simple linear regression - important things to know

Question 1

what is the formula for the least squares estimator (simple)?

Answer 1

∑(yᵢ - β₀ - β₁xᵢ)²

Question 2

how do we find β ̂₁ and β ̂₀ (simple)?

Answer 2

take 𝛿S/𝛿β₁ = 0 and 𝛿S/𝛿β₂ = 0 then solve

Question 3

what is ∑β₀ equivalent to

Answer 3

nβ₀

Question 4

what is (1/n)∑xᵢ equivalent to?

Answer 4

x ̅
(note similar for y)

Question 5

what is β ̂₀ (simple)?

Answer 5

y ̅ - β ̂₁x ̅

Question 6

what is β ̂₁ (simple)?

Answer 6

( ∑(yᵢ-y ̅)(xᵢ-x ̅) )/( ∑(xᵢ-x ̅)² )

Question 7

what is sₓₓ?

Answer 7

∑(xⱼ - x ̅)²

from j=1 to n

Question 8

what is E(εᵢ)?

Answer 8

0

Question 9

what is E(yᵢ)?

Answer 9

β₀ + β₁xᵢ

Question 10

what is E(β ̂₁)?

Answer 10

β₁

Question 11

what is V(yᵢ)?

Answer 11

σ²

Question 12

what is V(β ̂₁)?

Answer 12

σ²/sₓₓ

Question 13

what is what is E(β ̂₀)?

Answer 13

β₀

Question 14

what is V(β ̂₀)?

Answer 14

σ²( (1/n) - (x ̅ ² / sₓₓ) )

Question 15

what is V(y ̅)

Answer 15

V((1/n)∑yᵢ) = σ²/n

Question 16

for y = β ̂₀ + β ̂₁x, what is eᵢ?

Answer 16

yᵢ - y ̂ᵢ = yᵢ - (β ̂₀ + β ̂₁x)

∑eᵢ = 0
∑xᵢeᵢ = 0
∑y ̂ᵢeᵢ = 0