Question

Consider a two-variables (x, y) linear regression model
\[ y = \alpha + \beta x + \epsilon \] where x and y are the parameters, and \( \epsilon \) is the error term. The parameters are estimated using the Ordinary Least Squares (OLS) method. Let \( \hat{\beta} \) denote the estimated value of \( \beta \). If \( \hat{\beta} = 0 \), then which one of the following statements is CORRECT?

• A. \( R^2 \) can be any real number in \( (0, 0.5] \)
• B. \( R^2 \) can be any real number in \( (0.5, 1) \)
• C. \( R^2 \) is any positive real number greater than 1
• D. \( R^2 = 0 \)

Hint:

When \( \beta = 0 \), the model is just a horizontal line, and \( R^2 = 0 \), indicating that there is no relationship between \( x \) and \( y \).

Answer 1

The Correct Option is D

Step 1: Understanding the regression model
The linear regression model is given by:
\[ y = \alpha + \beta x + \epsilon \] where \( \alpha \) is the intercept, \( \beta \) is the slope, \( x \) is the independent variable, and \( \epsilon \) is the error term. In this model, \( \beta \) represents the effect of the independent variable \( x \) on the dependent variable \( y \).

Step 2: Analyze what happens when \( \beta = 0 \)
If \( \beta = 0 \), then the model becomes:
\[ y = \alpha + \epsilon \] This means that \( y \) is constant, and it does not depend on \( x \). Hence, there is no relationship between the independent variable \( x \) and the dependent variable \( y \).

Step 3: Implications for \( R^2 \)
The coefficient of determination, \( R^2 \), measures the proportion of the variance in the dependent variable \( y \) that is explained by the independent variable \( x \). If \( \beta = 0 \), there is no variation in \( y \) explained by \( x \), so the relationship is purely random, and the model does not explain any of the variability in \( y \).

Thus, when \( \beta = 0 \), the value of \( R^2 \) will be 0, meaning no explanatory power of the model.