IAI Actuarial Core Principles · Actuarial Statistics · Linear regression models
In a multiple linear regression with p parameters (including the intercept) fitted to n observations, H is the hat matrix X(XᵀX)⁻¹Xᵀ. Which statement is correct?
The hat matrix is symmetric and idempotent, and its trace equals p, the rank of the design matrix. The matrix I − H, which produces residuals, has trace n − p instead.
- AH is idempotent and its trace equals pCorrect
- BH is idempotent and its trace equals n − p
- CH is not symmetric but its trace equals p
- DH has trace n and is the identity matrix
- H is idempotent and its trace equals n
Explanation
H is symmetric and HH = H, so it is idempotent. Its trace equals the rank of X, which is p. The residual maker I − H has trace n − p, which is option B's value.
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