Estimating regressions and computing an unbiased estimate

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Rkryyrag_X

Economics

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It's an econometrics question estimating regressions and computing an unbiased estimate.

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3. The following sums were obtained from 16 pairs of observations on y on x : y} = 526;> x* = 657;/ Xiyi = 492; 4; = 64; x; = 96. Estimate the regression of y on x and compute an unbiased estimate of the error variance o?. If the random error term ui are independent drawings from a N(0,02) population, use statistical tables to calculate PCB-B
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Surname: 1
Name:
Course:
Lecturer:
Date:
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)

Surname: 2

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