Test on Introduction to Simple Regression Analysis

Question 1: When finding the adjoint of a 2x2 matrix using the matrix inversion method, elements along the main diagonal interchange positions.

A. Ano

B. Ne

Explanation: The study material explicitly states that for a 2x2 matrix, to find the Adjoint matrix, 'Elements along the main diagonal interchange positions while elements off the main diagonal interchange positions (Positives become negatives and vice versa).'. This confirms that main diagonal elements interchange positions.

Question 2: According to the provided study materials, which of the following statements correctly applies Cramer's Rule to determine the ordinary least squares (OLS) estimator for the slope coefficient, $\hat{\beta}$, in a simple linear regression model?

A. $\hat{\beta} = \frac{\left| \begin{array}{cc} n & \sum Y_i \\ \sum X_i & \sum X_i Y_i \end{array} \right|}{\left| \begin{array}{cc} n & \sum X_i \\ \sum X_i & \sum X_i^2 \end{array} \right|}$

B. $\hat{\beta} = \frac{\sum X_i^2 \sum Y_i - \sum X_i \sum X_i Y_i}{n \sum X_i^2 - (\sum X_i)^2}$

C. $\hat{\beta} = \frac{\left| \begin{array}{cc} \sum Y_i & \sum X_i \\ \sum X_i Y_i & \sum X_i^2 \end{array} \right|}{\left| \begin{array}{cc} n & \sum X_i \\ \sum X_i & \sum X_i^2 \end{array} \right|}$

D. $\hat{\beta} = \frac{n \sum X_i Y_i - \sum X_i \sum Y_i}{n \sum X_i^2 - (\sum X_i)^2}$

Explanation: Cramer's rule states that if $AX = b$, then $x_i = \frac{|A_i|}{|A|}$. For the slope coefficient $\hat{\beta}$, the study materials show its calculation using Cramer's Rule as $\hat{\beta} = \frac{\left| \begin{array}{cc} n & \sum Y_i \\ \sum X_i & \sum X_i Y_i \end{array} \right|}{\left| \begin{array}{cc} n & \sum X_i \\ \sum X_i & \sum X_i^2 \end{array} \right|}$. This is the formula where the second column of the coefficient matrix $A$ is replaced by the constant vector $b$. Evaluating the determinants, this expression simplifies to $\hat{\beta} = \frac{n \sum X_i Y_i - \sum X_i \sum Y_i}{n \sum X_i^2 - (\sum X_i)^2}$. Therefore, options 0 and 3 are correct representations of $\hat{\beta}$ using Cramer's Rule.

Question 3: An estimator is considered more efficient than another if it possesses a larger variance.

A. Ano

B. Ne

Explanation: An estimator is considered more efficient than another if it has a smaller variance, not a larger one. This is explicitly stated in the study materials under the minimum variance property and the Gauss-Markov Theorem.

Question 4: What does the acronym BLUE stand for in the context of OLS estimators, according to the Gauss-Markov Theorem?

A. Best Linear Unbiased Estimator

B. Beta-Lagrangian Universal Equation

C. Bias-Limited Underestimated Error

D. Basic Least-Squares Unconditional Estimator

Explanation: The study materials state that the Gauss-Markov Theorem declares OLS estimators to be BLUE, which is an acronym for Best Linear Unbiased Estimator.

Question 5: The Ordinary Least Squares (OLS) method estimates the sample regression function by minimizing the simple sum of the residuals.

A. Ano

B. Ne

Explanation: The study materials state that the OLS method minimizes the sum of the squares of the residuals (Residual Sum of Squares, RSS), not the simple sum of the residuals. Minimizing the simple sum of residuals would result in zero, as positive and negative residuals would balance out.