Remark Re also a linear combination of the yis hence 5 The sum of the coefficients in 4 is 1 Sampling distrib linear mean function Vindependence The new assumption means we can consx1 ID: 864440
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1 i -) yi Comment: If we also assume e|x
i -) yi Comment: If we also assume e|x (equivalently, Y|x) is normal with constant variance, then the least squares estimates are the same as the maximum l Remark: Re = , also a linear combination of the yi's, hence É 5) The sum of the coefficients in (4) is = = = 1. Sampling distrib (linear mean function) ¥ V (independence) The new assumption means we can cons |x1, É , xn. Expe
2 cted value of (as the y's vary): E(|x
cted value of (as the y's vary): E(|x1, É , xn) = E(|x1, É , xn) = "ci E(yi|x1, É , xn) = "ci E(yi|xi) ( Var(|x1, É , xn) = Var(|x1, É , xn) = "ci2 Var(yi|xi) (s = #2" (defi = For short: Var() = $ s.d.( ) = Comments: This is vaguely analogous to the sampling standard deviation for a mean : s.d. (estimator) = However, here the " i's to be _________ from their
3 mean will result in a more precise estim
mean will result in a more precise estimate of . (Assuming the linear model fits!) Expected value and variance of : Using the formula = , calculations (left to the interested student) similar t Covariance of and: Similar calculations (left to the interested student) will show ) is opposite that of . % : To use the variance ), the same for all i). First, some plausible reasoning: If we h
4 ad lots of observations from Y|xi, then
ad lots of observations from Y|xi, then we could use the univariate standard deviation of these m observations to estimate #2. (Here is the mean of , which would be our best estimate of ) just using ) We don't typically have lots of y's fr (where the expected value is over all samples of the yi's with the xi's fixed) Thus we use the estima to get an unbiased estimator for #2: E(|x1
5 , É , xn) = #2. [If you like to think
, É , xn) = #2. [If you like to think heuristically in terms of losing one degree of freedom for each calculation from data involved in the estimator, this makes sense: Both and need to be calculated from the data to get RSS.] Standard Errors for and: Using = as an estimate of # in the formulas for s.d () and s.d(), we obtain the standard errors s.e. () = and s.e.( ) = as esti