Appraisal of excess Kurtosis through outlier-modified GARCH-type models

Abstract The aim of this paper is to appraise if there is any improvement subtracting the effects of outliers from existing heteroscedastic models and whether this improvement makes difference with the existing models in achieving efficiency in capturing excess kurtosis in the returns series. The study employed both existing and outlier modified autoregressive conditional heteroscedastic (ARCH), generalized autoregressive conditional heteroscedastic (GARCH), exponential GARCH (EGARCH), Glosten, Jagnnathan and Runkle GARCH (GJR-GARCH) models with respect to normal and student-t distributions to assess the portion of excess kurtosis of the returns series expressed compare to the theoretical value of kurtosis. The data applied were the share prices of Union bank of Nigeria and Unity bank from January 3, 2006 to November 24, 2016, comprising 2690 observations and were obtained from Nigerian Stock Exchange. The results obtained revealed that the Outlier Modified GARCH-type models chosen were adequate and sufficiently reducing the value of excess kurtosis in close proximity to the theoretical value. Therefore, the modification of existing GARCH-type models by subtracting the effects of outliers seems to show a substantive improvement in the portion of excess kurtosis captured and thus proves that the Outlier Modified GARCH-type models make difference with the existing ones.


Introduction
It is a pertinent remark that one of the major setbacks to linear stationary models when applying to financial data (returns series) is their failure to account for changing variance. In other words, whenever the assumption of constant variance is violated, heteroscedasticity (changing variance conditional on the past information) has occurred, implying that the conditional distribution of the dependent variable has different degree of variability at different level. The relationship between the occurrence of heteroscedasticity in financial data and the violation of assumption of constant variance in linear time series has given birth to an extensive research area for professionals in Statistics, Economics and Finance.
outliers have great impact on the existing heteroscedasticity tests and the estimators of heteroscedastic model and such impact of outliers on the diagnostic tools for heteroscedasticity is well defined in Van Dijk, Franses, and Lucas (1999). They showed that both the asymptotic size and power properties of Lagrange (LM) test for ARCH/GARCH are adversely affected by outliers, particularly, additive outliers. Grossi and Laurini (2004) found that order of identification, t-statistics and corresponding p-values of the estimates of GARCH parameters are affected by outliers in an unexpected manner. Moreover, prior studies involving GARCH-type modeling in the presence of outliers include Urooj and Asghar (2017), Grane and Veiga (2014), Grane and Veiga (2010), Hotta and Tsay (2012), Van Dijk, Franses, and Lucas (1999), Bilen and Huzurbazar (2002), Carnero, Pena, and Ruiz (2001), Franses and Ghijsels (1999), and . So far, we identified that previous studies have tried in one way or the other to improve the forecasting ability of the GARCH-type models but failed to appraise the ability of the models to completely capture the heavy-tailedness (Excess kurtosis) of the data comparable to the theoretical value of kurtosis. Hence, this study seeks to improve (by subtracting the effects of outliers from existing models to achieve efficiency) the work of  that modeled the efficiency of heteroscedastic models by adjusted for the effects of outliers in the returns series of Nigerian Banks stocks without the corresponding and necessary modifications on the existing GARCHtype models to depict such adjustment.

Returns
The return series R t can be obtained given that P t is the price of a unit share at time, t, and P tÀ1 is the share price at time t À 1 as follows: The R t in equation (1) is regarded as a transformed series of the share price, P t meant to attain stationarity, that is, both mean and variance of the series are stable (Akpan and Moffat 2019; Moffat and Akpan 2018;Akpan and Moffat 2017). The letter B is the backshift operator.

Autoregressive integrated moving average (ARIMA) model
According to Box, Jenkins, and Reinsel (2008) the autoregressive integrated moving average process is the general form of model used to describe time series where u B ð Þ is the nonstationary autoregressive operator with d of the roots of u B ð Þ ¼ 0 equal to unity, that is, d unit roots. / B ð Þ is a stationary autoregressive operator and h B ð Þ is a moving average operator.

Heteroscedastic models
Heteroscedastic models are hybridized of both mean and variance equations. The mean equation is represented the ARIMA Model as shown in equation (3), where e t is a sequence of independent and identically distributed (i.i.d.) random variables with mean zero, that is E(e t ) ¼ 0 and variance, 1, while a 0 t is the standardized residual term and follows ARCH(q), GARCH (q, p), EGARCH(q,p) and GJR-GARCH(q,p) models in (5-7) and (8), respectively.
2.3.2. Generalized autoregressive conditional heteroscedastic (GARCH) model GARCH(q,p) model provides an alternative to ARCH model for the purpose of achieving parsimony and overcoming the weakness of ARCH model that requires many parameters to completely capture the heteroscedasticity (Bollerslev 1986). The model is defined as follows: 2.3.3. Exponential generalized autoregressive conditional heteroscedastic (EGARCH) model EGARCH(q,p) model applies the natural logarithm to ensure that the conditional variance is positive and thus overcome the requirement of parameter restrictions (Nelson 1991). The EGARCH (q, p) is defined as, c k is the asymmetric coefficient.

Glosten, Jagannathan and Runkle (GJR-GARCH) model
The GJR GARCH (q, p) model (Glosten, Jagannathan, and Runkle 1993) is a variant, represented by where I tÀ1 is an indicator for negative a tÀi , that is, and a i , c i , and b j are nonnegative parameters satisfying conditions similar to those of GARCH models. Also the introduction of indicator parameter of leverage effect, I tÀ1 in the model accommodates the leverage effect, since it is supposed that the effect of a 02 tÀi on the conditional variance r 02 t is different accordingly to the sign of a 0 tÀi (For more details, see Francq and Zakoian 2010; Tsay 2010).

Proposed modification
Different findings have indicated that integration of different models can be an effective way of improving upon their performances especially when the models in combination are quite different (Zhang 2003). However, efficiency of such hybridized model could still be threatened especially in the presence of outliers. Take for instance, the generalized autoregressive conditional heteroscedastic (GARCH-type) models that were introduced to account for heteroscedasticityand were specified based on the normal distribution for the innovations yet could not capture the heavy-tailed characterizations. Similarly, the student-t distribution which was traditionally stated to remedy the weakness of the normal distribution in accommodating the heavy-tailed property also failed in many applications to account for excess kurtosis and thus, the resulting estimates of GARCH models are not efficient. In addition, this heavy-tailed property indicates the presence of excess kurtosis which in turn is a measure of outliers. It is against this background that the following proposal is done to improve and adjust for outliers to obtain a more efficient model than the existing one.
where e t is a sequence of independent and identically distributed (i.i.d.) random variables with mean zero, that is E(e t ) ¼ 0 and variance 1, a Ã t is outlier adjusted residual term.
in equation (9) is an expression for multiple outliers introduced into the ARIMA model (mean for a temporary change, and s is the size of outlier (For more details on the types of outliers and estimation of the outliers effects (see Moffat and Akpan 2018;Moffat and Akpan 2017;Sanchez and Pena 2003;Box, Jenkins, and Reinsel 2008;Wei 2006;Chen and Liu 1993;Chang, Tiao, and Chen1988). Moreover, in financial time series, the residual series is assumed to be uncorrelated with its own past, so additive, innovative, temporary change and level shift outliers coincide, and where both the mean and variance equations evolves together. a Ã0 t is the outlier free standardized residual which now enters into the variance equation. Therefore, the idea behind this modification is to show whether there is an improvement after subtracting the effects of the outliers and if it thus makes difference with the existing model. Hence, a Ã0 t follows outlier modified; ARCH(q), GARCH (q, p), EGARCH(q,p) and GJR-GARCH(q,p) models in (11-13) and (14), respectively.

Outlier modified ARCH model
2.4.3. Outlier modified EGARCH model Alternatively, EGARCH (q, p) model with respect to student-t distribution can be represented by where c k is the asymmetric coefficient.
where I tÀ1 is an indicator for negative a Ã0 tÀi , that is, Going forward, this particular modification is based on the assumption that returns series are uncorrelated (no serial correlation). Serial correlations (a relationship between a variable and its lagged-value over a period of time) tend to exist in most financial series and these serial correlations are believed to be introduced by those in the time-varying heteroscedasticity process (Zhang, Wong, and Li 2016;Conrad and Karanasos 2015;Zhang et al. 2013;Tsay 2010;Dias 2017;Hong 1991).

Procedure for detecting outliers
The procedure for detecting outliers is based on the following steps: Step 1. Derive initial estimates of the model parameters.
Step 2. Given the parameter values, for any t and for each type of outliers, assume that an outlier has occurred at time T, and estimate its amplitude. If the largest absolute estimated amplitude is significant, that is, larger than a priori fixed sensitivity level, usually 3.5 if T 50 or !4 if T ! 450 times its estimated standard error which is called critical value, CV, identify an outlier of that type at that time; otherwise stop (L opez-de-Lacalle 2019; Kaya 2010). Moreover, where the critical value of 4 is not sufficient, critical value of 5 is considered in this study.
Step 3. Remove the effects of the identified outlier by subtracting its estimated amplitude from R t (and also correcting all subsequent observations according to the estimated model in case of innovational outlier).
Step 4. Estimate again the model parameters on the corrected series, and iterate step 2.

Results and discussion
Data collection was based on secondary source as documented in the records of Nigerian Stock Exchange. The documented data on the daily closing share prices of the sampled banks (Union bank and Unity bank) from January 3, 2006 to November 24, 2016 were purchased from the Nigerian Stock Exchange and delivered through contactcentre@nigerianstockexchange.com. The dataset used can be assessed as supplementary data. The time plots of share price series, ACF and PACF of share price series and time plots of the return series for all the banks were generated by Gretl version 1.10.1. The main analyses were carried out using R-project version 3.4.0.

Time plots
Figures 1 and 2 represent the share price series for the two banks. It could be observed that the share prices do not fluctuate around a common mean, which clearly indicate the presence of a stochastic trend in the share prices. This is also an evidence of non-stationarity of the series. Since the share price series is found to be non-stationary, the first difference of the natural logarithm of the series is taken to obtain a stationary series. The inclusion of the log transformation is to stabilize the variance. Figures 3 and 4 show that the return series appear to be stationary and they suggest that heteroscedasticity is quite evident in the series.

Modeling ARIMA processes of return series
From Table 1, ARIMA(1,1,0) model was each selected for the return series of the banks considered based on the grounds of significance of the parameters and minimum AIC.
Evidence from Ljung-Box Q-statistics in Table 2 showed that each of ARIMA(1,1,0) model was adequate at 5% level of significance. That is, the hypothesis of no autocorrelation was not rejected.

Identification of ARCH effects in the residuals of ARIMA models fitted to the return series of the banks
Evidence from Portmanteau-Q (PQ) statistics in Table 3 showed that heteroscedasticity does not exist given that the null hypothesis of no autocorrelation is not rejected at 5% significance level which is due to the presence of outliers. Meanwhile, Lagrange-Multiplier (LM) test statistics in Table 3 showed that heteroscedasticity exists given that the null hypotheses of no autocorrelation and no ARCH effect are rejected at 5% significance level.

Modeling ARIMA-GARCH-type processes of the return series of the banks
The following hybridized ARIMA-GARCH-type models with respect to both normal (norm) and student-t (std) distributions were considered and selected on the grounds of smallest information criteria (Table 4). The models were found to be adequate at 5% level of significance according to evidence provided by weighted Ljung-Box Q-statistics on standardized residuals, weighted Ljung-Box Q-statistics on standardized squared residuals, and weighted Lagrange-Multiplier statistics [see Table 5]. This implies that the hypotheses of no autocorrelation and no remaining ARCH effects were not rejected.

Identification of outliers in the residual series of ARIMA models fitted to return series of the banks
Several different outliers were identified to have contaminated the residual series of ARIMA models using the critical value (a priori fixed sensitive level), CV ¼ 5 on the condition that the sample size is very large: An observation is regarded as outlier if its corresponding t-statistic is greater than 5 times the estimated standard deviation in absolute value [see Tables 6 and 7]. Moreover, in financial time series, it is assumed that the error is uncorrelated with its past values, and then all the outliers are classified as innovation outliers with a unified effect.
Outlier adjusted series was obtained by removing the effects of outliers from the return series      3.6. Modeling the outlier-modified ARIMA-GARCH-type processes of the return series The following hybridized outlier-modified ARIMA-GARCH-type models with respect to both normal (norm) and student-t (std) distributions were considered and selected on the grounds of smallest information criteria (Table 8).
The Outlier Modified models were found to be adequate at 5% level of significance according to evidence provided by weighted Ljung-Box Q-statistics on standardized residuals, weighted Ljung-Box Q-statistics on standardized squared residuals, and weighted Lagrange-Multiplier statistics [see Table 9]. That is to say, the hypotheses of no autocorrelation and no remaining ARCH effect are not rejected.  The proximity of kurtosis value captured by the outlier modified GARCH-type model for each bank to the theoretical value of kurtosis in percentage is presented in Table 10. It is revealed that for Union bank and Unity bank, the respective values are 8.91% and 26.78%, which are above the theoretical value of kurtosis.

Conclusion
Outstandingly, our study revealed that there is an improvement after subtracting the effects of the outliers and it makes difference with the existing models. Hence, all the outliers modified models selected successfully provided the needed improvement on the work of  by efficiently capturing the excess kurtosis in proximity to the theoretical value of kurtosis in the returns series irrespective of the choice of the distribution of the innovations. Given that the modification in this work was based on the assumption that returns series are uncorrelated and the selected outlier modified models could not completely capture the excess kurtosis that is, arriving at exact 3, a value required for a normal distribution, showed that there is still room for improvement. One possible way of improving upon this study is to account for the existence of serial correlations which is believed to be introduced or caused by the presence of heteroscedasticity and thus forms the basis for further studies.