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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.