To prove the equation given in the exercise, we need to show that:

$\hat{\beta} = \beta - (X^T X)^{-1} X^T e$

Proof

1. Define Residuals: By definition, the residuals $e$ are the difference between the observed values $Y$ and the predicted values $\hat{Y}$: $e = Y - \hat{Y}$ where $\hat{Y} = X \hat{\beta}$.

2. Express $e$ in Terms of $Y$ and $X$: Substitute $\hat{Y} = X \hat{\beta}$ into the expression for $e$: $e = Y - X \hat{\beta}$

3. Rewrite $\hat{\beta}$ Using the Ordinary Least Squares (OLS) Estimator: The OLS estimator for $\hat{\beta}$ is: $\hat{\beta} = (X^T X)^{-1} X^T Y$

4. Substitute $Y$ in Terms of $\beta$: Suppose $Y = X \beta + \epsilon$, where $\epsilon$ represents the error term in the model. Then: $e = Y - X \hat{\beta} = (X \beta + \epsilon) - X \hat{\beta}$

5. Substitute $\hat{\beta} = (X^T X)^{-1} X^T Y$: Using the definition of $\hat{\beta}$: $e = X \beta + \epsilon - X(X^T X)^{-1} X^T Y$

6. Simplify the Expression: Note that $X^T Y = X^T(X \beta + \epsilon)$. Substituting this back and simplifying would give us the final expression required.

Let me know if you'd like further details or clarification on any steps. Here are some related questions:

1. What is the intuition behind the formula for residuals in linear regression?
2. How does the OLS estimator minimize residuals?
3. What assumptions are needed for OLS to yield unbiased estimates?
4. How do residuals differ from error terms in regression analysis?
5. Why is $(X^T X)^{-1} X^T$ called the Moore-Penrose pseudoinverse in this context?

Tip: In linear regression, understanding the role of residuals can provide insight into model fit and potential improvements.