Use the Gaussian model with second moments given in the previous exercise to compute a pseudo log likelihood l 0 ( , ) l 0 ( , ) for the parameter ( , ) ( , ) Show that the pseudo log likelihood differs from the correct log likelihood l ( , ) n log 1 2 ( y ) 2 2 l ( , ) n log 1 2 ( y ) 2 2 by terms...

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Use the Gaussian model with second moments given in the previous exercise to compute a pseudo-log likelihood

Question:

Use the Gaussian model with second moments given in the previous exercise to compute a pseudo-log likelihood $l_{0} (β, σ)$ for the parameter $(β, σ)$ . Show that the pseudo log-likelihood differs from the correct log likelihood

$l (β, σ) = - n \log σ - \frac{1}{2} \sum (y - β)^{2} / σ^{2}$

by terms that are relatively small for large $n$ , so that ${\hat{β}}_{0} = \hat{β}$ and ${\hat{σ}}_{0} - \hat{σ} = O_{p} (n^{- 1})$ . What precisely does 'relatively small for large $n$ ' imply about the magnitude of the difference $l_{0} (β, σ) - l (β, σ)$ ?