Chapter 11, Problem 39E

(a)

To determine

To find: A table with the estimated standard errors, t- statistics, regression coefficients, and P- values of the different regression model.

Answer to Problem 39E

Solution: The required table is shown below:

Model	Explanatory variable	regression coefficients	standard errors	t- statistics	P values
Model.	Intercept	$6.5269$	$0.6464$	$10.096$	$<0.001$
	GINI	$-0.0074$	$0.01689$	$-0.4285$	$0.675$
Model II	Intercept	$-3.8077$	$1.1211$	$-3.396$	$0.0011$
	GINI	$0.039116$	$0.01184$	$3.3034$	$0.0015$
	LIFE	$0.11788$	$0.01187$	$9.927$	$<0.05$
Model-III	Intercept	$-2.9671$	$1.07834$	$-2.7516$	$0.0076$
	GINI	$0.03662$	$0.01109$	$3.3016$	$0.0015$
	LIFE	$0.09530$	$0.01304$	$7.3074$	$<0.05$
	DEMOCRACY	$0.2077$	$0.0630$	$3.2956$	$0.0015$
Model-IV	Intercept	$-2.3139$	$1.01313$	$-2.2839$	$0.0255$
	GINI	$0.04470$	$0.010529$	$4.246$	$<0.001$
	LIFE	$0.07821$	$0.01287$	$6.0775$	$<0.05$
	DEMOCRACY	$0.05261$	$0.07390$	$0.71187$	$0.0949$
	CORRUPT	$0.19414$	$0.0571$	$3.4000$	$0.0011$

Explanation of Solution

Calculation: On the basis of the GINI, LIFE, CORRUPT, and DEMOCRACY explanatory variables, one can estimate the multiple regression coefficient, standard error, t- statistics, and corresponding P- values by using excel.

The regression equation is written as:

$LSI={\beta }_0+{\beta }_{1}GINI+{\beta }_{2}LIFE+{\beta }_{3}DEMOCRACY+{\beta }_{4}CORRUPT+\epsilon$

The assumption of the error term is normally distributed with average 0 and a constant standard deviation, and it can be written as $\epsilon ~N(0,\sigma )$.

The multiple regression analysis Model I can be obtained by using Minitab and following the steps given below:

Step 1: Open the Minitab worksheet.

Step 2: Go to Stat > Regression > Regression

Step 3: Select “LSI” in Response and select “GINI” in Predictors.

Step 4: Click “OK.”

The multiple regression analysis Model II can be obtained by using Minitab and following the steps given below:

Step 1: Open the Minitab worksheet.

Step 2: Go to Stat > Regression > Regression

Step 3: Select “LSI” in Response and select “GINI and LIFE” in Predictors.

Step 4: Click “OK.”

The multiple regression analysis Model III can be obtained by using Minitab and following the steps given below:

Step 1: Open the Minitab worksheet.

Step 2: Go to Stat > Regression > Regression

Step 3: Select “LSI” in Response and select “GINI, LIFE, and DEMOCRACY” in Predictors.

Step 4: Click “OK.”

The multiple regression analysis Model IV can be obtained by using Minitab and following the steps given below:

Step 1: Open the Minitab worksheet.

Step 2: Go to Stat > Regression > Regression

Step 3: Select “LSI” in Response and select “GINI, LIFE, DEMOCRACY, and CORRUPT” in Predictors.

Step 4: Click “OK.”

Interpretation: Hence, the estimated regression coefficients, t-statistics, standard errors, and P-values vary from the different regression model and its significant level changes according to the model.

(b)

To determine

To test: The change of regression coefficients and P-values in the different models.

Answer to Problem 39E

Solution: Some t-statistics are significant and some are not significant depends on the model, for example, t- statistics for the GINI coefficient grows from $t=-0.42(P=0.675)$ to $t=4.25(P<0.005)$. The DEMOCRACY t-statistics drop from the third model to the fourth model as $t=3.295(P<0.005)$ to $t=0.712(P=0.095)$.

Explanation of Solution

Calculation: On the basis of the GINI, LIFE, CORRUPT, and DEMOCRACY explanatory variables, one can estimate the multiple regression coefficients, standard error, t-statistics, and corresponding P-values by using excel. The regression equation is written as:

$LSI={\beta }_0+{\beta }_{1}GINI+{\beta }_{2}LIFE+{\beta }_{3}DEMOCRACY+{\beta }_{4}CORRUPT+\epsilon$

The assumption of the error term is normally distributed with average 0 and a constant standard deviation, and it can be written as $\epsilon ~N(0,\sigma )$.

Conclusion: Hence, some t-statistics are significant and some are not significant depends on the model, for example, t-statistics for the GINI coefficient grows from $t=-0.42(P=0.675)$ to $t=4.25(P<0.005)$. The DEMOCRACY t- statistics drops from the third model to the fourth model as $t=3.295(P<0.005)$ to $t=0.712(P=0.095)$.

(c)

To determine

To test: The choice of best model and summarize the results on the basis of four model.

Answer to Problem 39E

Solution: A best model to predict the LSI is GINI, LIFE, and CORRUPT as explanatory variables due to the high value of $R^2=0.70$ and the estimated regression equation is $\widehat{y}=-2.354+0.04619GINI+0.22034LIFE+0.07934CORRUPT$.

Explanation of Solution

Calculation: The multiple regression analysis can be obtained by using Minitab, follow the steps below:

Step 1: Open the Minitab worksheet.

Step 2: Go to Stat > Regression > Regression

Step 3: Select “LSI” in Response and select “GINI, LIFE, and CORRUPT” in Predictors.

Step 4: Click “OK.”

Conclusion: Therefore, best model to predict the LSI is GINI, LIFE, and CORRUPT as explanatory variables due to the high value of $R^2=0.70$ and the estimated regression equation is $\widehat{y}=-2.354+0.04619GINI+0.22034LIFE+0.07934CORRUPT$, where $F=52.92$ and $P<0.005$ with a high $R^2=0.70$ value. So, the model is fit to predict LSI on the basis of the GINI, LIFE, and CORRUPT as explanatory variables. The assumption of the error term is normally distributed with average 0 and a constant standard deviation, and it can be written as $\epsilon ~N(0,\sigma )$.