Standard GARCH model

The standard GARCH model (1,1) proposed by ^{Bollerslev (1986)} in literature on volatility, is a generalized alternative of ^{Engle's ARCH model (1982)}. In this context, the model of the conditional measure and the conditional variance is governed by:

where μ is the conditional measure, h _t expresses the conditional variance that depends on the last innovation of the square residues ε² _t−1 also known as the ARCH effect and the previous conditional variance h _t-1 , ω is a deterministic term, and its function allows for the conditional variance to reach a positive level as long as the level of persistence determined by α + β is lower than 1. The term AR(1) or autoregressive of the 1st order is aggregated to the equation of the conditional measure, given that the present yields are highly correlated with the distant yields in time.

One of the deficiencies of the GARCH model is that it does not allow differentiating the patterns of decline of the persistence in short and long-term volatility, so that a model that is able to capture the high persistence in volatility is recommended.

Two-component CGARCH model

^{Engle and Lee (1999)} propose the CGARCH model as an alternative to capture the property of high persistence in volatility of the financial yields. The approximation allows decomposing the conditional volatility in two components and properly describes the behavior of decline of the persistence in short and long-term volatility.

The specification of the CGARCH model (1,1)⁶ is defined as

where h _t indicates the short-term volatility level that captures innovations, fed through exogenous events related to economic, geopolitical and even speculative aspects, and which fluctuate in a cyclical manner; q _t represents the long-term volatility or tendency, which converges at the level of the unconditional volatility ω to the velocity of (α+β) < δ < 1. The term (ε² _t−1 − h _t−1 ) works as the dynamic power for the movements of the tendency and the difference between the conditional variance and its tendency, (h _t-1 - q _t-1 ) is the temporary component that converges to zero at a velocity (α + β).

The CGARCH model collects the effects of short and long-term persistence, but its capacity is reduced before the presence of asymmetric effects. Due to the fact that negative shocks have a different impact on the volatility than positive shocks of the same magnitude, not only in the short-term, but in the long-term as well.

Asymmetric CTGARCH model

The flexibility of the CGARCH model allows for the capturing of asymmetric effects in the short and long-term, by only adding asymmetry parameters to its econometric structure, i.e., using the results of the TGARCH specification proposed by ^{Glosten et al. (1993)}. The asymmetric structure or CTGARCH model captures the asymmetry effects in the following manner:

where the dummy variable is governed by the Heaviside indicator function I (•), which is equal to 1 if ε _t-1 < 0 and zero in any other case. The asymmetry effect is observed if γ > 0 and ψ > 0, which indicates a greater impact than bad news with values (α + γ) and (ρ + ψ) on residuals ε² _t−1 in the short and long-terms and the effect of the optimistic news is measured by α and ρ.

CEGARCH asymmetric models

Another extension of the CGARCH model for the capture of asymmetric effects in the short and long-terms, is proposed in this study under ^{Nelson's EGARCH structure (1991)}.

The CEGARCH model has the following form:

where the asymmetry parameters γ and ψ are negative unlike the CTGARCH model, which indicates a greater impact of the bad news in the short and long-term volatilities than the good news in the same magnitude. The total effect in the short and long-term volatilities is of (α−γ) |ε _t−1 | and (ρ − ψ) |ε _t−1 | if ε _t-1 < 0 or (α + γ) |ε t−1| and (ρ + ψ) |ε t−1| when ε _t-1 > 0

Evaluation of the predictive performance of the volatility models

In this section we describe the measures of errors for the predictive performance evaluation of the volatility models. In general, this process is carried out outside the sample because the participants in the equity markets are more interested in the capacity of reaction of the models when new information arrives to the market.

The predictive error measures are classified in symmetric and asymmetric. Among the more common symmetric measures are the mean squared error (MSE) and the mean absolute error, which are defined in the following manner:

where T indicates the number of predictions, h _t is a proxy variable for the no observable volatility, which is generally obtained from the square yields, and ĥ _t is the estimated volatility through the different GARCH specifications.

In an analytical study, ^{Patton (2011)} showed that these measures are strong in order to minimize the predictive error. However, none of them provide the additional information on the asymmetry in the errors; that is, when the models underestimate or overestimate the no observable volatility.

According to ^{Brailsford and Faff (1996)}, the asymmetric error measures give a different weight to the underestimated and overestimated predictions of the volatility with a similar magnitude, and are defined in the following manner:

where U and O represents the underestimations and overestimations, and their sum indicates the total number of predictions T.

Recent investigations on the prediction of volatility outside the sample have empirically demonstrated the power of the error measures in the generation of information regarding the evaluation of the estimated models. However, one of the disadvantages of the measures is that, unlike the contrast hypothesis tests, it does not allow for a strong analysis in a statistical framework. This is due to the fact that it cannot be concluded that the predictive performance between the two estimated models is significantly different from a statistical point of view by only comparing their predictive errors. In order to alleviate this problem, this work uses ^{Hansen's (2005)} superior predictive ability test (SPA). This strong statistical test allows for the evaluation of the performance of two or more estimated models, unlike the ^{Diebold-Mariano (1995)} and ^{White (2000)} statistical tests.

In this context, the predictive evaluation of the models is carried out based on the measurements of errors, given that the base model (best approximation) is chosen by the measurement with smallest predictive error. The SPA statistic test consists in determining the model with the best predictive performance. In the period t, the superior predictive performance of the alternative model k in relation with the base model is defined as:

where L _0,t is the prediction error as determined in Eqs. (12), (13), (14) and (15) for the base model M ₀ and L _k,t is the prediction error associated with each alternative model M _k .

Under the assumption that the vector d _k,t is strictly stationary, the null hypothesis of interest, that none of the alternative models reach a higher predictive performance in relation to the base model, can be presented as:

Here, the use of the estimator μ _k ≡ E [d _k,t ] allows for the reduction of the impact of the models with a weak predictive performance, but at the same time it controls the impact of the alternative models with μ _k = 0 as documented by ^{Hansen (2005)}.

where 1_{⋅} is an indicator function. Furthermore, an immediate result of the assumption of seasonality is that the selection of the threshold √2 log log n guarantees the consistency of the estimator μ_k ^c for a sufficiently large n, even for the alternative models with μ _k = 0.

Consequently, the statistic of the null hypothesis is defined by:

where ŵ _k ² is a consistent estimator of ω _k ² ≡ .

For the estimation of ω _k ² ≡ and the probability of the statistic T _n ^SPA , ^{Hansen (2005)} suggests the use of the stationary bootstrap procedure based on ^{Politis and Romano (1994)} in order to obtain the empirical distribution of the contrasting statistic under the null hypothesis, defined by the following expression:

where b = 1,...,B determines the number of bootstrap samples of the vector d _k,t for k = 1,..., m and . In order to obtain reliable results that do not affect the current samples, B must be rather large.

The probability value of the SPA test is calculated in two stages. First of all, the values of the statistic T _b,n ^SPA * are obtained for each of the bootstrap samples b = 1,...,B, which is defined as:

Finally, the values of the statistics T _n ^SPA and T _b,n ^SPA * are compared in order to obtain the bootstrap probability value, i.e.,

The null hypothesis is rejected when the probabilities reach small values.

Description and preliminary analysis of the data

Even if the equity markets of the emerging countries are characterized by experiencing high volatility, effects of asymmetry and an elevated degree of persistence, their study has been concentrated in the temporal behavior of the characteristics common of volatility, and therefore there is the need for a study on the long-term effects of asymmetry and persistence on validity. This work investigates the long-term effect of asymmetry and persistence in the prediction of conditional volatility using the daily prices of the six most important equity markets of the Latin American region: Argentina, Brazil, Chile, Colombia, Peru and Mexico.⁷ The analysis covers the period from January 2nd, 1992, to December 31st, 2014, with a sample of approximately 5951 daily yields. The price series of the stock indexes were obtained from the Bloomberg database.

Table 1 shows the basic statistics of the stock yields. Apparently, all of the stock indexes show properties very similar with average positive yields, which is justified by the common tendency of the rise of the stock prices during the period of analysis, as shown in Fig. 1. Nevertheless, the standard deviation of the yields is relatively high, which implies a greater exposure to risks for the participants in these stock markets, particularly in Argentina and Brazil.

Fig. 1: Dynamic behavior of the Latin America equity markets. 

All the financial series, with the exception of Argentina, show the characteristics typical of positive asymmetry and an excess of kurtosis, which indicates that the positive shocks are more frequent than the negative and leptokurtic yield distributions, with longer and broader tails than the normal distribution, in particular the upper tail. The phenomenon of normality of the distribution is also confirmed by the Jarque-Bera statistic value. Regarding the statistics of Ljung-Box Q(10) for serial correlation, the results show that the hypothesis that there is no autocorrelation of the 10 order is rejected by the small probability values, which confirms the presence of linear dependence in the stock yields. Likewise, the statistically significant serial correlation in the squared yields, determined by Q² (10), imply that there is non-linear dependence in the series of the yields of all the equity markets.

Furthermore, there is the strong presence of conditional heteroscedasticity or ARCH effects in all the series of the financial yields, which is supported by the significance of the statistic of the Lagrange Multiplier test with a level of 5%. This characteristic common in the financial yields is widely supported by Fig. 2, where one can appreciate strong evidence of validity in agglomerations. It can also be observed that the volatility intensity is more persistent when the prices of the stock indexes tend to decrease than when they increase.

Table 1: Descriptive statistics for each daily equity returns series. 

Notes: The descriptive statistics of Latin America equity market returns are expressed as percentages for the period from 2 January 1992 to 31 December 2014. The numbers in parentheses are p-values of Jarque-Bera, ARCH effect, and Ljung-Box statistic tests of the return and squared return series.

Fig. 2: Returns of the Latin America equity markets. 

In conclusion, the preliminary analysis of the data recommends the use of GARCH processes of two components that manage to capture the short and long-term effects of asymmetry and the phenomenon of persistence in the innovations of the stock yields.

Estimation results of the volatility models

In this paper, the CGARCH model of ^{Engle and Lee (1999)} is expanded in order to investigate if the characteristics of long-term asymmetry and persistence exercise effects in the prediction of conditional volatility of the yields of the equity markets of the Latin American region. The parameters of the volatility models are estimated within sample using the period of study from January 2nd, 1992, to December 31st, 2009, and assuming that the residuals are independent and identically distributed under a normal distribution.

The results of the parameters estimated under the four GARCH structures and their diagnostic tests on the simple and squared standardized residuals are reported in Table 2 for the indexes of Argentina, Brazil and Mexico and Table 3 for the indexes of Chile, Colombia and Peru. The estimators of μ that correspond to the specification of the conditional mean, are statistically significant at a level of 1%, except for the asymmetric CGARCH models of the equity markets of Argentina and Colombia. All the parameters ϕ of the autoregressive process of the 1st order are positive and significant at a level of 1%. This fact implies that the tendency in the changes of the stock prices is maintained in the same direction in the following period.

Table 2: Estimates of the volatility models for Brazil, Argentina and Mexico countries. 

Notes: Q(36) and Q² (36) denotes the Ljung-Box Q-statistics of order 36 computed on the standardized residuals and squared standardized residuals, respectively. p-Values are reported in square brackets. The numbers in parentheses are standard errors of the estimations. *Denotes significance at the 1% level. **Denotes significance at the 5% level. ***Denotes significance at the 10% level.

Table 3: Estimates of the volatility models for Chile, Colombia and Peru countries. 

Notes: Q(36) and Q2(36) denotes the Ljung-Box Q-statistics of order 36 computed on the standardized residuals and squared standardized residuals, respectively. p-Values are reported in square brakets. The numbers in parentheses are standard errors of the estimations. *Denotes significance at the 1% level. **Denotes significance at the 5% level. ***Denotes significance at the 10% level.

Regarding the parameters of the process of conditional variance, all the models successfully capture the dynamic patterns of the short-term conditional volatility: its estimated parameters are positive and statistically significant at the conventional levels, with the exception of the CTGARCH model of the stock indices of Argentina and Mexico. The sum of the parameters α and β less than the unit indicates a considerable persistence in the volatility of the temporal component, especially in the traditional GARCH model. In fact, it can be observed that the persistence coefficients α + β reach values of 0.9655, 0.9770, 0.9772, 0.9787 and 0.9865 for Brazil, Argentina, Chile, Mexico and Peru, respectively. Although the results of the two component models based on the CGARCH and CTGARCH specifications confirm the opposite. The estimations between 0.9555 and 0.9881 of the parameters δ, for the six equity markets of the Latin American region, clearly reveal that the component of the long-term volatility is more persistent and declines at a slower rhythm than the component of the short-term volatility. This is attributed to the fact that the values of the persistence coefficient are lower than the values of δ, e.g., 0.9266 versus 0.9874 for Chile and 0.8521 versus 0.9881 for Peru for the CGARCH and CTGARCH models, respectively. In contrast, the results of the CEGARCH model, in all the stock indices, denote that the temporal component of volatility is the most persistent.

Considering the statistical significant and the sign of the parameters of asymmetry that capture the impact of the news in the short and long-terms associated with the financial crises, stock market crashes and/or economic booms. The results under the CEGARCH model are mixed, as effects of asymmetry in the short and long-term volatility can be observed for the equity markets of Mexico and Peru, and only the long-term component of volatility in the stock indices of Chile and Colombia. In the case of the CTGARCH model, the positive and significant parameter at a level of 1% indicates that there is only asymmetry in the response of temporal volatility in the presence of financial crises and stock market crashes for all the countries of the Latin American region. Consequently, the implementation of the asymmetric CGARCH models is clearly justified by empirical results.

Finally, the diagnostic of the simple and squared standardized residuals is reported at the end of Tables 2 and 3. The results of the Ljung-Box tests indicate that the null hypothesis of the absence of 36th order autocorrelation in the standardized residuals is impossible to reject at a level of significance of 5%, which implies sufficient evidence in favor of the correct specification of the conditional mean to explain the behavior of the yields of the stock indices of Argentina and Mexico. In the case of the squared standardized residuals, the insignificance of the Ljung-Box statistics confirms the good performance of the volatility models in correcting the 36th order autocorrelation in the equation of the conditional variance of the financial yields of Argentina and Mexico. These facts indicate that there is statistically significant evidence of specification error in the GARCH, and the symmetric and asymmetric CGARCH models in order to describe the heteroscedasticity exhibited in these equity markets.

Evaluation outside the sample based on the test of superior predictive ability

In this section, the evaluation outside the sample of the precision and efficiency of the GARCH, CGARCH, CEGARCH and CTGARCH models is carried out in the period from January 4th, 2010, to December 31st, 2014. The parameters of the equation of conditional variance are re-estimated using a mobile window of approximately 4957 observations, i.e., from January 2nd, 1992, to December 31st, 2009, which implies that the most remote observation is removed and the most recent observation is added to the sample. The prediction obtained is compared with the non-observable variance or proxy in order to calculate the prediction error. The process is repeated until obtaining the prediction of the conditional variance of December 31st, 2014, for the equity market. Thus, the sample size is kept fixed during the re-estimation of the volatility models and the predictions outside the sample do not overlap.

Table 4 shows the results of the measurements of prediction errors of MSE, MAE, MME(U), MMM(O) and the probabilities of the SPA statistic test, which were estimated on a base of 10 000 stationary bootstrap samples of the empirical test under the different measurements of predictive errors. In this case, the highest probabilities reached by any base model indicate that the null hypothesis that the predictive performance outside of the sample of the alternative models is widely surpassed by the base model cannot be rejected. The first column of the Table constitutes the name of the base model that will be compared with the other three alternative models.

Table 4: Results of the superior predictive ability test. 

Notes: The values in bold face refer to the highest p-values of superior predictive ability test (SPA) under four criterions of the loss functions, i.e., MSE, MAE, MME(U), MME(O). The null hypothesis is that none of the alternative models have best performance than the benchmark model. The number of bootstrap replications to calculate the p-values is 10 000.

For the case of Argentina, Brazil and Mexico, the CEGARCH and CTGARCH models allow for more exact volatility predictions outside of the sample unlike the GARCH and CGARCH models under the four measurements MSE, MAE, MME(U) and MME(O). These findings are supported by the small values reached in the measurements of symmetric and asymmetric errors. Therefore, the empirical results clearly suggest that the volatility of the equity markets of Argentina, Brazil and Mexico respond in a different manner to good and bad news, which in turn implies that the negative shocks in these stock markets have a greater impact in the short-term, but limited effects in the long-term.

For their part, the probabilities of the SPA statistic test pointedly indicate that the CEGARCH model shows the best predictive performance outside of the sample than the alternative models based on the MSE, MAE and MME(O) models for the equity markets of Argentina and Mexico. In the case of the MME(U), the CTGARCH model provides the highest probability value of the SPA statistic test for all the predictions outside of the sample considered. This empirical finding is attributed to the fact that the asymmetric measurement MMM(U) penalizes the underestimated predictions of volatility, which in this case represent approximately 73.42 and 77.28% of the sample for Argentina and Mexico, respectively. Nevertheless, it is important to note that the probability of the SPA statistic test of the CEGARCH model is above the significance level of 5%, which means that it can still be considered an excellent base model in predicting future volatility in the equity market of Mexico, whereas in the case of the equity market of Argentina, model CGARCH is available as a second alternative. This finding is justified by the insignificance of the asymmetry parameter for both short and long-term.

The information generated by the measurements of asymmetric errors is relevant for the coverage strategies with contracts of options on stock indices, because there is a positive relation between volatility and the prime of the option. In this sense, the high percentage in the underestimated predictions of volatility can provide bias in the primes of the options, which would directly benefit the institutional investors who maintain long financial positions of replica portfolios on stock indices, and that in an uncertain environment have the need to protect them through put options, but at the same time harms option buyers.

On the other hand, the yields of the equity markets of the Latin American region possess similar characteristics. Nevertheless, the ability of the symmetric and asymmetric CGARCH models is not enough to provide precise estimations of future volatility in the equity markets of Brazil, Chile, Colombia and Peru. By analyzing the statistical results of Table 4, one can observe that the CTGARCH model reaches the best predictive performance in accordance with the small values of the measurements of symmetric and asymmetric errors. However, the results of the SPA statistic test contradict said empirical findings because the GARCH and CGARCH models provide the highest probabilities under the MSE, MAE and MME(O) criteria, respectively. This is attributed to the fact that none of the volatility models managed to eliminate the autocorrelation observed in the simple and squared standardized residuals in the equity market of Brazil, which leads to underestimate or overestimate future volatility.⁸