Sensitivity , Persistence and Asymmetric Effects in International Stock Market Volatility during the Global Financial Crisis

Financial market volatility is an important element when setting up portfolio management strategies, option pricing and market regulation. The Subprime crisis affected all markets around the world. Daily data of twelve stock indexes for the period of October 1999 to June 2011 are studied using basic GARCH type models. The data were then divided into three different sub-periods to allow the behavior of stock market in different sub-periods to be investigated. The following sub-periods are identified: Dot-Com crisis, Quiet and Subprime crisis. This paper revealed that the Subprime crisis turned out to have bigger impact on stock market volatility, namely at sensitivity, persistence and asymmetric effects.

Efectos de sensibilidad, persistencia y asimetría en la volatilidad de los mercados bursátiles internacionales en el entorno de la crisis financiera global RESUMEN La volatilidad de los mercados financieros es un importante elemento para la estrategia de carteras de inversión y para la regulación de los mercados.La crisis subprime afectó a los mercados bursátiles mundiales.
Los resultados demuestran la influencia de la crisis financiera global en el comportamiento de la volatilidad del mercado bursátil, sobre todo en cuanto a la sensibilidad, la persistencia y la asimetría.

INTRODUCTION
According to Claessens et al. (2010), Bekaert et al. (2011) and Lin and Treichel (2012), the current financial crisis is the first global crisis and the most severe since the Great Depression.Although the crisis had its origin in the United States, particularly in subprime credit, it would be transmitted to other economic sectors as well as other developed and emerging economies.
The quantification of risk, as a financial variable, has represented a major challenge for researchers, regulators and financial professionals.In modern finance theory, Markowitz (1952) considers the volatility of asset's returns as a measure of risk.According to Lin (1996), the risk is usually associated with volatility.When the volatility of a financial asset rises, so does the risk.
However, volatility measures only the magnitude, but not the direction.The financial markets volatility is an important indicator of the dynamic fluctuations in asset prices (Raja and Selvam, 2011).
Understanding stock markets volatility is also an important element to calculate the cost of capital and to support investment decisions.Volatility is synonymous with risk.Bollerslev et al. (1992) argue that volatility is a key variable for a large majority of financial instruments, playing a central role in many areas of finance.Bala and Premaratne (2003) consider that substantial changes in financial market volatility can cause significant negative effects on risk aversion, and make markets more unstable, increasing the uncertainty for market players, particularly in their predictions and their income.
Usually, financial series reveal some enigmatic empirical regularity.These regularities are called stylized facts and correspond to observations so consistent, confirmed in many contexts, markets and instruments, which are eventually accepted as truth (Cont, 2001 and2005).Thus, the stylized facts are based on a common denominator, which results from the properties observed in multiple studies, about markets and instruments.Due to its general nature, the stylized facts reveal a qualitative dimension, but not accurate enough to distinguish between different parametric models (Coolen, 2004;Ding et al., 1993).Several studies have confirmed some of the most common stylized facts, including volatility clustering and asymmetric effect (Brock and de Lima, 1996;Campbell et al., 1996;Mandelbrot and Hudson, 2006).The first is related to autocorrelation.According to Mandelbrot (1963) and Engle (1982), if volatility is high at a given moment, it tends to continue high in the next period.If volatility is low in a given moment, it tends to continue low in the next periods, because the new information that arrives to the market is correlated in time.For its part, the asymmetric effect results from the diverse reaction of volatility to the arrival of news in the market, reflecting the effect of good and bad news on volatility, which results in a negative correlation between lagged returns and volatility.The asymmetric effect was first observed by Black (1976).
In this work conditional heteroskedasticity models are applied, in order to analyze the impact of global financial crisis on conditional volatility, sensitivity, persistence and asymmetric effect in the international stock markets.This study is structured as follows: Section 2 presents information about the data and the methodology chosen, Section 3 shows the empirical results, while Section 4 summarizes the main conclusions.

DATA AND METHODOLOGY
In order to analyze the evolution of daily volatility stock markets, twelve indices were selected, evolving European, non-European, developed and emerging indices, according to the Morgan Stanley Capital International classification, representing about 62% of world stock market capitalization, in 2010, as can be seen in Table 1.The set of developed markets included European and non-european markets.From the European continent, Germany (DAX 30), France (CAC 40), UK (FTSE 100), Spain (IBEX 35), Ireland (ISEQ Overall), Greece (ATG) and Portugal (PSI 20) were selected.The set of non-European developed markets included the U.S. (Dow Jones), Japan (Nikkei 225) and Hong Kong (Hang-Seng).Additionally, Brazil (Bovespa) and India (Sensex) were selected as emerging stock markets.
We believe that the use of a large set of stock market indexes (emerging and developed), in different regions, with different capitalization levels, including some of the major stock markets of the world and the European markets under sovereign debt support program, it helps to understand the consequences of the global financial crisis.To estimate the conditional volatility, GARCH (1,1) and EGARCH (1,1) models were considered.GARCH models were proposed by Bollerslev (1986) and they are consistent with the phenomenon of volatility clustering.
The GARCH (p, q) specification is given by: where: is the set of the available information at time 1 − t , t z is a vector of explanatory variables, q is the order of the ARCH process and p is the order of the GARCH process, t ε corresponds to the vector of estimated residuals; j α represents the short-term persistence shocks (ARCH effect) and i β represents the long-term persistence shocks.( ) are the basic conditions for the conditional variance to be positive ( ) is the stationarity condition of the GARCH models.Verifying this condition ensures that conditional variance is not finite, while the conditional volatility varies in time, being positive and stationary.According to Alexander (2008), in a GARCH (1,1) model, the 1 α parameter measures the conditional volatility reaction to unexpected market shocks.When this parameter is relatively high (above 0.1), volatility is very sensitive to market events.The volatility persistence is considered usually as the sum of 1 α and β parameters.An alternative measure to evaluate persistence is volatility half-life.Engle and Patton (2001) define half-life as the median time spent by volatility to move halfway, back to its unconditional mean.This parameter provides a more appropriate description of persistence, representing the longest period in which the market shock will die.In a GARCH model, the half-life market shock is given by ( ) ( ) To accommodate the asymmetric effect, Nelson (1991) proposed the EGARCH model, also called exponential GARCH.In this model, the conditional variance is described by an asymmetric function of past values of t ε .
The EGARCH (p, q) model specification is given by: where: k γ measures the asymmetric effect; is the set of information available at the time 1 − t , t z is a vector of explanatory variables, q is the order of the ARCH process and p is the order of the GARCH process, t ε is the vector of estimated residuals.According to (McAleer, 2005), if 1 1 < β , the conditional variance is finite.
As stated above, in the EGARCH (1,1) model, the asymmetric effect is captured by coefficient γ .The negative sign of this coefficient indicates the existence of an asymmetric effect; that is, it indicates a negative relationship between return and volatility.When the coefficient is negative, the positive shocks produce less pronounced volatility than negative shocks of equal size.This has been detected in several empirical studies, concluding that small investors are panicking about the impact of negative shocks and leave their market positions in order to avoid more pronounced losses.
Consequently, there is an increase in volatility.
To verify the correct specification of the estimated models, the Ljung-Box and ARCH-LM tests were performed.Under the null hypothesis, , the Ljung-Box test assumes that quadratic residues are not correlated.
( ) , where k represents the number of estimated parameters.
The ARCH-LM test is considered under the null hypothesis , where q is the order of the process.The test statistic is given by 2 NR , following an asymptotic distribution of 2 χ , with q degrees of freedom, where 2 R is the determination coefficient and N the number of observations.
To conclude if stock markets volatility has increased, two types of tests are applied.The first involves the equality of means, using the t-test and the analysis of variance with one factor; the second test, the equality of variances by applying the F statistic and the Bartlett test.These tests are presented briefly below.

Tests for equality of means
The t-test is calculated based on The test is compared with Student-t distribution, where the number of degrees of freedom is given by: The test for equality of means, by analysis of variance with one factor, allows to evaluate the statistical significance of the difference between means, for a specific probability level, involving the calculation of the F statistic, which is based on the variability within and among sub-periods.
The test statistic is given by: where: is the average sum of squares between sub-periods; is the average sum of squares within sub-periods.
whereas SSE is the sum of squares between sub-periods, SSD is the sum of squares within subperiods, k is the number of sub-periods and N is the total number of observations.
In both tests, the null hypotheses and alternative hypotheses are:

Test for equal variances
The F test for equality of variances is given by , where ( ) is the estimated variance of the sub-period with higher (lower) value.
The Bartlett's test is used to test equality/homogeneity of variances among groups versus the alternative of variances being unequal, for at least two groups.
The test statistic is given by: where: ( ) ( ) where i n is the sample size of the p-th group, 2 i S is the sample variance of the p-th group, N is the sample size and 2 p S is the pooled variance.
In both tests, the null hypotheses and the alternative hypotheses are:

EMPIRICAL RESULTS
Figure 1 shows the daily returns series in the full period.The visual analysis indicates the tendency for volatility clustering in certain periods.The second sub-period was relatively quiet.However, the remaining sub-periods showed great turbulence and volatility, suggesting volatility clustering, as we will see later on.The year of 2008 revealed the highest volatility concentration as the result of the emergence of the global financial crisis.Table 2 presents the descriptive statistics of conditional volatility for the three sub-periods and for the twelve markets, generated by the GARCH (1,1) models.The values shown in Table 2 allow the conclusion that the estimated conditional volatility reveals signs of deviation from normality assumption, taking into account the skewness and kurtosis coefficients.In order to confirm the appropriateness of the adjustment to the normal distribution, in each of the sub-periods and for the twelve series, the Jarque-Bera test was considered.The statistics of this test is given in Table 2. Based on the results, we conclude that all the series are statistically significant at a significance level of 1%, clearly rejecting the hypothesis of normality.
In Dot-Com sub-period, the BOV index showed the highest average conditional volatility, three times higher than ISEQ and PSI indices, as the least volatile markets.For its part, the DAX index showed the greatest degree of variability, measured by the standard deviation.
In the quiet sub-period, Sensex and BOV indices showed higher average of conditional volatility.The remaining markets showed lower levels of volatility.In either case, the recorded values were below those seen during Dot-Com sub-period.Regarding conditional volatility variability, the Sensex index showed the greatest variability.Conversely, DJ and PSI were the least variable.
During global financial crisis sub-period, the differences in volatility levels of various indices were not as pronounced as in the previous sub-periods.HANG-SENG index recorded the highest average conditional volatility, followed by ATG and ISEQ indices.For its part, DJ and PSI indices were the least volatile.Some estimates are somehow unexpected.This is what happens with the Portuguese market, which has registered the lowest volatility between all the markets, although it is a small developed market, and especially for being under foreign financial assistance since 2011.
Figure 2 shows the graphical evolution of the conditional volatility of each of the twelve daily indices in the full period, estimated according to GARCH (1,1) and EGARCH (1,1) specifications.
To verify the variance persistence, the ARCH-LM test was applied.The results are shown in Table 4.The analysis of the coefficients and its respective probability values indicates that they are not statistically different from zero.Testing coefficients in the group, the probability (F-statistic) is significant, so the null hypothesis is accepted.There is reason to believe that estimated models have the ability to model conditional heteroskedasticity.The results also indicate that, during the three sub-periods, the volatility of daily returns proved to be quite persistent, especially in the last sub-period.Half-life was particularly high in HANG-SENG (130) and SENSEX (744) indices.In this sub-period, NIKKEI and PSI indices had recorded the lowest half-life, with a value of 24.In both cases, an unanticipated shock in the daily returns produces, on average, effects on volatility for 24 days.

Tests for equality of means and variances
A visual analysis of Figure 2 leads to a first conclusion: Dot-Com and Global Financial Crisis subperiods were characterized by a higher concentration of volatility and showed peaks of volatility.The quiet sub-period reveals that volatility levels were much lower than that in the other two sub-periods.
The Sensex index was the exception, which showed peaks of volatility in the quiet sub-period.

DAX
The results shown in Table 5 allow several conclusions.Comparing global financial crisis and Dot-Com sub-periods, we conclude that the average conditional volatility indicates statistical differences, at a significance level of 1%, with the exception of the BOV, CAC and DAX indices.The BOV index showed a statistical difference at a significance level of 10%.The CAC index revealed no statistical difference, whereas the DAX index showed a decreasing average of conditional volatility, at a significance level of 1%.Additionally, the test of equality of variances, applied to the conditional volatilities comparing the first and the third sub-periods, supports the conclusion that all the reported indices increase, at a significance level of 5%.
The comparison of the last sub-periods allows the conclusion that all the daily average volatilities recorded strong increases, with statistical significance at a significance level of 1%.In some cases, increases were greater than 300%.This happened with the ISEQ (409%), PSI (362%), Hang-Seng (338%) and DJ (303%) index.The Brazilian market increased by 58%.Moreover, increases on average volatility were complemented by increases in variability and evidenced by testing the equality of variances, which in all the cases were significant at a significance level of 1%.The results indicate the occurrence of a generalized increase in conditional volatility.This increase was not restricted to the U.S. market (which led to the subprime crisis) or the euro area markets (in the epicenter of the sovereign debt crisis), revealing a global scale.

Asymmetric effect
To analyze the asymmetric effect, EGARCH (1.1) models were estimated, from the returns of the twelve indices.The estimated results are shown in Table 6.Estimates show that all the γ coefficients had a negative sign.Additionally, in the three subperiods, these coefficients were statistically different from zero, at a significance level of 1%.The exceptions were the HANG-SENG index in the quiet sub-period, which was statistically significant at a significance level of 5%, and the PSI index in the quiet sub-period, where asymmetry coefficient was not proved to be statistically different from zero.The high significance of the asymmetry coefficient clearly shows the existence of asymmetric shocks in the volatility process.In this sense, one can conclude that in the three sub-periods, "bad news" was more impactful than "good news".
A comparison of the asymmetry coefficients in the three sub-periods, allows the conclusion that a rising trend of these values has been verified.From the first to the second sub-period, eight indices reported an increase in the asymmetry coefficient (in absolute value).From the second to the third subperiod, there was an increase in nine asymmetry coefficients.When comparing the first and the third sub-periods, the same happens in nine markets.The results showed that markets are, in general, more sensitive to "bad news" than to "good news", especially during the global financial crisis.
The Ljung-Box test does not accept the null hypothesis for BOV (quiet sub-period), DAX (Dot-Com sub-period) and HANG-SENG (quiet sub-period) indices at the significance level of 5%.For the remaining indices, there is a strong evidence of acceptance of the null hypothesis, concluding that the standardized residues are not correlated because the results of the test showed that the p-value is very above the significance level of 5%.The LM test results (see Table 7) confirmed the previous conclusions.The group test (F-Statistic) showed that the probability is not significant in the cases mentioned above, rejecting the null hypothesis.

SUMMARY, CONCLUSIONS AND LIMITATIONS
In this work, we have studied the current financial crisis.According to several authors, this crisis is the most severe after the Great Depression and the first global financial crisis the world has known.
To analyze the crisis, various stock markets were considered, which all together represent about 62% of the world stock market capitalization, in order to understand the impact of global financial crisis on the level of volatility, sensitivity, persistence and asymmetric effect.For this purpose, we studied the period from October 4 th 1999 to June 30 th 2011, which was divided into three sub-periods: One corresponding to the Dot-Com crisis; other relative to a phase of rise and accumulation for global indices; and finally, one corresponding to the global financial crisis.To estimate the market volatility, generalized and exponential autoregressive conditional heteroskedasticity models were considered.
The findings confirm that, in most cases, the conditional volatility in the global financial crisis sub-period experienced a significant increase compared with the previous two sub-periods, but particularly in relation to the quiet sub-period.Note that the PSI index showed, in all sub-periods analyzed, lower levels of conditional volatility, which is somehow surprising if we take into account the small size of this market.Additionally, the model estimation confirms, in general, a higher persistence in volatility during the financial crisis sub-period; it is the same with sensitivity.Similarly, all the markets considered in the analysis revealed an asymmetric effect; in other words, their volatilities were more influenced by "bad news" than by "good news", especially during the global financial crisis.
Several limitations of our analysis should be noted.First, the sample period covers only the first years of the global financial crisis, but financial markets are suffering with this crisis because it has not finished yet.Second, this study considered only twelve stock markets, including some major capitalizations and markets directly related to sovereign debt crisis.For more robust conclusions, future work may cover the full period of the global financial crisis and consider a large set of developed and emerging markets.

Figure
Figure 1.Evolution of daily returns -Com sub-period, are not statistically significant, at a significance level of 10%.Both the DAX coefficient ( ) 0 α in Dot-com sub-period and the HANG-SENG index in Dot-Com and Global

Figure 3 .
Figure 3. Half-life estimates in the three sub-periods

Table 1 :
Market capitalization as a percentage of global capitalization Source: World BankThe data used in this study were obtained from EconoStats and cover the period from October 4 th 1999 to June 30 th 2011, which was subdivided into three sub-periods.To analyze the Dot-Com crisis, the period from 10/04/1999 to 03/31/2003 was considered.The latest episode of crisis, which

Table 2 .
Descriptive statistics from conditional volatility estimates in each sub-period.

Table 3
presents the GARCH (1,1) estimation results.All the coefficients of the estimated models showed the expected signals, except for β parameter for BOV index during the Dot-Com subperiod, which has a negative coefficient (-0.538).The remaining coefficients are non-negative, ensuring that the conditional variance is positive.

Table 5 .
Mean and variance equality tests and their p-values
Notes: This table presents the EGARCH (1,1) model estimations, applied to the daily returns of the twelve indices studied in the three sub-periods.All the estimates are based on Maximum Likelihood.