Necessary Conditions for the Application of Moving Average Process of Order Three

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 1 1 5 1 5 4 4 − − − < < ρ , 2 0.5 0.5 − < < ρ and 1 0.5 0.5 − < < ρ .


Introduction
Moving average processes (models) constitute a special class of linear time series models.A moving average process of order q ( ( ) MA q process) is of the form: where 1 2 , , , q θ θ θ  are real constants and t e , t Z ∈ is a sequence of independent and identically distributed random variables with zero mean and constant variance.These processes have been widely used to model time series data from many fields [1]- [3].The model in (1.1) is always stationary.Hence, a required condition for the use of the moving average process is that it is invertible.Let m t t m B e e − = , then the model in (1.1) is invertible if the roots of the characteristic equation 1 0 lie outside the unit circle.The invertibility conditions of the first order and second order moving average models have been derived [4] [5].
Ref. [6] used a moving average process of order three (MA (3) process) in his simulation study.Though, higher order moving average processes have been used to model time series data, not much has been said about the properties of their autocorrelation functions.This study focuses on the invertibility condition of an MA (3) process.Consideration is also given to the properties of its autocorrelation coefficients of an invertible moving average process of order three.

Consequence of Invertibility Condition on the Parameters of an MA (3) Process
For 3 q = , the following moving average process of order 3 is obtained from (1.1): The characteristic equation corresponding to (2.1) is given by Dividing (2.2) by 3 θ yields It is important to know that (2.2) is a cubic equation.Detailed information on how to solve cubic equations can be found in [7] [8] among others.It has been a common tradition to consider the nature of the roots of a characteristic equation while determining the invertibility condition of a time series model [9].As a cubic equation, (2.2) may have three distinct real roots, one real root and two complex roots, two real equal roots or three real equal roots.The nature of the roots of (2.2) is determined with the help of the discriminant [8] and where θ is measured in radians and 2) has only real root given by [1] as The other roots are [8] ( ) ( ) ( ) 2) has two equal roots.The roots of (2.2) in this case, are the same as (2.7), (2.8) and (2.9).For 0 D = and 1 2 0 D D = = , (2.2) has three real equal roots.Each of these roots is given by [8] as For (2.1) to be invertible, the roots of (2.2) are all expected to lie outside the unit circle and 3 1 θ < .In the following theorem, the invertibility conditions of an MA (3) process are given subject to the condition that the corresponding characteristic equation has three real equal roots.
Theorem 1.If the characteristic equation Since each of the roots lie outside the unit circle, the absolute value of their product must therefore be greater than one.Hence,

Identification of Moving Average Process
Model identification is a crucial aspect of time series analysis.A common practice is to examine the structures of the autocorrelation function (ACF) and partial autocorrelation function (PACF) of a given time series.In this regard, a time series is said to follow a moving average process of order q if its associated autocorrelation function cut off after lag q and the corresponding partial autocorrelation function decays exponentially [10].Authors using this method, believe that each process has unique ACF representation.However, the existence of similar autocorrelation structures between moving average process and pure diagonal bilinear time series process of the same order makes it difficult to identify a moving average process based on the pattern of its ACF.Furthermore, a careful look at the autocorrelation function of the square of a time series can help one determine if the series follows a moving average process.If the series can be generated by a moving average process, then its square follows a moving average process of the same order [11] [12].The conditions under which we use the autocorrelation function to distinguish among processes behaving like moving average processes of order one and two have been determined by [13] [14] respectively.These conditions are all defined in terms of the extreme values of autocorrelation coefficients of the processes.

Intervals for Autocorrelation Coefficients of a Moving Average Process of Order Three
As stated in Section 3, knowledge of the extreme values of the autocorrelation coefficient of a moving average process of a particular order can enable us ensure proper identification of the process.It has been observed that for a moving average process of order one, ρ − ≤ ≤ [5].In order to generalize about the range of values of q ρ for a moving average process of order q , it is worthwhile to determine the range values of 3 ρ for a moving aver- age process of order three.The model in (2.1) has the following autocorrelation function [10]: We can deduce from (4.1) that the autocorrelation function at lag one of the MA (3) process is Using the Scientific Note Book, the minimum and maximum values of 1 ρ are found to be − respectively.For the autocorrelation function at lag two, we have The extreme values of 2 ρ are equally obtained with the help of the Scientific Note Book.To this effect, 2 ρ has a minimum value of −0.5 and a maximum value of 0.5.
From (4.1), we obtain Based on the result obtained from the Scientific Notebook, 3 ρ has a minimum value of −0.5 and a maxi- mum value of 0.5.However, the intervals for 3 ρ can easily be obtained analytically and this result is genera- lized in Theorem 2 for q ρ of the MA ( ) q process.The partial derivatives of 3 ρ with respect to 1 θ , 2 θ and 3 θ are ( ) The critical points of 3 ρ occurs when ) At ( ) Therefore, ( ) 0, 0, 1 − is a local minimum.The value of 3 ρ at this point is 0.5 − .

Conclusion
We have established necessary conditions for the parameters of an invertible MA (3) process.When the characteristic equation has three real equal roots, the conditions are 2 This completes the proof.The invertibility region of a moving average of order three with equal roots of the characteristic Equation (2.2) is enclosed by triangle OAB in Figure1.

Figure 1 .
Figure 1.Invertibility region of an MA (3) process when the characteristic equation has three real equal roots.
[15] while for a moving average process of order two of the partial derivatives in (4.5), (4.6) and (4.7) to zero, we obtain maximum values of a function occur at it critical points.To determine which of the critical points is a local minimum, local maximum or a saddle point, we shall apply the second derivative test.The second derivative test for critical points of a function of three variables

3 ρ
represents a local maximum.A critical point that is neither a local minimum nor a local maximum is called a saddle point.Though has four critical points, it is not defined at ( ) focus on the classification of the two remaining critical points.
is easily seen that for the MA ( )