Simple Linear Regression
Foundation
Prof. Maria Tackett
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General form of model
Y=f(X)+ϵ
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General form of model
Y=f(X)+ϵ
Y: response variable
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General form of model
Y=f(X)+ϵ
Y: response variable
X: predictor variable
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General form of model
Y=f(X)+ϵ
Y: response variable
X: predictor variable
f: fixed but unknown function
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General form of model
Y=f(X)+ϵ
Y: response variable
X: predictor variable
f: fixed but unknown function
ϵ: random error
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Simple linear regression
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Simple linear regression
Y=Model+Error=f(X)+ϵ=μY|X+ϵ=β0+β1X+ϵ
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Y=β0+β1X+ϵ
where the errors are independent and normally distributed ϵ∼N(0,σ2ϵ)
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Y|X∼N(β0+β1X,σ2ϵ)
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Y|X∼N(β0+β1X,σ2ϵ)
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Y|X∼N(β0+β1X,σ2ϵ)
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Regression standard error
Once we fit the model, we can use the residuals to calculate the regression standard error
ˆσϵ=√n∑i=1(yi−ˆyi)2n−2=√n∑i=1e2in−2
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Standard error of ˆβ1
SEˆβ1=ˆσϵ√1(n−1)s2X
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