Suppose that the true model is

$Y_i = \beta_0 + \beta_1X_i + \beta_2X_i^2 + \epsilon_i, 1 \leq i \leq n,$

where $ \{\epsilon_i\}$ are i.i.d normal random variables with mean zero and variance $ \sigma^2$. However, we make a mistake, and fit a simple linear regression model

$Y_i = \beta_0^* + \beta_1^*X_i + \epsilon_i^*$.

(i) Write down the ordinary least squares estimator of the slope parameter, denoted by $ \hat\beta_1^*$.

(ii) Calculate $E( \hat\beta_1^* )$ and the variance of $ \hat\beta_1^*$. When it is unbiased?


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