Necessary Conditions for the Application of Moving Average Process of Order Three
Affiliation(s)
1 Department of Statistics, Michael Okpara University of Agriculture, Umudike, Nigeria. 2 Department of Statistics, Federal University of Technology, Owerri, Nigeria. 3 Department of Mathematics, Computer Science and Informatics, Federal University, Otueke, Nigeria.
1 Department of Statistics, Michael Okpara University of Agriculture, Umudike, Nigeria. 2 Department of Statistics, Federal University of Technology, Owerri, Nigeria. 3 Department of Mathematics, Computer Science and Informatics, Federal University, Otueke, Nigeria.
ABSTRACT
Invertibility is one of the desirable properties of moving average processes. This study derives consequences of the invertibility condition on the parameters of a moving average process of order three. The study also establishes the intervals for the first three autocorrelation coefficients of the moving average process of order three for the purpose of distinguishing between the process and any other process (linear or nonlinear) with similar autocorrelation structure. For an invertible moving average process of order three, the intervals obtained are
, -0.5<ρ2<0.5 and -0.5<ρ1<0.5.
Invertibility is one of the desirable properties of moving average processes. This study derives consequences of the invertibility condition on the parameters of a moving average process of order three. The study also establishes the intervals for the first three autocorrelation coefficients of the moving average process of order three for the purpose of distinguishing between the process and any other process (linear or nonlinear) with similar autocorrelation structure. For an invertible moving average process of order three, the intervals obtained are
, -0.5<ρ2<0.5 and -0.5<ρ1<0.5.
KEYWORDS
Moving Average Process of Order Three, Characteristic Equation, Invertibility Condition, Autocorrelation Coefficient, Second Derivative Test
Moving Average Process of Order Three, Characteristic Equation, Invertibility Condition, Autocorrelation Coefficient, Second Derivative Test
Cite this paper
Okereke, O. , Iwueze, I. and Johnson, O. (2015) Necessary Conditions for the Application of Moving Average Process of Order Three. Applied Mathematics, 6, 173-181. doi: 10.4236/am.2015.61017.
Okereke, O. , Iwueze, I. and Johnson, O. (2015) Necessary Conditions for the Application of Moving Average Process of Order Three. Applied Mathematics, 6, 173-181. doi: 10.4236/am.2015.61017.
References
[1] Moses, R.L. and Liu, D. (1991) Optimal Nonnegative Definite Approximation of Estimated Moving Average Covariance Sequences. IEEE Transactions on Signal Processing, 39, 2007-2015.
http://dx.doi.org/10.1109/78.134433 [2] Qian, G. and Zhao, X. (2007) On Time Series Model Selection Involving Many Candidate ARMA Models. Computational Statistics and Data Analysis, 51, 6180-6196.
http://dx.doi.org/10.1016/j.csda.2006.12.044 [3] Li, Z.Y. and Li, D.G. (2008) Strong Approximation for Moving Average Processes under Dependence Assumptions. Acta Mathematica Scientia, 28, 217-224.
http://dx.doi.org/10.1016/S0252-9602(08)60023-5 [4] Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994) Time Series Analysis: Forecasting and Control. 3rd Edition, Prentice-Hall, Englewood Cliffs. [5] Okereke, O.E., Iwueze, I.S. and Johnson, O. (2013) Extrema of Autocorrelation Coefficients for Moving Average Processes of Order Two. Far East Journal of Theoretical Statistics, 42, 137-150. [6] Al-Marshadi, A.H. (2012) Improving the Order Selection of Moving Average Time Series Model. African Journal of Mathematics and Computer Science Research, 5, 102-106. [7] Adewumi, M. (2014) Solution Techniques for Cubic Expressions and Root Finding. Courseware Module, Pennsylvania State University, Pennsylvania. [8] Okereke, O.E., Iwueze, I.S. and Johnson, O. (2014) Some Contributions to the Solution of Cubic Equations. British Journal of Mathematics and Computer Science, 4, 2929-2941.
http://dx.doi.org/10.9734/BJMCS/2014/10934 [9] Wei, W.W.S. (2006) Time Series Analysis, Univariate and Multivariate Methods. 2nd Edition, Pearson Addision Wesley, New York. [10] Chatfield, C. (1995) The Analysis of Time Series. 5th Edition, Chapman and Hall, London. [11] Palma, W. and Zevallos, M. (2004) Analysis of the Correlation Structure of Square of Time Series. Journal of Time Series Analysis, 25, 529-550.
http://dx.doi.org/10.1111/j.1467-9892.2004.01797.x [12] Iwueze, I.S. and Ohakwe, J. (2011) Covariance Analysis of the Squares of the Purely Diagonal Bilinear Time Series Models. Brazilian Journal of Probability and Statistics, 25, 90-98.
http://dx.doi.org/10.1214/09-BJPS111 [13] Iwueze, I.S. and Ohakwe, J. (2009) Penalties for Misclassification of First Order Bilinear and Linear Moving Average Time Series Processes.
http://interstatjournals.net/Year/2009/articles/0906003.pdf [14] Okereke, O.E. and Iwueze, I.S. (2013) Region of Comparison for Second Order Moving Average and Pure Diagonal Bilinear Processes. International Journal of Applied Mathematics and Statistical Sciences, 2, 17-26. [15] Montgomery, D.C., Jennings, C.L. and Kaluchi, M. (2008) Introduction to Time Series Analysis and Forecasting. John Wiley and Sons, New Jersey. [16] Sittinger, B.D. (2010) The Second Derivative Test. www.faculty.csuci.edu/brian.sittinger/2nd_Derivtest.pdf
[1] Moses, R.L. and Liu, D. (1991) Optimal Nonnegative Definite Approximation of Estimated Moving Average Covariance Sequences. IEEE Transactions on Signal Processing, 39, 2007-2015.
http://dx.doi.org/10.1109/78.134433 [2] Qian, G. and Zhao, X. (2007) On Time Series Model Selection Involving Many Candidate ARMA Models. Computational Statistics and Data Analysis, 51, 6180-6196.
http://dx.doi.org/10.1016/j.csda.2006.12.044 [3] Li, Z.Y. and Li, D.G. (2008) Strong Approximation for Moving Average Processes under Dependence Assumptions. Acta Mathematica Scientia, 28, 217-224.
http://dx.doi.org/10.1016/S0252-9602(08)60023-5 [4] Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994) Time Series Analysis: Forecasting and Control. 3rd Edition, Prentice-Hall, Englewood Cliffs. [5] Okereke, O.E., Iwueze, I.S. and Johnson, O. (2013) Extrema of Autocorrelation Coefficients for Moving Average Processes of Order Two. Far East Journal of Theoretical Statistics, 42, 137-150. [6] Al-Marshadi, A.H. (2012) Improving the Order Selection of Moving Average Time Series Model. African Journal of Mathematics and Computer Science Research, 5, 102-106. [7] Adewumi, M. (2014) Solution Techniques for Cubic Expressions and Root Finding. Courseware Module, Pennsylvania State University, Pennsylvania. [8] Okereke, O.E., Iwueze, I.S. and Johnson, O. (2014) Some Contributions to the Solution of Cubic Equations. British Journal of Mathematics and Computer Science, 4, 2929-2941.
http://dx.doi.org/10.9734/BJMCS/2014/10934 [9] Wei, W.W.S. (2006) Time Series Analysis, Univariate and Multivariate Methods. 2nd Edition, Pearson Addision Wesley, New York. [10] Chatfield, C. (1995) The Analysis of Time Series. 5th Edition, Chapman and Hall, London. [11] Palma, W. and Zevallos, M. (2004) Analysis of the Correlation Structure of Square of Time Series. Journal of Time Series Analysis, 25, 529-550.
http://dx.doi.org/10.1111/j.1467-9892.2004.01797.x [12] Iwueze, I.S. and Ohakwe, J. (2011) Covariance Analysis of the Squares of the Purely Diagonal Bilinear Time Series Models. Brazilian Journal of Probability and Statistics, 25, 90-98.
http://dx.doi.org/10.1214/09-BJPS111 [13] Iwueze, I.S. and Ohakwe, J. (2009) Penalties for Misclassification of First Order Bilinear and Linear Moving Average Time Series Processes.
http://interstatjournals.net/Year/2009/articles/0906003.pdf [14] Okereke, O.E. and Iwueze, I.S. (2013) Region of Comparison for Second Order Moving Average and Pure Diagonal Bilinear Processes. International Journal of Applied Mathematics and Statistical Sciences, 2, 17-26. [15] Montgomery, D.C., Jennings, C.L. and Kaluchi, M. (2008) Introduction to Time Series Analysis and Forecasting. John Wiley and Sons, New Jersey. [16] Sittinger, B.D. (2010) The Second Derivative Test. www.faculty.csuci.edu/brian.sittinger/2nd_Derivtest.pdf