Twelve steps were employed for the development of the model.
Step 1: VAR white noise estimation was efficient but weak in modelling heteroscedasticity, the weakness were as reported in Table 4.8 to Table 4.16.
Step 2: EGARCH estimation modelled heteroscedasticity without leverage effect efficiently but weak in modelling the leverage effect in the heteroscedasticity, the weakness were as reported in Table 4.8 to Table 4.16.Therefore, the data that exhibited heteroscedasticity were simulated, estimated and the graphs of the estimated standardized residuals with unequal variances and zero mean were considered in this study to resolve the leverage effect challenges.
Step 3: The simulated data of 200 sample size were estimated to obtain the standardized residuals in graphical form. The graphs of standardized residuals displayed the error terms of these models for the purpose of this study. The error terms have the characteristics of heteroscedasticity with leverage effect(unequal variances), which made up the conditional variance challenges in the estimation for 200 sample size different values of leverages and different values of skewness as reported in Figure 4.1 to Figure4.3. Similar results were obtained when standardized residuals graphs of 250 and 300 sample sizes were conducted.
Step 4: The standardized residuals graphs of unequal variances were decomposed (rearranged and grouped) manually into equal variances (white noise) series to overcome the leverage effect which were examined by displaying in graphical form. The decomposition for low leverage and low skewness have forty equal variances, the
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low leverage and moderate skewness have forty four equal variances and the low leverage and high skewness have forty three equal variances. The decomposition for moderate leverage and low skewness have forty equal variances, the moderate leverage and moderate skewness have forty five equal variances and the moderate leverage and high skewness have forty one equal variances. The decomposition for high leverage and low skewness have forty one equal variances, the high leverage and moderate skewness have forty four equal variances and the high leverage and high skewness have forty three equal variances. Maximum likelihood estimation method was applied on each equal variance to obtain the log-likelihood.
Step 5: The Log-Likelihood
The log-likelihood was maximized by the maximum likelihood estimation method for the number of equal variances in each standardized residual. The estimation of maximum likelihood was employed to optimize the parameters for sufficiency, consistency, efficiency and invariance parameterization of the equal variances (white noise) series. The log-likelihood values were reported in Appendix C.
Based on the log-likelihood obtained, the number of equal variances (white noise) from each standardized residuals were fitted into linear model by MLE and BIC for modelling each equal variance. This revealed the equal variance model called white noise (WN) model.
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Low Leverage and Low Skewness
-4 -3 -2 -1 0 1 2 25 50 75 100 125 150 175 200 U V Time
Low Leverage and Moderate Skewness
-3 -2 -1 0 1 2 3 4 25 50 75 100 125 150 175 200 U V Time
Low Leverage and High Skewness
U V represented Unequal Variances
Figure 4.1.Graphs of Standardized Residuals for Low Leverage and Different Values of Skewness
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Moderate Leverage and Low Skewness
Time -4 -3 -2 -1 0 1 2 3 25 50 75 100 125 150 175 200 225 250 U V Time
Moderate Leverage and Moderate Skewness
-4 -3 -2 -1 0 1 2 3 4 25 50 75 100 125 150 175 200 225 250
Moderate Leverage and High Skewness
U
V
Time
U V represented Unequal Variances
Figure 4.2.Graphs of Standardized Residuals for Moderate Leverage and Different Values of Skewness
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Hgh Leverage and Low Skewness
-4 -3 -2 -1 0 1 2 3 25 50 75 100 125 150 175 200 U V Time
High Leverage and Moderate Skewness
-3 -2 -1 0 1 2 3 4 25 50 75 100 125 150 175 200 U V Time
High Leverage and High Skewness
U V represented Unequal Variances
Figure 4.3. Graphs of Standardized Residuals for High Leverage and Different Values of Skewness
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Step 6: Fitting Linear Model and BIC
The linear model was fitted into the series of equal variances (WN) by MLE and BIC to obtain the fitted WN models. In fitting these linear models, each WN model has mean zero and variance one (constant) and each model was significant. White noise assumed zero mean and constant variance. Therefore, WN models with zero mean and constant variance confirmed the WN. The standardized residual graphs have zero mean. The Bayesian model averaging was used for model selection.
Step 7: Bayesian Model Averaging (BMA)
There were 2K certainty and uncertainty models to account for, and for this study K
were the numbers of equal variances in step 5 that were transformed to models in step 6by fitting the linear model. Some models were selected out of 2K uncertainty and
certainty models. The best models were determined by the lowest BIC and highest posterior probability (the correct model) in BMA output. The computer outputs were in Appendix D.
The Appendix D detail were: The column “p!=0” indicated the probability that the coefficient for a given predictor is not zero. This indicated that at least one of the best models considered in the row directly under the column “p!=0”. The column “EV” displayed the BMA posterior distribution mean for each coefficient and the column “SD” displayed the BMA posterior distribution standard deviation for each coefficient. The posterior probabiliy of quantity of interest was determined by each of the models considered when the posterior propability was correct, given that one of the considered models was correct.The best five models (discribed as model 1, model 2, model 3,
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model 4 and model 5)were displayed. The predictors (independent variables)to be included in a regression model were determined by BMA. Two best predictors were displayed in Appendix D (number 10, number 11 and number 12).
Appendix D (number 1 to number 9) displayed the numbers of predictors for 200 sample size. Similar results were computed for 250 and 300 sample sizes. Appendix D (number 10) summarized the BMA for 200 sample size. The low leverage and low skewness revealed that predictor A has the best model which was in the third model discribedas model 3 with minium BICand highest posterior probability values. Predictor B has the best model in model 4 which was the best model. The low leverage and moderate skewness revealed that predictor Chas the best model inmodel 4 with minium BIC and highest posterior probability values. Predictor D has the best model in model 3 which was the best model. The low leverage and high skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor E. Predictor F has the best model in model 2 which was the best model.
The moderate leverage and low skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor G. Predictor H has the best model in model 2 which was the best model. The moderate leverage and moderate skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor I. Predictor J has the best model in model 3 which was the best model. The moderate leverage and high skewness revealed the best model was in model 2 with minium BIC and highest posterior
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probability values as predictor K. Predictor L has the best model in model 3 which was the best model.
The high leverage and low skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor M. Predictor N has the best model in model 2 which was the best model. The high leverage and moderate skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor P. Predictor Q has the best model in model 3 which was the best model. The high leverage and high skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor R. Predictor S has the best model in model 3 which was the best model.
Appendix D (number 11) summarized the BMA for 250 sample size. The low leverage and low skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor A1. Predictor B1has the best model in model 2 which was the best model. The low leverage and moderate skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor C1. Predictor D1 has the best model in model 4 which was the best model. The low leverage and high skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor E1. Predictor F1 has the best model in model 3 which was the best model.
The moderate leverage and low skewness revealed the best model was in model 1 with minium BIC and highest posterior probability values as predictor G1. Predictor H1 has the best model in model 2 which was the best model. The moderate leverage and
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moderate skewness revealed the best model was in model 4 with minium BIC and highest posterior probability values as predictor I1. Predictor J1 has the best model in model 3 which was the best model. The moderate leverage and high skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor K1. Predictor L1 has the best model in model 3 which was the best model.
The high leverage and low skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor M1. Predictor N1 has the best model in model 2 which was the best model. The high leverage and moderate skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor P1. Predictor Q1 has the best model in model 2 which was the best model. The high leverage and high skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor R1. Predictor S1 has the best model in model 3 which was the best model.
Appendix D (number 12) summarized the BMA for 200 sample size. The low leverage and low skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor A2. Predictor B2has the best model in model 2 which was the best model. The low leverage and moderate skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor C2. Predictor D2 has the best model in model 3 which was the best model. The low leverage and high skewness revealed the best model was in model 1
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with minium BIC and highest posterior probability values as predictor E2. Predictor F2 has the best model in model 2 which was the best model.
The moderate leverage and low skewness revealed the best model was in model 4 with minium BIC and highest posterior probability values as predictor G2. Predictor H2 has the best model in model 3 which was the best model. The moderate leverage and moderate skewness revealed the best model was in model 3 with minium BIC and highest posterior probability values as predictor I2. Predictor J2 has the best model in model 2 which was the best model. The moderate leverage and high skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor K2. Predictor L2 has the best model in model 3 which was the best model.
The high leverage and low skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor M2. Predictor N2 has the best model in model 3 which was the best model. The high leverage and moderate skewness revealed the best model was in model 2 with minium BIC and highest posterior probability values as predictor P2. Predictor Q2 has the best model in model 3 which was the best model. The high leverage and high skewness revealed the best model was in model 3 with minium BIC and highest posterior probabilityvalues as predictor R2. Predictor S2 has the best model in model 2 which was the best model.
Step 8: Fitting Linear Regression with Autoregressive Errors
Fitting linear regression with autoregressive errors of which 200 were the numbers of sample size, with zero mean and variance one for each model to confirm that the white
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noise models were invertible to AR models (Higgins & Bera 1992). The estimated values of the fitted linear regression with autoregressive errors, based on significant code asterisk showed the best models for different values of leverages and skewness as reported in Appendix E (number 1 to number 9) for 200 sample size. Similar results were computed for 250 and 300 sample sizes. The best two models were summarized in Appendix E (number 10, number 11 and number 12).These confirmed the best selected models by BMA.
P-values revealed the significant values of each best two models in different values of leverages and skewness. The more significant of each of the two models indicated the dependent variable for the combine white noise (CWN) in step 12. When the two models were having equal significant values as models G and H in Appendix E (number 10), models
R
1 andS
1 in Appendix E (number 11), modelsA
2 andB
2 in Appendix E (number 12) of the three different sample sizes respectively, the best which has the minium BIC and highest posterior probability values in step 7 were considered as dependent variable.Appendix E (number 10) displayed the fitted linear regression with autoregressive errors for 200 sample size.The low leverage and low skewness shown that Model Awas the dependent variable because its value was more significant than model B. The low leverage and moderate skewness shown that Model D was considered as dependent variable because its value was more significant than model C. The low leverage and high skewness shown that Model F was the dependent variable because its value was more significant than model E.
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The moderate leverage and low skewness shown that Model G and model H were having the same value of significant figures, model Hwas considered as dependent variable in step 12 because predictor Hvalue was in model 2 step 7 as reported in Appedix D (number 10) with minium BIC and high posterior probability values.The moderate leverage and moderate skewness shown that Model I was considered as dependent variable because its value was more significant than model J.
The high leverage and low skewness shown that Model M was considered as dependent variable because its value was more significant than model N. The high leverage and moderate skewness shown that Model P was considered as dependent variable because its value was more significant than model Q. The high leverage and high skewness shown that Model S was considered as dependent variable because its value was more significant than model R.
Appendix E (number 11) displayed the fitted linear regression with autoregressive errors for 200 sample size. The low leverage and low skewness shown that Model A1 was considered as dependent variable because its value was more significant than model B1. The low leverage and moderate skewness shown that Model C1 was considered as dependent variable because its value was more significant than model D1. The low leverage and high skewness shown that Model E1 was considered as dependent variable because its value was more significant than model F1.
The moderate leverage and low skewness shown that Model G1 was considered as dependent variable because its value was more significant than model H1. The moderate leverage and moderate skewness shown that Model J1 was considered as
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dependent variable because its value was more significant than model I1. The moderate leverage and high skewness shown that Model K1 was considered as dependent variable because its value was more significant than model L1.
The high leverage and low skewness shown that Model N1 was considered as dependent variable because its value was more significant than model M1. The high leverage and moderate skewness shown that model Q1 was considered as dependent variable because its value was more significant than model P1. The high leverage and high skewness shown that model R1 and model S1 were having the same value of significant figures, model R1was considered as dependent variable in step 12 because predictor R1 value was in model 2 in step 7 with minium BIC and high posterior probability values.
Appendix E (number 12) displayed the fitted linear regression with autoregressive errors for 200 sample size. The low leverage and low skewness shown that model A2 and model B2 were having the same value of significant figures, model B2 was considered as dependent variable because predictor B2 value was in model 2 in step 7 with minium BIC and high posterior probability values. The low leverage and moderate skewness shown that Model C2 was considered as dependent variablebecause its value was more significant than model D2. The low leverage and high skewness shown that Model F2 was considered as dependent variable because its value was more significant than model E2.
The moderate leverage and low skewness shown that Model H2 was considered as dependent variable because its value was more significant than model G2. The
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moderate leverage and moderate skewness shown that Model I2 was considered as dependent variable because its value was more significant than model J2. The moderate leverage and high skewness shown that Model K2 was considered as dependent variable because its value was more significant than model L2.
The high leverage and low skewness shown that Model M2 was considered as dependent variable because its value was more significant than model N2. The high leverage and moderate skewness shown that Model P2was considered as dependent variable because its value was more significant than model Q2. The high leverage and high skewness shown that Model R2 was considered as dependent variablebecause its value was more significant than model S2.
The SARIMA models were used for the lag selection of autoregressive order of the models with Y as dependent variable of a model.
Step 9: The Regression Model with ARIMA Errors
Firstly, regress with the models obtained in step 8, and then run the following ACF of the models. The ACF spike of the first lag signified autoregressive (AR) of order one which was significant, while the rest lags were close to zero which signified that the orders were zero. SARIMA (1, 0, 0) indicated AR (1) converge with short iteration. Therefore, SARIMA (1, 0, 0) were considered as the best.
The confirmation of two models from the result of BMA in step 7by fitting the linear regression with autoregressive errors in step 8 revealed that the first columns for the