Question

Suppose the true equation is given by:

y = 4 + (B₁ * x1) - (6 * x₂) + E
the estimate equation is given by:
y = 3 + (8 * x₁)
and the true relationship between x1 and x2 is given by:
y = -3 + (2 * x₁) + u
What is the true value of B₁ (amount of bias) in the first equation?

Answer 1

The bias of B₁ in the first equation is calculated by subtracting the true coefficient of x₁ (which is 2) from the estimated coefficient (which is 8), resulting in a bias of 6. This means the estimation equation overestimates x₁'s effect by 6 units.

Step-by-step explanation:

Understanding the Bias in an Estimation Equation

We are given a true equation: y = 4 + (B₁ * x1) - (6 * x₂) + E, an estimate equation: y = 3 + (8 * x₁), and the true relationship between x1 and x2: y = -3 + (2 * x₁) + u. We need to determine the true value of B₁. Bias in an estimate arises when the expected value of the predictor does not equal the true value of the parameter it is estimating. To find the bias, we can subtract the true coefficient of x₁ in the true relationship from the coefficient of x₁ in the estimated equation.

The true relationship tells us the coefficient of x₁ is 2; however, in the estimated equation, the coefficient of x₁ is 8. Therefore, the bias that B₁ has in the first equation is 8 - 2, which equals 6.

This means that the estimate of B₁ in the estimation equation overestimates the true effect of x₁ by 6 units.