Asymmetry and Spillover Effects in the North American Equity Markets

In this paper we examine the issue of asymmetry in the return and volatility spillover effects from the US equity market into the Canadian and Mexican equity markets. We model the conditional volatility of the returns in each of the three markets using the asymmetric power model of Ding, Granger and Engle (1993). The empirical findings indicate that the US market has a significant impact on the returns in the Canadian and Mexican markets. However, the findings for Canada vary considerably from those for Mexico. In particular, the empirical results indicate that volatility spillover effects, but not return spillover effects, exhibit an asymmetric behavior, with negative shocks from the US equity market impacting on the conditional volatility of the Canadian and Mexican equity markets more deeply than positive shocks. Moreover, while the impact of positive shocks from the US equity market is not much different between the two markets, this is not the case with negative shocks, which affect the volatility of the Mexican market more intensely than the volatility of the Canadian market. --


Introduction
There is ample evidence that national equity markets have become more interdependent in recent years. 1 The developments in the liberalization of capital movements and financial reforms, coupled with advances in computer technology and information processing, have reduced the isolation of national equity markets and increased their ability to react promptly to news and shocks originating from the rest of the world. 2 Evidence of increased linkages between national equity markets has also been found following the October 1987 market crash, and the Asian and Russian financial crises. 3 In general, most of the research has documented four stylized facts: 1) correlations across stock markets are time-varying; 2) returns in major markets tend to be more correlated when volatility is high; 3) all major episodes of high volatility are associated with market drops; and, finally, 4) correlations in volatility and returns appear to be causal from the US market. 1 There is a substantial literature investigating the mechanisms through which returns and volatilities in one market are transmitted to other markets. See, e.g., Ng, Chang and Chow (1991), Lin, Engle and Ito (1994), Karolyi (1995), Kim and Rogers (1995), and Booth, Martikainen and Tse (1997). 2 These developments in information technology also indicate that the use of low frequency data is exceedingly restrictive and that it is essential to use high frequency data to examine the issue of spillover effects in any meaningful manner. 3 These studies include the important contributions by Eun & Shim (1989), Von Furstenberg and Jeon (1989), King and Wadhwani (1990), Schwert (1990), Hamao, Masulis and Ng (1990), King, Sentana and Wadhwani (1994), Arshanapalli and Doukas (1993), and Longin & Solnik (1995). 4 investing and ownership of foreign stocks. Trends toward greater economic interdependence and increased financial integration in the North American continent dilute the long-term diversification benefits available to market participants. Hedging strategies depend on shocks to stock markets being relatively isolated, idiosyncratic events, and if shocks to returns and volatilities in the US travel quickly across the Canadian and Mexican borders, the benefits of diversification may be undermined.
Apart from the focus on North America, this paper differs from previous research on spillover effects on two grounds. First, a key concern in the paper is the manner in which spillover effects from the US are transmitted across the North American markets.
Specifically, our interest is in documenting (a) whether stock returns in an advanced, mature market (Canada) react differently from stock returns in an emerging market (Mexico) to return and volatility shocks from the US stock market, and (b) whether these spillover effects display nonlinear and asymmetric characteristics. The latter concern is motivated by the common observation that rising and declining patterns of a process frequently display nonlinear, asymmetric characteristics. This generality in the modeling of spillover effects has thus far been scant in the literature studying dependencies in national stock markets. Clearly, failure to properly account for these asymmetries, if they are present in the data, is likely to lead to incorrect inferences concerning the nature of the US spillover transmission across the North American markets. Second, whereas previous research on spillover effects has, with few exceptions, such as Bae and Karolyi (1994), used the standard Bollerslev's GARCH (1, 1) specification, we model the volatility of the equity returns of the three North American markets using the asymmetric power APARCH model proposed by Ding, Granger and Engle (1993). The main advantage of the APARCH specification is its functional flexibility. The APARCH model does not impose a common and uniform structure on the conditional volatility of the three North American equity market returns; rather, it uses a Box-Cox power transformation of the conditional standard deviation process and the asymmetric absolute residuals. This permits a virtually infinite range of transformations inclusive of any positive value. 6 In addition, it accommodates asymmetries in the volatility of the idiosyncratic shocks in equity returns.
The empirical analysis proceeds through a two-step approach. First, we estimate an AR (1)-APARCH (1, 1) model for the US equity market returns in order to identify the volatility shocks from the US equity market. Second, we estimate an augmented AR (1)-APARCH (1, 1) model for the returns of the Canadian and Mexican equity markets that incorporates an asymmetric specification of the return and volatility spillover effects from the US market. Return spillovers occur when returns of the US market enter significantly in the anticipated part of the Canadian and Mexican returns. Volatility spillovers, on the other hand, occur when innovations in the US market have a significant effect on the unanticipated component of the Canadian and Mexican returns. It should be mentioned that the two-step approach is not without some conceptual limitations, as it excludes the possibility of reverse spillover effects from the Canadian and Mexican markets. Still, the focus on the US equity market as the reference, and hence as the center 6 Ding, Granger and Engle (1993) show that the functional form of the APARCH (p, q) model encompasses seven other GARCH extensions as special cases, including, in addition to the standard class of Engle's ARCH models (Engle 1982) and Bollerslev's GARCH models (Bollerslev, 1986), the Taylor (1986) and Schwert (1990) GARCH in standard deviation model, the log-ARCH of Geweke (1986) and Pentula (1986), the threshold ARCH (TARCH) model of Zakoian (1994), the GJR-GARCH model of Glosten, Jaganathan and Runkle (Glosten et al., 1993), and the nonlinear ARCH (NARCH) model of Higgins and Bera (1992). See Ding, Granger and Engle (1993) for details. of the international transmission of stock returns and volatility, is plausible based on the existing empirical evidence. 7 Using this approach, we find strong statistical evidence that indicate that volatility spillover effects, but not in return spillover effects, exhibit an asymmetric behavior, with negative shocks from the US equity market impacting on the conditional volatility of the Canadian and Mexican equity markets more deeply than positive shocks.
The rest of the paper is organized as follows. Section 2 outlines the specification of the model. The data and their statistical properties are described in Section 3. In Section 4, we present the empirical results. A summary of the findings and some concluding remarks follow in Section 5.

A Spillover Model with Asymmetric Effects
Let t k R , represent the return on the national equity index of country k (k = Canada, Mexico), and let t US R , denote the return on the US equity index. The dynamics of the return on the US equity index t US R , is specified in equations (1) through (5).
Following Bekaert and Harvey (1997), Ng (2000) and Baele (2005), we assume that US equity index returns consists of a predictable part,  Masih and Masih (2001) and Eun and Shim (1989). This is consistent with the partial adjustment model of stock returns of Amihud and Mendelson (1987) as well as the nonsynchronous trading hypothesis of Lo and MacKinlay (1990 In equation (5) (5) and can be given by any positive values. In particular, Ding, Granger and Engle (1993) conclude that when δ = 1 the long memory property of stock returns is the strongest compared to other values of δ .
The equity index return t k R , in country k, (k = Canada, Mexico) is specified in equations (6) through (10). As in the case of the US equity return, the return t k R , is also decomposed into a predictable part, 1 , − t k µ , and an unpredictable part, t k , ε , However, as shown in equations (7) and (8) ε are extended to capture the asymmetric spillover effects from the US stock market. First, the AR (1) specification of the predictable part of the Mexican and Canadian stock returns is augmented to capture the asymmetric return spillover effects originating from US stock market: , 0). The parameters + k θ and − k θ measure the return spillover intensities of positive and negative US returns at time t-1, respectively, on the returns of Canada and Mexico, respectively, at time t. Obviously, by definition, for every t. Note that the specification of the asymmetric return spillover process utilizes two different filters, one for positive return spillover effects, + −1 ,t US R and one for negative return spillover effects − −1 ,t US R . It follows that when + k θ = − k θ = k θ the return spillover process is symmetric, and the sequence 1 − k,t µ reduces to: In equation (11) the returns of country k are determined by local past information and past information arriving from the US market (the predictable or expected component of the returns) as well as by contemporary idiosyncratic shocks, initiated both locally and in the US market (the unpredictable or unexpected component of the returns). We further assume that the idiosyncratic shocks t k e , are conditionally normal distributed and are mutually uncorrelated, as well as uncorrelated with the US idiosyncratic shock, US,t e , and have time-varying conditional standard deviations that are described by an APARCH (1, 1) process, i.e., , for all k . Note that the model specified in equations (1) through (15) accommodates three kinds of asymmetries. The first is defined through the idiosyncratic shocks of each of the three markets and is captured by the leverage term in the APARCH (1, 1) model. The second is the asymmetry in the return spillover effects from the US market on the stock market of Canada and Mexico. This type of asymmetry is provided by the differential specification of the effects of the previous day positive and negative returns from the US stock market on the stock market of Canada and Mexico. Finally, the third kind of asymmetry refers to the asymmetry of the volatility spillover effects. This is accommodated by the differential impact of the contemporaneous positive and negative idiosyncratic shocks from the US stock market on the stock market of Canada and Mexico. The model implies that whenever shocks from the US market influence the unanticipated component of the Canadian and Mexican returns, they also contribute to their correlations. The implied time varying conditional variance, k,t h , of the unpredictable part of the return of Canada and Mexico, based on information available at time t-1, given by equation (16), depends only on the volatility of the unpredictable part of the US return and their own idiosyncratic volatility: (16) denotes the variability of the returns of Canada and Mexico explained by the variability of the US returns. Thus, the conditional variance of the unexpected return for Canada and Mexico depends not only on the US and its own idiosyncratic volatility, but also on whether that volatility is driven by positive or negative US shocks. Similarly, the implied time varying conditional covariance between the unexpected return of Canada and the volatility spillovers intensities: (16) and (17). We present in the Appendix the formal details of the derivation of equations (16) and (17). It is straightforward to verify that under the hypothesis that the volatility spillover effects have a symmetric impact on the unpredictable part of the returns of Canada and Mexico, equations (16)-(18) reduce to: (16), it follows that the variance ratio US k,t VR , i.e., the proportion of the variance of the unexpected returns of Canada and Mexico that is driven by US volatility is given by

Equation (18) follows directly from equations
under the assumption that volatility spillovers originating from the US equity market have symmetric effects on the volatility of the Canadian and Mexican returns, i.e., Similarly, the proportion of the variance of the unexpected returns of Canada and Mexico that is explained by their own local idiosyncratic shocks is = given by

The Data
The

c) Mexico
As Figure 1 shows, the returns exhibit volatility clustering: large (small) shocks, of either sign, tend to follow large (small) shocks, a characteristic associated with a time varying conditional variance. There is also some visual evidence that periods of high volatility tend to be common across the three North-American countries. In addition, it appears that the volatility of the returns in the Mexican stock market has increased after the ratification of NAFTA and the peso crisis of December 1994. Summary statistics relating to the distributional and time series properties of the return series are presented in Table 1.
The results indicate that the mean of the returns is higher for Mexico, as is the standard deviation, compared to Canada and the US.  Notes. In a normal distribution skewness is zero and kurtosis is 3. The Jarque-Bera statistics is distributed as 2 χ (2). The critical value at the 5% level is 5.99. Q(6) and Q 2 (6) are the Ljung-Box statistics based on the returns and the squared returns respectively up to the 6 th order. Both statistics are asymptotically distributed as 2 χ (6). The critical value at the 5% level is 12.59.
All three return series exhibit negative skewness in returns, although only for Canada is skewness significantly different from zero, and are consistently leptokurtic, which is a typical characteristic of financial time series. For all three series, we can conclude that the respective distributions underlying the data have fatter tails and are unlikely to have been drawn from a normally distributed sample. As a confirmation, the Jarque and Bera (1987) test for normality rejects the null of normality in the distribution of these series. Table 1 also reports tests for serial correlation, using the Ljung and Box (1978) Q statistic up to and including the 6 th lag. Also reported are tests for ARCH effects based on the methodology of Bollerslev (1986), which uses the Ljung and Box Q statistic applied to the squared returns.
The well-known facts that stock returns are leptokurtic, heteroskedastic, and not normally distributed are therefore confirmed by our findings.
However, it is worth emphasizing that there are significant changes in the non-normal distributions of the returns over sub-periods. In particular, kurtosis is substantially less in each of the two sub-periods compared to the whole sample period, and negative skewness is present only for Canada. Further, the hypothesis that the US returns are not serially correlated can be rejected, at the 5 per cent level, for the whole period but not for the two sub-periods.  The most isolated market in the North American region before October 1997 appears to be Mexico, which shows consistently lower correlations with the other two markets. As shown in Table 2, the cross-market correlations during the month of October 1997 are extremely high. This increase in cross-market correlations is consistent with the findings of King and Wadhwani (1990) and Lee and Kim (1993) and provides evidence of contagion (Forbes and Rigobon, 2001). Though cross-market correlations after the crash decrease, they still remain higher than the correlations before the crash. The markets of the North America continent thus appear to follow the same changes in correlation patterns similar to those found in emerging markets. See, e.g., Bekaert and Harvey (1997), Solnik, Bourelle and Le Fur (1996), Bekaert and Urias (1999) and Meric et al. (2001).

Empirical Results
The results of estimating the AR (1)-APARCH (1, 1) model for the US, Canada and Mexico, under the assumption of no spillover effects are presented in Table 3. The Hausman (1974), but all test statistics and t-values are computed using the quasimaximum likelihood methods (QML) described by Bollerslev and Wooldridge (1992), which are robust to distributional non-normalities. Notes. In parenthesis are the robust t-statistics based on Bollerslev and Wooldridge (1992). ARCH(1) is the value of the ARCH-LM test of order 1. The statistics is asymptotically distributed as 2 χ (1). The critical value at the 5% level is 3.84. Q(6) and Q 2 (6) are the Ljung-Box statistics based on the standardized residuals and the squared standardized residuals respectively up to the 6 th order. Both statistics are asymptotically distributed as 2 χ (6). The critical value at the 5% level is 12.59. The Jarque-Bera statistics is distributed as 2 χ (2). The critical value at the 5% level is 5.99.
The estimates of the conditional mean indicate that the autoregressive parameter is significant at the 1 per cent level or better only for Canada and Mexico. For the US, instead, the estimate is not significantly different from zero. Conversely, the estimates of the conditional variance are all significant at the 1% level or better. The estimate ofα , the ARCH coefficient, is the highest for Mexico and the lowest for the US, suggesting that idiosyncratic shocks tend to linger around longer in the Mexican stock market than in the US and Canadian markets. This may be an indication that the Mexican stock market is less efficient than the US and Canadian markets since the effects of idiosyncratic shocks take longer to dissipate. Interestingly, the estimate of β , the GARCH coefficient, is also not the same in the three markets. Rather, it is the highest for the US and the lowest for Mexico. The estimate ofγ , the asymmetry coefficient, is also the highest for the US, while for Canada and Mexico is approximately the same, and about half of that found for the US. The leverage effect does exist in the returns of all three markets, but has a different impact among the three markets. In each market a negative shock has a greater impact than a positive shock, but a negative shock in the US market has an even greater effect on the volatility of the US than a negative shock of equal magnitude in the Although the Jarque-Bera test for normality of the standardized residuals fails to accept normality, and there is some residual negative skewness and positive excess kurtosis, in comparison to the statistics for the returns in Table 1, the model appears to capture much of the non-normality of the data.
In Tables 4 and 5  the decomposition offers no advantage over the approach used by Bekaert and Harvey (1997), Baele (2005) and Ng (2000) in that the effects of return and volatility spillovers are symmetric. Notes. In parenthesis are the robust t-statistics based on Bollerslev and Wooldridge (1992). ARCH(1) is the value of the ARCH-LM test of order 1. The statistics is asymptotically distributed as 2 χ (1). The critical value at the 5% level is 3.84. Q(6) and Q 2 (6) are the Ljung-Box statistics based on the standardized residuals and the squared standardized residuals respectively up to the 6 th order. Both statistics are asymptotically distributed as 2 χ (6). The critical value at the 5% level is 12.59. The Jarque-Bera statistics is distributed as 2 χ (2). The critical value at the 5% level is 5.99. Notes. In parenthesis are the robust t-statistics based on Bollerslev and Wooldridge (1992). ARCH(1) is the value of the ARCH-LM test of order 1. The statistics is asymptotically distributed as 2 χ (1). The critical value at the 5% level is 3.84. Q(6) and Q 2 (6) are the Ljung-Box statistics based on the standardized residuals and the squared standardized residuals respectively up to the 6 th order. Both statistics are asymptotically distributed as 2 χ (6). The critical value at the 5% level is 12.59. The Jarque-Bera statistics is distributed as 2 χ (2). The critical value at the 5% level is 5.99.
In both Table 4 and Table 5, and for all models, the specification tests in terms of the Q(12) and Q 2 (12) and ARCH(1) statistics indicate that the series are adequately modeled without any remaining serial correlation or residual ARCH effect. The autocorrelations of the standardized residuals and squared standardized residuals all lie within the asymptotic bounds of N 2 ; however, the structure of the standardized residuals still reflects a significant amount of kurtosis, although significantly decreased in the case of Canada. As expected, the Jarque-Bera test rejects the normality of standardized residuals. As a further specification test, we checked the correlations of the idiosyncratic shocks. A correct specification of the model requires that the idiosyncratic shocks Comparison of the parameter estimates for Canada and Mexico presented in Table   4 indicates few substantial differences from extending the model to include US spillover effects. For all four models, the estimate of the power coefficient for Canada increases, but still remains significantly different from 2 but not from 1, and the estimate of the power coefficient for Mexico increases, but still remains significantly different from 1 but not from 2. Thus, the estimate of the power coefficient is invariant to the specification of the model. Similarly, the estimate of the autoregressive parameter in the conditional mean is slightly reduced, but remains highly significant. The remaining coefficients estimates are also approximately the same, with one exception. As a result of the inclusion of the US spillovers, the estimate of the asymmetry coefficient for Canada becomes insignificant. For Mexico, the coefficient estimates of the US return spillover effects are also statistically insignificant, regardless of the maintained hypothesis.
Conversely, the return from the US market has a significant and positive effect on the Canadian market, implying that an episode of weakness in the US market today leads to a drop in the Canadian stock market tomorrow. However, the Wald test fails to reject the hypothesis that +   9 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

b) Mexico
The US volatility spillover effects make up approximately between 6 % and 80% of the conditional variance of the unexpected returns of Canada. The mean of the variance ratio for Mexico lower than the mean for Canada, and is approximately 20%, but the standard deviation is about the same. Over the whole sample period the US volatility spillover effects make up between 0.4 % and 70% of the conditional variance of the unexpected return of Mexico. There are two features of the variance ratio series in Figure 2 that stand out. The first is that in the Canadian equity market the variance ratios series appears to be more stable.  1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 impact of the US market. This is further highlighted by the conditional correlations between the unexpected return of the US and that of Canada and Mexico. These are shown in Figure 3. We can see that the conditional correlation between the two largest and more mature markets, the US and Canada, is the most stable. The correlations between the US and Canadian returns fluctuate within a narrow band around 0.6.
Conversely, the correlations between the US and the Mexican returns fluctuate around 0.3 in the period prior to the October 1997, while in the following period they begin to increase substantially, peaking at more than 0.8 in 2001. impacting on the conditional volatility of the Canadian and Mexican equity markets more deeply than positive shocks. Moreover, while the impact of positive shocks is not much different between the two markets, this is not the case with negative shocks, which impact on the volatility of the Mexican stock market more intensely than on the volatility of the Canadian stock market. This finding is generally consistent with the results of Bekaert and Harvey (1995), Karolyi and Stulz (1996) and Bekaert and Harvey (1997)  Proof. Using the formulas for the moments of the truncated normal distribution (see, e.g., Maddala, 1983, p. 365) and applying the law of iterated expectations yields

Figure 3. Correlations between Unexpected Returns Computed Using the
Similarly,