To solve question 2.3 based on the provided image, let's first summarize what it's asking for:

Question 2.3 Explanation:

• Context: The question states that if the data suffers from Perfect Multicollinearity, the Ordinary Least Square (OLS) method will not be able to estimate the regression coefficients.
• Request: The task is to provide reasoning on why OLS fails to estimate regression coefficients when there is perfect multicollinearity. Essentially, we need to explain why the estimation problem arises in this situation.

Perfect Multicollinearity and OLS:

Perfect Multicollinearity occurs when one predictor variable in a multiple regression model can be perfectly predicted by one or more of the other predictor variables. In other words, there is an exact linear relationship between the independent variables.

When perfect multicollinearity is present:

1. Indeterminacy of Regression Coefficients: The design matrix $X$ becomes singular, meaning it does not have full rank. This leads to the matrix $X^TX$ being non-invertible (its determinant is zero). Since OLS requires inverting $X^TX$ to compute the coefficients $\hat{\beta} = (X^TX)^{-1}X^Ty$, the presence of perfect multicollinearity makes it impossible to find a unique solution for the coefficients $\hat{\beta}$. Hence, OLS fails.

2. Interpretation Difficulty: When perfect multicollinearity exists, the regression model can't distinguish the individual contribution of the correlated variables, which leads to instability in the estimated coefficients. Any small change in the data can cause large variations in the coefficient estimates.

3. Practical Example: If you have two variables $X_1$ and $X_2$ such that $X_2 = cX_1$ (where $c$ is a constant), the regression model can't decide how much weight to assign to $X_1$ and $X_2$ because they provide redundant information.

Summary:

• OLS fails in the presence of perfect multicollinearity because the necessary matrix operations to compute the coefficients $\hat{\beta}$ are not mathematically possible (due to the singularity of $X^TX$).
• The model lacks enough independent information to separate the effects of the collinear variables, leading to indeterminate or infinitely many possible solutions for the coefficients.

---

Would you like further details on how this affects the interpretation of regression results or how to address multicollinearity in practice?

Related Questions:

1. What are some techniques to detect multicollinearity in a dataset?
2. How can multicollinearity be resolved or mitigated in regression analysis?
3. Why does the singularity of $X^TX$ specifically cause problems for OLS?
4. What are some real-world examples where multicollinearity might occur?
5. How does partial multicollinearity differ from perfect multicollinearity?

Tip:

If perfect multicollinearity exists, consider using regularization techniques like Ridge or Lasso regression, which can handle collinear variables by adding penalties to the regression coefficients.