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Does OLS produce inconsistent and/or bias estimators?

The first example has the beta raised to a power. $$ y = β_0 + (β_1^2 x_1) + (β_2 x_2) + u $$ The second example has (beta + 1)estimator. $$ y = β_0 + ((β_1+1) x_1) + (β_2 x_1) + u $$

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    $\begingroup$ Is this for a class assignment? If so, please add the [self-study] tag & read its wiki. $\endgroup$ – gung - Reinstate Monica Oct 7 '14 at 4:30
  • $\begingroup$ 1. In your second example, should the second $x_1$ be $x_2$ instead? $\ $ 2. What are the models actually being fitted by OLS and how are you computing the parameter estimates for these parameters from that? $\endgroup$ – Glen_b -Reinstate Monica Oct 7 '14 at 9:00
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    $\begingroup$ The examples can be unified, considerably clarifying the nature of the question, by asking "Suppose $\hat\beta$ is the OLS estimate for the model $Y=X\beta+\varepsilon$ with iid, zero-mean $\varepsilon$, and let $f:\mathbb{R}^{p+1}\to\mathbb{R}^{p+1}$. For what $f$ is $\mathbb{E}(f(\hat\beta))=f(\beta)$?" In example (1), $f(\beta_0,\beta_1,\beta_2)=(\beta_0,\sqrt{\beta_1},\beta_2)$ and in example (2), $f(\beta_0,\beta_1,\beta_2)=(\beta_0,\beta_1-1,\beta_2)$. $\endgroup$ – whuber Oct 7 '14 at 15:46

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