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
Page 3
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
Page 3
