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Question:

Having in mind the main research question, select the best linear regression model, using the least squares method with backwards stepwise variable elimination, at α = 1%. Describe step by step your analysis providing in each step the relevant results.

Give the interpretation of the regression coefficients of the selected model. Challenge the feasibility of the sign and magnitude of the coefficients in your model and if necessary try and propose an alternative.

For the selected model, calculate the coefficient of determination and give its interpretation in terms of the given research question.

Using the natural logarithm transformation of all but the dummy variables, repeat the same exploration as the one described in 1.1, but this time at α = 4%. Once more, describe step by step your analysis providing in each step the relevant results.

Give the interpretation of the regression coefficients of the selected model. Challenge the feasibility of the sign and magnitude of the coefficients in your model and if necessary try and propose an alternative.

Answer:

Subject 1:

  • The process starts with description of co linearity among  independent variables. The independent variables and correlation between them can be depicted here:
Table 1: Correlation between the independent variables.

 

WAGES

KCAPITAL

Labor

D1

D2

WAGES

1

 

 

 

 

KCAPITAL

0.905554

1

 

 

 

Labor

0.564246

0.250203

1

 

 

D1

0.025988

0.028247

-0.02952

1

 

D2

0.028428

-0.02534

0.073159

0.06072

1

The highlighted correlation is greater then 0.8. Therefore, the variable has to be removed from the dataset and it can be said that the rest of the variables are not dangerously correlated. Regression analysis on the dependent variable and the rest three of the independent variable is given below:

 
Table 2: Regression table.

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.82712

 

 

 

 

 

 

 

R Square

0.684128

 

 

 

 

 

 

 

Adjusted R Square

0.681428

 

 

 

 

 

 

 

Standard Error

17644.38

 

 

 

 

 

 

 

Observations

473

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

4

3.16E+11

7.89E+10

253.403

1.2E-115

 

 

 

Residual

468

1.46E+11

3.11E+08

 

 

 

 

 

Total

472

4.61E+11

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 99.0%

Upper 99.0%

Intercept

-518.847

1524.07

-0.34044

0.733682

-3513.71

2476.02

-4460.66

3422.97

X Variable 1

0.74864

0.026659

28.08157

2E-102

0.696253

0.801027

0.679689

0.817591

X Variable 2

147.2564

21.67842

6.792765

3.35E-11

104.6573

189.8555

91.1879

203.325

X Variable 3

842.2054

1694.082

0.497145

0.61932

-2486.74

4171.155

-3539.33

5223.738

X Variable 4

7993.062

1896.699

4.214195

3.01E-05

4265.96

11720.16

3087.485

12898.64

It can be said from the table that the regression fit is good fit but the co-efficient table shows that variable 3  has a p-value higher then 0.01. Therefore, the variabl that is D1 has to deleted from the data table. Regression test with the same dependent variable and with those same independent variables other than D1 is given below:

Table 3: Regression table.

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.827019333

 

 

 

 

 

 

 

R Square

0.683960977

 

 

 

 

 

 

 

Adjusted R Square

0.681939405

 

 

 

 

 

 

 

Standard Error

17630.21504

 

 

 

 

 

 

 

Observations

473

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

3

3.15E+11

1.05E+11

338.3313

7E-117

 

 

 

Residual

469

1.46E+11

3.11E+08

 

 

 

 

 

Total

472

4.61E+11

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 99.0%

Upper 99.0%

Intercept

13.95975517

1082.726

0.012893

0.989719

-2113.63

2141.554

-2786.35

2814.271

X Variable 1

0.749167279

0.026617

28.14623

8.4E-103

0.696864

0.801471

0.680326

0.818008

X Variable 2

146.7921723

21.64091

6.783087

3.56E-11

104.267

189.3173

90.82115

202.7632

X Variable 3

8054.211397

1891.187

4.258812

2.48E-05

4337.963

11770.46

3162.934

12945.49

It can be said from the table that the regression fit is quite good here and the p-values of the co-efficient falls under 0.01. The regression analysis can be interpreted as the ultimate model here with all the variables falling in line. Therefore, the required regression equation is :

Y= (0.75)*KCAPITAL + (146.79)*Labor + (8054.21)*D2.  

  • Co-efficient of KCAPITAL is the average increase in the dependent variable with the per unit increase in KCAPITA with Labor keft fixed. Co-efficient of Labor is the average increase in the dependent variable with the per unit increase in Labor keeping KCAPITAL fixed. D1 is categorical variable. Therefore, coefficient of D1 is the average change in y with every category of D1. The coefficient of KCAPITAl can be challenged here since it can be said that capital has a much larger effect in business. Again, the sign can be challenged here regarding Labor since a large number of Labor can have a negative impact.  The coefficient can also be lowered regarding Labor. The model can be challenged in the lights of these arguments and a new model can be proposed like:

Y= (5)*KCAPITAL - (90)*Labor + (8054.21)*D2.

  • Co-efficient of determination is defined as the proportion of variation in the dependent variables that is being interpreted from independent variables.  It can be interpreted here that 68% of variation in industrial production can be explained through Labor, KCAPITAL and D1.

Subject 2:

2.1  The process starts with description of co linearity among  independent variables. The independent variables and correlation between them can be depicted here:

Table 4: Correlation table

 

WAGES

KCAPITAL

Labor

D1

D2

WAGES

1

 

 

 

 

KCAPITAL

0.844151

1

 

 

 

Labor

0.960251

0.751036

1

 

 

D1

0.027968

-0.03644

0.004177

1

 

D2

0.12812

-0.07081

0.155761

0.06072

1

The highlighted correlation is greater then 0.8. Therefore, the variable has to be removed from the dataset and it can be said that the rest of the variables are not dangerously correlated. Regression analysis on the dependent variable and the rest three of the independent variable is given below:

Table 5: Regression table.

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.971118892

 

 

 

 

 

 

 

R Square

0.943071903

 

 

 

 

 

 

 

Adjusted R Square

0.942585338

 

 

 

 

 

 

 

Standard Error

0.131050435

 

 

 

 

 

 

 

Observations

473

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

4

133.1499

33.28748

1938.224

1.2E-289

 

 

 

Residual

468

8.037533

0.017174

 

 

 

 

 

Total

472

141.1875

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 96.0%

Upper 96.0%

Intercept

0.728946993

0.045026

16.18956

3.92E-47

0.640469

0.817425

0.636217

0.821677

X Variable 1

0.745283949

0.017807

41.85416

2.6E-160

0.710293

0.780275

0.708611

0.781957

X Variable 2

0.302633323

0.020118

15.04322

5.29E-42

0.263101

0.342165

0.261201

0.344065

X Variable 3

0.000311127

0.012578

0.024736

0.980276

-0.0244

0.025027

-0.02559

0.026215

X Variable 4

0.291151384

0.014821

19.64427

4.23E-63

0.262027

0.320276

0.260627

0.321675

It can be said from the table that the regression fit is good fit but the co-efficient table shows that variable 3  has a p-value higher then 0.01. Therefore, the variabl that is D1 has to deleted from the data table. Regression test with the same dependent variable and with those same independent variables other than D1 is given below:

Table 6: Regression table.

Regression Statistics

 

 

 

 

 

 

 

Multiple R

0.971119

 

 

 

 

 

 

 

R Square

0.943072

 

 

 

 

 

 

 

Adjusted R Square

0.942708

 

 

 

 

 

 

 

Standard Error

0.130911

 

 

 

 

 

 

 

Observations

473

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

 

 

Regression

3

133.1499

44.3833

2589.817

2.2E-291

 

 

 

Residual

469

8.037544

0.017138

 

 

 

 

 

Total

472

141.1875

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 96.0%

Upper 96.0%

Intercept

0.729187

0.043918

16.60335

4.95E-49

0.642887

0.815488

0.638739

0.819636

X Variable 1

0.745264

0.01777

41.93906

8.4E-161

0.710345

0.780183

0.708667

0.781862

X Variable 2

0.302649

0.020087

15.06726

4E-42

0.263178

0.342119

0.261281

0.344016

X Variable 3

0.291168

0.01479

19.68681

2.48E-63

0.262105

0.320231

0.260708

0.321628

It can be said from the table that the regression fit is quite good here and the p-values of the co-efficient falls under 0.01. The regression analysis can be interpreted as the ultimate model here with all the variables falling in line. Therefore, required regression equation is :

Y = 0.73 + 0.74*KCAPITAL + 0.30*Labor + 0.29*D2.

2.2. Co-efficient of KCAPITAL is the average increase in the dependent variable with the per unit increase in KCAPITA with Labor keft fixed. Co-efficient of Labor is the average increase in the dependent variable with the per unit increase in Labor keeping KCAPITAL fixed. D1 is categorical variable. Therefore, coefficient of D1 is the average change in y with every category of D1. The coefficient of KCAPITAl can be challenged here since it can be said that capital has a much larger effect in business. Again, the sign can be challenged here regarding Labor since a small number of Labor can have a negative impact.  The coefficient can also be increased regarding Labor. The model can be challenged in the lights of these arguments and a new model can be proposed like:

Y= (5)*KCAPITAL - (90)*Labor + (0.29)*D1.

Subject 3.

It can be checked from the residual plot and the normality plot that the necessary assumptions of residual homoscadasticity and independence are not being met here regarding the log linear model but normality condition is being met. The normality and  homoscadasticity is not being met in the linear model but the residuals are independent here.. The residual plot and normality plot is attached below:

Residual plot for the log linear model.

Normality plot for log linear model.

Residual plot for linear model.

Normality plot for linear model.

  1. The independent variable should be choosen here.
 
References:

De Oliveira, A.B., Fischmeister, S., Diwan, A., Hauswirth, M. and Sweeney, P.F., 2017, March. Perphecy: Performance Regression Test Selection Made Simple but Effective. In Software Testing, Verification and Validation (ICST), 2017 IEEE International Conference on (pp. 103-113). IEEE.

Saha, R.K., Zhang, L., Khurshid, S. and Perry, D.E., 2015, May. An information retrieval approach for regression test prioritization based on program changes. In Software Engineering (ICSE), 2015 IEEE/ACM 37th IEEE International Conference on (Vol. 1, pp. 268-279). IEEE.

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