Notational terms

Question 1

Law of Iterated Expectations LIE

Answer 1

E(X)=E(E(X|Y))

Question 2

Covariance

Answer 2

Cov(X,Y)=EXY-EXEY
Cov(X,Y)=Cov(Y,X)
Cov(X,a)=0

Question 3

Variance

Answer 3

Var(X)=Cov(X,X)=E(X^2)-(EX)^2

Question 4

Var(βj^|X)=

Answer 4

σ^2/(1-Rj^2)Σ(xij-xjbar)^2, j=0,1,…,k

Question 5

Varβ^=

Answer 5

E(β^-β)(β^-β)’ = σ^2E(X’X)^-1

Question 6

S^kk=

Answer 6

((X’X)^-1)_(k,k)

Question 7

standard error of the regression formula:

Answer 7

s^2= (ε^’ ε^)/(n-K) or s^2=(Σ(Xi-X̄)^2)/(n-1)

Question 8

Approximate distribution of β^:

Answer 8

β^ –> N(β, (σ^2E(xi’xi)^-1)/n) (asymptotically)

Question 9

Law of Large Numbers LLN

Answer 9

If x1,…,xn is a random sample with finite mean μ and variance σ^2, then the sequence of sample means converges in probability to μ, so X̄–>μ (in probability)

Question 10

Weak Law of Large Numbers WLLN

Answer 10

X̄–>μ (in probability) when n–>∞

Question 11

Convergence in probability implies

Answer 11

convergence in distribution

Question 12

If Xn–>c (in distribution), then

Answer 12

Xn–>c (in probability)

Question 13

Taylor Expansion:

Answer 13

E(g(wi,θ))≈E(g(wi,θ0))+E(∂g(wi,θ0)/∂θ’)(θ-θ0)

Question 14

Jensen’s inequality:

Answer 14

E(g(h(y)))≤gE(h(y)) when g is concave

Question 15

The variance of the OLS estimator is increasing with … and …, decreasing with … and …

Answer 15

increasing with the variance of ε and Rj^2, decreasing with sample size and the sample variance of X

Question 16

The standard deviation of the sampling distribution of β^ is:

Answer 16

the standard error of β^

Question 17

An estimator is BLUE (Best Linear Unbiased Estimator) when:

Answer 17

var(β0^)-var(β^)≥0 is a positive semidefinite matrix for every other linear unbiased estimator β

Question 18

Var(aX+bY+c)=

Answer 18

Var(aX+bY)=a^2VarX+b^2VarY+2abCov(X,Y)

Question 19

Cov(aX+bY,cZ)=

Answer 19

acCov(X,Z) + bcCov(Y,Z)

Question 20

Correlation coefficient:

Answer 20

ϱ(X,Y)=Cov(X,Y)/√(VarXVarY) which is in [-1,1], it is the normalized covariance

Question 21

Var(X,Y)=

Answer 21

VarX+VarY+Cov(X,Y)+Cov(Y,X)

Question 22

Slutsky’s theorem

Answer 22

Suppose Xn–>c (in probability), where c is a constant and h is a real valued function continuous at c. Then h(Xn)–>h(c) (in probability)

Question 23

Xn converges in probability to a random variable X, if for any ε>0:

Answer 23

P(|Xn-X|>ε)–>0 as n–>∞

Question 24

Xn converges in distribution to a random variable X if:

Answer 24

Fn(x)–>F(x) for all x in R such that F is continuous in x, where Fn(x)=P(Xn≤X)

Question 25

Cramér Convergence Theorem:

Answer 25

1: Suppose Xn-->X (in distribution), Yn-->c (in probability), then: Xn+Yn-->X+c, XnYn-->Xc, Xn/Yn-->X/c 2: If Xn-->X (in probability), then Xn-->X (in distribution), the converse is true is X is a constant 3: If Xn-Yn-->0 (in probability) and Xn-->X (in distribution), then Yn-->X (in distribution)

Question 26

Continuous Mapping Theorem (CMT)

Answer 26

Suppose that Xn-->X (in distribution) and let h be a continuous function on a set χ with P(X∈χ)=1. Then h(Xn)-->h(X) (in distribution)

Question 27

Central Limit Theorem (CLT)

Answer 27

√n(X̄-μ)=(1/√n)Σ(Xi-μ)-->N(0,σ^2)