Advanced Econometrics
Bachelor Econometrics & Operations Research
Yellow background | All parts in this summary with a yellow background do not need to be known by
heart. Only the intuition behind it is required.
Review | Basic Econometrics
Properties of estimators
Quality Meaning
Unbiasedness 𝐸𝐸(𝑏𝑏) = 𝛽𝛽
Consistency plim 𝑏𝑏 = 𝛽𝛽
Efficiency For a certain class of estimators (i.e. within the group of unbiased or
biased estimators), it should hold that
var(𝑏𝑏∗ ) > var(𝑏𝑏)
where 𝑏𝑏∗ is any other estimator of the same class.
Matrix algebra
det(𝐴𝐴) = |𝐴𝐴|
𝑎𝑎 𝑏𝑏
det(𝐴𝐴) = � � = 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏
𝑐𝑐 𝑑𝑑
𝑎𝑎 𝑏𝑏 𝑐𝑐
𝑑𝑑 𝑓𝑓 𝑑𝑑 𝑒𝑒
|𝐴𝐴| = �𝑑𝑑 𝑒𝑒 𝑓𝑓� = 𝑎𝑎 �𝑒𝑒 𝑓𝑓
� − 𝑏𝑏 � � + 𝑐𝑐 � �
ℎ 𝑖𝑖 𝑔𝑔 𝑖𝑖 𝑔𝑔 ℎ
𝑔𝑔 ℎ 𝑖𝑖
det(𝐴𝐴) = 0 ↔ matrix 𝐴𝐴 is singular (has no inverse)
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,rank(𝐴𝐴): the largest number 𝑟𝑟 for which there exists an invertible, 𝑟𝑟 × 𝑟𝑟 square submatrix of 𝐴𝐴.
rank(𝐴𝐴) = 𝑟𝑟
min(𝑝𝑝, 𝑞𝑞) ≥ rank(𝐴𝐴)
The smallest of both, either 𝑝𝑝 or 𝑞𝑞, should be either
⋅ larger than 𝑟𝑟
⋅ equal to 𝑟𝑟 (full rank)
𝑏𝑏
⋅ If 𝑞𝑞 > 𝑟𝑟, then there exists ≠ 0 such that 𝐴𝐴𝐴𝐴 = 0. The 𝑞𝑞 columns of 𝐴𝐴 are then linearly
𝑞𝑞 × 1
dependent.
𝑏𝑏
⋅ If det(𝐴𝐴) = 0, then 𝑝𝑝 > rank(𝐴𝐴) and then there exists ≠ 0 such that 𝐴𝐴𝐴𝐴 = 0. The 𝑞𝑞
𝑝𝑝 × 1
columns of 𝐴𝐴 are then linearly dependent.
⋅ If 𝐴𝐴 is invertible (det(𝐴𝐴) ≠ 0), then the only vector 𝑏𝑏 for which 𝐴𝐴𝐴𝐴 = 0 is 𝑏𝑏 = 0.
Linear independent: if the only vector 𝑏𝑏 for which 𝐴𝐴𝐴𝐴 = 0 is given by 𝑏𝑏 = 0. Then rank(𝐴𝐴) = 𝑞𝑞 if
𝑝𝑝 ≥ 𝑞𝑞, i.e. full column rank.
Orthogonal complement: for vector 𝛼𝛼, its orthogonal complement is given as 𝛼𝛼⊥ such that 𝛼𝛼⊥′ 𝛼𝛼 = 0.
White noise errors
If errors 𝑢𝑢𝑡𝑡 are white noise (innovation process), it holds that
⋅ 𝐸𝐸(𝑢𝑢𝑡𝑡 ) = 0
⋅ 𝐸𝐸(𝑢𝑢𝑡𝑡 𝑢𝑢𝑡𝑡′ ) = Σ𝑢𝑢
⋅ 𝑢𝑢𝑠𝑠 and 𝑢𝑢𝑡𝑡 are independent for 𝑠𝑠 ≠ 𝑡𝑡
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,Stationary time series
Stationary time series is given by
𝑦𝑦𝑡𝑡 = 𝜌𝜌𝑦𝑦𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
where |𝜌𝜌| < 1 and 𝜀𝜀𝑡𝑡 ~iid(0, σ2 ).
Non-stationary time series (random walk) is given by
𝑦𝑦𝑡𝑡 = 𝑦𝑦𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
which is said to have a unit root.
Unit root process
Consider a discrete-time stochastic process
{𝑦𝑦𝑡𝑡 , 𝑡𝑡 = 1, … , = ∞}
and suppose that it can be written as an autoregressive process of order 𝑝𝑝:
𝑦𝑦𝑡𝑡 = 𝑎𝑎1 𝑦𝑦𝑡𝑡−1 + 𝑎𝑎2 𝑦𝑦𝑡𝑡−2 + ⋯ + 𝑎𝑎𝑝𝑝 𝑦𝑦𝑡𝑡−𝑝𝑝 + 𝜀𝜀𝑡𝑡
Here,
{𝜀𝜀𝑡𝑡 , 𝑡𝑡 = 0, … , ∞}
is a serially uncorrelated, zero-mean stochastic process with constant variance 𝜎𝜎 2 .
For convenience, assume 𝑦𝑦0 = 0.
Let the characteristic equation be given by:
𝐴𝐴(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑝𝑝−1 𝑎𝑎1 − 𝑧𝑧 𝑝𝑝−2 𝑎𝑎2 − ⋯ − 𝑎𝑎𝑝𝑝 = 0
If 𝑧𝑧 = 1 is a solution of the characteristic equation, then 𝑦𝑦𝑡𝑡 has a unit root, also known as:
⋅ integrated of order one 𝐼𝐼(1)
⋅ time series with stochastic trend
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,Kronecker product ⊗
𝑢𝑢𝑖𝑖1
⎡ 𝑢𝑢𝑖𝑖2 ⎤
⎢𝑥𝑥𝑖𝑖1 � ⋮ � 0 ⋯ 0 ⎥
⎢ 𝑢𝑢𝑖𝑖𝑖𝑖 ⎥
⎢ ⎥
𝑢𝑢𝑖𝑖1 𝑢𝑢𝑖𝑖1
𝑥𝑥𝑖𝑖1 0 ⋯ 0 ⎢ ⎥
𝑢𝑢𝑖𝑖2 ⎢ 𝑢𝑢𝑖𝑖2 ⎥
0 𝑥𝑥𝑖𝑖2 . 0 0 𝑥𝑥𝑖𝑖2 � ⋮ � . 0
𝑋𝑋𝑖𝑖 ⊗ 𝑢𝑢𝑖𝑖 = � �⊗� ⋮ �=⎢ ⎥
⋮ . ⋱ ⋮
𝑢𝑢𝑖𝑖𝑖𝑖 ⎢ 𝑢𝑢𝑖𝑖𝑖𝑖 ⎥
0 0 ⋯ 𝑥𝑥𝑖𝑖𝑖𝑖
⎢ ⋮ . ⋱ ⋮ ⎥
⎢ 𝑢𝑢𝑖𝑖1 ⎥
⎢ 𝑢𝑢𝑖𝑖2 ⎥
0 0 ⋯ 𝑥𝑥𝑖𝑖𝑖𝑖 � ⋮ �
⎢ ⎥
⎣ 𝑢𝑢𝑖𝑖𝑖𝑖 ⎦
Single-equation OLS
yi = 𝑥𝑥𝑖𝑖 𝛽𝛽 + 𝜀𝜀𝑖𝑖
Ordinary least square estimator
𝑁𝑁 −1 𝑁𝑁
β�𝑂𝑂𝑂𝑂𝑂𝑂 = �𝑁𝑁 −1
� 𝑥𝑥𝑖𝑖′ 𝑥𝑥𝑖𝑖 � �𝑁𝑁 −1
� 𝑥𝑥𝑖𝑖′ 𝑦𝑦𝑖𝑖 �
𝑖𝑖=1 𝑖𝑖=1
β�𝑂𝑂𝑂𝑂𝑂𝑂 = (𝑋𝑋′𝑋𝑋)−1 𝑋𝑋′𝑦𝑦
Unbiased and consistent if
⋅ 𝐸𝐸(𝑥𝑥𝑖𝑖′ 𝜀𝜀𝑖𝑖 ) = 0 (exogeneity assumed)
⋅ 𝐸𝐸(𝑥𝑥𝑖𝑖′ 𝑥𝑥𝑖𝑖 ) is non-singular or has full column rank (to avoid multicollinearity)
During the Advanced Econometrics course, we use system OLS, which consists of multiple single-
equations. We therefore make a difference between “equation by equation OLS” (which is OLS
performed on each single equation) and “system OLS” (which is the technique taught in this course to
perform OLS on the entire system of equations).
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, Topic 1 | Systems of Equations
Seemingly unrelated regressions (SUR)
Although each equation in the system has its own coefficient vector (seems unrelated), correlation
across the error terms in different equations can provide some links in estimation (actually related).
𝑦𝑦𝑖𝑖1 = 𝑥𝑥𝑖𝑖1 𝛽𝛽1 + 𝑢𝑢𝑖𝑖1
𝑦𝑦𝑖𝑖2 = 𝑥𝑥𝑖𝑖2 𝛽𝛽2 + 𝑢𝑢𝑖𝑖2
⋮
𝑦𝑦𝑖𝑖𝑖𝑖 = 𝑥𝑥𝑖𝑖𝑖𝑖 𝛽𝛽𝐺𝐺 + 𝑢𝑢𝑖𝑖𝑖𝑖
for 𝑖𝑖 = 1, … , 𝑁𝑁.
Properties
⋅ 𝐺𝐺: amount of seemingly unrelated equations
⋅ 𝑥𝑥𝑖𝑖𝑖𝑖 = �𝑥𝑥𝑔𝑔1,𝑖𝑖 , 𝑥𝑥𝑔𝑔2,𝑖𝑖 , … , 𝑥𝑥𝑔𝑔𝐾𝐾𝑔𝑔 ,𝑖𝑖 � are the 𝐾𝐾𝑔𝑔 regressors for a single equation 𝑔𝑔, which can differ
per equation
⋅ 𝛽𝛽𝑔𝑔 is 𝐾𝐾𝑔𝑔 × 1
SUR in matrix form
𝑌𝑌𝑖𝑖 = 𝑋𝑋𝑖𝑖 𝛽𝛽 + 𝑢𝑢𝑖𝑖
𝑦𝑦𝑖𝑖1 𝑢𝑢𝑖𝑖1
𝑦𝑦𝑖𝑖2 𝑢𝑢𝑖𝑖2
𝑌𝑌𝑖𝑖 = � ⋮ � 𝑢𝑢𝑖𝑖 = � ⋮ �
𝑦𝑦𝑖𝑖𝑖𝑖 𝑢𝑢𝑖𝑖𝑖𝑖
𝛽𝛽1 𝑥𝑥𝑖𝑖1 0 0 ⋯ 0
⎡0 𝑥𝑥𝑖𝑖2 0 . 0⎤
𝛽𝛽2
𝛽𝛽 = � � ⎢ ⎥
⋮ 𝑋𝑋 = ⎢ 0 0 𝑥𝑥𝑖𝑖3 . 0⎥
𝛽𝛽𝐺𝐺 ⎢ ⋮ . . ⋱ ⋮ ⎥
⎣0 0 0 ⋯ 𝑥𝑥𝑖𝑖𝑖𝑖 ⎦
System OLS: parameter estimation
𝑁𝑁 −1 𝑁𝑁
β�𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = �𝑁𝑁 −1 � 𝑋𝑋𝑖𝑖′ 𝑋𝑋𝑖𝑖 � �𝑁𝑁 −1
� 𝑋𝑋𝑖𝑖′ 𝑌𝑌𝑖𝑖 �
𝑖𝑖=1 𝑖𝑖=1
𝛽𝛽̂𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = (𝑋𝑋′𝑋𝑋)−1 𝑋𝑋′𝑦𝑦
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