Find CI for mean of linear regression with variance unknownFor the simple linear regression model:Yi=β⋅Xi+ϵiI want to find CI for βx which is E(Yi) when xi=x.I find that β^∼N(β,σ2∑(Xi2)), so the distribution of β x is N(βx,X2σ2∑(Xi2)). If the variance is known I can use P(−zα2≤β^x−βxxσ∑(xi2)≤zα2).If the variance is unknown, what unbiased estimator should I use for σ2? Is it the sample variance? What is it in this case?

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gil001q4wq · Accepted Answer

Suppose number of predictors is p and number of observations is N. Let X be N×p matrix, and the specific x be 1×p. Thenxβ−xβ^σ^x(XTX)−1x∼tN−pwhere σ^2=RSSN−p=(Y−Xβ^)T(Y−Xβ^)N−p.In the above t stands for t distribution.For simple regression p=1, so σ2 turns out to be sample variance. However, you need to use t distribution, not Gaussian.

tralhavahr9c · Accepted Answer

If your model is Yi=βxi+ϵi, then the unbiased estimator for var(ϵi)=σ2 is given by s2=∑ei2n−1, where ei=yi−y^i.

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