Discussion
If the regression estimator of \(\bar{Y}\) is \(\bar{y} + b(\bar{X} - \bar{x})\) where \(b = \dfrac{S_{xy}}{S_{x}^{2}}\), then an exact expression for bias of regression estimator is :
Answer: D
The exact bias is \(-Cov(b, \bar{x})\).
\(\bar{y}\) is unbiased for \(\bar{Y}\), so the bias of the regression estimator is \(E[b(\bar{X} - \bar{x})]\). Using \(E(b\bar{x}) = Cov(b, \bar{x}) + E(b)E(\bar{x})\) and \(E(\bar{x}) = \bar{X}\):
\[ \begin{aligned} E[b(\bar{X} - \bar{x})] &= \bar{X}E(b) - E(b\bar{x}) \\ &= \bar{X}E(b) - Cov(b, \bar{x}) - \bar{X}E(b) \\ &= -Cov(b, \bar{x}) \end{aligned} \]
The bias comes from the randomness of \(b\) being correlated with \(\bar{x}\); with a fixed \(b\) the estimator would be unbiased. Options involving \(\bar{y}\) are wrong because \(\bar{y}\) enters linearly and contributes no bias, and option C has the sign reversed.
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