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AshBorg
25.01.2020 •
Mathematics
Consider the simple linear regression model yi=β0+β1xi+ϵi, where ϵi's are independent n(0,σ2) random variables. therefore, yi is a normal random variable with mean β0+β1xi and variance σ2. moreover, yi's are independent. as usual, we have the observed data pairs (x1,y1), (x2,y2), ⋯⋯, (xn,yn) from which we would like to estimate β0 and β1. in this chapter, we found the following estimators β1^=sxysxx,β0^=y−β1^x¯¯¯. where sxx=∑i=1n(xi−x¯¯¯)2,sxy=∑i=1n(xi−x¯¯¯)(yi−y). show that β1^ is a normal random variable. show that β1^ is an unbiased estimator of β1, i.e., e[β1^]=β1. show that var(β1^)=σ2sxx.
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Ответ:
See proof below.
Step-by-step explanation:
If we assume the following linear model:
And if we have n sets of paired observations
the model can be written like this:
And using the least squares procedure gives to us the following least squares estimates
for
and
for
:
Where:
Then
is a random variable and the estimated value is
. We can express this estimator like this:
Where
and if we see careful we notice that
and ![\sum_{i=1}^n a_i x_i =1](/tpl/images/0469/7606/59740.png)
So then when we find the expected value we got:
And as we can see
is an unbiased estimator for ![\beta_1](/tpl/images/0469/7606/9cc95.png)
In order to find the variance for the estimator
we have this:
And we can assume that
since the observations are assumed independent, then we have this:
And if we simplify we got:
And with this we complete the proof required.
Ответ:
blah blah
Step-by-step explanation:u suck