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The model is $Y=XB+Zy+e$ where $B$ and $y$ are unknown parameters and $e\sim \text{N}(0,σ^2 I)$.

Using ordinary least squares and ignoring $Z$ in parameter estimation, I need to find the distribution of SSE obtained from the misspecified model. Can anyone show me the process? I am struggling with how to set up the problem or go about solving it.

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If the specified model is correct, you can estimate the parameters, derive the distribution of the statistics, etc.. because the correctly specified model is unique.

For the misspecified model, it is not unique. So it is impossible to do any kind of statistical analysis.

We know that SSE/$\sigma^2$ follows chi square distribution when the model is specified correctly. It is hard or impossible to derive the distribution of SSE obtained from the misspecified model.

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  • $\begingroup$ Suppose Y = Xβ+Zγ+ε, where β and γ are unknown parameters and ε ∼ N(0, σ2I). (a) Suppose that ordinary least squares (OLS) is employed, but that Z is ignored in the parameter estimation. Find the distribution of SSE obtained from this misspecified model. $\endgroup$ – Zack Oct 19 '18 at 17:31
  • $\begingroup$ That is the problem I've been assigned. I am confused as to how to even go about setting this up. I struggle a lot with theory and concepts, I don't know if it is too abstract for me or what but when it comes to application I generally understand it, but with this I am lost $\endgroup$ – Zack Oct 19 '18 at 17:32
  • $\begingroup$ My answer is: It is impossible. At least I am unable to find it. Let see if others can find it. Or wait to assigner's answer. $\endgroup$ – user158565 Oct 19 '18 at 17:42

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