I´m trying to deduce the profile log-likelihood function in "Rule-Based and Case-Based Reasoning in Housing Prices" (page 14) by Gayer et al. (2007).

Given is: $$\mathbf{y}=\alpha \mathbf{1} + \mathbf{S}^{-1}\mathbf{\varepsilon}$$,

where $\mathbf{y}$ and $\mathbf{\varepsilon}$ are $n \times1$ vectors, with $\varepsilon \sim N(0, \sigma^2 I)$. $\mathbf{S}$ is a $n \times n$ matrix, depending on a vector $\mathbf{w}$ and $\mathbf{1} $ is a $n \times 1$ vector whose entries are all 1. Additional $\mathbf{S1}=\sqrt{n}e_1$. The unknown parameters are $\alpha, \sigma^2 $ and $\mathbf{w}=(w_1, \dots, w_m)$. They define $$\mathbf{H}=\frac{\mathbf{S}'\mathbf{S}}{\sigma^2}$$ and the log-likelihood is $$l(\theta)= -\frac{n}{2}log(2\pi)+\frac{1}{2} ~log~ det(\mathbf{H})-\frac{1}{2}(\mathbf{y}-\alpha \mathbf{1})'\mathbf{H}(\mathbf{y}-\alpha \mathbf{1}),$$which is just the log of the multivariate normal distribution as $\mathbf{y} \sim N(\alpha1, \mathbf{H}^{-1})$. The MLE of $\alpha$ is $$\hat{\alpha}=(1'\mathbf{H}1)^{-1}1'\mathbf{H}y=\bar{Y}_n$$. Finally they define $\mathbf{Sy}-\sqrt{n}\bar{Y}_ne_1=\mathbf{S}_0\mathbf{y}$ and give the proflie log-likelihood

$$l_P(\mathbf{w})=-\frac{n}{2}[log(2\pi)+1-log(n)]-\frac{n}{2} log(\mathbf y'S_0'S_0y)+\frac{1}{2} ~log~ det(S'S)$$ Especially I'm wondering where the log in $-\frac{n}{2} log(\mathbf y'S_0'S_0y)$ comes from. The log-likelihood I'm getting is $$-\frac{n}{2}log(2\pi)-\frac{n}{2} log(\sigma^2)+\frac{1}{2}log~det(S'S)-\frac{1}{2\sigma^2}(y'S_0'S_0y)$$


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