Condition for Successful Square Transformation in Time Series Modeling

J. Ohakwe, O. Iwuoha, E. L. Otuonye

Department of Statistics, Faculty of Biological and Physical Sciences, Abia State University, Uturu, Nigeria.
Department of Statistics, Faculty of Science, Federal University Otuoke, Yenagoa, Nigeria.

DOI: 10.4236/am.2013.44093                

Abstract

In this study we establish the probability density function of the square transformed left-truncated N(1,σ²) error component of the multiplicative time series model and the functional expressions for its mean and variance. Furthermore the mean and variance of the square transformed left-truncated N(1,σ²) error component and those of the untransformed component were compared for the purpose of establishing the interval for σ where the properties of the two distributions are approximately the same in terms of equality of means and normality. From the results of the study, it was established that the two distributions are normally distributed and have means ≌1.0 correct to 1 dp in the interval 0 < σ < 0.027, hence a successful square transformation where necessary is achieved for values of σ such that 0 < σ < 0.027.

Keywords

Error Component; Multiplicative Time Series Model; Square Transformation; Moments

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	I. S. Iwueze, “Some Implications of Truncating the N (1, ?2) Distribution to the Left at Zero,” Journal of Applied Sciences, Vol. 7, No. 2, 2007, pp. 189-195. doi:10.3923/jas.2007.189.195
[2]	C. Chatfield, “The Analysis of Time Series: An Introduction,” Chapman and Hall, CRC Press, Boca Raton, 2004.
[3]	D. B. Percival and A. T. Walden, “Wavelet Methods for Time Series Analysis,” Cambridge University Press, Cam bridge, 2000.
[4]	M. B. Priestley, “Spectral Analysis and Time Series Analysis,” Vol. 1-2, Academic Press, London, 1981.
[5]	G. E. P. Box, G. M Jenkins and G. C. Reinsel, “Time Series Analysis, Forecasting and Control,” 3rd Edition, Prentice-Hall, Englewood Cliffs, 1994.
[6]	W. W. Wei, “Time Series Analysis: Univariate and Multivariate Methods,” Addison-Wesley Publishing Company, Inc., Redwood City, 1989.
[7]	M. G. Kendal and J. K. Ord, “Time Series,” 3rd Edition, Charles Griffin, London, 1990.
[8]	I. S. Iwueze, E. C. Nwogu, J. Ohakwe and J. C. Ajaraogu, “Uses of the Buys-Ballot Table in Time Series Analysis,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 633-645. doi:10.4236/am.2011.25084
[9]	M. S. Bartlett, “The Use of Transformations,” Biometrica, Vol. 3, No. 1, 1947, pp. 39-52. doi:10.2307/3001536
[10]	G. E. P. Box and D. R. Cox, “An Analysis of Transformations,” Journal of the Royal Statistical Society, Series B, Vol. 26, No. 2, 1964, pp. 211-243.
[11]	A. C. Akpanta and I. S. Iwueze, “On Applying the Bartlett Transformation Method to Time Series Data,” Journal of Mathematical Sciences, Vol. 20, No. 3, 2009, pp. 227-243.
[12]	C. R. Nwosu, I. S. Iwueze and J. Ohakwe, “Distribution of the Error Term of the Multiplicative Time Series Model under Inverse Transformation,” Advances and Applications in Mathematical Sciences, Vol. 7, No. 2, 2010, pp. 119-139.
[13]	E. L. Otuonye, I. S. Iwueze and J. Ohakwe, “The Effect of Square Root Transformation on the Error Component of the Multiplicative Time Series Model,” International Journal of Statistics and Systems, Vol. 6, No. 4, 2011, pp. 461-476.
[14]	J. Ohakwe, O. A. Dike and A. C. Akpanta, “The Implication of Square Root Transformation on a Gamma Distributed Error Component of a Multiplicative Time Series Model,” Proceedings of African Regional Conference on Sustainable Development, Vol. 6, No. 4, 2012, pp. 65-78.
[15]	R. V. Hogg and A. T. Craig, “Introduction to Mathematical Statistics,” 6th Edition, Macmillan Publishing Co. Inc., New York, 1978.