Question

A study considered a sample of 50 observations used to predict SALES. Included in the analysis were 9 predictors variables, ( Independent Variables).

1. Based on the correlation matrix shown below, is there any concern about Multicollinearity?

Correlations

	X 1	X 2	X 3	X 4	X 5	X 6	X 7	X 8	X 9
X 2	0.804
X 3	0.625	0.443
X 4	0.032	0.032	0.231
X 5	0.159	0.214	0.177	-0.194
X 6	0.319	0.373	0.308	0.054	0.293
X 7	-0.016	0.030	0.079	0.168	-0.309	0.067
X 8	-0.026	0.103	0.015	0.151	-0.311	0.059	0.912
X 9	0.169	-0.027	-0.104	0.017	-0.248	0.114	0.174	0.223
SALES	0.764	0.630	0.756	0.149	0.171	0.426	0.145	0.141	-0.068

• If YES, list all pairs of variables involved:

• What limit did you use?

• What action must be taken to eliminate multicollinearity? Be very specific.

• Which variable appears to have the weakest relationship with SALES?

2. The following is the initial Regression Printout. Variable X2, and X8 were not included in the regression.

• Why?

• Use the F test and a 0.05 level of significance to determine whether the Regression model is significant.

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value
Regression	7	145157	20736.7	19.62
Error	42	44385	1056.8
Total	49	189543

• What is the value of the F Critical Point?

• How many degrees of freedom did you use?

• Compute the R-Square:

• Show your work and compute the Adjusted R-Square:

Term	Coef	SE Coef	T-Value
Constant	-15.8	22.1	-0.71
X 1	11.93	2.34
X 3	5.75	1.77	3.24
X 4	0.000023	0.000096	0.24
X 5	-0.59	2.84
X 6	4.08	2.07
X 7	0.0891	0.0530	1.68
X 9	-0.0302	0.0150	-2.02

• Based on the Table above, what is the regression equation?

Sales =

• Three T-Values are missing. Compute them. The T-values are:

For X1 =                               X5 =                                X6 =      

Use the t test and a 0.05 level of significance to determine the significance of each independent variable.

• What is the value of the T Critical Point?
• How many degrees of freedom did you use?
• Would you eliminate any variable from the model?
• Which variable?

Answer 1

Based on the correlation matrix shown below, is there any concern about Multicollinearity?

Yes , there any concern about Multicollinearity

This pairs of variables involved : Pair ( X1, X2) and Pair ( X7, X8 )

For Pair ( X7 , X8 ) the correlation is 0.912 , which implies they are highly correlated with each other.

Also Pair ( X1 ,X2 ) has correlation = 0.804 , so these pair is also highly correlated

-What limit did you use?
Limit we have used is " > | 0.70 | " for any pair

- What action must be taken to eliminate multicollinearity?  Be very specific.  

The only think can be done is to remove any of one variables of each pair which have lowest relationship with SALES .

Like in pair ( X7 , X8 ) X8 has lowest relationship with SALES ( which is 0.141 ) , so it is better to remove it .

-Which variable appears to have the weakest relationship with SALES?
X9 variable have the weakest relationship with SALES ( which is -0.068 )

-The following is the initial Regression Printout. Variable X2, and X8 were not included in the regression. Why?  

Pair ( X1, X2) and Pair ( X7, X8 ) are highly correlated , among pair ( X1, X2) X2 have the weakest relationship with SALES and among pair ( X7, X8 ) X8 have the weakest relationship with SALES .So remove X2 and X8

Source	DF	Adj SS	Adj MS	F-Value
Regression	7	145157	20736.7	19.62
Error	42	44385	1056.8
Total	49	189543

Use the F test and a 0.05 level of significance to determine whether the Regression model is significant.

To test

H0 : bi = 0 , i = 1,3,4,5,6,7,9    { Model is not significant }

H1 : bi 0    for atleast one i    { Model is significant }

Test Statistics F:

F = MS_R / MS_RES = 20736.7 / 1056.8 = 19.62216

Thus calculated F- value is 19.62

What is the value of the F Critical Point

It is given by

is F-distributed with df1= 7 and df2=42 degree of freedom and =0.05,

It can be computed from statistical book or more accurately from any software like R,Excel

From R

> qf(1-0.05,df1=7,df2=42)
[1] 2.23707

Thus value of the F Critical Point is 2.23707

We reject null hypothesis if calculated F-value is less than

How many degrees of freedom did you use?

is F-distributed with df1= 7 and df2=42 degree of freedom and =0.05,

Conclusion -

Since F- value = 19.62 > 2.23707 i.e F- value >

So we reject null hypothesis at 5% of level of significance at hence conclude that model is significant.

-Compute the R-Square:

Formula :

R-Square = 1 - SS_RES / TSS

                = 1 - 44385 / 189543

            = 0.7658315

Thus R-Square = 0.7658315

-Show your work and compute the Adjusted R-Square:

Formula :

Adjusted R-Square: = 1 - SS_RES/ df(Error) / TSS / df(Total)

        or = 1 - SS_RES/ (n-k ) / TSS / (n-1)         

Here k = 7        ( number of regressor )

also   n-k = 42    ( given)     

               = 1 - ( 44385 / 42 ) / ( 189543 / 49 )

               = 0.7268034

Thus Adjusted R-Square = 0.7268034

Based on the Table above, what is the regression equation?

Sales = -15.8+11.93*X1+5.75*X3+0.000023*X4-0.59*X5+4.08*X6+0.0891*X7-0.0302*X8

Three T-Values are missing. Compute them. The T-values are:

t-VALUE = coef / SE coef

For X1 =  11.93/2.34 = 5.098291              

For X5 =   -0.59/2.84 =-0.2077465

For X6 =     4.08/2.07 = 1.971014

Term	Coef	SE Coef	T-Value
Constant	-15.8	22.1	-0.71
X 1	11.93	2.34	5.098291
X 3	5.75	1.77	3.24
X 4	0.000023	0.000096	0.24
X 5	-0.59	2.84	-0.2077465
X 6	4.08	2.07	1.971014
X 7	0.0891	0.0530	1.68
X 9	-0.0302	0.0150	-2.02

-Use the t test and a 0.05 level of significance to determine the significance of each independent variable.

-What is the value of the T Critical Point?

value of the T Critical Point is given by

Here n-(k+1) = 42

is t-distribute with df=42 degree of freedom and =0.05,

It can be computed from statistical book or more accurately from any software like R,Excel

From R

> qt(1-0.05/2,42)
[1] 2.018082

Thus value of the T Critical Point is 2.018082

-How many degrees of freedom did you use?

42 degree of freedom      ( defined in above part )

-Would you eliminate any variable from the model?

Here hypothesis to test are

H0 : bj = 0        { j th variable is not significant }

vs

H1 : bi 0    { j th variable contributes significantly to model }

Test Statistics is t-value

where t-value = = coef / SE coef

t-value are obtained for each variables .

We reject null hypothesis if calculated absolute t-value is greater than T Critical Point 2.018082

i.e is | t-value | > 2.018082

We can see variable X1, X3 , X9 have t-value greater than 2.01

Hence we conclude that variable X1 , X3 , X9 contributes significantly to our model .

Other remaining variables can be removed from our model .

-Which variable?

We remove variable X3 , X4 , X5 , X6 and X7