Quiz 4 (10-15 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: \(0<\sum_t X_t <n\)

Determine whether the following statements are true or false:

• \(\mathbb{E}\left(\varepsilon_t|X_t\right)=\mathbb{E}\left(\varepsilon_t\right)\).
• \(\mathbb{E}\left(\varepsilon_t\right)=0\).
• \(\mathsf{Var}\left(\varepsilon_t|X_t\right)=\mathsf{Var}\left(\varepsilon_t\right)\).
• \(\mathsf{Var}\left(\varepsilon_t\right)\) is constant for all \(t\).
• Assumptions 3.1 to 3.3 are satisfied.
• Assumptions 3.4 is satisfied.
• Assumption 3.5 is satisfied.

Answers and remarks

• All are true except for the last statement.
• As always, whatever your answers are, you SHOULD be able to explain why the statement is true or false.
• This exercise is related to Exercise 3.12 of the main textbook. In the exercise found in the main textbook, the information is incomplete and you need to make more assumptions to be able to give a complete answer.
• Try answering the questions again but this time, you remove Assumption 4.
• You can also use the data generating process in Slide 60 of the Lectures 8-12 as a starting point and answer the questions again.