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.