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It's an econometrics question estimating regressions and computing an unbiased estimate.
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Estimating Regressions and Computing an Unbiased Estimate
∑ 𝑦𝑖2 = 526; ∑ 𝑥𝑖2 = 657; ∑ 𝑥𝑖 𝑦𝑖 = 492; ∑ 𝑦𝑖 = 64; ∑ 𝑥𝑖 = 96
𝑖
𝑖
𝑖
𝑖
𝑖
For the estimation with the summations, we apply the following rules:
𝑛
∑ 𝑎 = 𝑛𝑎
𝑖=1
𝑛
𝑛
∑ 𝑎 𝑥1 = 𝑎 ∑ 𝑥𝑖
𝑖=1
𝑖=1
𝑛
𝑛
𝑛
∑(𝑥𝑖 + 𝑦𝑖 ) = ∑ 𝑥1 + ∑ 𝑦1
𝑖=1
𝑖=1
𝑖=1
Operating with the normal equations, we get:
𝑛
𝑛
̂1 + 𝛽
̂2 ∑ 𝑥𝑖
∑ 𝑦𝑖 = 𝑛𝛽
𝑖=1
𝑛
(1)
𝑖=1
𝑛
𝑛
̂1 ∑ 𝑥𝑖 + 𝛽
̂2 ∑ 𝑥𝑖 2
∑ 𝑦𝑖 𝑥𝑖 = 𝛽
𝑖=1
𝑖=1
𝑖=1
Dividing equation (1) with 𝑛, we get:
̂1 + 𝛽
̂2 𝑥̅
𝑦̅ = 𝛽
Therefore,
̂1 = 𝑦̅ − 𝛽
̂2 𝑥̅
𝛽
̂2 we get:
Solving for 𝛽
(2)
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