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Estimation and Testing for Linear Models Questions

Estimation and Testing for Linear Models Questions.

I’m studying for my Statistics class and need an explanation.

Consider the simple linear regression model
Yi = β0 + β1xi + i
for i = 1, . . . , n, where 1, . . . , n
iid∼ N (0, σ2
). For this model we write the maximized likelihood as
f1(y1, . . . , yn) = max
β0,β1,σ2
L(y1, . . . , yn, β0, β1, σ2
)
with the likelihood L that we characterized in our lecture.
For the alternative model
Yi = β0 + i
for i = 1, . . . , n, where 1, . . . , n
iid∼ N (0, σ2
), we write the corresponding maximized likelihood as
f0(y1, . . . , yn) = max
β0,σ2
L(y1, . . . , yn, β0, σ2
)
with likelihood
L(y1, . . . , yn, β0, σ2
) = Yn
i=1
1

2π σ
exp

1
2
(yi − β0)
2
σ
2
.
For some given observations y1, . . . , yn and x1, . . . , xn, which is larger, f0(y1, . . . , yn) or f1(y1, . . . , yn)? The
model with the larger maximized liklihood can be said to better ‘fit’ the data. Is the model with the better ‘fit’
necessarily going to be the ‘true’ model? Explain.

Estimation and Testing for Linear Models Questions

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