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
To find the ordinary least squares (OLS) estimates of the coefficients (Bs) in this study, you'll need to use the given matrix of sums of squares and cross-products adjusted for means.
Let's denote the coefficients (Bs) as B0, B1, and B2, where B0 is the intercept and B1 and B2 are the coefficients for X1 and X2, respectively.
The formula for the OLS estimates is:
B = (X'X)^(-1) * X'Y
Where:
- B is the vector of coefficient estimates,
- X is the matrix of predictor variables (which includes a column of 1s for the intercept),
- Y is the vector of dependent variable values.
In this case, the matrix X and vector Y can be constructed as follows:
X = [ 1 X1 X2 ],
[ 1 X1 X2 ],
[ 1 X1 X2 ],
...
[ 1 X1 X2 ],
Y = [ C ],
[ C ],
[ C ],
...
[ C ],
Where C is the vector of total costs.
Now, let's calculate the matrix (X'X) and the vector (X'Y).
First, transpose the X matrix:
X' = [ 1 1 1 ... 1 ],
[ X1 X1 X1 ... X1 ],
[ X2 X2 X2 ... X2 ],
To calculate (X'X), multiply X' by X:
(X'X) = X' * X
(X'X) = [ Σ(1) Σ(X1) Σ(X2) ],
[ Σ(X1) Σ(X1^2) Σ(X1X2) ],
[ Σ(X2) Σ(X1X2) Σ(X2^2) ],
Next, calculate (X'Y) by multiplying X' by Y:
(X'Y) = X' * Y
(X'Y) = [ Σ(C) ],
[ Σ(CX1) ],
[ Σ(CX2) ],
Since the matrix (X'X) is a 3x3 matrix and its values are given, we can directly substitute those values into the equation to calculate it.
(X'X) = [ 100 50 40 ],
[ 50 50 70 ],
[ 40 -70 900 ],
Similarly, the vector (X'Y) can be calculated using the given values.
(X'Y) = [ Σ(C) = 100 * C + 100 * C + ... + 100 * C = 100 * Σ(C) ],
[ Σ(CX1) = 50 * (C * X1) + 50 * (C * X1) + ... + 50 * (C * X1) = 50 * Σ(CX1) ],
[ Σ(CX2) = 40 * (C * X2) - 70 * (C * X2) + ... + 900 * (C * X2) = 900 * Σ(CX2) ],
Now, substitute the values into the equation for the OLS estimates:
B = (X'X)^(-1) * X'Y
To find the inverse of (X'X), you'll need to use matrix inversion techniques.
Once you have the inverse, multiply it by (X'Y) to get the estimates of the coefficients (Bs).
Note: It's important to remember that to obtain accurate results, the matrix (X'X) needs to be full-rank (i.e., non-singular). If it is not, additional steps or techniques may be necessary to get valid estimates.