Question

Consider the following regression model \[ y_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \epsilon_t, t = 1, 2, ...., 100, \] where $\alpha_0, \alpha_1, \alpha_2$ are unknown parameters and $\epsilon_t$'s are independent and identically distributed random variables each having $N(\mu, 1)$ distribution with $\mu \in \mathbb{R$ unknown. Then which one of the following statements is/are true?}

• A. There exists an unbiased estimator of $\alpha_1$.
• B. There exists an unbiased estimator of $\alpha_2$.
• C. There exists an unbiased estimator of $\alpha_0$.
• D. There exists an unbiased estimator of $\mu$.

Hint:

- In linear regression models, the OLS estimators of the parameters are unbiased if the error term has zero mean and constant variance.
- The parameter $\mu$ represents the mean of the error term, which is assumed to be zero, and thus cannot be estimated directly.

Answer 1

The Correct Option is A

1) Analyzing statement (A):
For $\alpha_1$, which is the coefficient of $t$, there exists an unbiased estimator because $\epsilon_t$ has zero mean, and the ordinary least squares (OLS) estimator is unbiased in this linear model.
2) Analyzing statement (B):
Similarly, for $\alpha_2$, the coefficient of $t^2$, there exists an unbiased estimator using the same reasoning that the OLS estimator is unbiased in the linear regression model.
3) Analyzing statement (C):
For $\alpha_0$, the intercept term, there also exists an unbiased estimator using OLS.
4) Analyzing statement (D):
However, $\mu$ is not an identifiable parameter from the model because it represents the mean of the error term $\epsilon_t$, which is assumed to have zero mean. Therefore, there is no unbiased estimator for $\mu$.
Thus, the correct answer is (A) and (B).