Forecasting anything but the future
in Latin American Markets

Carlos Virgilio Zurita*

August 31, 2006.







Abstract

This paper analyses the volatility phenomena in Latin American Equity Markets. An asymmetric
time varying approach is developed and applied to the three major indices.
A strong evidence of returns asymmetry and autocorrelation is found and modelled through
impact surfaces. Furthermore, the out of sample forecast relationship between the realized
volatility and the asymmetric model is considerably higher than the realized volatility and the
sample variance model relationship, the later being negative.
This results leads to particular remarks about financial policy and risk management applying
traditional models in emerging economies.


Resumen

El presente trabajo analiza el fenómeno de la volatilidad en mercados de renta variable
de América Latina. Un enfoque dinámico asimétrico es desarrollado y aplicado a tres
índices principales.
Se encuentra una fuerte evidencia de asimetría y autocorrelación en retornos, la cual es
modelada a través de superficies de impacto. Para pronósticos fuera de la muestra, la relación
entre varianza realizada y varianza del modelo asimétrico es marcadamente superior a la
relación entre varianza realizada y varianza histórica, siendo incluso la última negativa.
Estos resultados conducen a particulares implicancias sobre la aplicación de modelos
tradicionales de volatilidad para la política financiera y la administración de riesgo en economías
emergentes.


JEL Classification: C5, G1.

* UCEMA-ROFEX; email: cvz04@cema.edu.ar. The author wishes to thank Enrique Kawamura (UDESA) and Juan
Pacheco (INDEC) for helpful suggestions.

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I. Risk and volatility in finance.

The use and forecast of a degree of uncertainty plays a key role in today's financial stability and
management of risk. Whether in microeconomic applications, such as the pricing and hedging
of derivatives instruments, or macroeconomic analysis, such as inflation rate trends, volatility
measures define agents paths of decisions.
Due to the fast growth of emerging markets and the development of new and more complex
financial instruments the need for theoretical and empirical risk models becomes a priority for
policy makers. Furthermore, since Basle Accord II, financial institutions and insurance
companies are suggested to estimate volatility forecasts and value at risk techniques to
reinforce their risk systems and prevent credit events, leading to a new field in the
econometrics of finance.
Consequently, the relevance of a consistent measure of volatility in asset pricing, risk
management and monetary policy making is clearly an important issue though it is not possible
to erase risk, but it is certainly preferable to have an accurate measure than a static one.
In brief, a volatility model should be able to forecast the volatility of the variable with a rational
approach. Thus, estimation, forecasting and testing becomes a unified experiment.
Unfortunately, the answer about how to define volatility is non trivial. Traditionally, the square
root of sample variance of returns around its mean value has been used as a benchmark.
Sample variance, although an acceptable measure of disturbance in statistics, has specific
drawbacks when applied to financial time series analysis. First of all, sample variance is an
equally weighted measure of selected observations, and thus gives the same relative
importance to recent and old events. Second, this estimator is a memory less function of time,
which means that for a particular selected group, gives zero relative importance to
observations outside that group1. Third, it is a symmetric measure of risk since its definition of
second moment disturbance around its mean the sign of changes becomes meaningless when
squared.
The objective of this work is to introduce a flexible measure of volatility, adapted from the
Variance Heteroskedastic ARCH models of Engle (1982) and Bollerslev (1986) to allow for
higher order asymmetry than Glosten (1993), a common pattern found in Emerging Markets.
The presented model covers the three main pitfalls of the traditional historical volatility
measure used as a standard concept of volatility by practioners and policy makers.
With ten year historical series of daily returns, it is performed an out of sample experiment of
one day ahead forecasts and compared with the ex post realized volatility to obtain error
predictions measures. The same analysis is applied to the benchmark model, namely, the
historical sample variance of returns.
The hypothesis of clustering in returns first proposed by Engle, and the negative correlation of
returns and variances accounted by Black (1976) and Christie (1982) is tested in Latin
American Stock Market indices of Argentina and Brazil and the developed market of USA.
The analysis is summarized through impact surfaces, which also allows for asymmetry in
returns and persistence of past variance forecasts, an extended version of impact curves of
Engle (1993).
The paper is organized as follows: next section deals with volatility concepts, standard models,
variance heteroskedastic models, and develops the asymmetric conditional variance model,
which is exposed with an impact surface depicting persistence and skewness of returns.

1 Indeed, the first two issues create a trade-off between the use of long sets, giving a relative high weight to past
events, and short sets, adding noise to the output needed, i.e., the one day ahead forecast of conditional variance.
This topic is discussed in Section II.
1

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Section III outlines the implications of the econometric model and the forecast of variance in
finance policy and decision making. Section IV presents the empirical evidence found in Latin
America for Argentina and Brazil stock indices and for the Dow Jones 500 of USA. Then
comparison of the proposed model with the traditional one is made. Finally, Section V leads to
some concluding remarks regarding the forecast of uncertainty in emerging markets.


II. A definition for volatility. Old standards, new insights.

The dynamics of financial time series require a consistent approach to forecast short run
uncertainty in the light of volatility modelling as a major topic in finance.
The primary input in the modelling of variance are indices returns defined from daily index
price as y = ln( p ) − ln( p ) where y stands for index price return from day t to t-12.
t
t
t 1
−
t
Traditional expected volatility return is explained by the sample variance as
n
−
v
= (n −
1
1)
*
y − y 2
)
t + 1
∑ ( t

(1)
t = 1
Where v stands for the forecast of volatility, the root squared variance, for period t+1
t 1
+
based on the information set available of log returns y on period t.
t
While the temporal behavior of returns and its volatility forecast can be estimated with almost
any disturbance method, several features concerning the underlying asset must be taken into
account.
Looking at equation 1 several issues might be stylized. Though it is certainly known that closer
days are more relevant than farther days in order to forecast a value, equation 1 gives the
same weight to all days included in the set, i.e.,
1
(n
)
1 −
−
. As stated above, this issue creates a
trade off between relevance of updated events and noise added because of too little
information set.
A second fact is how expected volatility reacts to news arrivals. Equation 1 states that a k
percent increase in returns must lead to the same expected volatility as a k percent decline in
asset returns. However, bad news in equity markets, that is, price drops, increases volatility in
a higher degree than the opposite price change. Therefore, volatility reacts asymmetrically to
asset information arrivals3.
One possible argument for this issue would be debt to equity ratio (D/E) variation. Maintaining
debt firm constant for short time horizon, a declining price situation produces an increment in
debt to equity ratio, and thus firm becomes more leveraged, increasing risk to equity holder,
leading to a higher expected volatility4.

2 Since the goal is to forecast unexpected changes, indices are computed with dividends adjustments.

3 While this idea is test in the presented work for equity markets, similar argument might hold for interest rate
instruments, this is, a rise in interest rate decreases bond prices and augments bond expected volatility as pointed in
Brenner (1996).

4 A similar argument should hold for rising prices and D/E ratio deterioration with a finally lower volatility. This
issue is tested to the indices rejecting the hypothesis of deleveraged events. See Section IV.

2

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While several deviations from the preceding equation might be used instead, those including
squared returns around zero, semi variance, etc., they still have the same statistical
misconceptions.
What is observed in real market data is that volatilities are positively auto correlated, the so
called clustering behavior, which means that the impact of past and current volatility on the
expected value tends to prevail several periods in the future.
In addition, the precise difference with risk and traditional volatility measure must be explicitly
incorporated, while the former is aware about a rise or a fall in the asset value, the later does
not.
The following figure depicts index value and continuously compound returns series for the
Argentinean Merval Stock Index5. As it can be seen, variation of returns comes in clusters,
there are periods of relative calm followed by periods of high variation of returns.



Also, large declines in asset prices generates more expected risk than large price increases,
signaling that investors particularly fears downside movements.
Due to issues presented above, a model that incorporates the observed situation should be
useful in forecasting an appropriate measure of risk.
The plan is to model these particular features, explain it with an impact surface, forecast the
next period conditional variance and then compare it with the traditional forecast model.
Therefore, the proposed model should be consistent with the autocorrelation of returns,
where ARIMA modelling might contribute, and squared returns, where GARCH modelling can
include the situation previously exposed.




5 Price-Returns figures for the Brazilian Stock Exchange Bovespa and for the American Dow Jones 500 presented in
Appendix A.

3

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Given the available information set y | Ω of realized values, the conditional mean is
t 1
−
modelled as an ARMA(p,q) approach
Φ (L)y = Θ (L ε
)






(2)
p
t
q
t
ε = η * v with η ~ F( )
1
,
0





(3)
t
t
t
t
Where Φ (L) is the AR polynomial, Θ (L) is the MA polynomial, L is the Lag operator and
p
q
F is a probability distribution with zero mean and unit variance.
The conditional expected variance is governed by the following process6
v
= ω + Z (ε ) * ε + β * v
t + 1
t
t
(4)
Where ω stands for the unconditional variance while ε and v are past residual and variance
t
t
respectively. Second term Z (ε ) equals α *ε for the standard Generalized Autoregressive
t
Conditional Heteroskedastic GARCH(1,1) model, and
−1
(−)
Z (ε ) equals α * ε +ψ * ε
* ε
for
t
t
t
the Modified GARCH(1,1), which allows for asymmetry. The ψ parameter allows for variation
in returns to affect asymmetrically since (−)
ε stands for negative returns. Finally, α stands for
t
the short run persistence factor, while (α + β ) stands for the long run persistence.
Non-negativity of forecast variance requires ω > ,
0 α ≥ ,
0 β ≥ ,
0 ψ ≤ 0 .
To obtain the variance forecast it is necessary to specify a probability density function in order
n
to optimize the log likelihood function, namely, L(θ ) = ∑log f (ε ,v
t
t ) with conditional mean
t =1
and variance recursively calculated from (2)-(4) equation set. The Gaussian probability function
is applied in this model.
Then, next period variance is due to unconditional variance,ω , last squared residual, 2
ε , and
t
last forecast variance, v . Adding the asymmetric residual term, ψ , forecast variance reacts
t
differently to news arrivals.
With equation 4 as the process for expected variance, clusters and leverage effects can be
explained with the following figure named impact surface for the Merval Index7. The specific
shape of the surface is due to the parameters estimation of model equations, which are
obtained in Section IV from optimization procedures. This figure permits to intuitively find the
relationship between how returns affects future volatility and how persistent is the shock
imposed.
X-axis (ε shows the differential of impact between good and bad news. While good news
t )
contributes to expected variance, Z-axis (v
, with (
2
α *ε , bad news contributes variance
t )
t 1
+ )
with (α *ε 2 +ψ *ε .
t
t )
Y-axis (vt ) shows the persistence of current and past volatility on the expected value.


6 This paper uses a first order lag in both the autoregressive and the moving average terms of general GARCH(p,q)
model since extra lags parameters does not proves to be significant. For a complete survey of ARCH Models see
Granger (2003).

7 Impact surfaces for the remaining indices presented in Appendix F.
4

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An interesting point to remark due to results obtained in Section IV is the clustering and
leverage effects across markets. Particularly, the Argentina Stock Index, compared to Brazil
and the United States, has the highest impact of news on expected volatility, and the lowest
persistence of past volatility on the expected one with a lower slope pointing that current
estimate of volatility remains few periods in the future. These features might indicate that the
market is very sensitive to news while the effect of news tends to deteriorate faster.


III. Model implications in economics.
The aim of risk modelling in order to make intertemporal decisions is to provide a volatility
measure and the success of this modelling is related to its out of sample forecast.
Clustering and asymmetry doe not seem to be included in traditional models. Since conditional
variance phenomena described the situation in financial markets and outlined issues concerning
what drives volatility through next day, a model is proposed to account for all this features.
However, it is mandatory to evaluate these ideas in order to obtain several conclusions.
It is a reasonable assumption that investors especially dislike downside returns. Since
symmetric models, neither homoskedastic nor heteroskedastic, does not accounts for sign
shocks, they finally underpredicts the volatility following bad news. Opposite argument holds
for good news arrivals.

5

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An accurate forecast of volatility has many fields where can be a useful input. In asset pricing
models such as CAPM, expected returns might be better explained with asymmetric GARCH
models than historical sample variances, since its relation with expected market volatility, with
different models implying different market risk premiums. This would also imply that beta
coefficient might not reflect the entire relationship between market and stock volatilities.
Following with equity valuation, the standard benchmark Sharpe ratio, defined as asset return
spread over treasury times reciprocal volatility would not account for good and bad news
differently.
Also, the asymmetric term might be used as a proxy for implicit leverage of firms when short
debt information is not periodically reported.
In derivatives pricing, the volatility of the underlying asset play a crucial role in finding the fair
option price, Hull (1987) finds that option contract might be substantially misspriced when
symmetric static volatility is used. Moreover, sensitivities like vega will be influenced by
different volatilities forecasts. Also, dynamic hedging strategies relative to options and futures
might be an important user of different predictability of volatility models.
Monetary institutions such as the Federal Reserve or the Bank of England collects volatility of
stocks, bonds and currencies in establishing its monetary policy, with the former two assets
showing strong evidence of asymmetry in returns.
Nowadays, private institutions include modifications to traditional sample variance. The
Riskmetrics division of J. P. Morgan publishes a simplified GARCH approach called EWMA,
which stands for exponentially weighted moving average, differentiating from the equally
weighted moving average rolling variance, which is the standard benchmark. Though symmetric
and simple in nature because of its one parameter model structure, EWMA has the ability to
capture clusters.
Value at Risk confidence bands improves its accuracy using asymmetric conditional variances
due to under predictions of bad news in symmetric models.
In addition, contagion effects of volatility through markets might be detected in advance with
clustered processes. There is evidence of spillover effects of conditional volatility supported by
Hamao (1990), where market uncertainty is transferred from the United States to United
Kingdom.
These features as a whole study are summarized in the impact surface shape, even though
developed markets show signs of asymmetry in some cases, it is particularly useful for
emerging markets where news has a greater impact on expected volatility. Besides, despite
clustering effect is found, volatility shocks tend to be less persistence relative to developed
markets.
Finally, as pointed in Section II, due to debt to equity ratio, if falling prices increases volatility,
rising prices might have some decreasing effect in volatility. This would lead to an extra term in
Equation 4.3 that incorporates only good news8.
Thus, anything including expected volatility as an argument might be better explained with
conditional asymmetric variance models than using static sample variances as a proxy of risk.



8 This idea is tested adding a term that captures the deleverage effect. Since resulted parameters are not significant,
the hypotheses are rejected (not reported).

6

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IV. The clustering and the asymmetry of returns in Latin
America and above.
Once described the situation in financial markets and outlined issues concerning what drives
volatility, four models are applied and tested to account for all these features.
Descriptive statistics depict that returns exhibits kurtosis higher than normal, which might be
explained by clustering in returns over time. Also negatively skewed returns might indicate that
bear markets generates higher expected volatility than rising markets9.
Then, the analysis in order to arrive to a final volatility forecast model is structured in three
stages:
First it is tested whether returns exhibits autocorrelation or not, and model them trough an
ARMA process. Since the objective of this paper focuses on variance forecast rather than mean
modelling, simple ARMA models are selected due to presence of correlation in returns10.
Secondly, the idea of clustering, measured as autocorrelation in squared residuals is tested
with the ARCH Lagrange multiplier test to look for heteroskedaticity. Since GARCH models
are designed to capture this issue, they are applied to the selected series.
Third, although GARCH model accounts for clustering in most cases, it does not distinguish
between good news and bad news. Third step deals with this problem, which consists in
testing the asymmetry with the previous symmetric GARCH model. Doing so, residuals and
squared residuals are related with one period lag with cross-correlograms. With asymmetry in
returns detected, two models incorporating skewness are used, namely, GJR due to Glosten et
al. and Modified GARCH models.
Thus, equation 4 might be decomposed in the following GARCH(1,1), GJR(1,1)11 and Modified
GARCH(1,1) models, respectively
v
= ω + α *
2
ε
+ β * v
t +
(4.1)
1
t
t
2
2
v
= ω + α * ε
+ β * v + d * γ * ε
t +

(4.2)
1
t
t
t
2
(− )
v
= ω + α * ε
+ β * v + ψ * ε

(4.3)
t + 1
t
t
t
It is important to note that even though the process is separated in three steps, the model
calculates the parameters in a recursive fashion, that is, coefficients of equation set (2)-(4) are
obtained as a unified framework12.
The interest of this work is focus on out of sample forecasts, hence models are estimated with
a truncated time frame at 2600 observations, leaving more than 100 days to evaluate forecasts.
Since realized variance it is not observable in advance, five days a head expost variance
measure is selected as the realized model13.
Thus, with four models, three GARCH models, and the rolling historical sample variance
benchmark model, forecasts are made and examined with expost realized variance.

9 See Appendix A for stock indices descriptive statistics.

10 Nevertheless, it is found that adding extra terms to mean equation does not necessarily improves variance
forecasts.

11 With d = 1 if residuals are negative and zero otherwise.

12 This is due to in some cases a mean model might not result significant in parameters when the ARMA-GARCH
framework is analyzed. In all cases, the Marquardt algorithm is used for optimization procedures.

13 Whether five days ahead is a rather arbitrary number of observations, the final set does not change results.
7

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Comparison is made with several error prediction measures, namely, root mean squared error
(RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE). Finally, the
forecasted series are analyzed with a correlation matrix for the cited models.
The sample selected corresponds to daily close to close logarithmic returns for the January 3,
1995 – December 30, 2005 interval. The time frame applied is due to particular issues
concerning number of observations, Ng (2003) suggests a sample observation longer than 2000
in order to obtain consistent estimators.

Results
Following table summaries the results for the Merval Index and shows that, regarding mean
equation of GARCH and GJR, all coefficients are significant at 1% level significance.

Dependent Variable: Merval_RET
Method: ML - ARCH (Marquardt)
Sample(adjusted): 11 2600
Model
GARCH(1,1)
GJR(1,1)
Mod. GARCH(1,1)
Mean Equation

Coefficient
Coefficient
Coefficient
AR(1)
0.10109
0.11872
0.13484
Std. Error
0.02187
0.02175
0.02233
Prob.
0.00000
0.00000
0.00000
AR(10)
0.04133
0.04372
0.05207
Std. Error
0.01952
0.01864
0.01694
Prob.
0.03425
0.01901
0.00212




Variance Equation

Coefficient
Coefficient
Coefficient
Constant
0.00002
0.00002
0.00002
Std. Error
0.00000
0.00000
0.00000
Prob.
0.00000
0.00000
0.00000
ARCH Term
0.11970
0.05193
0.08400
Std. Error
0.00810
0.00863
0.00877
Prob.
0.00000
0.00000
0.00000
GARCH Term
0.85102
0.84565
0.73326
Std. Error
0.01012
0.01128
0.01558
Prob.
0.00000
0.00000
0.00000
ASYM Term
-
0.13365
-0.00810
Std. Error
-
0.01411
0.00069
Prob.
-
0.00000
0.00000




Log likelihood
6347.95212
6375.62388
6506.55870
Schwarz criterion
-4.88672
-4.90505
-5.00616
Akaike criterion
-4.89803
-4.91863
-5.01974
8

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For the Merval Index of Argentina, due to correlograms of returns, series is modelled with an
AR(1) AR(10) model. For the Bovespa Index an AR(1) process is selected, while for the DJ500
Index raw returns are selected due to non significance of AR and MA correlograms14.
Looking at ARCH LM test, modelling conditional variance is needed due to autocorrelation
found in squared residuals of the ARMA model. This feature is found in the three markets
studied15.
Regarding negative returns effect on expected volatility, cross correlograms indicates that
symmetric GARCH model cannot solve the situation, with asymmetric GJR model partially
incorporating it to the model. However, asymmetry effect does not remain with Modified
GARCH16.
Due to heteroskedasticity in squared residuals and asymmetry of lagged residuals, the three
ARCH models are applied to the series.
Looking at the preceding table, Log likelihood indicates that Modified GARCH has a greater
accuracy in modelling the three indices series. Akaike and Schwarz Information criteria support
this finding in all markets.
Long run persistence in Latin markets shows lesser impact on future volatility than developed
markets, as evidenced by parameters for Merval (0.73) Bovespa (0.78) and Dow Jones 500
(0.83). In addition, long run – short run ( β − α ) parameter difference indicates that
importance of new events is higher in Merval (0.64) relative to Bovespa (0.75) and DJ500
(0.81).
Relative to short run persistence measured by α coefficient, Brazilian and American indices
shows significantly lower persistence (0.02 and 0.03 respectively) relative to the Argentinean
market (0.08).
It shall be noted that asymmetric coefficient for Merval and Bovespa doubles American Dow
Jones 500. These finding supports the sensitivity of emerging markets to news arrivals relative
to mature markets already expressed.
Since the occurrences of extremely low daily returns (below negative 10%) are rare as data
indicates, Modified GARCH has the ability to capture asymmetry of observed returns more
clearly than GJR. It is found that 99.23% of the time frame selected presents daily returns
higher than -7.5%, a threshold in which MGARCH model contributes to higher volatility than
GJR model. Then, while both models allows for differences in sign returns, the former allows
ψ
for higher levels of sensitivity to bad news in the forecast variance, with
> 1 for the
γ *ε
three markets.
Evaluating the forecast performance with error measures, although not highly significant,
RMSE, MAE and MAPE indicates greater accuracy when GARCH models are used instead of
sample variance, especially when MAPE measure is used, this conclusion stands for the three
markets studied17.



14 See Appendix D including resulting coefficients for Bovespa and DJ500 models.

15 See Appendix B for Heteroskedasticity tests in the three markets.

16 See Appendix C for Squared Residuals – Lagged Residuals Cross Correlograms.

17 For the remaining indices see Appendix E for Correlation Matrices and Error Measures.
9

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