Moving average processes (models) constitute a special class of linear time series models. A moving average process of order (process) is of the form:

where are real constants and, 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, then the model in (1.1) is invertible if the roots of the characteristic equation

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.

For, 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 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]

where

and

If, (2.2) has the following distinct roots [7]

and

where is measured in radians and.

When, (2.2) has only real root given by [1] as

The other roots are [8]

If, and, then and (2.2) has two equal roots. The roots of (2.2) in this case, are the same as (2.7), (2.8) and (2.9). For and, (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. 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 has three real equal roots, then the moving average process of order three is invertible if

, and.

Proof

For invertibility, we expect each of the three real equal roots to lie outside the unit circle. Thus,

or

Solving the inequality, we obtain

For, we have

Since each of the roots lie outside the unit circle, the absolute value of their product must therefore be greater than one. Hence,

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 Figure 1.

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 if its associated autocorrelation function cut off after lag 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.

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, [15] while for a moving average process of order

two and [5] . In order to generalize about the range of values of for a

moving average process of order, it is worthwhile to determine the range values of for a moving average 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 are found to be and respectively. For the autocorrelation function at lag two, we have

The extreme values of are equally obtained with the help of the Scientific Note Book. To this effect, 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, has a minimum value of −0.5 and a maximum value of 0.5. However, the intervals for can easily be obtained analytically and this result is generalized in Theorem 2 for of the MA process.

The partial derivatives of with respect to, and are

The critical points of occurs when,. Equating each of the partial derivatives in (4.5),

(4.6) and (4.7) to zero, we obtain

From (4.10), we have

Using (4.8), we obtain

or

Substituting into (4.11) yields

For, (4.9) becomes

If we also substitute into (4.9), we obtain

When we substitute and into (4.11), we have. It is also clear that if and, then. Similar result is obtained when and.

Hence, the critical points of are, , and.

The minimum and 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 focuses on the Hessian matrix:

where

Let be a critical point of. Then is called a local minimum point if at

, , and [16] . If, and at,

then 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 and. We then focus on the classification of the two remaining critical points.

At

Hence, , and.

Therefore, is a local minimum. The value of at this point is.

For the critical points, we have

Consequently,

and

We therefore conclude that is a local maximum. The maximum value of obtained at is 0.5.

We can deduce from the result in this section and other previous works that for MA (1) process, while for MA (2) process and MA (3) process and respectively.

In what follows, we establish the bounds for, where is order of the moving average process.

Theorem 2.

Let be an MA process. Then,.

Proof

It is easily seen that for the MA process,

Partial derivatives of with respect to are as follows

Equating each of the partial derivatives to zero yields

From (4.24), we obtain

Since for an MA process, it is obvious that the equations preceding (4.24) are only satisfied if. Substituting into (4.25) leads to. The two critical points of are then and.

At, while at,. It then follows that.

Remark: For an invertible MA (3) process,. Hence, , and.

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, and. Also the intervals for the autocorrelation coefficients of an invertible moving average process of order three are estab-

lished. These are, and. It is also noteworthy that the

condition on for an invertible MA (3) process is generalized for of the invertible MA process. That is for the invertible MA process,. These results can now be used to compare other linear and nonlinear processes that have similar autocorrelation structures with the MA (3) process.