A Proposed Method for Estimating Parameters of Non-Gaussian Second Order Moving Average Model

Transcription

1 International Journal of Statistics and Systems ISSN Volume 11, Number 2 (2016), pp Research India Publications A Proposed Method for Estimating Parameters of Non-Gaussian Second Order Moving Average Model Mohammed Qadoury Abed Assistant Professor, Al-Mansour University College mqadory2@yahoo.com & moqa7040@gmail.com Abstract: The importance of time Series in researches and Scientific Studies is manifested through its dependence on the values of the observation of the phenomenon on studying the way these values change at periods of equal time on observation noted concerning the phenomenon, the reasons causing it and the factors affecting it with the object of setting up a suitable mathematical model to express the time series of the phenomenon. Because of the model is non-linear in its parameters, the researcher proposed a method to estimate these parameters by depending on one of the non-linear procedures as adopted by Newton-Raphson. Among the most important realized by the researcher are: For all distribution (continuous, discrete) and for all the samples sizes the values of MAPE for both parameters ( ), by Using proposed method are greater than the values of MAPE for both parameters ( ) by using Iterative process method when the sign ( ) is negative and ( ) is positive. The Values of MAPE for the parameter ( ) decrease whenever the sample size increases according to proposed method for continuous distributions when the sign of ( ) is positive and ( ) is negative. The values of MAPE for the parameter ( ) increase whenever the sample size increases if the sign of ( ) is negative and ( ) is positive according to proposed method for discrete distributions. Key words: Moving Average, Newton-Raphson, Iterative Process, Non- Linear, Non-Gaussian, Simulation, Time Series, Distributions. Contribution/Originality Because the moving average model is non-linear in its parameters, suggesting a new method for estimating these parameters of a model of the second order may be fruitful in solving problems researchers may encounter. Respectively, the researchers can

2 104 Mohammed Qadoury Abed develop this method to include the parameters of moving average models of higher orders. This suggested method is compared with a new method for verifying its application. 1- Introduction Time series is a group of recorded measures for one variable or more organized according to the time of their occurrence, i.e The independent time variable (t) and the corresponding values which are the dependent variable (y) and each value in (t) time has equivalent values for the dependent variable (y). So, (y) is a function (t) time, so: y=f (t). The importance of time series in researches and scientific studies shows itself through its dependence on the observed values of the phenomenon and on the studying the way of change of these values at the range of equal time series to know the reality of variations that arise on noted observations of the phenomenon, and to get acquainted with its causes and its effectives factors. The objective is to build a suitable mathematical model to express the time series of the phenomenon. The moving average is one of the oldest and wide spread technical Analysis and the more abundant. It is mainly used as a tool for trend following. In 2001, in a simulation study, Al-Khudhairy (1) used the Iterative process method to estimate the parameter of a Non-Gaussian seasonal moving average model of the first order. In 2005 Al-Nassir & Al-Khudhairy (2) estimated the parameter of the mixed model ARMA (1,1) by using a method based on non-linear Newton Raphson procedure, and comparing it with non-linear least square method. In 2011, Mohammed & Wadhah (9), by simulation, studied a Non-Gaussian moving average model of the second order. This research aims at proposing anew method to estimate the parameters of the second order moving average by using non-linear Newton-Raphson procedure and comparing it with Iterative process method by assuming that random errors of the model follows a Non-Gaussian distribution (continuous, discrete), for different sample sizes and different initial values. 2- The second order moving average model By using back shift factor B in the following formula: (3) (5) (12) And that can be written as follows: Therefore the general formula for the second order moving average model MA (2) will be:

3 A Proposed Method for Estimating Parameters 105 Therefore: ( ): Time series observations values. ( ): moving average model s parameters. ( ): random errors To understand inevitability and the theoretical aspects of the model see (4) (6) (7) (8) (10) 3- Estimation of the model parameters: (1) (9) a- Iterative process method (IP) This method is used to estimate the non-linear parameters of the second order moving average and as follows: We estimate ( ) according to the following formula: We give the (3.1) equation an initial value equal zero for both parameters ( ), and this means that we make ( ) and then we estimate the parameters according to the following formula: So: And we continue in repeating this till we get stationary status in parameters ( ) values. b- Proposed method (PM): Since the equation is non-linear in its parameters, the researcher propose this method by relying on one of the non-linear procedures which is Newton-Raphson to estimate the parameters of the model and as follows:

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5 A Proposed Method for Estimating Parameters 107 So: ( ) are initial values for the model s parameters. 4- Simulation: Four simulation experiments are designed to estimate the parameters of the second order moving average model by using two methods, the Iterative process and the proposed method and then by having a comparison between them to determine the goodness and efficiency of the proposed method. The comparison is done by using mean absolute percentage error MAPE for the parameters and supposing that random error of the model MA(2) is distributed as continuous (Beta, Weibull) and as discrete (Binomial, Poisson) and for the initial values of the parameters (0.5,-0.1)(-0.5,0.1) and different sample sizes (30,75,125,200). Each experiment is repeated 1000 times, to understand and apply the generation methods by Monte Carlo procedure, see (11). Table (1): MAPE values when ( ) follow Beta distribution for MA(2) N IP PM IP PM MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) Table (2): MAPE values when ( ) follow weibull distribution for MA(2) N IP PM IP PM MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) Table (3): MAPE values when ( ) follow Binomial distribution for MA(2) N IP PM IP PM MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( )

6 108 Mohammed Qadoury Abed Table (4): MAPE values when ( ) follow Poisson distribution for MA(2) N IP PM IP PM MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) MAPE( ) Conclusions a- For all distributions (continuous, discrete) and for all samples size: a.1: When the sign of ( ) is negative, so the values of MAPE ( ) by using the two methods Iterative process (IP), proposed method (PM) will be greater than the values of MAPE when the sign of the latter is positive. a.2: When the sign of ( ) is negative, so the values of MAPE ( ) by using the two methods Iterative process (IP), proposed method (PM) will be smaller than the values of MAPE when the sign of the latter is positive. b- For all distribution (continuous, discrete) and for all the sample sizes the values of MAPE for both parameters ( ) by using proposed method are greater than the values of MAPE for both parameters ( ) by using Iterative process method and the sign ( ) is negative and ( ) is positive. c- For Iterative process method: c.1: The values of MAPE are decrease for the parameter ( ) whenever the sample size increases. c.2: The values of MAPE for the parameter ( ) are increase whenever the sample size increases. d- For the proposed method, the values of MAPE for both parameters ( ) increase whenever the sample size increases. e- The values of MAPE for the parameter ( ) decrease whenever the sample size increases according to proposed method for the continuous distributions when the sign of ( ) is positive and ( ) is negative. f- The values of MAPE for the parameter ( ) increase whenever the sample size increases if the sign of ( ) is negative and ( ) is positive according to the proposed method and for discrete distributions. g- The values of MAPE for the parameter ( ) and for continuous distributions are smaller than the values of MAPE for the parameter ( ) and for discrete distributions according to Iterative process method. h- The values of MAPE for the parameter ( ) and for continuous distribution are greater than the values of MAPE for the parameter ( ) and for discrete distributions according to the Iterative process and the proposed method. i- The values of MAPE for the parameter ( ) and for continuous distribution are greater than the values of MAPE for the parameter ( ) and for discrete distributions according to proposed method if the sign of ( ) is negative and ( ) is positive.

7 A Proposed Method for Estimating Parameters 109 j- The values of MAPE for the parameter ( ) and for continuous distribution are smaller than the values of MAPE for the parameter ( ) and for discrete distributions according to the proposed method if the sign of ( ) is positive and ( ) is negative. References: [1] Al-Khudhairy, M.Q.(2001)"The used of Iterative process method to estimate a parameter of a Non-Gaussian seasonal moving average of the first order: A simulation study", Journal of the college of Administration and Economy, Issue(8), No.25, University of Baghdad, pp( ),(in Arabic). [2] Al-Nassir,A.H. & Al-Khudhairy,M.Q.(2005)"Estimating parameters of Non- Gaussian mixed model ARMA(1,1) by using a non-linear Newton-Raphson procedure-simulation study", Al-Mansour Journal, 5 th year- Issue(8),Iraq,pp( ). [3] Al-Nassir, A.H. & Jumma,A.A.(2013) Introduction to Applied Time Series Analysis, 1 st ed., Al-Jazeera printing and publishing, Baghdad, Iraq. [4] Anderson, T.W.(2011) The Statistical Analysis of Time Series, John Wiley & Sons, Inc, New York. [5] Box, G. E. P. & Jenkins, G.M. & Reinsel, G. C. (2013) Time Series Analysis, 4 th ed., John Wiley & Sons, Inc, New York. [6] Brockwell, P. & Davis, R. (2009) Introduction to Time Series: Theory and Methods, 2 nd ed., Springer, New York. [7] Fuller, A.W.(2008) Introduction to Statistical Time Series, John Wiley & Sons, Inc, New York, published online. [8] Kirchgassner, G. & Wolters, J.(2007) Introduction to Modern Time Series Analysis, Springer-Verlag, Berlin, Heidelberg. [9] Mohammed, Q.A. & Wadhah, S.I.(2011)" Non-Gaussian Moving Average Model from Second Order-simulation study", Magazine of College of Administration & Economics: For Economics & Administration & Financial Studies, Babylon University, No.1, Iraq, pp(57-67), (in Arabic). [10] Shumway, R.H. & Stoffer, D.S.(2005) Time Series Analysis and Its Applications, Springer, New York. [11] Robert, C.P. & Casella, G.(2004) Monte Carlo Statistical Methods, 2 nd ed., Springer, New York. [12] Wei, W. W. S. (1990) Time Series Analysis: Univariate and Multivariate Methods, Addison-Wesley publishing company Inc, New York. [13] Yaffee, R. A. & McGreen, M. (2000) Introduction to Time Series Analysis and Forecasting, Academic Press, San Diego.

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