# Chapter 10 The Aitken Model

• Orthogonal Matrices: A square matrix $$P$$ is said to be orthogonal if and only if $$P'P = I$$
• Spectral Decomposition Theorem: An $$n\times n$$ symmetric matrix $$H$$ may be decomposed as $$H = P\Lambda P' = \sum_{i=1}^n \lambda_i p_ip_i'$$
• $$P$$ is an $$n\times n$$ orthogonal matrix whose columns $$p_1,\ldots, p_n$$ are the orthonormal eigenvectors of $$H$$
• $$\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_n)$$ is a diagonal matrix whose diagonal entries are the eigenvalues of $$H$$
• Symmetric Square Root Matrix: $$H$$ is an $$n\times n$$ symmetric, NND matrix. Then there exists a symmetric, NND matrix $$B$$ such that $$BB = H$$ where $$B = P\Lambda^{1/2}P'$$.
• Aitken Model: $$y = X\beta + \epsilon$$, $$E(\epsilon) = 0$$, $$Var(\epsilon) = \sigma^2 V$$ where $$V$$ is a known positive definite variance matrix.
• $$V^{-1/2}y = V^{-1/2}X\beta + V^{-1/2}\epsilon$$
• With $$z = V^{-1/2}y$$, $$W = V^{-1/2}X$$ and $$\delta = V^{-1/2}\epsilon$$, we have $$z = W\beta + \delta$$, $$E(\delta) = 0$$ and $$Var(\delta) = \sigma^2I$$, which is a Gauss-Markov Model.
• $$\hat z = V^{-1/2}X(X'V^{-1}X)^-X'V^{-1}y$$ so that $$\hat y = X(X'V^{-1}X)^-X'V^{-1}y$$.
• If $$C\beta$$ is estimable, we know the BLUE is the OLS estimator. $$C(W'W)^-W'z = C(X'V^{-1}X)^-X'V^{-1}y \equiv C\hat\beta_{V}$$ is called a Generalized Least Squares (GLS) estimator.
• $$\hat\beta_V = (X'V^{-1}X)^-X'V^{-1}y$$ is a solution to the Aitken Equations: $$X'V^{-1}X\beta = X'V^{-1}y$$. When $$V$$ is diagonal, $$\hat\beta_V$$ is called weighted least squares estimator.
• An unbiased estimator of $$\sigma^2$$ is $\frac{z'(I-P_W)z}{n - rank(W)} = \frac{\|(I- V^{-1/2}X(X'V^{-1}X)^-X'V^{-1/2})V^{-1/2}y\|^2}{n - rank(V^{-1/2}X)} = \frac{\|V^{-1/2}(y - X\hat\beta_V)\|^2}{n - r}$