Question

To determine the relationship between y and x1, Rohit estimated two different OLS models. In the first model, Rohit regressed y on x1 and x2 as given below: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u \quad (1) \] In the second model, Rohit regressed y only on x1 as given below: \[ y = \delta_0 + \delta_1 x_1 + \nu \quad (2) \] The estimated coefficients of x1 in the above two models are exactly the same. From this observation we can state conclusively that \begin{itemize} \item (i) \( \text{Cov}(x_1, y) = 0 \) \item (ii) \( \hat{\beta_2} = 0 \) \item (iii) \( \text{Cov}(x_2, x_1) = 0 \) \end{itemize} where \( \hat{\beta_2} \) is the estimated coefficient of x2 in the equation (1).

• A. Only (i) is true
• B. Only (ii) is true
• C. Either (ii) or (iii) or both are true
• D. Neither (ii) nor (iii) is true

Hint:

When the coefficients of a variable remain unchanged in different model specifications, it typically indicates that the variable is uncorrelated with the other variables in the model.

Answer 1

The Correct Option is C

We are given two models where \( y \) is regressed on \( x_1 \) and \( x_2 \) in the first model and only on \( x_1 \) in the second model. The key observation here is that the estimated coefficients of \( x_1 \) are the same in both models. This suggests that the presence of \( x_2 \) does not affect the relationship between \( x_1 \) and \( y \), meaning that \( x_1 \) and \( x_2 \) must be uncorrelated. Step 1: Analyzing the relationship between \( x_1 \) and \( x_2 \)
Since the coefficients of \( x_1 \) are the same in both models, it suggests that \( x_1 \) and \( x_2 \) are not correlated. This means that \( \text{Cov}(x_2, x_1) = 0 \), which implies that the independent variable \( x_2 \) does not provide any additional explanatory power in the first model. Thus, \( \hat{\beta_2} = 0 \). Step 2: Verifying the statements
- (i) \( \text{Cov}(x_1, y) = 0 \): This statement is incorrect. Since the estimated coefficient of \( x_1 \) is non-zero in both models, we cannot conclude that the covariance between \( x_1 \) and \( y \) is zero.
- (ii) \( \hat{\beta_2} = 0 \): This is true. The fact that the coefficient of \( x_1 \) is the same in both models implies that \( x_2 \) does not affect the relationship between \( x_1 \) and \( y \), meaning \( \hat{\beta_2} = 0 \).
- (iii) \( \text{Cov}(x_2, x_1) = 0 \): This is true. Since the coefficients for \( x_1 \) remain the same in both models, \( x_1 \) and \( x_2 \) must be uncorrelated.
Step 3: Conclusion
Therefore, the correct answer is (A), which states that only statement (i) is true, as it directly reflects the condition implied by the estimated coefficients in both models.
Final Answer: \boxed{(A) \text{Only (i) is true}}