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Econometrics Summary - ENDTERM UVA EBE $8.62   Add to cart

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Econometrics Summary - ENDTERM UVA EBE

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This document is a summary of everything you need to know for the endterm (and midterm) of the course 'Econometrics' (6012B0453Y) at the University of Amsterdam, taught by Hans van Ophem. This document includes the following topics: log and ln, expected value, variance, covariance, estimators, simp...

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By: wesleytebiesebeek • 10 months ago

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By: Helena0207 • 10 months ago

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Econometrics Summary
Rules for log and ln
log 𝑏 (𝑐 ) = 𝑘 → 𝑏𝑘 = 𝑐
ln(𝑐 ) = log𝑒 (𝑐 ) = 𝑘 → 𝑒𝑘 = 𝑐

ln(𝑥𝑦) = ln(𝑥 ) + ln⁡(𝑦) ln(𝑒) = 1 1
𝑥 𝑥 −1 =
ln ( ) = ln(𝑥 ) − ln(𝑦) ln(1) = 0 𝑥
𝑦 1 √𝑥 = 𝑥 1/2
ln ( ) = −ln⁡(𝑥)
ln(𝑥 𝑦 ) = 𝑦 ∙ ln⁡(𝑥) 𝑥
Expected value
𝐸 (𝑋 + 𝑌 ) = 𝐸 (𝑋 ) + 𝐸 (𝑌 )
𝐸 (𝑋𝑌) = 𝐸 (𝑋) ∙ 𝐸(𝑌) → need independence!
𝐸 (∑𝑛𝑖=1 𝑥𝑖 ) = ∑𝑛𝑖=1 𝐸(𝑥𝑖 )⁡
𝐸 (𝑐 ∙ 𝑋) = 𝑐 ∙ 𝐸(𝑋)

Law of iterated expectations: 𝐸(𝛽̂1 ) = 𝐸 (𝐸(𝛽̂1 |𝑥)) = 𝐸 (𝛽1 ) = 𝛽1

Variance and covariance
𝜎𝑌2 = 𝑣𝑎𝑟(𝑌) = 𝐸 ((𝑌 − 𝑌̅ )2 ) = 𝐸 (𝑌 2 ) − 𝑌̅ 2
1 2
𝑆𝑌2 = ∑𝑛𝑖=1(𝑌𝑖 − 𝑌̅ )
𝑛−1

𝜎𝑋𝑌 = 𝑐𝑜𝑣(𝑋, 𝑌) = 𝐸((𝑋 − 𝑋̅)(𝑌 − 𝑌̅)) = 𝐸 (𝑋𝑌) − 𝑋̅ 𝑌̅
1
𝑆𝑋𝑌 = ∑𝑛𝑖=1((𝑋𝑖 − 𝑋̅)(𝑌𝑖 − 𝑌̅))
𝑛−1 Dependent
on scale
𝑣𝑎𝑟(𝑋 + 𝑌) = 𝑣𝑎𝑟(𝑋) + 𝑣𝑎𝑟(𝑌) + 2 ∙ 𝑐𝑜𝑣(𝑋, 𝑌)
𝑣𝑎𝑟(𝑐 ∙ 𝑋) = 𝑐 2 ∙ 𝑣𝑎𝑟(𝑋)
𝑣𝑎𝑟(𝑎 ∙ 𝑋 + 𝑏 ∙ 𝑌) = 𝑎2 ∙ 𝑣𝑎𝑟(𝑋) + 𝑏2 ∙ 𝑣𝑎𝑟(𝑌) + 2𝑎𝑏 ∙ 𝑐𝑜𝑣(𝑋, 𝑌)

𝑐𝑜𝑣(𝑋, 𝑋) = 𝑣𝑎𝑟(𝑋)
𝑐𝑜𝑣(𝑎 ∙ 𝑋, 𝑏 ∙ 𝑌) = 𝑎𝑏 ∙ 𝑐𝑜𝑣(𝑋, 𝑌)
𝑐𝑜𝑣(𝑎𝑋 + 𝑏𝑌 + 𝑐, 𝑤) = 𝑎 ∙ 𝑐𝑜𝑣(𝑋, 𝑤) + 𝑏 ∙ 𝑐𝑜𝑣(𝑌, 𝑤)

Estimators → are random!
We try to estimate a population parameter. This is usually unknown, except in a Monte Carlo
Analysis.
• Unbiasedness: 𝐸 (𝑋̅) = 𝜇
• Consistency: 𝑣𝑎𝑟(𝑋̅) → 0 as 𝑛 → ∞
AND the estimator is asymptotically (“as 𝑛 → ∞”) unbiased!
Simple regression
𝑌𝑖 = 𝛽0 + 𝛽1 𝑋𝑖 + 𝑢𝑖 (population)
→ 𝛽1 measures the unit change in 𝑌, per unit change in 𝑋

We estimate 𝛽0 and 𝛽1 by min ∑𝑒𝑖2
𝑌̂𝑖 = 𝛽̂0 + 𝛽̂1 𝑋𝑖 (fitted value)
𝑒𝑖 = 𝑢̂𝑖 = 𝑌𝑖 − 𝑌̂𝑖 = 𝑌𝑖 − 𝛽̂0 − 𝛽̂1 𝑋𝑖
2
min ∑𝑒𝑖2 = min ∑(𝑌𝑖 − 𝛽̂0 − 𝛽̂1 𝑋𝑖 )
1. Take the first derivative with respect to 𝛽0 and/or 𝛽1
2. Set equal to 0 and solve for 𝛽0 or 𝛽1

𝛽̂0 = 𝑌̅ − 𝛽̂1 𝑋̅
𝑠 𝑠𝑎𝑚𝑝𝑙𝑒⁡𝑐𝑜𝑣(𝑌,𝑋) ∑(𝑋𝑖−𝑋̅)(𝑌𝑖−𝑌̅ )
𝛽̂1 = 𝑠𝑌𝑋
2 = 𝑠𝑎𝑚𝑝𝑙𝑒⁡𝑣𝑎𝑟(𝑋) =
𝑋 ∑(𝑋 −𝑋̅)2
𝑖

Least Squares Assumptions
1) 𝜀𝑖 is a random variable with 𝐸 (𝜀𝑖 |𝑋) = 0
2) (𝑌𝑖 , 𝑋𝑖 ) are i.i.d.
3) Large outliers are unlikely → finite nonzero 4th moments → kurtosis is finite


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