Discuss the below:
Q: Consider the standard simple regression model y = â0 + â1x + u under Gauss-Markov assumptions.The usual OLS estimators bâ0 and bâ1 are unbiased for their respective population parameters. Let eâ1 bethe estimator of â1 obtained by assuming the intercept is zero.
a) Find E(eâ1) in terms of xi, â0 and â1. Verify that eâ1 is unbiased for â1 when the populationintercept (â0) is zero. Are there other cases when eâ1 is unbiased?
b) Find the variance of eâ1 (Hint: The variance does not depend on â0).
c) Show that the V ar(eâ1) <= V AR(bâ1). Hint: For any sample of data,Px2i>=P(xi . x)2, withstrict inequality when unless x.
d) Comment on the trade-off between bias and variance when choosing between eâ1 and bâ1.