# Fitting a quadratic model using simple linear regression

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?