Long Memory Volatility Models in R: Application to a Regional Blue Chips Index

1. Introduction

The volatility process is associated with the conditional variance of financial returns over time. It is a subject to be taken into consideration while assessing the risk of an investment in financial assets. A relatively high level of volatility means that the corresponding asset price tends to large variations over time, therefore the investor can record high profits or high losses. The realized volatility is a positive valued process with a highly persistent temporal dependency, indicated by a slow hyperbolic decay of the autocorrelation function. Granger and Joyeux (1980) and Hosking (1981) suggest the use of fractionally integrated autoregressive moving average (ARFIMA) model in order to capture the long term persistence in time series of many fields. Engle and Russell (1998) propose a different method for the analysis of data which arrive at irregular interval such as financial transaction series. The authors define a new statistical model called autoregressive conditional duration (ACD), which can be considered part of the generalized autoregressive conditional heteroskedasticity (GARCH) model, introduced by Bollerslev (1986). Engle (2002) extends the research suggesting the use of the multiplicative error model (MEM) for modelling non-negative processes. This model specifies an error that is multiplied times the conditional expectation that follows a GARCH dynamic. Brownlees, Cipollini and Gallo (2011) focus on the different possible specifications of the persistence in the conditional mean using applications on the volatility of components of the S&P100 index, following the logic of Engle and Lee (1999). Cipollini, Engle and Gallo (2013) suggest an extension to the vector MEM model defined by Cipollini, Engle and Gallo (2006), where the conditional expectation is given as a function of its past and the conditional expectations of other variables. The authors follow a semiparametric method lacking the necessity of specifying the probabilistic distribution function or the particular copula function. Barigozzi, Brownlees, Gallo and Veredas (2014) expand the MEM model introduced by Engle (2002), decomposing the conditional expectation as the product of a systematic trend and an idiosyncratic dynamic component. This model, named SeminonParametric Vector MEM (SPvMEM), can be used to estimate panels of market activity, risk and liquidity measures.

We suggest in this paper an interesting flexible MEM specification in which the process for realized volatility can be seen as a mixture of two Gamma MEMs with different coefficients for the conditional expectation and different shape parameters for the Gamma (De Luca and Gallo, 2007; Gallo and Otranto, 2014). We extend this framework even further adopting a mixed Gamma model for a MEM with fixed parameters which depend on the value of the past (Lanne, 2006; Ahoniemi and Lanne, 2013).

The paper is organized as follows. Section 2 establishes the notation by summarizing some of the features of the multiplicative error model. Section 3 contains features of the equations and some empirical results illustrating the behaviour of the two MEM specifications for the realized volatility of ‘STOXX Balkan 50 Equal Weight’ Index. Section 4 concludes.

2. Theoretical Framework

Let us consider to be the daily log return at time t with negligible mean. We use a common indicator of volatility , the daily squared return. The general structure of a multiplicative error model was outlined by Engle (2002) for realized volatility. A linear MEM(p, q) is given by:

where, conditionally to information : is a non-negative predictable process. Theevolution depends on a vector of unknown parameters θ, . follows a distribution with nonnegative support, with mean 1 and unknown variance . We have:

~ .

These definitions provide us:

Inizio modulo

In order to close the model, we adopt a density function for and specify an equation for the conditional mean. A common choice in many studies, is to use the exponential distribution, a particular case of the Gamma distribution. We have:

~ ,

where >0, and . The density function conditioned to the innovation is:

.

We can conclude that follows a Gamma conditioned distribution

~

with density function:

.

So, we have in this case:

.

With regards to, we consider the simplest specification, the GARCH(p, q).

.

We estimate the parameters included in, let’s call them θ. The log-likelihood function useful for the estimation is:

,

where c is an irrelevant constant to the estimation method depending on Φ. We maximize the function with respect to θ:

.

We obtain the our estimators using iterative procedures. We have:

The asymptotic distribution of the estimators under regularity conditions will be:

~

~ ,

where is the asymptotic variance. We have:

,

If order to verify the null hypothesis , we use the Student’s t test:

,

where is the i-th row vector and is the (i, i) element of the matrix.

3. MEM Equations and Application

Initially, we consider an interesting formulation of MEM models (Lanne, 2006; Ahoniemi and Lanne, 2013), where the innovation termfollows a mixed gamma distribution with constant parameters. The innovation mean is unitary respecting the establishment of the previous section. Let’s call the realized volatility of a generic stock. We have:

.

The innovation is defined as follows:

,

where and

~

~ .

The general form of the conditional mean equation is:

.

We consider also a model, belonging to the MEM family, with an innovation term distributed according to a mixture of Gamma with variable weights (De Luca and Gallo , 2007; Gallo and Otranto, 2014). The overall mean of the innovation term is always unitary. So, we have:

~

~

where. We suppose that weights depend on their past and on the previous value of realized volatility:

Imposing the unitary mean constraint, we have:

.

We call the two models respectively, LANNE and DLG in our analysis. After defining the regression models, it is important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters. In simple linear models the errors follow a normal distribution, but in some regressions based on mixture of distributions this hypothesis may not be valid. The method used in this paper is based on the so-called probability integral transformation of Rosenblatt (1952), which is a universally approach of transforming a set of dependent variables into independent uniform distributed variables. Dunn and Smyth (1996) define and apply the quantile residuals to regression models with independent responses.

Let be the cumulative distribution function of . If F is continuous, then the are uniformly distributed on the unit interval. The quantile residuals are defined by

where is the cumulative distribution function of the standard normal. Quantile residuals are independent and follow a standard normal distribution if the model is correctly specified and the parameters are consistently estimated.

We start with considering the realized volatility of ‘STOXX Balkan 50 Equal Weight’ Index, which represents the largest and most liquid companies across eight Balkan countries. The index covers 50 stocks from Bulgaria, Croatia, Greece, the Former Yugoslav Republic of Macedonia (FYROM), Romania, Serbia, Slovenia and Turkey. We consider the period from 1^st May 2009 to 30^th January 2015. The time series of daily realized volatility has been built as the square root of realized variance. Figures 1 displays the plot of realized volatility. Figure 2 shows the estimated global autocorrelation function and partial autocorrelation function of realized volatility which exhibit a very slow decay, that is a long-memory pattern.

Figure 1. Time series of ‘STOXX Balkan 50 Equal Weight’ realized volatility.

Figure 2. Global Autocorrelation Function and Partial Autocorrelation Function for ‘STOXX Balkan 50 Equal Weight’ realized volatility.

We try to model the realized volatility series with both possible specifications of MEM models. Let’s see what happens with the LANNE specification. This is the conditional mean equation, where p = 2 and q = 1:

.

We can see in the tables below the estimate, the standard error, the statistic test value and the observed p-value for each of the seven parameters. We simply observe the last column of the table where is calculated the p-value for the statistical test. We can fix a priori a p-value of 5% for acceptance of the null hypothesis. So, we accept ‘H0: The parameter is null’ if the observed p-value is greater than 5% and vice versa if it is less than 5%. As we can see from the table we refuse the null hypothesis in all cases, a positive indicator for the model evaluation.

Table 1. Parameters test, LANNE model.

Parameter	Estimation	p-value
	0.067	~0
	5.634	~0
	0.127	~0
	0.374	~0
	0.749	~0
	11.564	~0
	0.011	~0
	0.378	~0
	0.769	~0

Let’s check now if the model is specified correctly or not using the quantile residuals method mentioned above. We have represented in figure 3 (from the left to the right) the quantile residual graph, the quantile-quantile graph (if the distribution approximates the normal standard one), the global autocorrelation (ACF) and the partial autocorrelation (PACF) of the quantile residuals and of the squared quantile residuals for LANNE model. We observe that the approximation is generally good.

Figure 3. Quantile residuals graphs, LANNE model.

Let's try now to estimate the our time series through the other MEM formulation, the DLG one. This specification is a mixture of two Gamma MEMs with different coefficients for the conditional expectation. The equation for the conditional mean will be:

We use the method of Maximum Likelihood (ML) to estimate the parameters of interest.

Table 2. Parameters test, DLG model.

Parametro	Stima	p-value
	6.723	~0
	16.016	~0
	0.099	~0
	0.374	~0
	0.081	0.805
	0.631	~0
	0.905	~0
	0.983	~0
	17.106	~0
	0.419	~0

As we can see from the table, we refuse the H0 hypothesis in all cases, except for . The p-value in this case is greater than 5%, so we accept the null hypothesis that the coefficient is statistically equal to zero. We use even in this case the quantile residuals method. We check if residuals are independent and approximate the standard normal distribution. If this is true, the model is correctly specified and the parameters are consistently estimated. If not, quantile residuals are expected to exhibit different characteristics, i.e. dependence, autocorrelation or different distribution.

Figure 4. Quantile residuals graphs, DLG model.

We have represented in figure 4 the quantile residual graph, the quantile-quantile graph (if the distribution approximates the normal standard one), the global autocorrelation (ACF) and the partial autocorrelation (PACF) of the quantile residuals and of the squared quantile residuals for the estimated model. We observe that the points line up perfectly to the normal distribution. The global and partial autocorrelations are, in most cases, within the confidence bands. We conclude that the model is correctly specified and the estimates are consistent.

Now we compare our models using well known selection criteria as AIC (Asymptotic Information Criterion; Akaike, 1974) and BIC (Bayesian Information Criterion; Schwarz, 1978). The first criterion, AIC, is defined as:

,

where is the maximized value of the likelihood function for the estimated model, k is the number of parameters in the statistical model and T is the number of data points. BIC is defined as:

.

We have presented in the table below the maximized value of the likelihood function, AIC and BIC for the estimated models.

Table 3. Maximized value of the likelihood function, AIC and BIC criteria.

Lower values ​​of AIC and BIC, as well as higher values ​​of the maximized value of the likelihood function correspond to the most fitting model and vice versa. In this case, we can conclude that DLG model fits better the corresponding data sample.

4. Concluding Remarks

We consider in this study two possible innovative equations for the realized volatility of financial returns, part of the multiplicative error models family, called LANNE and DLG respectively. We estimate and compare these models using some applications on ‘STOXX Balkan 50 Equal Weight’ Index, for the period from 1 May 2009 to 30 January 2015. The error term of LANNE equation follows a gamma distribution with constant parameters, whereas the error term of DLG equation is distributed according to a mixture of Gamma with variable weights.

We compute the quantile residuals in R 3.1.3 software, in order to measure how well the specific MEM models fit the corresponding data sample. We observe positive signals of independence and a satisfactory approximation of residuals to the standard normal distribution. This implies that the equations fit fairly well the volatility of ‘STOXX Balkan 50 Equal Weight’ Index, and the corresponding coefficients are consistently estimated. We also calculate the maximized value of the likelihood function, and the AIC and BIC criteria for each model, concluding that DLG model fits better the data. This study can be enriched with supplemental analysis regarding other possible approaches of volatility estimation and other possible variables. Furthermore, we can test and compare these methods through the quantile residuals analysis, as well as through forecasts.

References