Math Problem Statement

Calculate cov(B,SSE) so that the vector of regression coefficients in the multiple linear regression model is equal to B and SSE is the sum of squared errors

Solution

In a multiple linear regression model, we are interested in calculating the covariance between the vector of regression coefficients (B\mathbf{B}) and the sum of squared errors (SSE). Here's how we approach this:

Multiple Linear Regression Model

Given the model: y=XB+ε\mathbf{y} = \mathbf{X} \mathbf{B} + \boldsymbol{\varepsilon} where:

  • y\mathbf{y} is the vector of observed responses,
  • X\mathbf{X} is the matrix of predictors (design matrix),
  • B\mathbf{B} is the vector of regression coefficients,
  • ε\boldsymbol{\varepsilon} is the vector of errors (assumed to be independent with zero mean and constant variance σ2\sigma^2).

The sum of squared errors (SSE) is: SSE=εTε=(yXB^)T(yXB^)\text{SSE} = \boldsymbol{\varepsilon}^T \boldsymbol{\varepsilon} = (\mathbf{y} - \mathbf{X} \hat{\mathbf{B}})^T (\mathbf{y} - \mathbf{X} \hat{\mathbf{B}}) where B^\hat{\mathbf{B}} is the estimated vector of regression coefficients.

Covariance Between B\mathbf{B} and SSE

The covariance between B\mathbf{B} and SSE, denoted cov(B,SSE)\text{cov}(\mathbf{B}, \text{SSE}), is zero. This is due to the fact that:

  1. B\mathbf{B} and SSE are independent: In ordinary least squares (OLS), the estimated coefficients B^\hat{\mathbf{B}} are derived from the predictors and are independent of the error term ε\boldsymbol{\varepsilon}. Since SSE depends entirely on the error term, it is orthogonal to the vector of estimated coefficients B^\hat{\mathbf{B}}.

  2. Least squares properties: In the OLS framework, B^\hat{\mathbf{B}} is the value that minimizes SSE. By construction, the estimated residuals (ε^=yXB^\hat{\boldsymbol{\varepsilon}} = \mathbf{y} - \mathbf{X} \hat{\mathbf{B}}) are uncorrelated with the predictor variables, meaning that B\mathbf{B} and SSE are uncorrelated.

Thus, the covariance is: cov(B,SSE)=0\text{cov}(\mathbf{B}, \text{SSE}) = 0

Summary

In the context of a multiple linear regression model, the covariance between the vector of regression coefficients (B\mathbf{B}) and the sum of squared errors (SSE) is zero because they are independent.


Do you want to explore any details or ask questions? Here are 5 related questions you could consider:

  1. How is the covariance between different variables in regression models calculated?
  2. Why is the sum of squared errors (SSE) minimized in OLS regression?
  3. What are the assumptions behind OLS regression that lead to cov(B,SSE)=0\text{cov}(\mathbf{B}, \text{SSE}) = 0?
  4. Can we calculate the covariance of B\mathbf{B} with other statistics in the model, like the total sum of squares (TSS)?
  5. What is the relationship between the variance of B\mathbf{B} and the error term?

Tip: In OLS regression, the error terms are assumed to have a constant variance (σ2\sigma^2), which is crucial for valid inference on B\mathbf{B}.

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Math Problem Analysis

Mathematical Concepts

Multiple Linear Regression
Covariance
Ordinary Least Squares (OLS)
Sum of Squared Errors (SSE)

Formulas

y = X * B + ε
SSE = (y - X * B_hat)^T * (y - X * B_hat)
cov(B, SSE) = 0

Theorems

OLS Estimation Properties
Uncorrelatedness of Residuals and Predictors

Suitable Grade Level

College / Advanced Undergraduate