OLS Estimate of Work Conflict and Family Conflict

where, x, y and ε are n×1 vectors, β₀ and β are scalars.

To prove :  

This has been proved using OLSE which fix a data in such a way where Residual Sum of Squares (RSS) is minimum.

RSS=

In order to find the value of β₀  and  the value of  has to be minimum, for which first order derivation of RSS with respect to β₀  and  has to be performed. The derivation is as follows:

This leads to,

,      and      

Taking derivative of this further results into

From these equations, two equations occurs such as:

                                                                          ……………(1)

                                                                    ……………..(2)

Here, equation (1) and (2) are normal equations. Solving to these two equation will give β₀  and  values.

Equation (1) can be calculated further as shown below:

This calculated value of β₀ has been further used to calculate the value of  in equation (2).

Thus the value of  has been achieved as desired. 

In order to find the relationship of correlation and  the following equation has been used. 

Where, is standard deviation for y, and, is standard deviation for x.

From the equation  it can be said that the value of  would be equal to , if x_i and y_i are such as =.

Write down an expression for cos θ, where θ is an angle between x and y, using the law of cosines. Explain each step clearly. Compare your result with rxy. Interpret your findings

 The cosine rule  which is used to show angle between two n-dimensional vectors and is as shown below,

The measure of distance from the origin can be presented using an n-dimensional vector which is     .

Following the similar way  is a measure of the magnitude by which a random variable deviates from its mean.     ……Consider this as (3) 

In case of discrete random variables,

Furthermore,  from previous section it is clear that

 which further can be written as based on (3).

This expression is similar to that of . While comparing the values of each  vector of is equal to in ; vector in is equal to  in .  can be seen as  , and  can be seen as .

.1 Using SAS/IML language compute the covariance matrix S by definition as in (3.38) of the textbook.

The covariance matrix computed using SA/IML is as shown in below table.

Cov matrix S
	WC	FC	IC	WS	FS	LS
WC	0.2859367	0.2015267	0.1586767	-0.263473	-0.189393	-0.262987
FC	0.2015267	0.3448667	0.1879667	-0.242573	-0.313293	-0.293947
IC	0.1586767	0.1879667	0.2437767	-0.174813	-0.201833	-0.236767
WS	-0.263473	-0.242573	-0.174813	0.2999867	0.2027067	0.2724933
FS	-0.189393	-0.313293	-0.201833	0.2027067	0.3197467	0.2641333
LS	-0.262987	-0.293947	-0.236767	0.2724933	0.2641333	0.3472667

Inverse Cov matrix P as S-1
	WC	FC	IC	WS	FS	LS
WC	1.3643869	-0.242122	-0.279302	0.4512048	-0.193688	0.1412677
FC	-0.242122	1.611781	-0.183561	0.049706	0.7425796	0.1332994
IC	-0.279302	-0.183561	1.2238563	-0.188901	0.0751249	0.2860141
WS	0.4512048	0.049706	-0.188901	1.5702381	-0.150868	-0.645472
FS	-0.193688	0.7425796	0.0751249	-0.150868	1.6462092	-0.424015
LS	0.1412677	0.1332994	0.2860141	-0.645472	-0.424015	1.7314349

As the covariance matrix is invertible, it is said to be defined.

The table shown below shows that the rank of matrix S is 18.

tol	rankSVD
4.627E-16	18

Calculating the Cosine of the Angle between X and Y

In order to find the relationship between work conflict and family conflict regression analysis has been run while keeping the variable interrole fixed. Here, work conflict is a dependent variable, on which the influence of independent variable family conflict has to be analyzed.

Number of Observations Read	6
Number of Observations Used	6

The total number of observations are 6 in the model. The table shown below shows that the independent variables work conflict explains 41% variation in the value of dependent variable work conflict, which is depicted by the value obtained of R² i.e 0.41 in the regression model. 

Root MSE	0.45849	R-Square	0.4119
Dependent Mean	0.11167	Adj R-Sq	0.2648
Coeff Var	410.59033

In addition to the model, the ANOVA table given below shows that the variable family conflict does not have any significant impact on work conflict, when the variable interole is kept constant. This inference of no significant impact has been measured using the significance value p=0.1695 which is greater than α 0.10.

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > |t|
Intercept	1	0.07855	0.18822	0.42	0.6979
FC	1	0.58436	0.34916	1.67	0.1695
IC	1	0	0	.	.
RESTRICT	-1	0.24418	0.38542	0.63	0.6035*

Further, the next regression model has been run to determine the relationship between work satisfaction and family satisfaction, keeping the variable life satisfaction constant.  Here, work satisfaction is dependent variable and family satisfaction is an independent variable.

Here, the R² value=0.42 shows that the independent variable family satisfaction explains 42% variation in work satisfaction.

Root MSE	0.46298	R-Square	0.4284
Dependent Mean	0.16667	Adj R-Sq	0.2855
Coeff Var	277.78664

 In addition to the model, the table given below shows that family satisfaction has no significant impact on work satisfaction as the p-value=0.1584 is greater than the standard acceptance level α=0.10.

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > |t|
Intercept	1	0.08003	0.19552	0.41	0.7033
FS	1	0.63396	0.36616	1.73	0.1584
LS	1	1.11022E-16	0	Infty	<.0001
RESTRICT	-1	0.52522	0.37193	1.41	0.1826*

The covariance matrix of S1 is as given below.

S1
	WC	FC	IC
WC	0.2859367	0.2015267	0.1586767
FC	0.2015267	0.3448667	0.1879667
IC	0.1586767	0.1879667	0.2437767

The next table shows that the covariance matrix S1 has a rank 9.

tol	rankSVD
2.297E-16	9

The covariance matrix of S2 is as given below.

S2
	WS	FS	LS
WS	0.2999867	0.2027067	0.2724933
FS	0.2027067	0.3197467	0.2641333
LS	0.2724933	0.2641333	0.3472667

The next table shows that the covariance matrix S2 has also rank 9. 

tol	rankSVD
2.313E-16	9

Comparison between inverse covariance of both S1 and S2: 

Inverse Covariance of S1
	WC	FC	IC
WC	6.6095291	-2.617501	-2.283956
FC	-2.617501	6.0382586	-2.952108
IC	-2.283956	-2.952108	7.86502

Inverse Covariance of S2
	WS	FS	LS
WS	11.628985	0.4455668	-9.463937
FS	0.4455668	8.4313862	-6.762596
LS	-9.463937	-6.762596	15.449472

While  comparing the inverse covariance matrix S1 and S2 it has been analyzed that both gives unique matrix as the values for each element is different.

2.5 Compute their respective precision matrices using Gauss-Jordan Elimination. For S1 and S2 compute the following: α = − p12 √p11√p22 . (2) Show your work.

For covariance matrix S1 the calculated  is 0.641. This value has been calculated taking the value of p_12, p_11, and p₂₂ from covariance matrix S₁ using the following formula in MS-Excel

(0.2015267)/(SQRT(0.2859367)*SQRT(0.3448667))

Similarly, the value of α from S₂ has been calculated based on the p_12, p_11, and p₂₂  .i.e -0.654507, which has been calculated from (0.2027067)/(SQRT(0.2999867)*SQRT(0.3197467)) in Excel.

If the value of α depicting standard significance level 0.10 gets increased for first regression model from 0.10 to 0641, the variable family conflict will show significant impact on work conflict as the p-value is lesser than this new calculated standard significance level 0.41465. The value of β coefficient shows that 1 unit increase in work conflict brings 0.58 unit significant effect on work conflict as the p-value 0.16 is less than α=0.4146. On the other hand, in other regression model,  the new value of α gets decreased to value -0.65from 0.10, this shows that family satisfaction does not show any significant effect on work satisfaction as the p-value is greater than computed α-value -0.045447.

While adding the third variable in first regression model resulted into decreasing the value of adjusted R² which dropped down to 0.1181 from .2648. On the other hand, adding the independent variable life satisfaction in second regression model resulted into increasing the value of adjusted R² which increased to 0.52 from 0.41. However, none of the model showed any significant impact of any independent variables on dependent variables. 

Root MSE	0.50216	R-Square	0.4709
Dependent Mean	0.11167	Adj R-Sq	0.1181
Coeff Var	449.69241

Root MSE	0.37858	R-Square	0.7133
Dependent Mean	0.16667	Adj R-Sq	0.5222
Coeff Var	227.14581

Let         z1 = W S + F S + LS

                    z2 = W S + F S

                     z3 = W S

3.1 Find   and .

Variable	Mean	Std Dev
Z1 Z2 Z3 Z	0.4566667 0.3033333 0.1666667 0.9266667	1.5638627 1.0124953 0.5477104 3.0741026

Here, the mean and standard deviation of Z has been calculated along with calculating the mean value of z1, z2, and z3.  Looking at the standard deviation results it has been inferred that the members of Z group differs a lot from the mean value 0.927 of group as the standard deviation 3.074 is quite high than its mean.

Compare the variances of z1, z2 and z3 with the respective variances of the relevant original variables (ex., z3 with WS). Do the results meet your expectations? Explain your conclusions.

Variable	Variance
Z1 Z2 Z3 WS FS LS	2.4456667 1.0251467 0.2999867 0.2999867 0.3197467 0.3472667

While comparing the variance of Z1 with Family satisfaction it has been determined that the spread of Z1 2.44 is comparatively much higher than family satisfaction that is 0.32 approx. Similarly, the variance of Z2= W S + F S is also much higher than variance of life satisfaction that is 0.347. However, there is no difference among the variance of Z3 and work satisfaction that is 0.2999 for both. 

Var[X + Y ] = Var[X] + Var[Y ]

Answer:   The time when X and Y both are independent variable, and uncorrelated with each other, this condition occurs.