Question

Suppose the true model is

Y_i = B₀ + B₁ X_i + B₂Z_i + u_i

but you estimate the model:

Y_i = a₀ + a₁X_i+ u_i

If Z is positively correlated with Y and negatively correlated with X, a₁ will be a positively biased estimate of B₁.

Explain.

Answer 1

Z is positively correlated with Y, then B2 is positive.

Z is negatively correlated with X. With increase in Z, the value of X decreases.

In the estimated model, Z is not included. So the positive effect of Z on Y will be shown through the coefficient a1.

Thus, the average value of a1 will be larger than B1.

Therefore, E(a1) > B1

Bias in estimating B1 via a1 = E(estimator) - True value of parameter = E(a1) - B1 > 0.

Hence, a1 is a positively biased estimator of B1.

Hope this helps!