Quiz 3 (10 minutes)

Recall our setup for $\underset{\left(n\times1\right)}{Y}$, $\underset{\left(n\times\left(k+1\right)\right)}{\boldsymbol{\mathrm{X}}}$, and $\underset{\left(\left(k+1\right)\times1\right)}{\beta}$. Let $\beta^o$ be the optimal coefficients of the linear CEF, $\widehat{\beta}$ be the OLS estimator, $e$ be the residual vector, and $\varepsilon$ be the CEF error.

Determine whether the following statements are true or false:

• Under IID conditions, $Y=\mathbf{X}\beta^o+\varepsilon$ is a correctly specified linear regression model.
• $\left(Y-\boldsymbol{\mathrm{X}}\beta^o\right)^{\prime}\left(Y-\boldsymbol{\mathrm{X}}\beta^o\right) \leq \left(Y-\boldsymbol{\mathrm{X}}\widehat{\beta}\right)^{\prime}\left(Y-\boldsymbol{\mathrm{X}}\widehat{\beta}\right)$
• $\sum_{t=1}^n X_{3t} e_t = 0$
• $X_3^\prime e = 0$
• If $\mathbf{X}^\prime e \neq 0$, then Assumptions 3.1 and 3.2 do not hold.
• If Assumptions 3.1 and 3.2 hold, then $\mathbf{X}^\prime \varepsilon = 0$.

Answers

• True
• False
• False
• True
• False
• False

It is your turn to provide an explanation, whether or not you got the correct answer.