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Quiz 5 (15-30 minutes)

Suppose that you observe IID draws $\{\left(Y_t,X_t\right)\}_{t=1}^n$ from a data generating process satisfying the following assumptions:

• Assumption 1: $Y_t=\beta_0^o+\beta_1^o X_t +\varepsilon_t$ for some $\beta_0^o$, $\beta_1^o$, and some unobservable $\varepsilon_t$.
• Assumption 2: $X_t\in\{0,1\}$ is a binary/dummy random variable.
  
• Assumption 3: $\varepsilon_t \in \{-1,2\}$, where $\Pr\left(\varepsilon_t=-1\right)=2/3$ and $\Pr\left(\varepsilon_t=2\right)=1/3$.
• Assumption 4: $\varepsilon_t$ and $X_t$ are independent.
• Assumption 5: $0<\sum_t X_t <n$

Let $\widehat{\beta}_0$ and $\widehat{\beta}_1$ be the least squares estimator of $\beta_0^o$ and $\beta_1^o$. Suppose we condition on $\mathbf{X}$. Prove or disprove the following statements:

• The least squares estimators are unbiased estimators of $\beta_0^o$ and $\beta_1^o$.
• $2\widehat{\beta}_0+\widehat{\beta}_1$ is unbiased for $2\beta_0^o+\beta_1^o$.
• The least squares estimators are BLUE.