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My book states that for a standard regression through origin model, the Beta 2 hat, that is, the OLS Estimator is unbiased. Another estimator, say, Beta 2 tilda which is Summation of Yi/Summation of Xi which is also Mean of Y/Mean of X cannot be unbiased. For Beta 2 tilda, I tried substituing the PRF into the summation of Yi. The result gave me an unbiased estimator. I understand that the sum of residuals for No Intercept model may not be zero, but since we're only using the PRF for proving unbiasedness, this feature never comes into play. We're simply using the assumption of CLRM which states that the E(ui)=0. Application of this would give us an unbiased estimator. How is Beta 2 tilda 'never unbiased'? Please give me some hints as to where I'm going wrong.

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