It is currently puzzling to me, how to derive the likelihood function of the Spatial Autoregressive Model. When considering the following model

$$y=\rho Wy + Z\delta + \epsilon$$ with $\epsilon \sim N(0,\sigma^2I_n)$, I can rewrite the model to get

$$y=(I_n - \rho W)^{-1}Z\delta + (I_n - \rho W)^{-1}\epsilon$$ $$\epsilon = y-\rho Wy - Z\delta$$

and now I am struggling with the likelihood function, which looks as follows

$$L \propto (\sigma^2)^{-n/2}|I_n - \rho W|^{1/2}exp(-\frac{1}{2\sigma^2}\epsilon'\epsilon)$$

but actually I don't know from where the term $|I_n - \rho W|^{1/2}$ comes from. Can someone elaborate on this derivation a little bit?


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