Question

Assume that the true data generating process is Yi = € + Bxi + YZ + €

where y and x is observed data_ and the shocks are random normal variables with mean 0 and variance 1. You estimate the following regression: Yi = € + Bx; + Ui
The variable Z affects y but it is not observed (or you don't have the datal you forgot to include it in your regression). Assume that the correlation between x and z is p. The bias of an estimate is the difference between its expected value and its true value. Find the bias for the OLS estimate B in terms of the variance of Z, the variance of X, the correlation between x and z and the true coefficients B,Y:

Answer 1

The bias of the OLS estimate B can be calculated by comparing its expected value with its true value, taking into account the variance of Z, the variance of X, the correlation between x and Z, and the true coefficients B and Y.

Step-by-step explanation:

The bias of the OLS estimate B can be found by comparing its expected value with its true value. The true value of B is given in the data generating process as B, while the expected value is obtained through the OLS estimation.

Let's denote the true value of B as B_true and the expected value as E(B). The bias of B can be calculated as the difference between the expected value and the true value:

Bias = E(B) - B_true

To calculate the bias, we need to find the expected value of B. Since the variable Z affects y but is not included in the regression, it is treated as an omitted variable. The bias of B can be expressed in terms of the variance of Z, the variance of X, the correlation between x and Z, and the true coefficients B and Y:

Bias = p * (B_true + Y_true) * (Var(Z) / Var(X))