Stochastic models with heteroskedasticity : a Bayesian approach for Ibovespa returns

Current research compares the Bayesian estimates obtained for the parameters of processes of ARCH family with normal and Student’s t distributions for the conditional distribution of the return series. A non-informative prior distribution was adopted and a reparameterization of models under analysis was taken into account to map parameters’ space into real space. The procedure adopts a normal prior distribution for the transformed parameters. The posterior summaries were obtained by Monte Carlo Markov Chain (MCMC) simulation methods. The methodology was evaluated by a series of Bovespa Index returns and the predictive ordinate criterion was employed to select the best adjustment model to the data. Results show that, as a rule, the proposed Bayesian approach provides satisfactory estimates and that the GARCH process with Student’s t distribution adjusted better to the data.


Introduction
A large variety of models exists to estimate the volatility of financial assets return series.The most common in the literature are the Autoregressive Conditional Heteroskedasticity (ARCH) model, suggested by Engle (1982), and its extension, the Generalized ARCH (GARCH) models, proposed by Bollerslev (1986).The models characterize a nonlinear dependence among returns due to the serial dependency of conditional variance.
Since volatility at a specific time depends on the past values of the series, the determination of maximum likelihood estimators (MLE) of parameters of ARCH family models require the maximization of a non-linear function, and thus, estimates could only be obtained numerically.Engle (1982) suggested Newton's method as an iterative method to calculate maximum likelihood estimates.Such procedure relaxes the imposed restrictions to parameters (they must be positive and their sum must be less than one) which warrant stationary covariance.On the other hand, certain difficulties are involved in the determination of asymptotic characteristics of MLE with restrictions.It may lead to local maxima, since characteristics such as asymptotic normality do not allow restrictions.
Further, procedures for the identification, adjustment and diagnosis of models and prediction of econometric series values require the characteristics of asymptotic theory.Since models are distant from linearity, the estimators' asymptotic characteristics may only be verified for very long series and, in general, are more appropriate with symmetrical distribution for errors and with normal distribution for data.Bollerslev (2008) provided an extensive review of the models' characteristics.
Under a Bayesian approach, Geweke (1989) provided one of the first investigations for ARCH family models in which a special reparameterization case employed non-informative prior distributions.Estimates of parameters were obtained from Monte Carlo simulation algorithms.Nakatsuma (2000) used normal prior distributions for the parameters of ARMA-GARCH models and the Metropolis-Hastings algorithm to determine posterior summaries.Further, Polasek (2001) suggested a hierarchic structure for PAR-ARCH from a Bayesian approach by using Monte Carlo Markov Chain simulation methods.Within the context of unobserved component models, Giakoumatos et al. (2005) suggested a Bayesian approach for ARCH models with auxiliary variables (PITT; WALKER, 2005).
Whereas Ausín and Galeano (2007) recently suggested a Bayesian approach for GARCH models with errors generated by Gaussian mixtures, Barreto et al. (2008) compared Bayesian and Maximum Likelihood methods by simulated series, following ARCH processes, with different orders and under conditions of finite and infinite variance.Moreover, Andrade and Oliveira (2011) presented a Bayesian approach for ARCH models with normal prior distributions for their respective parameters and compared credibility intervals with bootstrap intervals by employing index return series of the Brazilian financial market.
The Stock Exchange Index of São Paulo (Ibovespa) is the most important index for the average performance of the Brazilian market shares rates.Its relevance is due to the fact that Ibovespa portrays the behavior of the main stocks and shares negotiated at Bovespa and also its history.In fact, the Bovespa index maintains the integrity of a historical series and did not undergo methodological modifications since its establishment in 1968 1 .Ibovespa's basic aim is to be an average indicator of the main transacted shares and a profile of cash negotiations in the Bovespa exchange.Stocks with the highest participation (in terms of volume) on the exchange are selected to compose the index.Ibovespa's behavior has been widely investigated in the literature owing to its economical relevance and several authors have used Ibovespa return series for modeling and for comparing different models.Morettin (2008), Andrade andOliveira (2011), andOliveira andAndrade (2012) are among the many authors who investigated the above-mentioned index and who employed different time series models from a classical or Bayesian approach.
Since the context and the relevance of the ARCH family models in the solution of problems in the economical and financial areas due to their applicability and interpretation (the relations between returns and volatility) have been provided, current investigation compares the Bayesian estimates obtained for the parameters of AR(p)-ARCH(q), ARCH(q) and GARCH(q,r) models, taking into account normal and Student's t distributions for the conditional distribution of the Ibovespa's financial returns series.Non-informative prior distributions, foregrounded on Geweke (1989), were suggested and a reparameterization of the models studied was taken into account for each case to map the parameter's space on real space.The procedure adopts normal prior distributions for the transformed parameters.Posterior summaries were obtained by MCMC simulation methods.

Stochastic models with heteroskedasticity for time series AR-ARCH models
The regression model proposed by Engle (1982) with its mean non-zero and expressed as a linear combination of exogenous variables, exhibits the following structure:  4) may be interpreted by disturbances in linear regression which follow an autoregressive conditional heteroskedasticity of the order q.
So that model ( 1)-( 4) may be plausible ( 0 h t > during t), there must be , where is the sufficient condition so that the process has a stationary covariance. Let be a trajectory of the process y t and that x t involves only past "p" values of y t .If normality is held for t ε , the likelihood function of y t , T ,..., 1 q p t + + = , conditioned to q p + first observations (presumed to be known) is defined as: where: degrees of freedom, or rather, ( ) , conditioned to q p + first observations is given by (MORETTIN, 2008): and for some v.

ARCH models
If the regression model proposed by Engle (1982), defined by the expressions ( 1)-( 4), considers 0 β = , then the new structure may be summarized as ( ) where: z t represents return series; ( ) ⋅ P is the parametric distribution; The interpretation for the model defined in ( 8)-( 9) is that returns in linear regression follow an autoregressive conditional heteroskedasticity of the order q.
Similarly, so that model ( 8)-( 9) is plausible ( 0 h t > during t), there must be . Further, process z t has finite variance and therefore stationary covariance if, and only if, all the roots of the polynomial  = − q 1 j j j l 1 α lie outside the unit radius circle.It may be thus shown that the unconditional variance of z t is given by be a trajectory of the process z t .If normality holds for t ε , the likelihood , conditioned to q first observations (presumed to be known) is given by , conditioned to q first observations, is given by (MORETTIN, 2008): and for some v.

GARCH models
The regression model proposed by Engle (1982) and generalized by Bollerslev (1986), with means zero and expressed as a linear combination of exogenous variables, have a structure summarized as ( ) where: z t is a return series; ( ) ⋅ P is a parametric distribution; is the set of information available up to time 1 t − .When 0 r = , the process is reduced to an ARCH(q).Let z t be a process that satisfies the model is a sequence of independent randomized variables and identically distributed with mean zero and variance 1.An interpretation for the model defined in ( 13)-( 14) is that disturbances in linear regression follow a generalized ARCH process respectively of the orders q and r.
So that model ( 12)-(13) be plausible ( Thus, it may be demonstrated that unconditional variance of z t is given by , whose condition for stationary covariance is , conditioned to q+r first observations (presumed to be known), is given by: Presuming that t ε has a standard Student's t distribution, or rather, ( ) Student's t with v degrees of freedom, the likelihood function of z t , T ,..., 1 r q t + + = , conditioned to q+r first observations, is given by (MORETTIN, 2008): and for some v.
In Bayesian context, the employment of the conditioned likelihood function instead of the exact likelihood function may be undertaken without any great precision loss in the estimates.This is due to the fact that one of the main advantages of Bayesian inference lies in the possibility of adjusting models, even in small samples.

Bayesian approach
Taking into consideration returns trajectory , the Bayesian approach for the inference of parameters of AR(p)-ARCH(q), ARCH(q) and GARCH(q,r) processes starts from the joint likelihood functions defined by the trajectory with a prior density for parameters (that reflects previous knowledge on the distribution of these parameters) by Bayes's rule: The expression is called posterior distribution of parameter(s) of interest and explains how these randomized variables are distributed after data have been complied with.
In current research, a non-informative prior distribution is proposed for the parameters of AR(p)-ARCH(q) and ARCH(q) processes based on the suggestion by Geweke (1989), defined as: , it is Jeffreys' nonvariant prior distribution for the normal linear regression model.Reparameterization is also considered and consists of which may be also defined as may be chosen based on some prior information, for instance, previous studies on the series under analysis.
When reparameterization, defined in (18), is employed, where: Similarly, prior distribution for parameters of GARCH (q,r) processes based on the proposal of Geweke (1989) and on the reparameterization given by: ( ) ( ) where: It should be emphasized that in current research a j and b j were defined by analyses undertaken by Andrade and Oliveira (2011), in which Ibovespa returns series were widely debated and discussed.
Therefore, volatility h t as a function of model parameters, duly transformed into j φ , causes posterior joint distributions for φ, described in Table 1.
Posterior densities have forms that are only similar to those of density functions of known probability.Consequently, the analytic calculation of the parameters' quantities of interest, such as means, mode, medians, standard deviation and others becomes impossible (KOTZ et al., 2000).The issue may be solved by MCMC simulation methods, specifically the Metropolis-Hastings algorithm.Representative samples of posterior distributions of Table 1 may be produced and recover the parameters estimated by inverse transformation.

Metropolis-Hastings Algorithm
The Metropolis-Hastings (M-H) algorithm is an iterative scheme to produce indirectly a density sample when direct production from this density is unknown.Algorithm comprises a choice of nucleus, or rather, a transition density q, and generate from this nucleus by using an acceptance criterion p of the generated value to warrant that the sample obtained is representative of the generated sample (GAMERMAN;LOPES, 2006).
M-H algorithm in current research follows specifically the steps below: Step 1: Attribute arbitrary initial rates Step 2: Generate a new rate ' j φ from the transition nucleus or density ( ) ( ) . In this case, the non-informative prior distributions lead towards posterior ones which do not present known expression to generate candidate parameters.

Gaussian nucleus ( )
φ N is thus produced so that posterior densities may be written as Step 3: Calculate the probability of acceptance of the new value generated as ' Step 4: Generate a uniform randomized variable ( ) and make: Step 5: Increase l and repeat Steps 2 to 4 till convergence is warranted.Finally, recover the estimated parameters by inverse transformation.
Presupposing a quadratic loss function, parameters' Bayesian estimates should be expressed as an evaluation of hope of a function of interest ( ) (a quantity obtained by the predictive density of Monte Carlo estimate).Gamerman and Lopes (2006) give greater details on the subject.

Evaluation of parameters' estimate methods
Below are given the results obtained by implementing the modeling proposed for the parameter inference of the processes AR(p)-ARCH(q), ARCH(q) and GARCH(q,r), taking into account normal and Student's t distributions for t ε .The historical series under analysis gives information on the final indexes of the Bovespa (Ibovespa) registered between January 2, 1996 and February 1, 1999, a total of 651 information items.Choice of the period is due to its importance for Brazilian and world economy.In fact, events which generated behavioral changes with irregular frequencies occurred and produced economical and financial impacts during the period.Since current research estimates the parameters for the evaluation of methods of time series modeling (and not the prediction of time series), the period may not necessarily be a recent occurrence.
Let p t be the final result of the Bovespa index (Ibovespa) on a trading day.Since .Figure 1 illustrates Bovespa indexes and their respective returns.The graphs in Figure 2 show the behavior of returns series of Ibovespa, 2 t y .In fact, the series is correlated and such behavior is typically associated with that of the models of the ARCH family.
A chain of 50,000 iterations was simulated within the implementation of the Metropolis-Hastings algorithm.Moreover, 50% of values were discarded to decrease the effect of initial conditions.Values, spaced in fives, totaling a sample of 5,000 observations, were established.Algorithm convergence was verified by Geweke criterion at 5% significance, under the null hypothesis Ho (GEWEKE, 1992).Parameter convergence was established for values obtained by Geweke's diagnosis between -1.96 and 1.96.
Adjusted models to the Ibovespa series, according to POC, were respectively ARCH(3), AR(3)-ARCH(6) and GARCH(1,1).According to values obtained by POC by the adjustments of the proposed models, the model that best adjusts itself to the Ibovespa series was the GARCH(1,1) process with Student's t distribution for t ε , or rather, model M6, as Table 5 shows.Figure 3 shows estimated volatility from estimates obtained within the Bayesian approach with non-informative prior distribution by the best adjustment model, or rather, GARCH(1,1) with Student's t distribution for t ε .Several events (Russia's default in August 1998; Asian crisis in October 1998; devaluation of the Brazilian real in January 1999), which generated important behavioral changes, occurred during the period under analysis, and caused changes in the behavior of volatility which were entirely identified by the adjusted model.The above changes impacted the prices of stocks and their relationship with the market.

Conclusion
Although the above is a highly simplified representation of the data-generating process of conditional returns data, normal distribution is widely employed in the estimation of volatility models.However, in certain situations, it is more appropriate to presume that t ε has Student's t distribution, as may be seen in current case.
Results show that, as a rule, the Bayesian approach provides satisfactory estimates and is entirely viable in returns modeling.In fact, it makes feasible the incorporation of experts' experience in finance which is a highly relevant issue within the analysis of economical and financial series.
The proposal of non-informative prior distributions, coupled to a reparameterization of the models under analysis, provides a faster convergence of the inference process of parameters of ARCH family models by MCMC methods.

1
BM&FBOVESPA was established in May 2008 and combined the Bolsa de Mercadorias & Futuros (BM&F) and Bolsa de Valores de São Paulo (Bovespa).It became the biggest Stock Exchange institution in Latin America, the second in the Americas and the third in the world (PORTAL DO INVESTIDOR, 2011).
y t has finite variance and, therefore, stationary covariance if, and only if, all the roots of the polynomials  versa.It also decreases the rejection rate of the simulation algorithm MCMC and accelerates its convergence process.Values for for the integrity of the desired function may be obtained by Monte Carlo simulation.Several criteria exist to select models within a Bayesian context.The predictive ordinate criterion (POC) based on density was used in current research, which was built from distribution m T y + , conditioned to data Y and to the parameters of each model analyzed.POC comprises the choice of model l which presents the highest rate of ( ) l c ˆ

Figure 1 .
Figure 1.Price and return of the Ibovespa series.

Figure 2 .
Figure 2. Histogram, normal probability plot, autocorrelation and partial autocorrelation of 2 t y -Ibovespa series.Tables 2, 3 and 4 show Bayesian estimates for the adjusted models, where M1 and M2: ARCH processes; M3 and M4: AR-ARCH processes; M5 and M6: GARCH processes, with normal and Student's t distributions for t ε , respectively.

Table 5 .
Criterion of model selection.