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59 Cards in this Set

  • Front
  • Back
What is a martingale difference sequence?
E[Mt+1|Ft] = 0 for all t
and {Ft} and {Mt} are adapted.
What is the stationary distribution?

If x_o = CapPi_inf then {x_t} is identically distributed.
What is global stability?
Regardless of x_0 you end up at the CapPi_inf stationary distribution.
What is required for the LLN to hold in time series?
IID, or at least that Cov(x_t , x_t+j) ~ 0 when j is large, then it hold asymptotically.
Global stability in markov series is equivalent to what?
Ro < 1. Then Cov(x_t , x_t+j) ~ 0 when j is large. Asymptotically independent.
What is the continuous mapping theorem?
If g is a continuous function and xn converges in probability to x then g(xn) converges in probability to x.

If g is a continuous function and xn converges in distribution to x then g(xn) converges in distribution to x.
What is slutsky's theorem?
If Yn converges in probability to a matrix C and xn converges in distribution to a random matrix x, then

Ynxn -d-> Cx and Yn + xn -d-> C + x
What is the formula for f to be a density of x?
Integral B f(s)ds = P{x memberof B} for all B subset R^k.
What is OPT mark I?
There exists a solution to ^y := argmin ||y-z||

^y member of S
y - ^y orthogonal to S
Requirements for OPT Mark II?
Py member of S
y-Py orthogonal to S
||y||^2 = ||Py||^2 + ||y-Py||^2
||Py|| =< ||y||
Py = y if y member of S
y = Py + My
Define the inner product and norm
Inner product of x and y is
||xy|| = (x'y)
<x,y> := E[xy]

||xy|| = sqrt((x'y))
<x,y> := sqrt(E[xy])
What is L_2?
L_2 := { all RVs x with E[x^2] < inf }
What is ||| x - y ||| when x and y are RVs?
What are the requirements for S to be a linear subspace of L2?
If a and b are both real numbers, and x and y are members of S then ax + by is a member of S.
If y is g-measureable then
It can be written as a deterministic function of the contents of g.
What's L2(g)? What a property has it?
All of the g-measurable RVs in L2.The set is a linear subspace of L2.
OPT for random variables and some resulting properties
Given linear subspace S of L2 and y in L2, there is a unique y memberof S, such that

|||y-^y||| <= |||y-z||| for all z member of S

Py memberof S
y - Py orthogonal to S
Py = y iff y memberof S
Rigorously define E[y | g]
It's the unique element of L2 such that

E[y | g] is g-measurable
E[E[y | g]z] = E[yz] for all g-measurable z memberof L2
Define covariance and variance of random vectors x (and y) conditional on Z.
Cov: E[xy' | Z] - E[x | Z] E[y | Z]'
Var: E[xx' | Z] - E[x | Z] E[x | Z]'
What is the TSS, SSR, ESS, R^2?
TSS: ||y||^2
SSR: ||My||^2
ESS: ||Py||^2
R2: ||Py||^2/||y||^2
What is Weierstrass theorem?
Given continuous function f, there exists a polynomial function g such that g is arbitrarily close to f.
What is the first moment assumption and some conclusions from it?
E[u | X] = 0 for all X.

From this E[u] = 0
E[um | xnk] = 0
E[umxnk] = 0
Cov[um,xnk] = 0
What are some other expressions for u?
My and Mu. Useful.
What are the three main assumption necessary for OLS?
y = Xbeta + u
E[u | X] = 0
E[uu' | X] = sigma^2 I
Why does the trace of P = K?
Trace(P) = Trace(X(X'X)-1X') = Trace((X'X)-1X'X) = Trace(I) = K
What is necessary for vector: xn -p-> x ?
xnk -p-> xk for all k
What is a transition density?
p(.|s) := conditional density on xt+1 when xt = s.
What makes a sequence {Ft} a filtration? What makes a sequence {Ft} adapted to the RV sequence {mt}?
If Ft is always a subset of Ft+1.
If mt is always Ft measurable.
What is required for Markov stability?
g() is continuous, lambda < 1 and L is a positive constant
littlePhi is density with finite mean and positive probability at all points

||g(s)|| <= lambda ||s|| + L for all s memberof Rk

then there exists a unique globally stable stationary density pi_inf
What comes out of a CDF F being symmetric?
F(-s) = 1-F(s)
What are some conditions for a probability mass function and the formula?
all 0 <= pj <= 1 and the sum of pj = 1

F(s) = Sum ( 1(sj < s) * pj)
What are some conditions for a density f?
Integrates to 1 and is nonnegative at all points. One to one correspondence with an F.
What is the quantile function F-1(q)?
:= the unique s such that F(s) = q
0 <= q <= 1
What is an intuitive idea of conditional expectation in F2?
The E[y | g] is the closest g measurable RV to y.
What is the difference between the joint and marginal density.
Joint: F(s1,..., sn) = |P(x1<s1, ..., xn < sn)
Marginal: F(s1) = |P (x1 < s1)

Joint may not exist, is a large integral. It's not just the same.
What is the conditional density?
p(sk+1 etc | s1 ... sk) = p(sk+q etc) / p(s1 ... sk)
What is identical distribution and independence for distributions?
Fn = Fm for all n and m
F(s1, ..., sN) = F(s1) * ... * F(sn)
What is the Kth moment?
E[x^k] = Int s^k F(ds)
Expand: Var(alpha x + beta y)
alpha^2 Var(x) + beta^2 Var(y) + 2 * alpha * beta Cov(x,y)
Describe the CLT
If the second moment of an xn is finite and draws are IID, then sqrt(n) (xbarN - mu) -d-> Normal(0, Var(xn))
Where mu = E[xn]
What is the LLN
If xn is an IID sequence with finite second moment, then xbarN -p-> E[xn] = Int sF(ds) as N --> Inf

What is the mean square error?

And of an estimator?
E[(xn - x)^2]
Chebychev's inequality
P{|y| >= delta} <= E[y^2] / delta^2
What is asymptotic normality and what is the asymptotic variance?
sqrt(N) * (sigmahat - sigma) -d-> Normal(0, v(sigma))

v(sigma) is the asymptotic variance of the estimator
What is the principle of maximum likelihood?
Choose the parameters that would make the data you saw most likely.
What is a parametric versus nonparametric class?
More generally, a parametric class of densities is a set of densities pq indexed by a vector of parameters. This is a large set of densities that cannot be expressed as a parametric class. In such
cases, we say that the class of densities is nonparametric
What is the ECDF?
The empirical distribution of the sample is the discrete distribution that puts equal probability on each sample point. The cdf for the empirical distribution is called the empirical cumulative distribution function, or ecdf.
What is the plugin estimator?
The average of the realizations (variance) across a sample.
What is a test?
Formally, a test is a binary function f mapping the observed data x into {0, 1}.
What is the power function?
The power function associated with the test f is the probability that the test rejects when the data is generated by Mo.
Fact. If A idempotent, then what do we know about the trace?
rank(A) = trace(A)
(AB)-1 = B-1A-1?
Yes, if A and B are both invertible.
rank(A) := ?
span(X) := ?
sum of all akXk such that (a1, . . . , aK) members of RK
What is the kolmogorov distribution good for?
Producing asymptotic confidence sets of F.
What is the triangle inequality?
||x+y|| < ||x|| + ||y||
How is t-test distributed?
z = normal
Q = chisq
k = dof of chisq
How is F distributed?