1. Introduction

Much attention is usually given to the structures of the autocorrelation function (ACF) and partial autocorrelation function (PACF) during the identification stage of a time series model. The moving average model is a linear time series model with known structures of autocorrelation function and partial autocorrelation function ^[10]. Let be a sequence of independent and identically distributed random variables with zero mean and variance . Then is a non zero mean moving average process of order two (MA(2) process) if:

(1.1)

The moving average process in (1.1) has the following first and second moments ^[3]:

(1.2)

(1.3)

and

(1.4)

The autocorrelation function in (1.4) cuts off at lag two (^{[2, 3]}) .Other properties of the autocorrelation function are and ^[9].

A non linear time series model which competes with the moving average process in (1.1) in terms of autocorrelation function structure is the pure diagonal bilinear time series process of order two (PDB(2) process) defined by ^[4]:

(1.5)

where is a sequence of independent and identically distributed random variables with zero mean and constant variance, and are real constants. If and, then the first and second moments of the model in (1.5) are as follows ^[8]:

(1.6)

(1.7)

(1.8)

It is quite obvious that the ACFs in (1.4) and the one in (1.8) all cut off after lag two. This is indicative of the fact that a moving average process of order two and a pure diagonal bilinear time series process of order two have similar autocorrelation structures. As a result, there is a possibility of misclassifying a pure diagonal bilinear process of order two as a moving average process of order two. The ease with which linear models are fitted and the practice of approximating nonlinear models by linear models can also cause misspecification of the nonlinear pure diagonal bilinear process of order two.

From the foregoing, it is imperative to investigate the statistical implication of the aforementioned model misclassification. In this regard, we will focus on the penalty function associated with misclassification of a PDB(2) process as an MA(2) process.

2. Relationship between the Parameters of the Pure Diagonal Bilinear Process of Order Two and Moving Average Process of Order Two

Having observed that the moving average process of order two and pure diagonal bilinear process of order two have similar autocorrelation structures, it is worthwhile to derive the relationship between the parameters of the two models. These relationships will help us to obtain the penalty function for misclassifying the nonlinear model as the competing linear model. The method of moments which involves equating the first and second moments of the pure diagonal bilinear model to the corresponding moments of the non zero moving average process of order two shall be used for this purpose.

Equating means, we have

(2.1)

Equating variances, we obtain

(2.2)

Equating first order autocorrelations leads to:

(2.3)

Equating second order autocorrelations gives

(2.4)

Dividing (2.4) by (2.3), we have

(2.5)

From (2.5), we obtain

(2.6)

Substituting (2.6) into (2.3), we obtain

(2.7)

where

(2.8)

and

(2.9)

One way of solving (2.7) involves modifying it first to obtain

(2.10)

Here, a, b and k are constants such that

(2.11)

(2.12)

(2.13)

The value of k satisfying (2.11), (2.12) and (2.13) is obtained by solving the equation

(2.14)

Substituting into (2.14), we obtain

(2.15)

Following the methods of ^[5] and ^[1], the real solution of (2.15) is found to be

(2.16)

where

(2.17)

and

(2.18)

Hence,

(2.19)

Comparing (2.7) and (2.10), we have

(2.20)

It can be deduced from (2.20) that

(2.21)

or

(2.22)

Solving (2.21), we obtain

(2.23)

or

(2.24)

From (2.22), we have

(2.25)

or

(2.26)

When , we have

(2.27)

For , we obtain

(2.28)

Using , we have

(2.29)

With , we obtain

(2.30)

Simulation concerning (2.23), (2.24), (2.25), (2.26) and their corresponding values of namely , , and respectively shows that only and take on values satisfying the invertibility condition of a moving average process of order two.

3. Penalty Function for Misclassification of a PDB(2) Process as an MA(2) Process

Penalty function based on model misclassification in time series analysis is defined by ^[6] as a function of error standard deviations. Let be the standard deviation of the errors associated with a PDB(2) process. Suppose represents the standard deviation of the errors corresponding to an MA(2) process. Then the penalty function for the misclassification of PDB(2) as an MA(2) is given as

(3.1)

We can write (3.1) as

(3.2)

Using (2.2), we obtain

(3.3)

Substituting (3.3) into (3.2) leads to

(3.4)

and in (3.3) are as defined in (2.27) and (2.23) respectively. Table 1 contains the penalties (P) corresponding to various values of , , and .

Considering the complete table containing 2129 sets of values, we can see that the penalty function for misclassification of a PDB(2) process as an MA(2) process (P) takes on positive values for all values of , , and . The positive value of the penalty for misclassification of a PDB(2) process as an MA(2) process shows that this misclassification leads to increase in variance of the errors. This finding agrees with the results obtained by ^[6] with regard to misclassification of a PDB(1) process as an MA(1) process.

For predictive purposes, we have to find the relationship between P and and. First, we plot P against each of and. Figure 1 shows the plot of P against.

Table 1. Penalties for various Values of Parameters of MA(2) Process and PDB (2) Process

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Figure 1. Plot of P against λ₁

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Figure 2. Plot of P against λ₂

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It can easily be seen from Figure 1 that there is a curvilinear relationship between P and . In Figure 2, we have the plot of P against.

Figure 2 also reveals that there is a curvilinear relationship between P and .

Combining the information in Figure 1 and Figure 2, the regression model in (3.5) is suggested for the relationship between P and and .

(3.5)

where , , , and are the parameters of the regression equation (3.5) and is the associated error term. The least squares estimation of (3.5) based on the admissible values of . and leads to the predictive equation ^[7].

(3.6)

Table 2 contains the summary of the test for significance of each of the parameters of the model in (3.6).

Table 2. Test for Significance of the Parameters of the Regression Model for Penalty for Misclassication of a PDB(2) Process as an MA(2) Process

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Each of the parameters appears to be significant at level of significance since the corresponding p value is less than 0.05. Next, we test for significance of the overall regression using the analysis of variance technique as shown in Table 3.

Table 3. ANOVA Table for Testing for Significance of the Fitted Penalty Model

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The p value of 0.00 in the Table 3 implies that the fitted regression model is suitable for describing the relationship between P and and.

4. Conclusion

In this study, we determined the effect of misclassifying a pure diagonal bilinear process of order two as a moving average process of order two. A penalty function was defined and was used to compute penalties for misclassification of the pure diagonal bilinear process of order two as the moving average process of order two based on various sets of values of the parameters of the two processes. The computed penalties assumed positive values. This indicated increase in error variance due to misclassification of pure diagonal bilinear process of order two as a moving average process of order two. A quadratic regression model was found suitable for predicting the penalties based on the parameters of the pure diagonal bilinear process of order two.

References

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