1 THE EFFECT OF HIGH POSITIVE AUTOCORRELATION ON THE PERFORMANCE OF GARCH FAMILY MODELS

This study compared the performance of five Family Generalized Auto-Regressive Conditional Heteroscedastic (fGARCH) models (sGARCH, gjrGARCH, iGARCH, TGARCH and NGARCH) in the presence of high positive autocorrelation. To achieve this, financial time series was simulated with autocorrelated coefficients as ρ = (0.8, 0.85, 0.9, 0.95, 0.99), at different time series lengths (as 250, 500, 750, 1000, 1250, 1500) and each trial was repeated 1000 times carried out in R environment using rugarch package. And the performance of the preferred model was judged using Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE). Results from the simulation revealed that these GARCH models performances varies with the different autocorrelation values and at different time series lengths. But in the overall, NGARCH model dominates with 62.5% and 59.3% using RMSE and MAE respectively. We therefore recommended that investors, financial analysts and researchers interested in stock prices and asset return should adapt NGARCH model when there is high positive autocorrelation in the financial time series data.


Introduction
Financial time series analysis is mainly concerned with the theory and application of asset valuation over time, because of this financial time series analyses are very useful to financial analysts and portfolio managers (Tsay, 2005).Financial time series contains uncertainty, volatility, excess kurtosis, high standard deviation, high skewness and sometimes non normality (Pedroni (2001); Grigoletto & Lisi (2009)).
Volatility is a very important element of financial time series since its introduction by Engle (1982).Models such as Auto-Regressive Conditional Heteroscedastic (ARCH), Generalized Auto-Regressive Conditional Heteroscedastic (GARCH), multivariate GARCH, Stochastic volatitlity (SV) and various variants of the models have been proposed to handle volatility in financial time series (Lawrance, 2003).In fact the ARCH and GARCH models are now so widely used and they are referred to as the "workhorse of the financial industry" (Lee & Hansen (1994); Lange (2011)).
Many economic time series including financial time series are strongly autocorrelated and can be modeled by linear (near) unit root or I(1) processes such as Threshold Autoregressive (TAR) model (Lanne & Saikkonen, 2002).Autocorrelation leads to serious underestimation of standard error for regression coefficients and makes prediction intervals to be excessively wide and as such the presence of autocorrelation renders inference and decision making about the estimated parameters invalid (Adenomon, et al., 2015).This means that financial time series and its possible modeling techniques such as ARCH and GARCH models may not also be immune against the effects of autocorrelation.
The aim of this study is to compare performance of some family GARCH models in the presence of first order autocorrelation.

Brief Review of Literature of Modelling Financial Time Series with Autocorrelation
Autocorrelation can be defined as correlation between members of series of observations ordered in time (as in time series data) or space (as in cross-section data) (Adenomon & Micheal, 2017).Gujarati (2003) identified the following as causes of autocorrelation in time series data: inertia, specification Bias, excluded variables, incorrect functional form, cobweb phenomenon, lags, data transformation and manipulation, and non stationarity.Lanne and Saikkonen (2002) investigated economic time series that are strongly autocorrelated using Threshold Autoregression because TAR model can accommodated time series that are strongly autocorrelated or I(1) time series processes.Xiao et al. (2003) proposed a modification of local polynomial time series regression estimators that improves efficiency when the innovation process is autocorrelated which was based on pre-whitening transformation of the dependent variable that must be estimated from the time series data.Their proposed method and method was more efficient than the conventional local polynomial method.Their study is similar to the work of Su and Ullah (2006).Lee and Lund (2004) investigated the properties of ordinary and generalized least squares in a simple linear regression with stationary autocorrelated errors.They derived variances of the parameter estimators for some time series autcorrelation structures which include a first order autocorraltion and general moving averages.Lanne and Saikkonen (2005) investigated and proposed non-linear GARCH models for highly persistent volatility.They observed that conventional GARCH are inflexible to simultaneously first order autocorrelation of squares, persistence of shocks to volatility and excess kurtosis prevalent in financial return series.
Lawrance (2013) examined volatility in financial time series using exploratory graphs and observed that volatility can be confused with the effect of negative autocorrelation and can be distorted by positive autocorrelation.Lawrence concluded that volatility can only be visualized and analyzed for linearly uncorrelated or decorrelated series.

Model Specification
This study focuses on the GARCH models that are robust for forecasting the volatility of financial time series data in the presence of high positive autocorrelation; so GARCH model and some of its extensions are presented in this section

Autoregressive Conditional Heteroskedasticity (ARCH) Family Model
Every ARCH or GARCH family model requires two distinct specifications, namely: the mean and the variance equations (Atoi, 2014).The mean equation for a conditional heteroskedasticity in a return series, t y is given by where The mean equation in equation (1) also applies to other GARCH family models. (.) is the expected value conditional on information available at time t-1, while t ε is the error generated from the mean equation at time t and t φ is the sequence of independent and identically distributed random variables with zero mean and unit variance.
The variance equation for an ARCH(p) model is given by It can be seen in the equation that large values of the innovation of asset returns have bigger impact on the conditional variance because they are squared, which means that a large shock tends to follow another large shock and that is the same way the clusters of the volatility behave.
So the ARCH(p) model becomes: Where t ε ~N (0,1) iid , ω > 0 and 0 ≥ i α for i > 0. In practice, t ε is assumed to follow the standard normal or a standardized student-t distribution or a generalized error distribution (Tsay 2005).

Asymmetric Power ARCH
According to Rossi (2004), the asymmetric power ARCH model proposed by Ding, Engle & Granger (1993) given below forms the basis for deriving the GARCH family models Given that: This model imposes a Box-Cox transformation of the conditional standard deviation process and the asymmetric absolute residuals.The leverage effect is the asymmetric response of volatility to positive and negative "shocks".

GARCH(p, q) Model:
The mathematical model for the GARCH(p,q) model is obtained from equation (4) by letting 2 = δ and 0 is the continuously compounded log return series), and t ε ~N (0,1) iid , the parameter i α is the ARCH parameter and j β is the GARCH parameter, and (Rossi, 2004;Tsay, 2005 andJiang, 2012).
The restriction on ARCH and GARCH parameters ) , ( j i β α suggests that the volatility ( i a ) is finite and that the conditional standard deviation ( i σ ) increases.It can be observed that if q = 0, then the model GARCH parameter ( j β ) becomes extinct and what is left is an ARCH(p) model.
To expatiate on the properties of GARCH models, the following representation is necessary: , (i = 0, . . ., q) into Eq.( 4), the GARCH model can be rewritten as It can be seen that { t η } is a martingale difference series (i.e., E( t η ) = 0 and 0 ) , cov( = − j t t η η , for j ≥ 1).However, { t η } in general is not an iid sequence.
A GARCH model can be regarded as an application of the ARMA idea to the squared series 2 t a .
Using the unconditional mean of an ARMA model, results in this provided that the denominator of the prior fraction is positive.(Tsay, 2005) When p =1 and q =1, we have GARCH(1, 1) model given by:

GJR-GARCH(p, q) Model
The Glosten-Jagannathan-Runkle GARCH (GJRGARCH) model, which is a model that attempts to address volatility clustering in an innovation process, is obtained by letting Which is the GJRGARCH model (Rossi, 2004).Where which allows positive shocks to have a stronger effect on volatility than negative shocks (Rossi, 2004).But when , the GJRGARCH(1,1) model will be written as

IGARCH(1, 1) Model
The integrated GARCH (IGARCH) models are unit-root GARCH models.The IGARCH (1, 1) model is specified in Tsay (2005) and Grek (2014) as Where t ε ~ N(0, 1) iid , and The model is also an exponential smoothing model for the { 2 t a } series.To see this, rewrite the model as By repeated substitutions, we have which is the well-known exponential smoothing formation with 1 β being the discounting factor (Tsay, 2005).

TGARCH(p, q) Model
The Threshold GARCH model is another model used to handle leverage effects, and a TGARCH(p, q) model is given by the following: where γ , and j β are nonnegative parameters satisfying conditions similar to those of GARCH models, (Tsay, 2005).When , the TGARCH(1, 1) model becomes:

NGARCH(p, q) Model
The

SGARCH(p, q) Model
The SGARCH model can be written as: where t Y is the leading market return at time t, .
the market, and is hypothesized to be Gaussian.0 δ has to be positive and the remaining parameters nonnegative in order to ensure the positivity of 2 t η , (De Luca & Loperfido, 2012)

Simulation Procedure
The simulation procedure here considers the following equations of GARCH (1,1): The Case simulated is the case of financial time series where there are positive high autocorrelation coefficients as ρ = (0.8, 0.85, 0.9, 0.95, 0.99), at different time series lengths (as 250, 500, 750, 1000, 1250, 1500).The experiment is repeated 1000 times.The rugarch package of the R software was used to execute the simulation.

Forecast Assessment
The following are the criteria for Forecast assessments used: 1. Mean Absolute Error (MAE) has a formula  .This criterion measures deviation from the series in absolute terms, and measures how much the forecast is biased.This measure is one of the most common ones used for analyzing the quality of different forecasts.
2. The Root Mean Square Error (RMSE) is given as 2 (y y ) is the time series data and f y is the forecast value of y (Caraiani, 2010).
For the two measures above, the smaller the value, the better the fit of the model (Cooray, 2008) In this simulation study, and where N=1,000, is the number of iterations or replications in the simulation study.Therefore, the model with the minimum RMSE and MAE result will be the preferred model

Simulation Analysis Results
The results of the simulation carried out are presented in Table 1 to Table 12 below.Tables 3 and 4 presents the RMSE and MAE values, and their respective ranks from the fGARCH family model for autocorrelation value of 0.85 at different time series lengths    Tables 5 and 6 presents the RMSE and MAE values, and their respective ranks from the fGARCH family model for autocorrelation value of 0.9 at different time series lengths       For autocorrelation value of 0.8, iGARCH and gjrGARCH performed better than other models when the time series length (T) is 250, but at time series length (T) is 500, sGARCH performed better.However, NGARCH dominated as it outperformed the other models at the time series lengths (T) of 750, 1000, 1250 and 1500.
For autocorrelation value of 0.85, NGARCH outperformed the other models irrespective of the time series (T) length except at time series length of 750 where TGARCH performed better than others.
Coming to the autocorrelation value of 0.9, it can be seen that while NGARCH performed better at T = 250 and T = 1500, gjrGARCH performed better at T = 1000 and T = 1250, and sGARCH and iGARCH, respectively, outperformed the other models at T = 500 and T = 750.
For autocorrelation value of 0.95, it can clearly be observed that NGARCH outperformed the other models irrespective of the time series (T) length except at time series length of 1000 where TGARCH performed better than others.
For autocorrelation value of 0.99, while gjrGARCH performed better than other models at T = 250, iGARCH performed better than other models at T = 750, NGARCH outperformed the other For autocorrelation value of 0.8, gjrGARCH and iGARCH performed better than other models when the time series length (T) is 250, gjrGARCH again performed better than other models at T = 1000, sGARCH outperformed the other models at T = 500, while NGARCH performed better than the other models at the time series lengths (T) of 750, 1250 and 1500.
For autocorrelation value of 0.85, NGARCH dominated as it outperformed the other models irrespective of the time series (T) length except at time series length of 750 where TGARCH performed better than others.
And for the autocorrelation value of 0.9, it can be seen that whereas NGARCH performed better at time series length (T) = 250 and T = 1500, sGARCH performed better than the other models at T = 500, while iGARCH dominated at the other time series lengths, performing better at T = 750, T = 1000 and T = 1250.
For autocorrelation value of 0.95, it can be observed that NGARCH was the preferred model as it outperformed the other models irrespective of the time series (T) length except at time series length (T) of 1000 where TGARCH performed better than others.
And for autocorrelation value of 0.99, while NGARCH dominated the other models, performing the level of autocorrelation is unreliable means for model selection .In summary in Table 13, modeling financial time series with high positive autocorrelation values is dominated by NGARCH model which is in line with results obtained by Lanne and Saikkonen (2005).

Conclusion and Recommendation
This study investigated the performance of fGARCH family models in the presence of high positive autocorrelation values, the results concluded the dominance of NGARCH model.We therefore recommended that investors, financial analysts and researchers interested in stock prices and asset return should adapt NGARCH model when there is high positive autocorrelation in the financial time series data.
Nonlinear Generalized Autoregressive Conditional Heteroskedasticity (NGARCH) Model has been presented variously in literature by the following scholars: Hsieh & Ritchken (2005), Lanne & Saikkonen (2005), Malecka (2014) and Kononovicius & Ruseckas (2015).The following model can be shown to represent all the presentations: better at time series lengths (T) = 500, T = 750, T = 1250 and T = 1500, gjrGARCH performed better than other models at T = 250, and sGARCH performed better than other models at T = 1000.Using RMSE and MAE criteria, this study has shown that different models performed better at different autocorrelation coefficients and at different time series lengths.This is in line with previous studies: Atoi (2014) modelling the volatility of stock returns using daily closing data of Nigerian Stock Exchange, found that GARCH (1,1), PGARCH (1,1,1) and EGARCH(1,1) with student's t distribution, and TGARCH with GED were the four best fitted models based on Schwarz Information Criterion, and the conclusions inGrek (2014), Chen, Min and Chen (2013),Dijk, Franses andLucas (1999) and Demos (2000), that different models performed differently under different conditions; in this case, different autocorrelation coefficients and different time series lengths.Results are however in contrast toMikosch and Starica (2000) using GARCH (1,1) model to estimate log return of foreign exchange, said that Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 16 November 2018 doi:10.20944/preprints201811.0381.v1

Table 1 :
The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.8 at different time series lengths

Table 2 :
The Ranks of The RMSE and MAE values from the fGARCH family model for

Table 2
Tables1 and 2presents the RMSE and MAE values, and their respective ranks from the fGARCH family model for autocorrelation value of 0.8 at different time series lengths

Table 4 :
The Ranks of The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.85 at different time series lengths

Table 5 :
The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.9 at different time series lengths

Table 6 :
The Ranks of The RMSE and MAE values from the fGARCH family model at for autocorrelation value of 0.9 at different time series lengths

Table 7 :
The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.95 at different time series lengths

Table 8 :
The Ranks of The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.95 at different time series lengths

Table 8
8 presents the RMSE and MAE values, and their respective ranks from the fGARCH family model for autocorrelation value of 0.95 at different time series lengths

Table 9 :
The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.99 at different time series lengths

Table 10 :
The Ranks of The RMSE and MAE values from the fGARCH family model for autocorrelation value of 0.99 at different time series lengths

Table 12 :
Summary of the Performances of the fGARCH family models at different levels of

Table 13 :
Overall performance rating of the fGARCH models irrespective of the autocorrelation values and time series lengths Models RMSE