Área de estudio

The simulation of 300 sample size data with different values of leverages and skewness were used for the estimation of CWN, VAR, EGARCH and MA as reported in Table 4.14 to Table4.16.

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Table 4.14 to Table 4.16 reported that the regression standard errors values of CWN have minimum error when compared with the three models estimated in this study. The estimated log-likelihood parameter of CWN indicated highest values among the models. The information criteria with minimum values of AIC and BIC indicated the best fit of CWN among the models.

The root mean square error (RMSE) values, mean absolute error (MAE) values, mean absolute percentage error (MAPE) values and geometric root mean square error (GRMSE) values were considered as forecast error measures in this study. CWN provided the minimum values of forecast error measure. These showed that CWN models were the best among the models estimated as reported in Table4.14 to Table 4.16.

CWN has the best model fit in 300 sample size with low leverage and low skewness as reported in Table 4.14.MAE, MAPE and GRMSE values have minimum forecast error measures in low leverage and high skewness among CWN as reported in Table 4.14.

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Table 4.14

Sample Size of 300 with Low Leverage and different Values of Skewness

Estimation CWN VAR EGARCH MA

Low leverage and low skewness

Standard Error 0.1249 1.6713 1.6881 1.6691 Log-Likelihood 903.61 333.91 -555.95 -575.94 AIC -6.0041 -2.1532 3.7655 3.9096 BIC -5.9298 -2.0047 3.8522 3.8873 RMSE 0.1253 1.4725 1.5642 1.5639 MAE 0.0633 1.1327 1.1356 1.1352 MAPE 6.2777 112.60 99.750 98.940 GRMSE 0.0011 1.0105 0.5472 0.5515

Low leverage and moderate skewness

Standard Error 0.1191 1.3004 1.3061 1.2909 Log-Likelihood 426.74 -69.060 -488.75 -499.09 AIC -2.8143 0.5422 3.3161 3.3956 BIC -2.7401 0.6907 3.4027 3.3734 RMSE 0.1363 1.2940 1.2972 1.2941 MAE 0.0743 0.9416 0.9514 0.9525 MAPE 7.4340 102.46 101.42 104.77 GRMSE 0.0018 0.3622 0.4123 0.3765

Low leverage and high skewness

Standard Error 0.0911 1.2563 1.2575 1.2563 Log-Likelihood 434.35 -48.380 -479.92 -490.98 AIC -2.8652 0.4039 3.2570 3.3413 BIC -2.7909 0.5524 3.3436 3.3191 RMSE 0.1356 1.1774 1.2001 1.2000 MAE 0.0615 0.8983 0.9277 0.9278 MAPE 5.1043 107.55 103.50 101.14 GRMSE 0.0010 0.3545 0.3537 0.3514

Table 4.15showed that CWN has the best model fit in moderate leverage and low skewness, but standard error has the highest value. The best forecast evaluation values were in 300 sample size with moderate leverage and moderate skewness.

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CWN has the best model fit in 300 sample size with high leverage and moderate skewness. While the best forecast was in 300 sample size with high leverage and high skewness on average as reported in Table 4.16.

CWN has the best forecast in 300 sample size with high leverage and high skewness on average as the best described in Table 4.14 to Table 4.16.

Table 4.15

Sample Size of 300 with Moderate Leverage and different Values of Skewness

Estimation CWN VAR EGARCH MA

Moderate leverage and low skewness

Standard Error 0.1694 1.6470 1.6549 1.6439 Log-Likelihood 807.35 244.63 -550.26 -571.38 AIC -5.3602 -1.5560 3.7275 3.8420 BIC -5.2859 -1.4075 3.8141 3.8791 RMSE 0.1634 1.5276 1.5529 1.5398 MAE 0.0890 1.1025 1.1133 1.1138 MAPE 7.4251 104.61 114.95 97.940 GRMSE 0.0026 0.5706 0.5033 0.5672

Moderate leverage and moderate skewness

Standard Error 0.0978 1.2910 1.2898 1.2786 Log-Likelihood 493.96 -1.1150 -485.20 -496.23 AIC -3.2639 0.0878 3.2923 3.3765 BIC -3.1897 0.2363 3.3789 3.3542 RMSE 0.0789 1.2895 1.2911 1.2901 MAE 0.0309 0.9400 0.9486 0.9501 MAPE 3.8897 121.06 98.628 103.75 GRMSE 3.14E-05 1.0603 0.3730 0.3551

Moderate leverage and high skewness

Standard Error 0.1254 1.2753 1.2805 1.2563 Log-Likelihood 563.63 72.890 -477.14 -492.59 AIC -3.7300 -0.4100 3.2603 3.3373 BIC -3.6557 -0.2608 3.3473 3.3746 RMSE 0.1130 1.2533 1.2574 1.2971 MAE 0.0510 0.9349 0.9515 0.9829 MAPE 5.1075 106.46 101.41 111.67 GRMSE 0.0007 0.3539 0.4094 0.3424

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Table 4.16

Sample Size of 300 with High Leverage and different Values of Skewness

Estimation CWN VAR EGARCH MA

High leverage and low skewness

Standard Error 0.0249 1.6432 41.653 1.6425 Log-Likelihood 827.11 268.08 -544.49 -571.13 AIC -5.4924 -1.7130 3.6889 3.8428 BIC -5.4181 -1.5644 3.7755 3.8799 RMSE 0.0221 1.5142 1.5505 1.5391 MAE 0.0098 1.0737 1.1026 1.1044 MAPE 4.8911 104.61 113.34 97.180 GRMSE 2.54E-05 0.5160 0.4781 0.5359

High leverage and moderate skewness

Standard Error 0.0266 1.2942 1.2961 1.2846 Log-Likelihood 867.44 371.86 -481.41 -497.64 AIC -5.7621 -2.4071 3.2670 3.3488 BIC -5.6879 -2.2586 3.3536 3.3859 RMSE 0.0197 1.3090 1.3076 1.2959 MAE 0.0077 0.9424 0.9543 0.9539 MAPE 3.8896 98.960 99.840 116.05 GRMSE 0.0017 0.3185 0.3694 0.3734

High leverage and high skewness

Standard Error 0.0492 1.3103 1.3160 1.3113 Log-Likelihood 759.88 260.76 -478.54 -503.79 AIC -5.0426 -1.6639 3.2478 3.3899 BIC -4.9684 -1.5154 3.3344 3.4270 RMSE 0.0722 1.3542 1.3481 1.3304 MAE 0.0306 0.9926 0.9846 0.9659 MAPE 3.3643 108.96 102.56 122.20 GRMSE 0.0002 0.4078 0.3781 0.2552

CWN estimation outperformed VAR, EGARCH and MA using three sample sizes. CWN has the best fit in 300 sample size with low leverage and low skewness, while the best forecast was in high leverage and high skewness.

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4.5 Summary

In 200 sample size of simulated data with moderate leverage and moderate skewness results, CWN outperformed the VAR,EGARCH and MA with the values of least standard error, log-likelihood highest and minimum information criteria, AIC and BIC. This made the model to be the best fit among the low, moderate and high values of leverages and skewness of the 200 data simulated sample size. The best forecast for CWN results were in 200 data simulated sample size with high leverage and moderate skewness values of RMSE, MAE, MAPE and GRMSE.

In 250 sample size of simulated data with moderate leverage and high skewness results, CWN outperformed the EGARCH, VAR and MA. The minimum information criteria values of AIC, BIC, standard error and log-likelihood highest value revealed the best result. This made the model to be the best fit among the low, moderate and high values of leverages and skewness of the 250 simulated sample size. The best forecast for CWN results were in 250 data simulated sample size with low leverage and high skewness minimum values of RMSE, MAE, MAPE and GRMSE.

In 300 sample size of simulated data with low leverage and low skewness results, CWN outperformed the EGARCH, VAR and MA. The minimum information criteria values of AIC, BIC and log-likelihood highest value displayed the best result, but the lowest standard error value was in low leverage and high skewness. This made the model to be the best fit among the low, moderate and high values of leverages and skewness of the 300 data simulated sample size. The minimum forecast error measure

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values of RMSE, MAE, MAPE and GRMSE revealed the best forecast for 300 data simulated sample size with high leverage and high skewness values.

The CWN outperformed the VAR, EGARCH and MA estimation results. The CWN have the best result among the models estimated with different values of leverages and skewness using the three sample sizes as reported in Table 4.8 to Table 4.16

The overall best forecast model for CWN result was in 200 data simulated sample size with high leverage and moderate skewness which has minimum values of RMSE, MAE, MAPE and GRMSE.

The CWN outperformed the VAR estimation results as CWN were having the best results among the VAR models estimated with different values of leverages and skewness using the three sample sizes as reported in Table 4.8 to Table 4.16. CWN and VAR error terms are white noise. Therefore, CWN can be used to improve VAR using the three sample sizes with different values of leverages and skewness.

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CHAPTER FIVE

VALIDATION OF COMBINE WHITE NOISE (CWN) MODEL

USING REAL DATA

5.1 Introduction

This chapter explained the development and estimation of combine white (CWN) model using real data as described in Chapter Three, Sections 3.2 to 3.3 and Subsection 3.3.2. Real data that exhibit heteroscedastic errors were used to validate the performance of CWN model as compared to VAR, EGARCH and MA. The four sets of data that were used for the validations were United States gross domestic product (US GDP), United Kingdom gross domestic product (UK GDP), Australia gross domestic product (AU GDP) and France gross domestic product (GDP). These data sets were retrieved from DataStream of Universiti Utara Malaysia Library.

Section 5.2 described the type of real data. The twelve steps were employed in Section 5.3for the description of model development process. Followed by, Section 5.4 that described the performance of the validated models by comparison; the results were in Subsection 5.4.1. Subsection 5.4.2explained the reliability of the measurements of degree of relationship between the data distribution and using Levene‟s test of equal variances to solve the challenges of non-normality in the data distribution. Subsection 5.4.3 explained the combination of two variances. Then, Section 5.5 explained the different values of leverages and skewness. Section 5.6 summarized the findings based on the four sets of the real data in Sections 5.4 and 5.5.

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