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| Step | Matrix Operation | Description | |-----------------------|------------------------------------|---------------------------------------------------------------------| | 1. Compute \(X'X\) | \(\mathbf{X'X}\) | Multiply the transpose of \(X\) with \(X\) to form a \(k \times k\) matrix. | | 2. Compute \(X'y\) | \(\mathbf{X'y}\) | Multiply the transpose of \(X\) with \(y\) to form a \(k \times 1\) vector. | | 3. Invert \(X'X\) | \(\mathbf{(X'X)^{-1}}\) | Compute the inverse of \(X'X\), ensuring it has full rank and is non-singular. | | 4. Solve for \(\hat{\beta}\) | \(\mathbf{\hat{\beta} = (X'X)^{-1}X'y}\) | Multiply the inverse of \(X'X\) by \(X'y\) to estimate the coefficients (\(\beta\)). | | Step | Matrix Operation | Description | |-----------------------|------------------------------------|---------------------------------------------------------------------| | 1. Compute \(X'X\) | \(\mathbf{X'X}\) | Multiply the transpose of \(X\) with \(X\) to form a \(k \times k\) matrix. | | 2. Compute \(X'y\) | \(\mathbf{X'y}\) | Multiply the transpose of \(X\) with \(y\) to form a \(k \times 1\) vector. | | 3. Invert \(X'X\) | \(\mathbf{(X'X)^{-1}}\) | Compute the inverse of \(X'X\), ensuring it has full rank and is non-singular. | | 4. Solve for \(\hat{\beta}\) | \(\mathbf{\hat{\beta} = (X'X)^{-1}X'y}\) | Multiply the inverse of \(X'X\) by \(X'y\) to estimate the coefficients (\(\beta\)). | | Step | Matrix Operation | Description | |-----------------------|------------------------------------|---------------------------------------------------------------------| | 1. Compute \(X'X\) | \(\mathbf{X'X}\) | Multiply the transpose of \(X\) with \(X\) to form a \(k \times k\) matrix. | | 2. Compute \(X'y\) | \(\mathbf{X'y}\) | Multiply the transpose of \(X\) with \(y\) to form a \(k \times 1\) vector. | | 3. Invert \(X'X\) | \(\mathbf{(X'X)^{-1}}\) | Compute the inverse of \(X'X\), ensuring it has full rank and is non-singular. | | 4. Solve for \(\hat{\beta}\) | \(\mathbf{\hat{\beta} = (X'X)^{-1}X'y}\) | Multiply the inverse of \(X'X\) by \(X'y\) to estimate the coefficients (\(\beta\)). |
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