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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.

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

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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My Assignment Help (2020) Linear Regression Model Selection With Least Squares Method [Online]. Available from: https://myassignmenthelp.com/free-samples/mba60-advanced-quantitative-methods-for-managers1
[Accessed 16 June 2024].

My Assignment Help. 'Linear Regression Model Selection With Least Squares Method' (My Assignment Help, 2020) <https://myassignmenthelp.com/free-samples/mba60-advanced-quantitative-methods-for-managers1> accessed 16 June 2024.

My Assignment Help. Linear Regression Model Selection With Least Squares Method [Internet]. My Assignment Help. 2020 [cited 16 June 2024]. Available from: https://myassignmenthelp.com/free-samples/mba60-advanced-quantitative-methods-for-managers1.

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