Summary Econometrics 2 / Econometrie 2 (Skill sheet)

Econometrics 2 / Econometrie 2

Bachelor Econometrics & Operations Research


Skill sheet
A skill sheet is the best way to learn all required formulas and definitions by heart!
 In this skill sheet for Econometrics 2 (Econometrie 2), all the different subjects of this
course are divided over 7 different lists.
 Each list can be seen as a “list of questions”.
 If you can completely fill out all formulas and definitions from all of the 7 lists, you appear
to be ready for the final exam!
 For each list, a quick summary of all answers and formulas is given afterwards.
 If you are not ready just yet… You can also use this skill sheet to quickly practice all the
content!




Page 1

,Skill list #1

⋅ Seven assumptions
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉(𝑏𝑏), 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋 + 𝑎𝑎), 𝑉𝑉𝑉𝑉𝑉𝑉(𝑎𝑎𝑎𝑎), 𝑉𝑉𝑉𝑉𝑉𝑉(𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑌𝑌)
⋅ 𝑆𝑆𝑆𝑆𝑆𝑆
⋅ 𝑆𝑆𝑆𝑆𝑆𝑆
⋅ 𝑆𝑆𝑆𝑆𝑆𝑆
⋅ 𝑅𝑅 2
⋅ 𝑠𝑠 2
⋅ 𝑀𝑀 and properties
⋅ 𝐻𝐻 (or 𝑃𝑃𝑋𝑋 ) and properties
⋅ Frisch-Waugh definition (𝑏𝑏1 in regression of 𝑦𝑦 on 𝑋𝑋1 and 𝑋𝑋2 )
⋅ 𝐹𝐹-test
⋅ White SE
⋅ WLS
⋅ FWLS
⋅ Goldfeld-Quandt test
⋅ Breusch Pagan LM-test
⋅ White test




Page 2

, Skill list #1 answers

⋅ Seven assumptions

𝐴𝐴1: constant regressors 𝑥𝑥1 , … , 𝑥𝑥𝑘𝑘
𝐴𝐴1∗ : stability assumption (with stochastic 1
plim � 𝑋𝑋′𝑋𝑋� = Q
regressors) 𝑛𝑛
𝐴𝐴2: random disturbances 𝜀𝜀1 , … , 𝜀𝜀𝑛𝑛
𝐴𝐴3: homoskedasticity 𝑉𝑉𝑉𝑉𝑉𝑉(𝜀𝜀𝑖𝑖 ) = 𝜎𝜎 2
𝐴𝐴4: no autocorrelation 𝐶𝐶𝐶𝐶𝐶𝐶�𝜀𝜀𝑖𝑖 , 𝜀𝜀𝑗𝑗 � = 0
𝐴𝐴5: constant parameters 𝛽𝛽, 𝜎𝜎
𝐴𝐴6: linear model 𝑦𝑦 = 𝑋𝑋𝑋𝑋 + 𝜀𝜀
𝐴𝐴7: normally distributed error terms 𝜀𝜀~𝑁𝑁(0, 𝜎𝜎 2 )


2
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉(𝑏𝑏) = 𝐸𝐸 ��𝑏𝑏 − 𝐸𝐸(𝑏𝑏)� � = 𝐸𝐸 [(𝑏𝑏 − 𝛽𝛽)2 ]
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋 + 𝑎𝑎) = 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋)
⋅ 𝑉𝑉𝑎𝑎𝑎𝑎(𝑎𝑎𝑎𝑎) = 𝑎𝑎2 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋)
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉(𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏) = 𝑎𝑎2 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋) + 𝑏𝑏 2 𝑉𝑉𝑉𝑉𝑉𝑉(𝑌𝑌) + 2𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎(𝑋𝑋, 𝑌𝑌)
⋅ 𝑆𝑆𝑆𝑆𝑆𝑆 = ∑(𝑦𝑦𝑖𝑖 − 𝑦𝑦�)2 = 𝑆𝑆𝑆𝑆𝑆𝑆 + 𝑆𝑆𝑆𝑆𝑆𝑆
⋅ 𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑏𝑏 2 ∑(𝑥𝑥𝑖𝑖 − 𝑥𝑥̅ )2
⋅ 𝑆𝑆𝑆𝑆𝑆𝑆 = ∑ 𝑒𝑒𝑖𝑖2 = 𝑒𝑒′𝑒𝑒
𝑆𝑆𝑆𝑆𝑆𝑆
⋅ 𝑅𝑅 2 =
𝑆𝑆𝑆𝑆𝑆𝑆
𝑒𝑒′𝑒𝑒
⋅ 𝑠𝑠 2 =
𝑛𝑛−𝑘𝑘
⋅ 𝑀𝑀 = 𝐼𝐼 − 𝑋𝑋(𝑋𝑋 ′ 𝑋𝑋)−1 𝑋𝑋 ′ = 𝐼𝐼 − 𝐻𝐻 where 𝑒𝑒 = 𝑀𝑀𝑀𝑀, 𝑀𝑀𝑀𝑀 = 0, 𝑀𝑀 = 𝑀𝑀′ and 𝑀𝑀 = 𝑀𝑀2
⋅ 𝐻𝐻 = 𝑃𝑃𝑋𝑋 = 𝑋𝑋(𝑋𝑋 ′ 𝑋𝑋)−1 𝑋𝑋′ , where 𝑦𝑦� = 𝑋𝑋𝑋𝑋 = 𝐻𝐻𝐻𝐻, 𝐻𝐻𝐻𝐻 = 0, 𝐻𝐻 = 𝐻𝐻′ and 𝐻𝐻 = 𝐻𝐻2
⋅ Frisch-Waugh: 𝑀𝑀2 𝑦𝑦 = 𝑀𝑀2 𝑋𝑋1 𝛽𝛽1 + 𝜀𝜀 is the regression 'cleaned' for effects of 𝑋𝑋2 . This gives
𝑏𝑏1 = (𝑋𝑋1′ 𝑀𝑀2 𝑋𝑋1 )−1 𝑋𝑋1′ 𝑀𝑀2 𝑦𝑦 (because 𝑀𝑀2′ 𝑀𝑀2 = 𝑀𝑀2 )


⋅ 𝑭𝑭-test

𝑛𝑛 − 𝑘𝑘 𝑒𝑒𝑅𝑅′ 𝑒𝑒𝑅𝑅 − 𝑒𝑒′𝑒𝑒 (𝑒𝑒𝑅𝑅′ 𝑒𝑒𝑅𝑅 − 𝑒𝑒′𝑒𝑒)/𝑔𝑔
𝐹𝐹 = ∙ = ~𝐹𝐹(𝑔𝑔, 𝑛𝑛 − 𝑘𝑘)
𝑔𝑔 𝑒𝑒 ′ 𝑒𝑒 𝑒𝑒 ′ 𝑒𝑒/(𝑛𝑛 − 𝑘𝑘)

⋅ 𝑛𝑛: number of observations
⋅ 𝑘𝑘: number of parameters of the unrestricted model
⋅ 𝑔𝑔: number of parameter restrictions under the null hypothesis


⋅ White SE
𝑛𝑛

� (𝑏𝑏) =
var (𝑋𝑋 ′ 𝑋𝑋) −1
�� 𝜎𝜎𝑖𝑖2 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖′ � (𝑋𝑋′𝑋𝑋)−1
𝑖𝑖=1
𝜎𝜎𝑖𝑖2 → 𝑒𝑒𝑖𝑖2 from 𝑒𝑒𝑖𝑖 = 𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖 ′𝑏𝑏




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