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# simon2016bht/Econometrics_learning_note.md

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vec space def and ex

• set which allows linear combination of vectors to form another vector in the set

basis vecotrs def

• vectors which span vec space

linear combination

linear span

• all possible linear combination

linear transformation

• L(ax+by)=aL(x)+bL(y)

centering operation as idempotent operation

• I-(1/n)ii'

tr(ABC)=tr(BCA)=tr(CAB)

rank(A)=rank(A')

partitioned matrices

• make sure dimension consistency

vec operator

• transform a matrix into vector

In linear algebra, a symmetric (n x n) real matrix M is said to be positive definite if the scalar z'Mz is positive for every non-zero column vector z n real numbers.

symmetric with positive eigenvalues is positive definite

Hessian matrix :second derivative symmetric matrix

symmetric matrix has real eigenvalues and real orthogonal eigenvectors

AC=C$\Lambda$

• spectral composition symmetric A=C$\Lambda$C'

• diagonalize symmetric A, CAC' = $\Lambda$

• rank, tr, det is the same as A

• easy to multiply

for non-singular square matrix

• $A^{-1}$ exist

• det(A) != 0

• columns independent

• full rank

===

probability space

• sample space S

• set of elements
• power set ($\sigma$ field)

• set of all subset of S
• probability measure

• P: power set -> [0,1]

independence

• f(x,y) = f(x)·f(y)

no correlation

• cov(x,y) = 0 or corr(x,y) = 0

• indep -> var = 0

moment generating func

• $E(e^{tx})$

• E(x) = M'(0), E(x^2) = M"(0)

normal distribution

f(x1,x2)=...

===

good estimator

• unbias

• efficiency

• var(theta1) < var(theta2)

• exist cramer-rao low bound

• sufficiency: statistics can convey same information as parameter

• asymptotic consistency： convergence in probability

• asymptotic normality

method of estimation

• LSE

• GMM: combine 2nd and 4th moment

• MLE: choose theta to max L(theta, x) = joint pdf of f(x;theta)

===

parametric statistical steps

1. set model x~f(x;theta)
2. sample
3. estimate
4. test