# econometrics.txt

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1. What is Econometrics?
Unification of necessary 3 views, Statistics, Economics Theory, and Mathematics.

A field of economics that concerns itself with the application of mathematical statistics and the tools of statistics inference to the empirical measurement of relationships postulated by economics theory
2. Symmetric Matrix
M=M'
3. Idempotent Matrix
M=M*M
4. Vector Space
• Closed under scalar multiplication
5. Basis vectors
A linearly independent set of vectors that span a vector space
6. Linearly independent vectors
The only solution for Ax=0 is A=0
7. Singular vs. Non-singular matrices
Det(A)=0 <=> Singular
8. Properties of Determinant
• 1. one of row(colum)=0 => det=0
• 2. det(A')=det(A)
• 3. interchanging two rows(columns) => change the sign of det
• 4. If 2 rows (columns) are identical => det=0
• 5. If one row (column) is a multiple of another => det=0
• 6. Linearly independent of rows (colums) <=> det <>0
• 7. det(A*B)=det(A)*det(B)
9. Row Rank & Column Rank
the maximum number of linearly independent rows (columns)
10. Properties of Rank
• 1. rank(A*B)<=min(rank(A),rank(B))
• 2. rank(A)=rank(A'*A)=rank(A*A')
• 3. If A is full rank, then Ax<>0 for non-zero x
11. Inverse matrix
AA^(-1)=A^(-1)A=I
12. Properties of inverse matrix
• 1. det(inv(A))=1/det(A)
• 2. inv(inv(A))=A
• 3. inv(A)'=inv(A')
• 4. A is symmetric => inv(A) is symmetric
• 5. inv(ABC)=inv(C)inv(B)inv(A)
13. Characteristic roots & vectors
=Eigen values & vectors

• (A-lambdaI)*c=0
• >> lambda=eigen values
• >> c=eigen vectors
14. Properties of characteristic roots
1. Zero characteristic roots possible

• 2. Rank of symmetric matrix=# of non-zero characteristic roots
• => rank of any matrix = # of non-zero eigenvalues of A'A (symmetric)

3. det=product of its characteristic roots
15. Trace of a square matrix
Sum(aii) for all i=1,...,n
16. Properties of trace
• 1. tr(A)=tr(A')
• 2. tr(AB)=tr(BA)
• 3. tr(ABC)=tr(BCA)=tr(CAB)
• 4. A scalar = its trace
q=x'Ax for any non-zero x:

1. q>0 <=> positive definite <=> eigenvalues all +

2. q>=0 <=> positive semidefinite <=> some eigenvalues +, some 0

3. q<0 <=> negative definite <=> eigenvalues all -

4. q<=0 <=> negative semidefinite <=> some eigenvalues -, some 0

5. q<>0 <=> indefinite, some eigenvalues -, some +
18. Properties of a symmetric matrix A of a quadratic form
1. If A is positive (semi)definite, then det(A)>=0

2. If A is positive definite, then det(inv(A)) is also <=> characteristic roots of inv(A) are reciprocals of these of A

3. If (nXK) matrix has full (column) rank, then A'A is positive definite => xA'Ax >0
19. Compare size of matrices
• Q. definite of (A-B)
• => for all non-zero x,
• x'(A-B)x >0 or <0 ?
• positive definite or negative definite?
20. Randome variable
Continuous vs. Discrete
21. PDF vs. CDF
• PDF: f(X=x) continuous f(X=x)=0,
• CDF: F(c)=sum(integral) x<=c f(x)
22. Moments
• 1. r-th moment about the origin: E[Xr]
• 2. r-th moment about the mean of X: E[(X-E(X))r]
23. E(X)?
• sum f(x)*x
• integral f(x)*x dx
24. Properties of E(X)
• 1. E(b)=b, b is a scalar
• 2. Y=aX+b => E(Y)=aE(X)+b
• 3. if X and Y are independent, then E(XY)=E(X)*E(Y)
25. 2nd moment about the mean = variance
• a measure of dispersion
• sum f(x)*(x-E(x))2
26. E[(x-E(x))2]?
E[X^2]-(E[X]^2)
27. 3rd moment about the mean
• skewness
• if it>0 => positive skew (왼쪽에 봉우리)
• if it<0 => negative skew (오른쪽에 봉우리)
28. 4th moment about the mean
• kurtosis
• low kurtosis: fat tails
• high kurtosis: thin tails
29. Moment Generating Function (MGF)
E(exp(xt))=M(t)

=> M(n)(t)=E(Xn)
30. Normal Dist (mu,sigma2)
f(x)=memorize?!
31. Standard normal dist
• Z=(X-mu)/sigma
• when X~N(mu,sigma2)
32. Chi square dist(d)
• d=degrees of freedom
• Chi(d)=sum d of z2
33. t distribution (d)
• t=z/sqr(chi(d)/d)
• t->z as n->inf
34. F distribution (n1,n2)
• [Chi(n1)/n1]/[Chi(n2)/n2]
• e.g. F[n-1,n-k]=[R2/(n-1)]/[(1-R2)/(n-k)] when H0=all coefficients of CLRM are 0's
35. Joint Distribution
f(x,y)
36. Marginal probability
• fx(x)=sumyf(x,y)
• fy(y)=sumxf(x,y)
37. Independence of joint distribution
• 1. f(x,y)=fx(x)*fy(y)
• 2. for any functions g1(x) and g2(y),
• E[g1(x)g2(y)]=E[g1(x)]*E[g2(y)]
38. Covariance
• E[(x-E(x))*(y-E(y))]
• = E[xy]-E[x]*E[y]
39. What if X and Y are independent?
Cov=0
40. Correlatoin
Cov(x,y)/(st.dev(x) st.dev(y))
41. Q. Correlation=0 => independent?
No
42. Var-Cov matrix
• diagonal = var(xi)
• off-diagonal=Cov(xi,xj)
43. Conditional Distribution
f(y|x)=f(x,y)/fx(x)
44. Distributions of functions of r.v.s
• a. change of variables
• b. using MGF

• a. Assume that we know f(x) & y=g(x)
• 1. x=g-1(y)
• 2. dx/dy
• 3. domain of y
• 4. f(g-1(y))abs(dx/dy)
• or f(g-1(y))det(dx/dy)

b. using MGF e.g. E[exp(axt)]
45. Statistics
A function of r.v.s that does not dependent on unknown parameters

e.g. sample mean, median...
46. Random sample <=> iid (independently identically distributed)
A sample of n observations on one or more variables, x1, ..., xn, drawn independently from the same probability distribution f(x1,...,xn|theta)
47. Estimators vs. Estimates
Estimators (statistics) = A formula for using data to estimate a parameters

Estimates = the value you get by plugging data into estimators
48. Method of moments
• sample moments=popoulation moments
• e.g. sum(xi)/n = E[x]
49. Maximum likelihood estimation
: likelihood function & log-likelihood fn.
• cf. dist is known
• maximize L(theta|x1,...,xn) or lnL(.)
50. MLE procedures
• 1. Find L by multiplying f(xi)'s
• 2. Take the log (not necessarily)
• 3. Find the theta's to maximize lnL(.)
• 4. Use FOC=0
• 5. Check SOC: negative definite
51. Ways to evaluate estimators
• 1. Monte-Carlo Analysis
• 2. Pre-data anlysis (small/large sample properties)
52. Small Sample Properties
• 1. Unbiasedness
• 2. Variance (Precision)
• 3. Mean Square Error
• 4. Efficiency
53. Unbiased
E(theta_hat)=theta

Bias=E(theta_hat)-theta
54. Variance
We prefer an estimator with smaller variance
55. MSE (Mean Squared Error)
• theta_hat=t:
• MSE(t)=Var(t)+[Bias(t)]2
• =E[(t-E(t))2]+[E(t)-theta]2
• =E[(t-theta)2]
56. Efficiency
• Unbiased &
• the smallest variance

• => Cramer-Rao lower bound
• if the estimator is unbiased, the variance >=CRLB=[-E[SOC of lnL(.)]]-1

cf. sufficient condition, not necessary condition
57. Large sample property
=asymptotic property as the sample size -> inf

• 1. consistent
• 2. asymptotically efficient
58. Consistency
plim theta_hat=theta
59. Asymptotically efficient
consistent & the smallest asymptotic variance
60. Convergence in Probability
• . . p
• xn-->c
• limn->inf Pr(|xn-c|>eps)=0
• <=> limn->infPr(|xn-c|<eps)=1 for any eps>0
61. Mean Square Convergence
• . . ms
• xn--->c
• mun converges to c & sigma2n converges to 0 as n->inf
62. Mean Sq. Convergence => Convergence in Probability (not true conversely)
• Because of Chebyshev's inequality
• : Pr(|x-mu|>eps)<=(sigma2/eps2)

• e.g. x_bar (sample mean)
• E(sample mean)=mu
• Var(sample mean)=sigma2/n
• as n->inf, E(.)->mu & Var(.)->0, thus it is consistent
63. Khinchine's Weak Law of Large numbers
If x1,...xn is a random iid sample from a distribution with a finite mean E(xn)=mu, then plim(sample mean)=mu
64. Convergence in Distribution
• F(x): limiting distribution
• if limn->inf|Fn(xn)-F(x)|=0 at all continuity points of F(x)
• . . d
• xn-->x
65. Convergence in dist.
Q. xn converges to constant?
No. different form the convergence in probability

Convergence in dist. related to CLT
66. Lindberg-Levy univariate central limit theorem (Asymptotic normality)
• Sums of r.v.s (like, sample mean, weighted sum) are normally distributed in large samples, no matter the distribution of the original population
• Formal def: let x1,...,xn be a random sample from a probabilistic distribution with finite mean mu and finite variance sigma2. Then, sqrt(n)(sample mean of xn - mu) converges to the distribution N(0,sigma2)
67. Repeated sampling
Get samples from the identical population distribution
68. Difference b/w joint dist & likelihood fn.
• Joint dist = L(x1,..,xn|theta)
• Likelihood = L(theta|x1,...,xn)
69. Classical estimators vs. Bayesian approach
estimation is not one of deducing the values of parameter, but rather one of continually updating and sharpening our subjective beliefs about the state of the world

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 Author: lucia831124 ID: 78957 Filename: econometrics.txt Updated: 2011-04-12 07:09:15 Tags: econometrics Folders: Description: Econometrics Show Answers:

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