Searches related to Searches related to In a study of 100 firms,the total cost© was assumed to be dependent on the rate of output (X1) and the rate of absenteeism (X2).The means were: C=6 X1=3 X2=4.the matrix showing sums of squares and cross products adjusted for means is C 100 50 40 X1 50 50 70 X2 40 -70 900 Find the ols estimates of the Bs
Answer
To find the OLS estimates of the coefficients (Bs), we need to solve the equation:
(X'X)B = X'Y
Where:
- X is the matrix of independent variables (X1, X2)
- Y is the vector of the dependent variable (C)
- B is the vector of coefficients to be estimated
Given the matrix of sums of squares and cross products adjusted for means:
C 100 50 40
X1 50 50 70
X2 40 -70 900
And the means:
C = 6
X1 = 3
X2 = 4
We can set up the equation:
(X'X)B = X'Y
Plugging in the values:
C 100 50 40 B1 6
X1 50 50 70 B2 = 3
X2 40 -70 900 B3 4
To solve for B, we need to calculate the inverse of X'X:
(X'X)^-1 =
(50,000 - 3,500) (3,450 - 500) (-7,000 + 2,800)
(3,450 - 500) (250 + 900) (-500 - 180)
(-7,000 + 2,800) (-500 - 180) (900 + 36,100)
=
\
/ 46,500 2,950 -4,200 \
| 2,950 1,150 320 |
\ -4,200 320 37,000 /
Next, we need to calculate X'Y:
X'Y =
(100)(6) + (50)(3) + (40)(4)
(50)(6) + (50)(3) + (-70)(4)
(40)(6) + (70)(3) + (900)(4)
=
\ 890 /
| -40 |
\ 3,710 /
Now, we can multiply the inverse of X'X by X'Y to get the estimates for B:
B = (X'X)^-1 * X'Y
=
\
/ 46,500 2,950 -4,200 \
| 2,950 1,150 320 |
\ -4,200 320 37,000 /
\
/ 890 /
| -40 |
\ 3,710 /
=
\
/ (46,500 * 890) + (2,950 * -40) + (-4,200 * 3,710) \
| (2,950 * 890) + (1,150 * -40) + (320 * 3,710) |
\ (-4,200 * 890) + (320 * -40) + (37,000 * 3,710) /
Simplifying the equation will give us the OLS estimates for the coefficients (Bs).
To find the OLS estimates of the Bs in this regression model, we can use the formula for calculating the OLS estimates. The formula is:
b = (X^T * X)^(-1) * X^T * y
Where:
- b is the vector of OLS estimates of the coefficients (or Bs).
- X is the matrix of predictors, including a column of ones as the intercept.
- y is the vector of the dependent variable.
In this case, the matrix X contains the columns for X1 and X2, and the vector y contains the column for C. The intercept term can be included by adding a column of ones to the matrix X.
Let's calculate the OLS estimates of the Bs step by step using the given information.
Step 1: Set up the matrix X and the vector y:
X = [1 3 4; 1 3 4; 1 3 4; ...; 1 3 4] (a 100x3 matrix with the first column all ones, the second column all threes, and the third column all fours)
y = [6; 6; 6; ...; 6] (a 100x1 vector with all elements equal to six)
Step 2: Calculate X^T * X
X^T * X = [100 300 400; 300 900 1200; 400 1200 1600]
Step 3: Calculate the inverse of X^T * X
(X^T * X)^(-1) = [0.2 -0.1 0.05; -0.1 0.3 -0.15; 0.05 -0.15 0.1]
Step 4: Calculate X^T * y
X^T * y = [600; 1800; 2400]
Step 5: Calculate the OLS estimates of the Bs
b = (X^T * X)^(-1) * X^T * y
b = [0.2 -0.1 0.05; -0.1 0.3 -0.15; 0.05 -0.15 0.1] * [600; 1800; 2400]
The result will be a vector of OLS estimates of the Bs.