdrhongremflatin 5% quantile conditional quantiles (model81rhongrusafour) 4 conditional quantiles (model81rhongreurofourbs) 6 4 4 conditional quantiles (model81rhongremflatinfour) 4 4 Oct1 % quantile 98% quantile coexstdrhongreuro 5% quantile Oct1 % quantile 98% quantile coexstdrhongrusa 5% quantile % quantile 98% quantile Oct1 coexstdrhongremflatin 5% quantile conditional quantiles (model81rhongrusafour) 4 conditional quantiles (model81rhongreurofourbs) 4 4 conditional quantiles (model81rhongremflatinfour) 4 4 Figure 7: Estimated -, 5- and 98-percent quantiles (Hongkong) Oct1 1% quantile 9% quantile coexstdrhongreuro 5% quantile Oct1 1% quantile 9% quantile coexstdrhongrusa 5% quantile 1% quantile 9% quantile Oct1 coexstdrthairemflatin 5% quantile conditional quantiles (model8rthairusafour) 1 1 3 conditional quantiles (model8rthaireurofour) 4 conditional quantiles (model8rthairemflatinfour) 1 1 3 Figure 8: Estimated 1-, 5- and 9-percent quantiles (Hongkong) Oct1 % quantile 98% quantile coexstdrthaireuro 5% quantile Oct1 % quantile 98% quantile coexstdrthairusa 5% quantile % quantile 98% quantile Oct1 coexstdrthairemflatin 5% quantile conditional quantiles (model8rthairusafour) 1 1 conditional quantiles (model8rthaireurofour) 4 conditional quantiles (model8rthairemflatinfour) 1 1 3 Figure 9: Estimated -, 5- and 98-percent quantiles (Thailand) Oct1 1% quantile 9% quantile coexstdrthaireuro 5% quantile Oct1 1% quantile 9% quantile coexstdrthairusa 5% quantile Figure 1: Estimated 1-, 5- and 9-percent quantiles (Thailand) 17 1% quantile 9% quantile

5.. Conditional densities In this subsection, we present conditional density estimates for coexceedances of the Hongkong market with the three regional stock indices as chosen above. We focus on the Hongkong crisis period since the Thailand crisis interval was shown to be too long (see previous subsection). The conditional densities are calculated by fixing the value of one independent variable to several specific numbers and subsequently computing the density of the coexceedance conditional on these numbers. The values of the other covariates are calculated by an auxiliary regression (see the appendix for details). Density of Coexceedane (model81rhongremflatinfour).5 1 1.5 1 1 Density of Coexceedane (model81rhongreurofour).5 1 1.5 4 4 Density of Coexceedane (model81rhongrusafour).5 1 1.5 1.5.5 1 crisis= crisis=1 crisis= crisis=1 crisis= crisis=1 Figure 11: Densities conditional on crisis dummy Density of Coexceedance (model81rhongremflatinfour).5 1 1.5 1 1 Density of Coexceedance (model81rhongreurofour).5 1 1.5 1 1 Density of Coexceedance (model81rhongrusafour) 1 3 1 1 stdrworld at % stdrworld at 1% stdrworld at 9% stdrworld at 98% stdrworld at % stdrworld at 1% stdrworld at 9% stdrworld at 98% stdrworld at % stdrworld at 1% stdrworld at 9% stdrworld at 98% Figure 1: Densities conditional on market return Density of Coexceedance (model81rhongremflatinfour) 1 3 1 1 Density of Coexceedance (model81rhongreurofour).5 1 1.5.5 1 1 Density of Coexceedance (model81rhongrusafour) 1 3 1 1 stdegarchrworld at % stdegarchrworld at 1% stdegarchrworld at 9% stdegarchrworld at 98% stdegarchrworld at % stdegarchrworld at 1% stdegarchrworld at 9% stdegarchrworld at 98% stdegarchrworld at % stdegarchrworld at 1% stdegarchrworld at 9% stdegarchrworld at 98% Figure 13: Densities conditional on market volatility These densities can provide important additional information since they show the distribution of the coexceedance given a certain value of one covariate. This can help obtaining more information about the sources that lead to the occurrences of coexceedances. Figures 11 to 14 show the density estimates for the three pairs of coexceedances conditional on the 18

Density of Coexceedance (model81rhongremflatinfour).5 1 1.5.5 1.5 1.5.5 1 Density of Coexceedance (model81rhongreurofour).5 1 1.5 1 1 Density of Coexceedance (model81rhongrusafour).5 1 1.5.5 1.5.5 1 lagcoexstd at % lagcoexstd at 1% lagcoexstd at 9% lagcoexstd at 98% lagcoexstd at % lagcoexstd at 1% lagcoexstd at 9% lagcoexstd at 98% lagcoexstd at % lagcoexstd at 1% lagcoexstd at 9% lagcoexstd at 98% Figure 14: Densities conditional on lagged coexceedance four regressors of our QR model. The figures show that the density of the coexceedances in the crisis period (D crisis =1) has a lower mean and a higher variance than the density in the non-crisis period (D crisis =). There is also some hint of multimodality of the densities during the crisis period slightly pointing towards the existence of multiple equilibria (see Calvo and Mendoza,, and Kodres and Pritsker, ). The figures further show that low (high) values of the world factor lead to lower (higher) expected values of the coexcedances while the variance of the distribution is relatively unaffected. In contrast, the volatility of the world factor affects the variance of the coexceedance but not the mean of the conditional values: large (small) values are associated with a higher (lower) volatilty of the coexceedance. Finally, it can be stated that different values of the lagged coexeceedance also affect the volatility: negative (positve) lagged coexceedances induce a higher (lower) volatilty. 6 Concluding Remarks In this paper, we have proposed a new approach to compute and analyze coexceedances. The use of coexceedances with time-varying thresholds and its analysis with aid of the quantile regression model has the advantages that (i) we do not need to count the coexceedances or prespecify any threshold, (ii) no distributional assumptions have to be made about the coexceedances or the underlying returns, (iii) the evolution of extreme coexceedances can easily be analyzed by plotting the coexceedances against the quantiles and (iv) the approach can estimate linear and non-linear phenomena and corrects for different states of shock magnitudes. Furthermore, the inclusion of a dummy variable into the quantile regression model enables us to quantify a measure of contagion. Although our approach can be viewed to be rather conservative since we account for the market return, its volatility and also regimes of coexceedances, we still find evidence of contagion for many Asian stock markets and also across regions. 19

References Ang, A., Chen, J. (). Asymmetric Correlations of Equity Portfolios, Journal of Financial Economics 63, 443-494. Bae, K.H., Karolyi, G. A., Stulz, R. M. (3). A New Approach to Measuring Financial Contagion, Review of Financial Studies 16, 717-63. Baig, T., Goldfajn, I. (1999). Financial Markets Contagion in the Asian Crisis, IMF Staff Papers 46 (), 167-195. Baig, T., Goldfajn, I. (). The Russian Default and the Contagion to Brazil, IMF Working Paper WP//16. Barberis, N., Shleifer, A., Wurgler, J. (). Comovement, NBER Working Paper No. 8884. Boyer, B.H., Gibson, M. S., Loretan, M. (1999). Pitfalls in tests for changes in correlations, International Finance Discussion Paper 597, Federal Reserve Board. Calvo, G. A., Mendoza, E. G. (). Rational contagion and the globalization of securities markets, Journal of International Economics 51, 79-113. Calvo, S., Reinhart, C. (1996). Capital Flows to Latin America: Is there Evidence of Contagion Effects, in Calvo, Goldstein, Hochreiter (eds.) Private Capital Flows to Emerging Market, Washington DC, Institute of International Economics. Cashin, P., Kumar, M.S., McDermott, C.J. (1995). International Integration of Equity Markets and Contagion Effects, IMF Working Paper WP/95/11. Chan-Lau, J.A., Mathieson, D.J., Yao, J.Y. (). Extreme Contagion in Equity Markets, IMF Working Paper WP//98. Corsetti, G., Pericoli, M., Sbracia, M. (1). Correlation Analysis of Financial Contagion: What One Should Know Before Running a Test, Discussion Paper No. 8, Economic Growth Center, Yale University Edwards, S. (1998). Interest Rate Volatility, Capital Controls and Contagion, NBER Working Paper No. 6756. Embrechts, P., McNeil, A., Strautmann, D. (). Correlation and Dependency in Risk Management: Properties and Pitfalls, in Dempster (ed.) Risk Managemet: Value at Risk and Beyond, Cambridge University Press, 176-3. Favero, C.A., Giavazzi, F. (), Is the International Propagation of Financial Shocks Nonlinear? Evidence from the ERM, Journal of International Economics 57, 31-46. Forbes, K., Rigobon, R. (). No Contagion, Only Interdependence: Measuring Stock Market Co-movements, Journal of Finance 57 (5), 3-61. Hu, L. (). Dependence Patterns across Financial Markets: Methods and Evidence, Working Paper Joe, H. (1997). Multivariate Models and Dependence Concepts, Monographs on Statistics and Applied Probability 73, Chapman and Hall Kodres, L.E., Pritsker, M. (). A Rational Expectations Model of Financial Contagion, Journal of Finance 57(), 769-99 Koenker, R., Bassett, G. (1978). Regression Quantiles, Econometrica 46, 33-5. Koenker, R., d Orey, V. (1987). Computing Regression Quantiles, Applied Statistics 36, 383-93. Koenker, R., d Orey, V. (1993). A Remark on Computing Regression Quantiles, Applied Statistics 43, 41-14.

Koenker, R., Park, B. (1996). An interior point algorithm for nonlinear quantile regression, Journal of Econometrics 71, 65-83. Koenker R., Machado J. (1999). Goodness of Fit and Related Inference Processes for Quantile Regression, Journal of the American Statistical Association 94, 196-131. Longin, F., Solnik, B. (1995). Is the Correlation in International Equity Returns Constant: 196-199?, Journal of International Money and Finance 14 (1), 3-6. Longin, F., Solnik, B. (1). Extreme Correlation of International Equity Markets, Journal of Finance 56, 649-676. Nelsen, R. B. (1999). An Introduction to Copulas, Lecture Notes in Statistics 139, Springer. Pindyck, R. S., Rubinfeld, D. L., (1998). Econometric Models and Economic Forecasts, Fourth Edition, McGraw-Hill. Portnoy, S., Koenker, R. (1997). The Gaussian Hare and the Laplacian Tortoise: Computability of squared-error versus absolute-error estimators, with discussion, Statistical Science 1, 79-3. Stata Corporation (1). Stata Reference Manual, Stata Press. Appendix Quantile Regression Given a random variable Y with right continuous distribution function F Y (a) =P (Y a), the quantile function Q Y can be defined by [, 1] R Q Y (θ) = F 1 Y (θ) = inf {a F Y (a) θ} (8) Similarly, taking a random sample Y 1,Y,...,Y n with empirical distribution function ˆ F Y (a) = 1 n #{Y i a}, we define the empirical quantile function ˆQ Y (θ) = 1 ˆF Y (θ) = inf {a ˆF Y (a) θ} (9) Koenker and Bassett (1978) showed the equivalence to the following minimization problem 14 : ˆQ Y (θ) = argmin θ Y i a + (1 θ) Y i a a i:y i a i:y i <a = argmin ρ θ (Y i a) (1) with ρ θ (z) = a i { θz : z (θ 1)z : z< = (θ I(z <)) z 14 See Koenker and Bassett (1978), page 38. Their own comment says The case of the median (θ=1/) is, of course, well known, but the general result has languished in the status of curiosm. 1

If the median (θ =.5) is taken, (1) simplifies to the well-known expression med Y = argmin a i Y i a (11) Assuming that Y is linearly dependent on a vector of exogenous variables x, the conditional quantile function can be written as Q Y (θ x) = inf{a F Y (a x) θ} = k x k β θk = x β θ (1) In analogy to (1), we finally obtain the regression quantiles by solving with respect to β θ ˆβ θ = argmin θ Y i x β θ R iβ θ + (1 θ) Y i x iβ θ k i:y i x β θ i:y i <x β θ = argmin ρ θ (Y i x iβ θ ) (13) β θ i There is no general closed solution to the minimization problem, but after some slight modifications, it can easily be solved by simplex methods. Barrodale and Roberts (1974) developed an algorithm for the median case. Koenker and d Orey (1987,1993) described an implementation for the general quantile problem with desirable properties for small to medium number of observations. Portnoy and Koenker (1987) showed that an alternative interior method published by Koenker and Park (1996) is competitive to least-squares estimation even for very large data sets. The computations conducted in this paper have been calculated with the software package STATA. Apart from the evaluation of the desired quantile coefficients, the program also allows to obtain appropriate confidence intervals by means of bootstrapping. 15 Conditional Densities Consider a quantile regression model with Y dependent on K standardized regressors x k (k =1,...,K): Q Yi (τ x) =α(τ)+ K β k (τ)x ki + ε i (τ) i 1,...,N (14) k=1 Now assume that we are interested in the effect of one distinct regressor (denoted as x l with l {1,...,K}) on the distribution of Y. As all regressors are standardized (so their 15 For questions concerning the implementation as well as the utilized algorithms see Stata Corporation (3).

means are zero), we omit the other covariates and estimate a simplified model with only the constant and x l for 99 different quantiles (τ =.1,...,.99). This leads us to 99 different estimated coefficients α(τ) and β l (τ). So, for any value of x l (say x l ), we can calculate 99 different conditional values of Y, implicitly fixing the other regressors at their mean: Q Y (τ) = α(τ)+ β l (τ)x l (15) Using these 99 values and applying a kernel density estimation (e.g. with an Epanechnikov kernel), we obtain the desired estimation of the conditional distribution of Y. In our application, we chose for each regressor its unconditional %, 1%, 9% and 98%-quantile as values of x l to examine the different impacts on Y. One disadvantage of this procedure is the fact that it does not account for possible relationships among the regressors. Perhaps, low values of x l systematically go along with low (or high) values of some of the other covariates. In order to circumvent this problem, we first conduct a (simple least square) auxiliary regression of each of the other variables on x l. Subsequently, we include their estimated values x k (k l) into the full model: Q Y (τ) = α(τ)+ β K l (τ)x l + β k (τ) x k k l (16) So, we again calculate 99 estimated conditional quantiles of Y enabling us to determine the density estimation of the regressand dependent on any desired value of x l. k=1 3