Question

Suppose the true model is

Y_{i} = B_{0} + B_{1} X_{i} +
B_{2}Z_{i} + u_{i}

but you estimate the model:

Y_{i} = a_{0} + a_{1}X_{i}+
u_{i}

If Z is positively correlated with Y and negatively correlated
with X, a_{1} will be a positively biased estimate of
B_{1}.

**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!

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