Advanced Statistical Analysis of Economic Factors

Part (A) Simple Linear Regression Model

1) GDP and import expenditure, these two variables are positively associated according to the scatter plot diagram (Figure 1). CPI and expenditure, these two variables are also positively associated according to the scatter plot diagram (Figure 2).

2) The estimated regression 1 equation (Between GDP and import expenditure) (figure 3)

Import expenditure (Y) = -195427.95 + 133.36*GDP + 4.36

T-test

Null hypothesis: the slope of GDP is equal to zero

Alternative hypothesis: the slope of GDP is not equal to zero

T statistic (t) = 133.36/4.36 = 30.587 (Figure 3)

Degree of freedom = n-2 = 29

T critical value at 29 degree of freedom and 95% significance level is 1.699 which means t statistic > t critical value so the null hypothesis cannot be rejected. In conclusion, the slope of GDP is equal to zero.

3) Coefficient of determination

R2 = 5386872477779.96/ 5554186302314.39= 0.969876087 (Figure 4)

F test of equation 1

Null hypothesis: The fit of the intercept only model and the following regression 1model are equal.

Alternative hypothesis: The fit of the intercept-only model is significantly reduced compared to following model

P value for the F-test of overall significance test is less than 0.05 (95% confidence interval) (figure 4) so null hypothesis is rejected and it means that the model explaining better than intercept only model.

4) The estimated regression 2 equation (Between CPI and import expenditure) (Figure 5)

Import expenditure (Y) = - 595264 (α2) + 9545.72*CPI (β2) + 658.81

T-test

Null hypothesis: the slope of CPI is equal to zero

Alternative hypothesis: the slope of CPI is not equal to zero

T statistic (t) = 9545.72 /658.81 = 14.49 (Figure 5)

Degree of freedom = n-2 = 29

T critical value at 29 degree of freedom and 95% significance level is 1.699 which means t statistic > t critical value so the null hypothesis cannot be rejected. In conclusion, the slope of CPI is equal to zero.

5) F-test of equation 2

Null hypothesis: The fit of the intercept only model and regression 2 model are equal.

Alternative hypothesis: The fit of the intercept-only model is significantly reduced compared following model

According to the summary output regression table, P value for the F-test of overall significance test is less than 0.05 (95% confidence interval) (Figure 6) so null hypothesis is rejected and it means that the model explaining better than intercept-only model.

6) Prediction of Import expenditure using GDP

Import expenditure = -195427.95 + 133.36*GDP

Difference in actual and prediction is significant. The standard deviation between actual and forecasted import expenditure is 78141.9 (Figure 7).

7) The sign and size of a coefficient in an equation affect its graph. The future prediction of Import expenditure, the dependent variable is affected by the slope coefficient into both cases.

The estimated regression equation (ln 1) (Between Ln_GDP and Ln_Import expenditure) (Figure 8)

Ln_Import expenditure (Y) = 2.002+ 1.293 *Ln_GDP + 0.0249

From the equation it can be depicted that 100 percent increase in Ln_GDP is associated with 129 percent increase in natural logarithm of import expenditure.

The estimated regression equation (ln 2) (Between Ln_ CPI and Ln_Import expenditure) (Figure 10)

Ln_Import expenditure (Y) = 3.578+ 1.987 * Ln_CPI+ 0.0725

From the equation it can be depicted that 100 % increase in natural logarithm of CPI is associated with 199 % unit increase in natural logarithm of import expenditure.

8) T-test of regression equation (ln 1)

Null hypothesis: the slope of ln_GDP is equal to zero

Alternative hypothesis: the slope of Ln_GDP is not equal to zero

T statistic (t) = 1.293/ 0.0249 = 51.92 (Figure 8)

Degree of freedom = n-2 = 29

T critical value at 29 degree of freedom and 95% significance level is 1.699 which means t statistic > t critical value so the null hypothesis cannot be rejected. In conclusion, the slope of Ln_GDP is equal to zero.

T-test of regression equation (ln 2)

Null hypothesis: the slope of Ln_CPI is equal to zero

Alternative hypothesis: the slope of Ln_CPI is not equal to zero

T statistic (t) = 1.987/ 0.0725 = 27.40 (Figure 10)

Degree of freedom = n-2 = 29

T critical value at 29 degree of freedom and 95% significance level is 1.699 which means t statistic > t critical value so the null hypothesis cannot be rejected. In conclusion, the slope of Ln_CPI is equal to zero.

F-test of regression equation (ln 1 and 2)

Null hypothesis: The fit of the intercept only model and regression 2 model are equal.

Alternative hypothesis: The fit of the intercept-only model is significantly reduced compared following model

According to the summary output regression table, P value for the F-test of overall significance test is less than 0.05 (95% confidence interval) (Figure 9 and 11) so null hypothesis is rejected and it means that the model explaining better than intercept-only model.

Part (B) Multiple Regression Analysis

9) The estimated regression equation 3 (Between Ln_CPI, Ln_GDP and natural logarithm of import expenditure) (Figure 12)

Ln_Import expenditure = 1.409 + 1.85 *GDP- 0.873 *CPI +0.0249

It can be depicted from equation 3 that 100% increase in Ln_GDP is associated with 185% increase in Ln_Import expenditure and 100% increase in Ln_CPI is associated with 87.3 % decrease in Ln_Import expenditure.

10) Import expenditure = -195427.95 + 133.36*GDP + 4.36 (Equation 1)

Ln_Import expenditure = 1.409 + 1.85 *GDP - 0.873 *CPI +0.0249 (Equation 3)

In two equations, the coefficient of GDP in different as log-log model give used for understanding the amount of comparative change of the dependent variable explained by the independent variable. This model’s coefficient is described the percentage rises in x lead to percentage variations in y while the normal linear model is described the normal increase in x lead to continuous variations in y.

11) Coefficient of determination of the multiple regression model

R2 = 17.284/ 17.424= 0.992 (Figure 13)

Import expenditure (Y) = - 595264 + 9545.72*CPI + 658.81 (Equation 2)

Ln_Import expenditure (Y) = 1.409 + 1.85 *Ln_GDP - 0.873 *Ln_CPI +0.0249 (Equation 3)

In two equations, the coefficient of CPI in different as log-log model give used for understanding the amount of comparative change of the dependent variable explained by the independent variable. Especially in this case coefficient CPI in equation 2 is in positive form and coefficient CPI in equation 2 is in negative form because of the slope of the intercept in cases.

12) T-test of ln_GDP of multiple regression

Null hypothesis: the slope of ln_GDP of multiple regression is equal to zero

Alternative hypothesis: the slope of Ln_GDP of multiple regression is not equal to zero

T statistic (t) is 10.115 (Figure 12)

Degree of freedom = n-2 = 29

T critical value at 29 degree of freedom and 95% significance level is 1.699 which means t statistic > t critical value so the null hypothesis cannot be rejected. In conclusion, the slope of Ln_GDP is equal to zero.

T-test of ln_GDP of multiple regression

Null hypothesis: the slope of ln_GDP of multiple regression is equal to zero

Alternative hypothesis: the slope of Ln_GDP of multiple regression is not equal to zero

T statistic (t) is -3.0665 (Figure 12)

Degree of freedom = 29

T critical value at 29 degree of freedom and 95% significance level is 1.699 which means t statistic > t critical value so the null hypothesis cannot be rejected. In conclusion, the slope of Ln_GDP is equal to zero.

13) F-test of equation 3

Null hypothesis: The fit of the intercept only model and the following regression 1model are equal.

Alternative hypothesis: The fit of the intercept-only model is significantly reduced compared to following model

F statistics = 1737.19 (Figure 13)

Degree of freedom = 29

The critical value of F statistics at 29 degree of freedom and 95% confidence level is 1.860. P value for the F-test of overall significance test is less than 0.05 (95% confidence interval) so null hypothesis is rejected and alternative hypothesis is accepted.

14) Coefficient of determination of the multiple regression model (R2) is mainly used to judge the best model. According to the value of R2, the best model is natural logarithm simple regression model of GDP. According to the value the dependent variable is alone able to explain the 99.5% of the dependent variable. In multiple regression, there is huge scope of multicollinearity that’s why it best to use linear regression to understand the impact of the independent variable on dependent variable differently.

Appendix

Part C: Multicollinearity

1) Multicollinearity usually happens when there is very high correlation or association between two or more predictor variables of data set. In another way, if one predictor variable of a dataset can be used to predict the other, this creates redundant result, skewing the investigated result in a regression model. Examples of multicollinear predictor variables (also called correlated predictor) can be age and sales price of a car, an individuals’ height and weight or years of experience and annual income.

2) Pearson correlation coefficient is a technique for investigating the relationship between two quantitative, continuous variables. Pearson's correlation coefficient (r) is a degree of calculating the strength of the association among the two variables. When Pearson’s correlation value is greater than 0.5 and close to 1, it depicts a strong positive correlation between two variable. According to the result, it was found a strong, positive and significant correlation between GDP and CPI as the correlation coefficient value is reported as .98 which is near about 1 (Figure 1). It means that two variables are highly correlated to each other. Multicollinearity happens when the regression model consist of multiple factors which are correlated not just the independent variable, but also to another dependent variable. It can be concluded that there is a chance of multicollinearity as GDP and CPI both independent multiple factors are highly correlated to each other.

3) Consequences of multicollinearity

With particular linear relationships amongst the descriptive variables, the condition of precise collinearity or precise multicollinearity, occurs and the least squares (OLS) estimator is not well-defined.

For independent variables that are extremely correlated to one another (but not impeccably related), the Ordinary Least Squares (OLS) estimators have huge variances and related covariance, creating detailed estimation problematic.

The OLS estimator and its standard errors or inaccuracies can be delicate with small changes in the data. Alternatively, the consequences will not be robust.

Even extreme multicollinearity (as long as it is not faultless) presents in data, it does not interrupt all assumptions of OLS. OLS evaluations are unbiased and best Linear Unbiased Estimators (BLUE).

4) The estimated regression 4 equation (Between Ln_GDP and Ln_CPI) (figure 2)

Ln_CPI (Y) = -0.6784 + 0.638* Ln_GDP + 0.0143

The multiple regression which ran in part B suggest that multicollinearity is a problem as a strong, positive and significant correlation between GDP and CPI was found.

Coefficient of determination = R2 = 0.993 (Figure 3)

From the equation 4 it can be depicted that 100 percent increase in Ln_GDP is associated with 63 percent increase in natural logarithm of CPI. According to the value the dependent variable is alone able to explain the 99.3% of the dependent variable. All these proves there is a problem of multicollinearity.

F test of equation 4

Null hypothesis: The fit of the intercept only model and the following regression 4 model are equal.

Alternative hypothesis: The fit of the intercept-only model is significantly reduced compared to following model

P value for the F-test of overall significance test is less than 0.05 (95% confidence interval) (Figure 3) so null hypothesis is rejected and it means that the model explaining better than intercept only model.

Part D: Heteroscedasticity

5) Heteroscedasticity

Heteroscedasticity discusses to the error of variance, or dependency of scatter, within a least one of independent variable within a specific sample. These deviations can be used to compute the error amongst data sets, such as predictable consequences and actual consequences, as it arranges for a portion for the deviance of data sets as of the mean value. There are numerous reasons when the alterations of error term may be variable, some of which are:

Heteroscedasticity can likewise emerges accordingly of the presence of outliers. The incorporation or avoidance of those extreme observations, particularly when the sample size is very small, can generously modify the aftereffects of regression analysis.

Heteroscedasticity emerges from abusing the assumptions of Classical linear regression model (CLRM), that the regression model is not effectively determined.

Presence of skewness in the distribution of at least one regressers variable incorporated into the model is another wellspring of heteroscedasticity.

Improper data transformation, incorporating incorrect functional form (linear, liner-log, log-linear, log-log model) is also another source of heteroscedasticity

6) Graphical approach to inspect heteroscedasticity

A scatter plot is performed between predicted outcomes of the dependent variable in the context of squared residual outcomes. The scatter plot is attached in the appendix (Figure 4). From the scatter plot, it is hard to inspect heteroscedasticity that’s why white test was performed below.

7) White’s test of Heteroscedasticity

Heteroscedasticity tests imply the two following hypotheses:

H0 (null hypothesis): residual is homoscedastic.

Ha (alternative hypothesis): residual is heteroscedastic

White’s test is done by taken squared residual value as dependent variable in the context of two independent variable predicted value and squared predicted value. Multiple regression was performed and P value is noted. According to the summary output, p value is 0.226 which is greater than 0.05, so null hypothesis is accepted (Figure 5). It can be concluded that residual is homoscedastic.

8) Consequences of heteroscedasticity

The OLS estimators and regression predictors in light of them stays unprejudiced and reliable.

The properties of Best Linear Unbiased Estimators are violated by the OLS estimators since they are no longer productive, so the relapse forecasts will be wasteful as well.

Because of the irregularity of the covariance matrix of the assessed regression coefficients, the trial for testing hypothesis, (t-test, F-test) are no longer substantial.

9) In the part C of the analysis, a strong, positive and significant correlation between GDP and CPI as the correlation coefficient value is reported as .98 which is near about 1 is found and the regression Of GDP on CPI is proved that , multicollinearity is presented in the data set. In the part D of the analysis, white’s test of inspecting heteroscedasticity is performed. According to the result, the data is homoscedastic. All this proves that the multiple regression which was performed in part B did not violate the rules of Best Linear Unbiased Estimators.

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