econometrics.txt

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

M=M'

M=M*M

• Closed under scalar multiplication
• Closed under addition

A linearly independent set of vectors that span a vector space

The only solution for Ax=0 is A=0

Det(A)=0 <=> Singular

• 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)

the maximum number of linearly independent rows (columns)

• 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

AA^(-1)=A^(-1)A=I

• 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)

=Eigen values & vectors

• (A-lambdaI)*c=0
• >> lambda=eigen values
• >> c=eigen vectors

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

Sum(a_ii) for all i=1,...,n

• 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 +

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

• Q. definite of (A-B)
• => for all non-zero x,
• x'(A-B)x >0 or <0 ?
• positive definite or negative definite?

Continuous vs. Discrete

• PDF: f(X=x) continuous f(X=x)=0,
• CDF: F(c)=sum(integral) _x<=c f(x)

• 1. r-th moment about the origin: E[X^r]
• 2. r-th moment about the mean of X: E[(X-E(X))^r]

• sum f(x)*x
• integral f(x)*x dx

• 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)

• a measure of dispersion
• sum f(x)*(x-E(x))²

E[X^2]-(E[X]^2)

• skewness
• if it>0 => positive skew (왼쪽에 봉우리)
• if it<0 => negative skew (오른쪽에 봉우리)

• kurtosis
• low kurtosis: fat tails
• high kurtosis: thin tails

E(exp(xt))=M(t)

=> M⁽ⁿ⁾(t)=E(Xⁿ)

f(x)=memorize?!

• Z=(X-mu)/sigma
• when X~N(mu,sigma²)

• d=degrees of freedom
• Chi(d)=sum d of z²

• t=z/sqr(chi(d)/d)
• t->z as n->inf

• [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

f(x,y)

• f_x(x)=sum_yf(x,y)
• f_y(y)=sum_xf(x,y)

• 1. f(x,y)=f_x(x)*f_y(y)
• 2. for any functions g1(x) and g2(y),
• E[g1(x)g2(y)]=E[g1(x)]*E[g2(y)]

• E[(x-E(x))*(y-E(y))]
• = E[xy]-E[x]*E[y]

Cov=0

Cov(x,y)/(st.dev(x) st.dev(y))

No

• diagonal = var(x_i)
• off-diagonal=Cov(x_i,x_j)

f(y|x)=f(x,y)/f_x(x)

• 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)]

A function of r.v.s that does not dependent on unknown parameters

e.g. sample mean, median...

A sample of n observations on one or more variables, x1, ..., xn, drawn independently from the same probability distribution f(x1,...,xn|theta)

Estimators (statistics) = A formula for using data to estimate a parameters

Estimates = the value you get by plugging data into estimators

• sample moments=popoulation moments
• e.g. sum(x_i)/n = E[x]

• cf. dist is known
• maximize L(theta|x1,...,xn) or lnL(.)

• 1. Find L by multiplying f(x_i)'s
• 2. Take the log (not necessarily)
• 3. Find the theta's to maximize lnL(.)
• 4. Use FOC=0
• 5. Check SOC: negative definite

• 1. Monte-Carlo Analysis
• 2. Pre-data anlysis (small/large sample properties)

• 1. Unbiasedness
• 2. Variance (Precision)
• 3. Mean Square Error
• 4. Efficiency

E(theta_hat)=theta

Bias=E(theta_hat)-theta

We prefer an estimator with smaller variance

• theta_hat=t:
• MSE(t)=Var(t)+[Bias(t)]²
• =E[(t-E(t))²]+[E(t)-theta]²
• =E[(t-theta)²]

• 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

=asymptotic property as the sample size -> inf

• 1. consistent
• 2. asymptotically efficient

plim theta_hat=theta

consistent & the smallest asymptotic variance

• . . p
• x_n-->c
• lim_n->inf Pr(|x_n-c|>eps)=0
• <=> lim_n->infPr(|x_n-c|<eps)=1 for any eps>0

• . . _ms
• x_n--->c
• mu_n converges to c & sigma²_n converges to 0 as n->inf

• 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

If x1,...xn is a random iid sample from a distribution with a finite mean E(x_n)=mu, then plim(sample mean)=mu

• F(x): limiting distribution
• if lim_n->inf|F_n(x_n)-F(x)|=0 at all continuity points of F(x)
• . . _d
• x_n-->x

No. different form the convergence in probability

Convergence in dist. related to CLT

• 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 x_n - mu) converges to the distribution N(0,sigma2)

Get samples from the identical population distribution

• Joint dist = L(x1,..,xn|theta)
• Likelihood = L(theta|x1,...,xn)

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