Imoh Udo Moffat¹, Emmanuel Alphonsus Akpan²

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¹ Department of Mathematics and Statistics, University of Uyo, Uyo, Nigeria.
² Department of Mathematical Science, Abubakar Tafawa Balewa University, Bauchi, Nigeria.
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Abstract: The purpose of this study is to apply white noise process in measuring model adequacy targeted at confirming the assumption of independence. This ensures that no autocorrelation exists in any time series under consideration, and that the autoregressive integrated moving average (ARIMA) model entertained is able to capture the linear structure in such series. The study explored the share price series of Union bank of Nigeria, Unity bank, and Wema bank obtained from Nigerian Stock Exchange from January 3, 2006 to November 24, 2016 comprising 2690 observations. ARIMA models were used to model the linear dependence in the data while autocorrelation function (ACF), partial autocorrelation function (PACF), and Ljung-Box test were applied in checking the adequacy of the selected models. The findings revealed that ARIMA(1,1,0) model adequately captured the linear dependence in the return series of both Union and Unity banks while ARIMA(2,1,0) model was sufficient for that of Wema bank. Also, evidence from ACF, PACF and Ljung-Box test revealed that the residual series of the fitted models were white noise, thus satisfying the conditions for stationarity.

Keywords: Autocorrelation, Model Identification, Model Estimation, Diagnostic Checking, Time Series

Cite this paper: Moffat, I. and Akpan, E. (2019) White Noise Analysis: A Measure of Time Series Model Adequacy. Applied Mathematics, 10, 989-1003. doi: 10.4236/am.2019.1011069.

References

[1]   Shumway, R.H. and Stoffer, D.S. (2011) Time Series Analysis and Its Applications with R Examples. 3rd Edition, Springer, New York.

[2]   Cryer, J.D. and Chan, K. (2008) Time Series Analysis with Application in R. 2nd Edition, Springer, New York, 249-260.
https://doi.org/10.1007/978-0-387-75959-3

[3]   Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (2008) Time Series Analysis: Forecasting and Control. 3rd Edition, John Wiley & Sons, Hoboken.

[4]   Tsay, R.S. (2010) Analysis of Financial Time Series. 3rd Edition, John Wiley & Sons, New York.
https://doi.org/10.1002/9780470644560

[5]   Wei, W.W.S. (2006) Time Series Analysis Univariate and Multivariate Methods. 2nd Edition, Adison Westley, New York.

[6]   Pankratz, A. (1983) Forecasting with Univariate Box-Jenkins Models: Concepst and Cases. John Willey & Sons, New York.
https://doi.org/10.1002/9780470316566

[7]   McLead, A.I. (1993) Parsimony, Model Adequacy and Periodic Correlation in Forecasting Time Series. International Statistical Review, 61, 387-393.
https://doi.org/10.2307/1403750

[8]   Li, W.K. (2014) Diagnostic Checks in Time Series. In: Monographs on Statistics and Applied Probability, Volume 102, Chapman & Hall/CRC, New York.

[9]   Alsharif, M.H., Younes, M.K. and Kim, J. (2019) Time Series ARIMA Model for Prediction of Daily and Monthly Average Global Solar Radiation: The Case Study of Seoul, South Korea. Symmetry, 11, 240.
https://doi.org/10.3390/sym11020240

[10]   Iwundu, M.P. and Efezino, O.P. (2015) On Adequacy of Variable Selection Techniques on Model Building. Asian Journal of Mathematics & Statistics, 8, 19-34.
https://doi.org/10.3923/ajms.2015.19.34

[11]   Sarkar, S.K. and Midi, H. (2010) Importance of Assessing the Model Adequacy of Binary Logistic Regression. Journal of Applied Sciences, 10, 479-486.
https://doi.org/10.3923/jas.2010.479.486

[12]   Goldstein, M., Seheult, A. and Vernon, I. (2013) Assessing Model Adequacy. In: Environmental Modeling: Finding Simplicity in Complexity, John Wiley & Sons, Hoboken, NJ, 435-449.
https://doi.org/10.1002/9781118351475.ch26

[13]   Kheifets, I. and Velasco, C. (2012) Model Adequacy Checks for Discrete Choice Dynamic Models. In: Chen, X. and Swanson, N., Eds., Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis, Springer, New York, 363-382.
https://doi.org/10.1007/978-1-4614-1653-1_14

[14]   Chand, S. and Karmal, S. (2014) Mixed Portmanteau Test for Diagnostic Checking of Time Series Models. Journal of Applied Mathematics, 2014, Article ID: 545413.
https://doi.org/10.1155/2014/545413

[15]   Zapranis, A. and Refenes, A.P.N. (1999) Model Adequacy Testing. In: Principles of Neural Model Identification, Selection and Adequacy. In: Perspectives in Neural Computing, Springer, London.
https://doi.org/10.1007/978-1-4471-0559-6

[16]   Godfrey, L.G. and Tremayne, A.R. (1998) Checks of Model Adequacy for Univariate Time Series Models and their Application to Econometric Relationships. Econometric Reviews, 7, 1-42.
https://doi.org/10.1080/07474938808800138

[17]   Box, G.E.P. and Pierce, D. (1970) Distribution of Residual Autocorrelations in Autoregressive Integrated Moving Average Time Series Models. Journal of the American Statistical Association, 65, 1509-1526.
https://doi.org/10.1080/01621459.1970.10481180

[18]   Akpan, E.A., Lasisi, K.E., Adamu, A. and Rann, H.B. (2019) Application of Iterative Approach in Modeling the Efficiency of ARIMA-GARCH Processes in the Presence of Outliers. Applied Mathematics, 10, 138-158.
https://doi.org/10.4236/am.2019.103012

[19]   Akpan, E.A. and Moffat, I.U. (2017) Detection and Modeling of Asymmetric GARCH Effects in a Discrete-Time Series. International Journal of Statistics and Probability, 6, 111-119.
https://doi.org/10.5539/ijsp.v6n6p111

[20]   Akpan, E.A. and Moffat, I.U. (2019) Modeling the Effects of Outliers on the Estimation of Linear Stochastic Time Series Model. International Journal of Analysis and Applications, 17, 530-547.

[21]   Moffat, I.U. and Akpan, E.A. (2019) Selection of Heteroscedastic Models: A Time Series Forecasting Approach. Applied Mathematics, 10, 333-348.
https://doi.org/10.4236/am.2019.105024

[22]   Ljung, G. and Box, G.C. (1978) On a Measure of Lack of Fit in Time Series Models. Biometrica, 65, 265-270.
https://doi.org/10.1093/biomet/65.2.297

[23]   NSE Contact Center. The Nigerian Stock Exchange. Stock Exchange House.
https://www.contactcenter@nse.com