IAI Actuarial Core Principles · Actuarial Statistics · Linear regression models
In the multiple linear regression model Y = Xβ + ε, where X is an n × p design matrix of full column rank (including a column of ones) and the errors are uncorrelated with mean 0 and variance σ², which expression gives the least squares estimator of β?
The least squares estimator is (XᵀX)⁻¹Xᵀy. It solves the normal equations XᵀXβ = Xᵀy obtained by minimising the residual sum of squares. The expression X(XᵀX)⁻¹Xᵀy is instead the vector of fitted values.
- A(XᵀX)⁻¹XᵀyCorrect
- BX(XᵀX)⁻¹Xᵀy
- C(XXᵀ)⁻¹Xᵀy
- D(XᵀX)Xᵀy
- Xᵀ(XᵀX)⁻¹y
Explanation
Minimising the sum of squared residuals (y−Xβ)ᵀ(y−Xβ) gives the normal equations XᵀXβ = Xᵀy. With XᵀX invertible, β̂ = (XᵀX)⁻¹Xᵀy. Option B is the fitted values vector, not the coefficients.
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