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.