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Summary Econometrics
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Summary of Econometrics for BA3 Business Economics at the VUB.
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Summary Econometrics Class 1Linear Algebra
Regressand and Regressors
● In econometrics one has a dependent variable (the regresand) y and explanatory variables
𝑥! , … , 𝑥" (the regressor)
o 𝑦 = [𝑦! ⋮ 𝑦# ]𝑎𝑛𝑑 𝑋 = [𝑥!! … 𝑥!" ⋮ ⋱ ⋮ 𝑥#! … 𝑥#" ]
o In the following data matrix has n observations of the dependent variable
o And the n observations of the k explanatory variables
Introduction of a system of linear equations
● In econometrics one typically has a system of linear equations
o Underneath you can find a system of 3 linear equations with 4 variables
▪ {𝑦! = 𝑥!! 𝛽 + 𝑥!$ 𝛽 + 𝑥!% 𝛽 + 𝜀! 𝑦$ = 𝑥$! 𝛽 + 𝑥$$ 𝛽 + 𝑥$% 𝛽 + 𝜀$ 𝑦% =
𝑥%! 𝛽 + 𝑥%$ 𝛽 + 𝑥%% 𝛽 + 𝜀%
● You can write this system in a more ordered way, by using matrix notation
o Matrix notation:
▪ 𝑦 = 𝑋𝑏 + 𝜀
o Where:
▪ “n”
observations:
rows of the matrix X and elements of the vector y
▪ “k” variables: columns of the matrix X and the vector y
▪ “b”: k unknown parameters
▪ 𝜀: n error terms
● Definition matrix:
o It is a table of real numbers consisting of m rows and n columns, is denotes as 𝑚 × 𝑛
matrix
▪ A row vector is a matrix with only one row
▪ A column vector is a matrix with only one column
Basic matrix operations
● Matrix addition
o (𝐴 + 𝐵)𝑖𝑗 = 𝐴𝑖𝑗 + 𝐵𝑖𝑗
o Where, commutativity and associativity apply:
▪ A+B=B+A
▪ (A + B) + C = A + (B + C)
● Scalar product of a matrix
o (𝑘𝐴)𝑖𝑗 = 𝑘𝐴𝑖𝑗
o The following properties hold:
▪ (k + l)A = kA + lA
▪ “k”(A + B) = kA + kB
▪ “k”(lA) = klA
● Matrix multiplication
o 𝑐)* = (𝐴𝐵))* = 𝐴) 𝐵* = ∑+#,! 𝐴)# 𝐵#*
o Where i= row and j= column
o The matrix product is associate and distributive with respect to addition:
▪ (AB)C = A(BC)
▪ A(A + C) = AB + AC
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,Summary Econometrics Class 1
▪ (A + B)C = AC + BC
● Transpose
o Transpose swaps rows and columns
o The transpose of (𝑘 × 𝑙)- matrix A is denotes as 𝐴+ or A’ and is the (𝑙 × 𝑘)- matrix
where: (𝐴+ ))* = (𝐴)*)
o The following properties hold:
▪ (𝐴+ )+ = 𝐴
▪ (𝐴 + 𝐵)+ = 𝐴+ + 𝐵+
▪ (𝐴𝐵)+ = 𝐵+ 𝐴+ (keep in mind the order!)
● Vector
o If x is a 𝑛 × 1 column vector, then 𝑥 + is a 1 × 𝑛 row vector (transpose is applied here)
o 𝑥 + 𝑥 = ∑#),! 𝑥)$
● Norm of a vector
o The (Euclidean) norm of a vector x is defined as
▪ ‖𝑥‖ = (𝑥 + 𝑥)!/$ = (∑#),! 𝑥)$ )!/$
o This is often used to minimize a sum of squares
Special matrices
● Null matrix
o Is a matrix completely filled with zeros
● Square matrix
o Is a matrix with an equal number of rows and columns
o And a square matrix of order 1 is simply a number
● Symmetrix matrix
o Is a square matrix that coincides with its transpose
o 𝐴+ = 𝐴
● Diagonal matrix
o Is a square matrix with scalars on it the main diagonal and zeros elsewhere?
● Unit matrix
o Is a square (and diagonal) matrix with ones on the maid diagonal and a zero
elsewhere
o For dimension n it is written as 𝐼#
▪ 𝐼% = [1 0 0 0 1 0 0 0 1 ]
● Upper triangle matrix
o Is a square matrix with underneath the main diagonal zeros everywhere
● Lower triangle matrix
o Is a square matrix with above the main diagonal zeros everywhere
Linear independence and the rank of a matrix
● Definition linear independence
o The (row or column) vectors 𝑎! , . . , 𝑎. are linear independent if any linear
combination of 𝑎! , . . , 𝑎. (except of the zero combination is non-zero, i.e. if:
▪ ∑.*,! 𝜆* 𝑎* = 𝜆! 𝑎! + 𝜆$ 𝑎$ + ⋯ + 𝜆. 𝑎. ≠ 𝑂.
o For any combination of scalars 𝜆! , … , 𝜆. of which at least one is different from 0.
▪ 𝑂. is the 𝑙 × 1 null matrix
● If not, 𝐴! , … , 𝐴. are linearly dependent
o With other words if no combinations can be made to construct a vector in the matrix
▪ Then there’s linear independence
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,Summary Econometrics Class 1
● We will see that it is important not to have linear dependent in the columns of X for
regression models
● Example:
o [6 1 2 ], ⌈0 1 0 ⌉, [3 0 1 ]
▪ These vectors are linear dependent
● Since, first column is equal 2nd columns + 2 times 3rd column
● Definition rank
o It is the maximum number of linear independent row or columns
● Important properties
o For a matrix A of order 𝑘 × 𝑙: rank(A) ≤ min (k,l)
o Rank(A) = rank(A𝐴+ ) = rank(𝐴+ A)
o Rank(AB) ≤ min(rank(A), rank(B))
● Example:
o What is the rank of: [2 − 1 0 − 1 1 1 ]?
▪ There can not be any linear combination written, so the rank is 2
o What is the rank of: [2 − 1 1 − 4 2 − 2 ]?
▪ Row 2 is a linear combination of row 1, by -2 times row2
▪ So, the rank is equal to 1
Inverse of a matrix
● Remember!
o That the inverse of a matrix only exists when the matrix is linear independent, with
other words full rank
o Hence, no row or column is a linear transformation of one other row or column
● See on how to calculate the inverse of the matrix in math 2 course
● Inverse of a non-singular (𝑛 × 𝑛)- matrix
o With other words, a matrix with a full rank
o Is denoted by 𝑋 /!
o Important property
▪ 𝑋 /! 𝑋 = 𝑋𝑋 /! = 𝐼#
● Inverse matrix times the regular matrix is equal the unit matrix
● Determinant of the square matrix
o In general case if a square matrix has a determinant different from 0. Then it means
that the matrix is invertible
o Example how to calculate 2*2 matrix determinant
▪
o If you have 2 matrices that are both invertible, the following properties hold:
▪ (𝐴/! )+ = (𝐴+ )/!
▪ (𝐴𝐵)/! = 𝐵/! 𝐴/! (keep in mind the order!)
Pseudo-inverse of a matrix
● Explanation on why this matrix exists
o Sometimes there exist no inverse for example in an overdetermined system
▪ Picture for overdetermined system:
● Is when you have data all over the place, and you try to fit a linear
model through this data
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, Summary Econometrics Class 1
● But you realize that there is no linear model that drives through all
the points
o So new approach is using the pseudo-inverse
● Definition pseudo-inverse of a matrix X
o 𝑋 0 = (𝑋 + 𝑋)/! 𝑋 +
o A pseudo inverse of m*n matrix is defined by the unique n*m matrix satisfying the
following 4 criteria’s
▪ 𝑋𝑋 0 𝑋 = 𝑋
▪ 𝑋 0 𝑋𝑋 0 = 𝑋 0
▪ (𝑋𝑋 0 )+ = 𝑋𝑋 0
▪ (𝑋 0 𝑋)+ = 𝑋 0 𝑋
Positive (semi-) definite
● Definition:
o A symmetric n*n matrix B is positive semi-definite
▪ If, for each n*1 vector 𝑎(≠ 0# ), quadratic form 𝑎+ 𝐵𝐴 ≥ 0
o It is positive definite if 𝑎+ 𝐵𝐴 > 0 for each vector 𝑎(≠ 0# )
▪ Positive semi-definite matrices are important to determine issues such as the
smallest covariance matrix among different estimators
● It’s used to see which of (say) two estimators has the lowest covariance
o Example when it is used:
▪ Suppose you have a model that is estimated by
▪ 𝛽123)#425 .647+ 7894267 𝑎𝑛𝑑 𝛽.647+ 4:71.9+6 36;)4+)1#
▪ You can show that Cov(Beta_ols) and Cov(Beta_lad) is positive semi definite
● This means that covariance matrix for Beta_lab is larger than that for
Beta_ols
● Hence, Beta_ols is also more efficient!
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