Theoretical Properties of New Error Innovation Distribution on GARCH Model

Olayemi Michael Sunday; Olubiyi Adenike Oluwafunmilola

doi:doi:10.11648/j.ajtas.20211001.12

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Theoretical Properties of New Error Innovation Distribution on GARCH Model

Olayemi Michael Sunday, Olubiyi Adenike Oluwafunmilola

Published in American Journal of Theoretical and Applied Statistics (Volume 10, Issue 1)

Received: 21 December 2020 Accepted: 31 December 2020 Published: 12 January 2021

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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.

Published in	American Journal of Theoretical and Applied Statistics (Volume 10, Issue 1)
DOI	10.11648/j.ajtas.20211001.12
Page(s)	9-13
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Exponentiated Gumbel Distribution, Error Innovation, Maximum Likelihood Estimate, Volatility and Transformation

References

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[4]	Bollerslev, T. (1986). Generalized Autoregressive Conditionally Heteroskedasticity. Journal of Econometrics, 31, 307–327.
[5]	William, S. G. (1908). The probable error of a mean. Biometrika, 6 (1), 1-25.
[6]	Nelson, D. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, 59, 349-370.
[7]	O’Hagan, A., and Leonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints, Biometrika, 63, 201-2012.
[8]	Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of statistics, 12, 171-178.
[9]	Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica, 46, 199-208.
[10]	Azzalini, A. (2005). The skewed-normal distribution and related multivariate families. Scandinavian Journal of Statistics, 32, 159-188.
[11]	Fermandez, C., and Steel, M. F. J. (1998). On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93, 359-369.
[12]	Hansen, B. E. (1994). Autoregressive Conditional Density Estimation. International Economic Review, 35, 705-730.
[13]	Theodossiou, P. (1998) Financial data and the skewed generalized t distribution. Management Science, 44, 1650-1661.
[14]	Samson T. K, Onwukwe C. E and Enang E. I (2020). Modelling Volatility in Nigerian Stock Market: Evidence from Skewed Error Distributions; International Journal of Modern Mathematical Sciences, 2020, 18(1): 42-57.
[15]	Timothy Kayode Samson, EkaetteInyangEnang and Christian ElenduOnwukwe (2020). Estimating the Parameters of GARCH Models and Its Extension: Comparison between Gaussian and Non-Gaussian Innovation Distributions; Covenant Journal of Physical & Life Sciences (CJPL) Vol. 8 No. 1, June 2020 ISSN: p. 2354 – 3574 e. 2354–3485.
[16]	Nadarajah, S. (2006). The exponentiated Gumbel distribution with climate application. Environmetrics, 17 (1), 13-23.
[17]	Gupta, R. C., Gupta, R. D., and Gupta, P. L., (1998). Modelling failure time data by Lehman alternatives. Communications in Statistics-Theory and Methods, 27 (4), 887-904.

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Olayemi Michael Sunday, Olubiyi Adenike Oluwafunmilola. (2021). Theoretical Properties of New Error Innovation Distribution on GARCH Model. American Journal of Theoretical and Applied Statistics, 10(1), 9-13. https://doi.org/10.11648/j.ajtas.20211001.12

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ACS Style

Olayemi Michael Sunday; Olubiyi Adenike Oluwafunmilola. Theoretical Properties of New Error Innovation Distribution on GARCH Model. Am. J. Theor. Appl. Stat. 2021, 10(1), 9-13. doi: 10.11648/j.ajtas.20211001.12

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AMA Style

Olayemi Michael Sunday, Olubiyi Adenike Oluwafunmilola. Theoretical Properties of New Error Innovation Distribution on GARCH Model. Am J Theor Appl Stat. 2021;10(1):9-13. doi: 10.11648/j.ajtas.20211001.12

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@article{10.11648/j.ajtas.20211001.12,
  author = {Olayemi Michael Sunday and Olubiyi Adenike Oluwafunmilola},
  title = {Theoretical Properties of New Error Innovation Distribution on GARCH Model},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {1},
  pages = {9-13},
  doi = {10.11648/j.ajtas.20211001.12},
  url = {https://doi.org/10.11648/j.ajtas.20211001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211001.12},
  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.},
 year = {2021}
}

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TY  - JOUR
T1  - Theoretical Properties of New Error Innovation Distribution on GARCH Model
AU  - Olayemi Michael Sunday
AU  - Olubiyi Adenike Oluwafunmilola
Y1  - 2021/01/12
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211001.12
DO  - 10.11648/j.ajtas.20211001.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 9
EP  - 13
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211001.12
AB  - 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.
VL  - 10
IS  - 1
ER  -

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Author Information

Olayemi Michael Sunday

Statistics Department, Faculty of Science, Ekiti State University, Ado-Ekiti, Nigeria
Olubiyi Adenike Oluwafunmilola

Statistics Department, Faculty of Science, Ekiti State University, Ado-Ekiti, Nigeria

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APA Style

Olayemi Michael Sunday, Olubiyi Adenike Oluwafunmilola. (2021). Theoretical Properties of New Error Innovation Distribution on GARCH Model. American Journal of Theoretical and Applied Statistics, 10(1), 9-13. https://doi.org/10.11648/j.ajtas.20211001.12

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ACS Style

Olayemi Michael Sunday; Olubiyi Adenike Oluwafunmilola. Theoretical Properties of New Error Innovation Distribution on GARCH Model. Am. J. Theor. Appl. Stat. 2021, 10(1), 9-13. doi: 10.11648/j.ajtas.20211001.12

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AMA Style

Olayemi Michael Sunday, Olubiyi Adenike Oluwafunmilola. Theoretical Properties of New Error Innovation Distribution on GARCH Model. Am J Theor Appl Stat. 2021;10(1):9-13. doi: 10.11648/j.ajtas.20211001.12

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@article{10.11648/j.ajtas.20211001.12,
  author = {Olayemi Michael Sunday and Olubiyi Adenike Oluwafunmilola},
  title = {Theoretical Properties of New Error Innovation Distribution on GARCH Model},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {1},
  pages = {9-13},
  doi = {10.11648/j.ajtas.20211001.12},
  url = {https://doi.org/10.11648/j.ajtas.20211001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211001.12},
  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.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Theoretical Properties of New Error Innovation Distribution on GARCH Model
AU  - Olayemi Michael Sunday
AU  - Olubiyi Adenike Oluwafunmilola
Y1  - 2021/01/12
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211001.12
DO  - 10.11648/j.ajtas.20211001.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 9
EP  - 13
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211001.12
AB  - 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.
VL  - 10
IS  - 1
ER  -

Copy | Download