Theoretical Properties of New Error Innovation Distribution on GARCH Model

Olayemi Michael Sunday,Olubiyi Adenike Oluwafunmilola

doi:10.11648/j.ajtas.20211001.12

Olayemi Michael Sunday, Olubiyi Adenike Oluwafunmilola

Open Access

https://doi.org/10.11648/j.ajtas.20211001.12

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Abstract

In the last decades, many error innovations have been introduced based on different modification techniques. One of the vital methods in estimating the true parameter of any volatility models is error innovation distribution, since volatility is affected by reaction from the stock market because of political recession, insecurity, constant power failure, war, political disorder, and other economic crises. In modelling of volatility in a financial investment, error innovation distribution was found advantageous. In this paper, the researcher provided a new error innovation distribution that will serve as a competitive to other existing error innovation. The theoretical properties of the standardized exponentiated Gumbel error innovation distribution is provided and the method of estimating its parameters, by maximum likelihood estimator was proposed. The exponentiated Gumbel distribution were standardized and then converted to the new error innovation through the method of transformation. The newly established error innovation which was obtained through the method of transformation in econometrics was applied on Generalized Autoregressive Conditional Heteroskedasticity (GARCH 1,1) model. For the partial derivative of the shape and volatility parameters were unable to get the exact solution of the parameters. Therefore, a method of numerical solution BFGS was applied to obtain the estimated values of the parameters.

Highlights

In the last decades, many error innovations have been introduced based on different modification techniques
One of the vital methods in estimating the true parameter of any volatility models is error innovation distribution, since volatility is affected by reaction from the stock market because of political recession, insecurity, constant power failure, war, political disorder, and other economic crises
This could trigger variation to stock prices falling yielding to high leptokurtic, Standardized exponentiated Gumbel error innovation distribution was proposed by the researchers to address such reaction

Summary

Introduction

The researchers will be standardizing and convert an existing distribution known and called Exponentiated Gumbel Distribution introduced by Nadarajah [16] into an error innovation distribution through the method of Maximum Likelihood Estimation and be applied to GARCH model. To obtain the PDF of the above CDF is by taking the differential equation with respect to x to give:. Proposed Standardized Exponentiated Gumbel Error Innovation Distribution (SEGEID). In order to obtain the standardize exponentiated Gumbel Error Innovation Distribution Let '(be substitute in equation 5 where − which gives the standardize exponentiated Gumbel Distribution as;. The Standardized Exponentiated Gumbel distribution (SEGD) is given as the equation (6) To obtain the Error innovation of the SEDG. The above equation (12) is the Standardized Exponentiated Gumbel Error Innovation Distribution (SEGEID)

Likelihood Function

Discussion

Conclusion