




































American Research Journal of Economics, Finance and Management 

Volume 11 Issue 4, October-December 2023 

ISSN: 2836-9416 

Impact Factor: 5.57 

Journal Homepage: https://americaserial.com/Journals/index.php/ARJEFM, 

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64 | P a g e  

A HISTORICAL PERSPECTIVE ON VOLATILITY IN THE 
MEXICAN STOCK MARKET INDEX 

 
 

María Rodríguez González 
Cristóbal Colón University- Veracruz, Mexico, Third year student of Doctoral Program in 

Management Science 
 

Abstract: The predictability of stock price movements has long been a subject of debate in academia 
and the finance sector. This study delves into the age-old question: how can historical stock price data 
be leveraged to forecast future behavior? This question has spurred two competing theories: the 
chartist theory and the theory of random walks. 
Chartists, who primarily follow Dow Theory and Technical analysis, share a common belief that 
historical asset price data holds valuable information for predicting future behavior. They posit that 
patterns observed in past prices tend to recur in the future, suggesting that history repeats itself. 
Consequently, analysts following this approach use historical patterns to forecast future price 
movements, aiming to boost their expected profits (Murphy, 2000). 
In stark contrast, the theory of random walks contends that stock prices follow a random walk, 
making it impossible to predict future prices based on historical data (Fama, 1965). This theory 
asserts that price changes are random variables that are independent and identically distributed 
(Johnston & Dinardo, 1997). In essence, past prices do not offer any information that can be used to 
anticipate future prices. Fama (1970) introduced three levels of market efficiency, with the weak form 
suggesting that the history of stock prices does not contain information capable of generating yields 
beyond what a random portfolio would yield. 
Keywords: stock prices, chartist theory, random walks, market efficiency, historical data, 
forecasting. 
  
1. - Introduction  
For many years, in the academia and finance sector the following question has been subject of 
discussion and controversy: how to extent, the historical data of stock prices to predict their behavior?  
The answers to this question have been addressed firstly by the chartist theory and the theory of random 
walks. The focus of chartists’ theories (primarily Dow Theory and Technical analysis) based on the same 
assumptions; assume that the historical data of asset price have much information that can be used to 
predict their behavior. The pattern of past prices tends to recur in the future, that is, history repeats 
itself.   
Thus, an analysis based on this approach to identify patterns can be used to predict future behavior of 
prices and thus increase expected profits (Murphy, 2000). 

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65 | P a g e  

In contrast, theory of random walks says that stock prices are determined in a random walk, its mean, 
cannot predict future prices from past prices (Fama, 1965). In statistical terms, theory says that the 
price changes are random variables, independent and identically distributed (Johnston & Dinardo, 
1997). That is, past prices do not provide such information that can be used to predict future prices.  
Fame (1970) proposed three levels of market efficiency with respect to the information reflected in 
prices. The weak form holds that the history of stock prices does not contain information that can be 
used to obtain yields above which gives a random portfolio.   
Therefore, in an efficient market in its weak form, prices follow a random walk. In semi-strong market 
efficiency prices even reflect public available information. With strong market efficiency prices add 
private information.  
Fame (1970) proposes three levels of markets efficiency in relation to the information of prices. The 
weak form sustains that the track record of stock prices does not contain information that could be used 
to obtain higher yields than those obtained by a portfolio of stocks, even taken at random. Hence it is 
written that in an efficient market in its weak form, prices follow a random walk. A market have 
semistrong efficiency, whether in addition the above, the prices reflect the publicly available 
information. An efficient market is strong, whether in addition the above, the prices reflect private 
information.  
In order to verify these assertions (despite the difficulty to test the semi-strong and strong efficiency) 
some statistical tests were developed to determine whether stock prices follow a random walk, example 
of this is: serial correlation analysis, unit root test and the variance ratio. The majority of the researches 
for developed countries and Latin American markets (Worthington and Higgs, 2006; López, 1998, 
2004; Ramírez, 2002) have found the presence of autocorrelation of stock returns, so random walk 
hypothesis it is rejected. In the last twenty years, several time series models have been proposed to 
represent the information of generation process on the basis of characteristics of the series of interest, 
the empirical evidence about behavioral and financial time series modeling show a more outstanding 
characteristic (Fama, 1965; Lamoureux & Lastrapes, 1990; Hassan, Islam & Abul, 2000; Geng, 2006; 
Balaban, Bayar & Faff, 2003; Ebeid & Bedeir, 2004), following it describes.  
i) Variance of series is not constant along the time, which violates the assumption of 
homoscedasticity from traditional models.  
ii) Substantial changes and conditioned volatility clusters in which the volatility is followed by high 
volatility and low volatility is followed by low volatility. This behavior suggests that the volatility of 
financial series is determined by an autoregressive process (serial correlation) and non-constant 
variance  
(Heteroscedasticity).  
iii) The distribution of gains although centered in zero show lightweights bias and the kurtosis value 
is greater relative to standard normal (excess kurtosis), implying biased results when assuming 
normality.  
iv) Volatility effect on stock prices. Bad news and good news affect the volatility in different ways: 
the first ones have greater effect than the seconds’ ones, therefore, if the volatility incremented, prices 
fall and vice versa, which allows observing a negative correlation.  

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The above results show the importance of modeling the financial series incorporating volatility (risk). 
From Engle (1982) famous work: Autoregressive conditional heteroscedasticity with estimates of the 
variance of United Kingdom inflation, a family of models of autoregressive conditional 
heteroscedasticity (ARCH) have been developed, mostly used to explain the volatility of financial yields.  
With regard to empirical evidence, there are a large number of studies that used ARCH models to 
analyze the behavior of volatility. In particular, we identify models that are more predictive for ex-post 
forecasts. 
Geng (2006) applies ARCH family models to analyze the characteristics of the volatility of the Chinese 
stock market. Compares several models by using standard criteria and concludes that the model 
EGARCH (1,1) and EGARCH-M model (1.1) have the same efficiency to forecast series. Ebeid & Bedeir 
(2004) evaluates the capacity of ARCH models to forecast stock market yields from Egypt and 
concludes that the GARCH (1,1) model is more appropriate for modeling the volatility of the market 
price index. Ludlow and Mota (2006) analyzes the financial markets volatility based on three indexes 
(IPC, Nasdaq, S & P500) using a multivariate model GARCH (1,1) and conclude that shocks derived 
from bad news have an impact with more force on market deeper.  
The work of Lopez (2004) evaluates the contribution of three models of the family ARCH (GARCH (1,1), 
TARCH (1,1), EGARCH (1,1)), for modeling the behavior of the Mexican stock market based in the 
IPC.The criterion used for the evaluation was; verify if the forecasted values of the IPC, once modeled 
the volatility, is capable of reproducing the first four moments of the distribution of the IPC. Concludes 
that the EGARCH (1,1) has better qualities to predict. 
The purpose of this work is to analyze the features of the volatility of the stock market in Mexico and 
identify the model with the best predictive ability. To this end, we apply the ARCH family models using 
daily returns of the Mexican stock market index. To identify the model with the highest predictive 
ability, it follows closely the work of Lopez (2004).  
The preparation of this exercise was justified by the great attention given the volatility term, not only 
by academics but also by investors in the stock market in Mexico which requires a measure of risk when 
making transactions. The investment portfolio design, the fixing of prices, calculation of value at risk, 
and financial strategies, justify the importance of modeling and forecasting the conditional volatility of 
returns. The main conclusion of this paper presents empirical evidence for EGARCH models for 
forecasting effects of ex-post.  
The work is organized as follows: second section presents the theoretical conceptualization of ARCH 
and GARCH models of Engle (1982), Bollerslev (1986), Zakoian (1990) and Nelson (1991). Third section 
presents the results of the estimation, statistical tests and forecasts. Fourth section presents the main 
conclusions. 
2. Models   
To capture volatility characteristics, Robert Engle (1982) propose to model the conditional volatility 
through a conditional heteroscedasticity autoregressive process (ARCH) in which the process mean is 
zero and variance dependent on conditional the random errors passed.  
However, empirical evidence shows that it requires a large amount of remnants to capture the dynamics 
of the conditional variance.   

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67 | P a g e  

Bollerslev (1986) developed a special technique using an ARMA process to the conditional variance of 
errors called GARCH (p, q) which has an autoregressive component and moving average on 
heteroscedastic variance.   
The advantage of GARCH about the ARCH model is that the GARCH model could to have more 
parsimony and representation that it is easier identify and estimate. The GARCH model is denoted as 
follows: 
y t x t t                                                   (1)  

t vt t 2 
                                                    (2)   
Where:  
  

v2 1 ,                                                      (3)  
  
And to model GARCH (1,1)  
  

t2 0 1 t2 1 1 t2 1                                       (4) 
v t 
Like in the ARCH model   , is white noise process which is independent  

t i from the past achievements of . The conditions over the parameters that guarantee  
0 0, 1 1 1 the stability of the model are:  . 
0 

The conditional variance depends upon three terms: the mean , information on the volatility of the 
previous period measured by the lag of the squared residuals (ARCH term) obtained from equation 1); 
and the last period forecast variance (GARCH term). 
GARCH model (p, q) is:  
  

v 2 t t t 
                                                   (5)  
q p 

t2 0 i t2 i j t2 j 
i 1 i 1                                    (6)  
  
The conditions on parameters are:  
  

0 0, i 0, j 0 1 2 q 1 2 p 1 
,           (7)  
  
Despite the fact that GARCH model is used to forecast volatility and determine the behavior of prices, 
there are situations that GARCH model cannot explain. The biggest problem is that standard GARCH 
models assume that the error terms positive and negative have the same symmetrical effects on 

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68 | P a g e  

volatility. That is, good news and bad news have the same effect on volatility in this model. The 
exponential GARCH (EGARCH) and Threshold ARCH model (TARCH) describe asymmetric market 
response under positive and negative impacts. 
EGARCH model of Nelson (1991) is denoted as  
q p 
ln( t ) 0 i g(vt i ) j ln( t j ) 
i 1 i 1                                           (8) 

g(vt ) ivt    
                                                              (9) 

0 0 
When , the effects of information are asymmetric. When  there is  
presence of effect “leverage”. There are not restrictions on parameters, which is an advantage of 
EGARCH model compared with the GARCH model.  
TARCH Model proposed by Zakoian (1990) have the follow expression:   
  
q p 

t 0 i t2 i t2 1dt 1 j t j i 1 i 1                    (10)  
  
Where d is a latent variable   

1 t   0 d  
0 t 0                       (11)  

  
t 0) and bad news t 0  have different effects In this model, good news ( on conditional variance. 

Good news has an impact of while bad news has an  
0 0 

impact . If  there is leverage effect. If  the impact of news is asymmetric.  
  
3. Estimation and Testing  
  
3.1 Data  
  
To carry out the estimation of the ARCH model, index of the Mexican Stock Market (BMV) are used. 
The daily yields are calculated as:  
  
Rt ln(IPCt / IPCt 1).  
  

t  

t  

t 

t 
E  

 

 

 

 
 

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Time series of IPC as representing the "price" of the stock market is used due to IPC is the main indicator 
of yields offered by investment in shares traded on the BMV and it consists of a sample of stocks with 
higher trading volume weighted on the basis of the proportion they have on market capitalization.   
Data used were daily closing of the IPQ, including all days of operations of the BMV for the period 
January 3th, 2000 to July 11th, 2008. We didn´t considere successive values in order to avoid stock 
return volatilities due to the impact of the recent global financial crises called the first largest crisis after 
the recession of 1930s (Ali and Afzal, 2012).   
In total 2136 observations, published in http://mx.finance.yahoo.com/. For data processing excel 
spreadsheet was used and to estimate the E-views 3 package was used.  
To identify the model which allows us to explain with more accuracy the historical performance of the 
IPC, and to obtain forecasts, the following methodology is used:  
a) 2127 sample observations are used to adjust the models.  
b) Volatility is modeled and the best model based on standard evaluation criteria (Akaike criterion, 
Schwarz criterion and value of the maximum likelihood function) is selected.  
c) With the resulting models, nine periods forward (July 1st to July 11th, 2008) are forecast and 
the results with real data from the same period, are compared.  
d) Accuracy measures are calculated (mean square error, mean absolute error, theil inequality 
coefficient and decomposition of the mean square error). This, because they allow evaluating the 
forecasts, and the model in which the difference between the forecasted values and real values is 
minimal, is chosen.   
3.2) Description of the Information  
This work begins obtaining a graph of the IPC in order to consider appropriate initial transformations. 
In graph 1 it can be observed an increasing tendency of the IPC over time, starting from an intercept. 
The graph is very similar as a random walk with trend.  
Graph 1: Historical IPC  

 

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Daily yield of the IPC over the sample period are presented in Graph 2 and it is possible to observe that 
series consistently fluctuates around zero, it´s variance is not constant through time and shows an 
autocorrelation behavior.   
Graph 2: Yields of IPC  

 
   

  
  
The estimated distribution of the yields (Kernel Epanechnikov, with Silverman bandwidth value h = 
0.0022) is shown in Graph 4, in which is again observed the excess kurtosis. The Jarque Bera test rejects 
the normal distribution in yields of the IPQ (Jarque Bera = 682.4052).   
Graph 4: Estimated Distributions of the IPQ Yields  
Kernel Density (Normal, h = 0.0022) 

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RIPC 
Statistically, the risk or volatility is the dispersion of the yields. Daily performance (average 
profitability) for the period under review is 0.000670 and the risk (standard deviation) per day is 
0.014114, which reflects the high volatility of the Mexican Stock Index. Table 1 summarizes the 
descriptive statistics in yields of the IPC.   
Table 1: Descriptive Statistics in Yields of the IPC   
Statistics   Value   
Mean   0.000670   

Median   0.001179   
Standard Deviation   0.014114   
Bias   -0.105974   
Kurtosis   5.767421   
Source: own   
3.3 Unitary Root Test 
In order to detect if the IPC yields series covariance is stationary, unit root test is carried out with the 
statistical Dickey-Fuller. Results are shown in Table 2.  
Table 2. Unitary Root Test   
Augmented  Dickey – Fuller  Unit Root  Test on RIPC  
ADF Test Statistic  -21.56508      1%   Critical Value* -3.4364  
      5%   Critical Value -2.8634       10% Critical Value -2.5678  
*MacKinnon critical values for rejection of hypothesis of a unit root.  

Source: own  
It was confirmed that the series RIPQ has not unit root, therefore, we can apply the conventional 
methodology of analysis for time series stationary in covariance.   
3.4 Models   
For adjusting the model series RIPQ, the AR and MA terms defining the process are identified, using 
2127 observations.   

0  

10  

20  

30  

40  

-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06  

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Different estimates were carried out by ordinary least squares and based on the Akaike, Schwarz and 
Log Likelihood criteria, the following ARMA (5,1) was chosen. The results are summarized in table 3.  
RIPQt 0 1RIPQt 1 5RIPQt 5 1 t 1 t                     (12)  
  
Rescuing the residual of the model, is verified through the ARCH-LM test (Lagrange multipliers) In 
order to get the residual of model, we may identify trough test ARCH-LM (Lagrange multiplications), 
if the squared errors (conditional variance are correlated with its past.   
Test result shows that errors are correlated with their immediate past, which indicates that we have 
ARCH effects (1) in the ARMA model. The RIPQ series is volatile.  
These results suggest modeling the effect of the autoregressive conditional volatility, therefore, a 
GARCH (1,1) model is proposed, thus we obtain the following equation:  
  
             RIPQt 0 1RIPQt 1 5RIPQt 5 1 t 1 t                  (13)   
2 2 2 
            0 1 t 1 1 t 1                         (14)   
  
When estimating jointly the equations 1) and 2) by the method of maximum likelihood, it is observed 
that the parameters of the terms AR (1), AR (5), MA (1), ARCH (1) and GARCH (1) is significant 
according to statistical z.  
As shown, the value of the sum of the coefficients of ARCH and GARCH term (0.081392 + 0.89368 = 
0.975072) is very close to 1, which shows the existence of high volatility. This suggests that given the 
possibility of a shock expected market the fluctuations do not stop in the near term. This is a signal of 
high risk. Similarly, considering that the sum of both parameters is less than unity, the proposed 
GARCH (1,1) model is a stationary process.   
In order to verify this, the WALD test was performed. The test ARCH-LM shows that are not taken 
problems of heteroscedasticity in the model.  
For describing the asymmetric market response under positive and negative shock, we proceed to 
estimate the EGARCH (1,1) model and TARCH (1,1) model.  
The first results show that skewness coefficient is statistically different from zero (z = -6.104619) and 
negative (-0.117746) which gives evidence of leverage effect on the Mexican Stock market for the sample 
period. This is, volatility caused by negative shocks is greater than those caused by positive shock.   
Results of the second model show that the asymmetry coefficient is statistically different from zero (z = 
5.776392) and positive (0.163874), which also shows evidence of leverage effect. This is consistent with 
the bulk of the research.  
Table 3: Parameter Estimates and Diagnostics of the Models  

  Models      
Terms ARMA   AR(1),   

AR(5),   
MA(1)   

AR(1),   
AR(5),   
MA(1)   

AR(1),   
AR(5),   
MA(1)   

AR(1),   
AR(5),   
MA(1)   

Parameters   ARMA(5,1)   GARCH(1,1)   EGARCH(1,1)   TARCH(1,1)   

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0   
  
  

1   
  

5   
  

1   
  
ARCH(1)   
GARCH(1)   

1   
  
Standard 
error of 
regression  
log 
likelihood  
Akaike 
criterion   
Schwarz criterion  
F-statistic   
ARCH-LM test   
  
Observations   

0.000660   
(0.000312)   
2.114489   
-0.378290   
(0.131065)   
-2.886274   
-0.055764   
(0.020027)   
-2.784496   
0.481759   
(0.124842)   
3.858934   
  
  
  
0.013906   
6060.702   
-5.711176   
-5.700502   
12.03360   
F= 38.89053   
  
2127   

0.00128   
(0.000253)   
5.071421   
-0.315002   
(0.120769)   
-2.608306   
-0.059082   
(0.021191)   
-2.788073   
0.404675   
(0.116990)   
3.459058   
0.081392   
(0.013212)   
6.160275   
0.899368   
(0.016351)   
55.00287   
  
0.013932   
6242.034   
-5.879334   
-5.860656   
5.137749   
F= 
0.403016   
  
2127   

0.000786   
(0.000260)   
3.019394   
-0.300079   
(0.121641)   
-2.466918   
-0.048604   
(0.020582)   
-2.361544   
0.392119   
(0.118283)   
3.315092   
0.128319   
(0.020068)   
6.394098   
0.965980   
(0.008411)   
114.8426   
-0.117746   
(0.019288)   
-6.104619   
0.013923   
6286.126   
-5.919968   
-5.898621   
4.951903   
F=0.20484   
  
2127   

0.000877   
(0.000260)   
3.376556   
-0.343568   
(0.123114)   
-2.790641   
-0.050880   
(0.021108)   
-2.410490   
0.436423   
(0.119088)   
3.664721   
0.006874   
(0.012468)   
0.551294   
0.877360   
(0.016949)   
51.76405   
0.163874   
(0.028370)   
5.776392  
0.013922   
6279.839   
-5.914039   
-5.892692   
4.995343   
F=0.360630   
  
2127   

Note: the values in each box are the coefficients of each term, their standard errors and the value of z-

statistic, respectively.    
Source: own  
It is observed in table 4, when comparing the values of the mean square error and the mean absolute 
error associated with the prediction of each of the models, that EGARCH model (1.1) has the minimum 
value, which suggests that the best model to forecast the IPC. Similarly, the coefficient value of 
inequality of Theil associated to  
EGARCH (1,1) is minimum, indicating that is the best fit model. However, the decomposition of the 
mean square error given by the bias ratio variance and covariance shows that GARCH (1,1) overcomes 
the other models. According to the value of the errors of each of the models, the forecast errors 

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74 | P a g e  

(proportion of bias and proportion of variance) are very large, up from 0.1 for the case of bias, which 
means the presence of a systematic bias (Pindyck, 2001).  
Table 4: Assessment of Predictive Ability of the Models  

Indicator   GARCH(1,1)   EGARCH(1,1)   TARCH(1,1)   
Root mean square error   1345.00   1260.474   1274.684   

Mean Absolute Error   1234.881   1158.502   1171.084   
Theil inequality coefficient   0.023207   0.021778   0.022018   
Proportion of bias   0.842957   0.844744   0.844056   
Proportion of variance   0.064700   0.091298   0.085941   
Proportion of covariance   0.092344   0.063958   0.070004   

Source: own   
4. Conclusions   
To determine which is the model that explain more precisely the behavior of the Mexican Stock Market 
Index for the period January 3th, 2000 to July 11th, 2008 has been used an ARMA model (5.1) in which 
the ARCH effects are identified, so the process is modeled through the models from the ARCH / GARCH 
family.   
The findings of this research provide evidence on the existence of highly persistent volatility of returns 
of the IPC. The significance of the asymmetry parameters about TARCH and EGARCH models suggest 
the existence of the leverage effect. The volatility of returns of the IPC caused by the bad news is greater 
than those caused by the good news. The above evidence is consistent with the bulk of research on the 
volatility of returns of stock prices and stock indices. 
GARCH (1,1), EGARCH (1,1) and TARCH (1,1) models, allow us suitably adjust the IPC series.   
However, the EGARCH (1,1) model shows to be the best model for estimating according to the standard 
evaluation criteria. The forecasting results by using the three models, give evidence that the EGARCH 
(1,1) model has the best predictive capability.  
Finally we can say that the forecasts obtained from these results, show the rapidity with they tend to 
their unconditional mean, which indicates, that the forecast horizon of these models is very short, 
provides immediate predictions (2 -3 days) before stabilizing .   
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