The societal level: Discursive analysis of national documents

This chapter gives the summary, conclusion and recommendations on the study.

The contribution to knowledge and suggestion for further studies are also highlighted.

5.1 SUMMARY OF FINDINGS

Problems involving eigenvalues of the matrix of the explanatory variables in estimating the ridge parameters for solving multicollinearity problem in multiple regressions have been considered in this research work. This is a situation where the eigenvalues of the explanatory variable are skewed or among the eigenvalues, there is an outlier. Methods of estimating the ridge parameter were proposed taking into consideration the skewness and the outlier among the eigenvalues. A computer program in R was written for the implementation of the proposed methods and its comparison with some existing methods via Monte Carlo simulation. Mean square error (MSE) and prediction sum of square (PRESS) were used as criteria for comparing the estimators. The comparison was made under the same sample size n, the random error  , number of explanatory variable p, and correlation coefficient  .

From the results obtained from this simulation study, it is evident that the performance of ridge regression depends on the random error and the correlation among the explanatory variables. Also increase in the standard deviation of the random error,  increases the mean square error and PRESS of an estimator.

The ridge estimator studied have shown to be better than OLS method in all the

cases in terms of smaller mean square error. They exhibit a substantial reduction

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in MSE in all the cases and PRESS in some of the cases. Also the simulation study has shown that for  =0.9 and  = 0.99, (a severe multicollinearity) ridge regression yields minimum mean square error where as in a situation of low multicollinearity (  =0.3 and   0 . 5 ) the results coincide with OLS. This indicates that ridge regression is efficient when multicollinearity is high and present that is when the explanatory variables are highly collinear.

The ridge regression was also applied to an economic data on Nigerian Eonomic indicators using GDP as dependent variable and ten other explanatory variables obtained from CBN Statistical Bulletin (2010) to illustrate in real life situation the ridge regression estimation. The data were highly collinear, as high as r 8.10 = 0.99. from the estimated ridge regression coefficients obtained, we observed that the coefficient of b 1 (k) change sign from positive to negative and that some of the ridge regression parameter estimates recorded high values of b _j (k) while some showed reduction in values of b j (k)

The eigenvalues of the matrix ~ ~ )

( X  X is skewed with a skewness value of 2.23.

Further observations were that some of the estimators recorded high values of k (k ₅ , k ₁₉ ) and some recorded smaller values of k (k ₁₁ , k ₁₂ and k ₁₅ ). These smaller values of k yielded minimum MSE and smaller PRESS. Amongst all the estimators considered, k 11 is the best in terms of minimum mean square and smaller PRESS with the value of k 11 =0.0324.

Furthermore, Q-test for detection of outlier in the eigenvalues of the matrix

~ )

( X ~  X , the maximum eigenvalue is found to be an outlier. In view of this, the

method of geometric mean is used in the eigenvalues to obtain the ridge

parameter and their corresponding ridge regression coefficients. The

performances of these estimators are encouraging when compared with the

existing ones. We also observed that amongst the proposed estimators, k ₁₇ , k ₁₉ ,

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and k 21 performed well in terms of minimum mean square error and smaller PRESS, and k 17 with the value of k 17 = 0.003 seems to be the best.

5.2 CONCLUSION

The following are the conclusions made on this research work:

1. The proposed estimators yielded smaller mean square error and predicted sum of square than the existing estimators.

2. In the simulation study, increasing the correlation between the explanatory variables has negative effect on the MSE and PRESS.

3. Increasing the number of regressors (p) has positive effect on the mean square error and Prediction sum of square.

4. When the sample size increases, the mean square error decreases even when the correlation between the explanatory variables is large.

5. The performance of the estimator depends on the error variance of the distribution.

6. The proposed estimators k 11 , k 12 , k 15 , k 17 , k 19 and k 21 are better estimators than the existing ones. Amongst these proposed estimators, K

 max   w

has a smallest MSE and PRESS respectively, in all the simulation studies.

7. Based on the real life application, we observed that

(i) when the ridge regression was introduced, the regression coefficients b

changed sign due to the presence of multicollinearity.

(ii) the size of the regression coefficients were either reduced or enlarged.

In conclusion, when the eigenvalues of the matrix of the explanatory variables

are skewed, the proposed ridge estimator k 11 performed better than existing ones

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in terms of PRESS and MSE. Whereas when there is an outlier among the eigenvalues of the matrix of the explanatory variables the proposed ridge estimator k 17 performed better than other existing ridge estimators in terms of PRESS and MSE.

5.3 RECOMMENDATIONS

Based on the findings from this work, we make the following recommendations:

 When the eigenvalues of a correlation matrix are skewed, K

 max   w

will be used to estimate the ridge parameter.

 In the presence of outlier among the eigenvalues of the correlation matrix,

  v

K

 max will be used to estimate the ridge parameter.

 Since in both cases, K

exhibits smaller mean square error (MSE) and

prediction sum of square (PRESS), it will be used to estimate the ridge

parameter.

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