# econometrics.txt

 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 Symmetric Matrix M=M' Idempotent Matrix M=M*M Vector Space Closed under scalar multiplicationClosed under addition Basis vectors A linearly independent set of vectors that span a vector space Linearly independent vectors The only solution for Ax=0 is A=0 Singular vs. Non-singular matrices Det(A)=0 <=> Singular Properties of Determinant 1. one of row(colum)=0 => det=02. det(A')=det(A)3. interchanging two rows(columns) => change the sign of det4. If 2 rows (columns) are identical => det=05. If one row (column) is a multiple of another => det=06. Linearly independent of rows (colums) <=> det <>07. det(A*B)=det(A)*det(B) Row Rank & Column Rank the maximum number of linearly independent rows (columns) 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 Inverse matrix AA^(-1)=A^(-1)A=I Properties of inverse matrix 1. det(inv(A))=1/det(A)2. inv(inv(A))=A3. inv(A)'=inv(A')4. A is symmetric => inv(A) is symmetric5. inv(ABC)=inv(C)inv(B)inv(A) Characteristic roots & vectors =Eigen values & vectors (A-lambdaI)*c=0>> lambda=eigen values>> c=eigen vectors 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 Trace of a square matrix Sum(aii) for all i=1,...,n 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 Quadratic form & definiteness 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 + 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 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? Randome variable Continuous vs. Discrete PDF vs. CDF PDF: f(X=x) continuous f(X=x)=0,CDF: F(c)=sum(integral) x<=c f(x) Moments 1. r-th moment about the origin: E[Xr]2. r-th moment about the mean of X: E[(X-E(X))r] E(X)? sum f(x)*xintegral f(x)*x dx Properties of E(X) 1. E(b)=b, b is a scalar2. Y=aX+b => E(Y)=aE(X)+b3. if X and Y are independent, then E(XY)=E(X)*E(Y) 2nd moment about the mean = variance a measure of dispersionsum f(x)*(x-E(x))2 E[(x-E(x))2]? E[X^2]-(E[X]^2) 3rd moment about the mean skewnessif it>0 => positive skew (왼쪽에 봉우리)if it<0 => negative skew (오른쪽에 봉우리) 4th moment about the mean kurtosislow kurtosis: fat tailshigh kurtosis: thin tails Moment Generating Function (MGF) E(exp(xt))=M(t) => M(n)(t)=E(Xn) Normal Dist (mu,sigma2) f(x)=memorize?! Standard normal dist Z=(X-mu)/sigmawhen X~N(mu,sigma2) Chi square dist(d) d=degrees of freedomChi(d)=sum d of z2 t distribution (d) t=z/sqr(chi(d)/d)t->z as n->inf 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 Joint Distribution f(x,y) Marginal probability fx(x)=sumyf(x,y)fy(y)=sumxf(x,y) 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)] Covariance E[(x-E(x))*(y-E(y))]= E[xy]-E[x]*E[y] What if X and Y are independent? Cov=0 Correlatoin Cov(x,y)/(st.dev(x) st.dev(y)) Q. Correlation=0 => independent? No Var-Cov matrix diagonal = var(xi)off-diagonal=Cov(xi,xj) Conditional Distribution f(y|x)=f(x,y)/fx(x) Distributions of functions of r.v.s a. change of variablesb. using MGF a. Assume that we know f(x) & y=g(x)1. x=g-1(y)2. dx/dy3. domain of y4. f(g-1(y))abs(dx/dy)or f(g-1(y))det(dx/dy) b. using MGF e.g. E[exp(axt)] Statistics A function of r.v.s that does not dependent on unknown parameters e.g. sample mean, median... 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) Estimators vs. Estimates Estimators (statistics) = A formula for using data to estimate a parameters Estimates = the value you get by plugging data into estimators Method of moments sample moments=popoulation momentse.g. sum(xi)/n = E[x] Maximum likelihood estimation : likelihood function & log-likelihood fn. cf. dist is knownmaximize L(theta|x1,...,xn) or lnL(.) MLE procedures 1. Find L by multiplying f(xi)'s2. Take the log (not necessarily)3. Find the theta's to maximize lnL(.)4. Use FOC=05. Check SOC: negative definite Ways to evaluate estimators 1. Monte-Carlo Analysis2. Pre-data anlysis (small/large sample properties) Small Sample Properties 1. Unbiasedness2. Variance (Precision)3. Mean Square Error4. Efficiency Unbiased E(theta_hat)=theta Bias=E(theta_hat)-theta Variance We prefer an estimator with smaller variance 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] Efficiency Unbiased &the smallest variance => Cramer-Rao lower boundif the estimator is unbiased, the variance >=CRLB=[-E[SOC of lnL(.)]]-1 cf. sufficient condition, not necessary condition Large sample property =asymptotic property as the sample size -> inf 1. consistent2. asymptotically efficient Consistency plim theta_hat=theta Asymptotically efficient consistent & the smallest asymptotic variance Convergence in Probability . . pxn-->climn->inf Pr(|xn-c|>eps)=0<=> limn->infPr(|xn-c|0 Mean Square Convergence . . msxn--->cmun converges to c & sigma2n converges to 0 as n->inf 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)=muVar(sample mean)=sigma2/nas n->inf, E(.)->mu & Var(.)->0, thus it is consistent 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 Convergence in Distribution F(x): limiting distributionif limn->inf|Fn(xn)-F(x)|=0 at all continuity points of F(x). . dxn-->x Convergence in dist. Q. xn converges to constant? No. different form the convergence in probability Convergence in dist. related to CLT 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 populationFormal 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) Repeated sampling Get samples from the identical population distribution Difference b/w joint dist & likelihood fn. Joint dist = L(x1,..,xn|theta)Likelihood = L(theta|x1,...,xn) 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 Authorlucia831124 ID78957 Card Seteconometrics.txt DescriptionEconometrics Updated2011-04-12T07:09:15Z Show Answers