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Flashcards in Stochastic processes Deck (18)
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1
Q

Stochastic process

A

Def: sequence of random variables A realization of the random process is a realization of different random variables

2
Q

First order markov process

A

For all t, the conditional distribution of w_t depends only on the first lag f(w_t|w_t-1,w_t-2…)=f(w_t|w_t-1)

3
Q

Strictly stationary process

A

For each set of indices t1, …, tn, the joint distribution of w_t1, w_t2, w_t3… depends only on the differences between the indices.

  • Implications:
  • marginal distributions are stationary
  • moments of the w_t are time independent
  • Covariances between any two variables only depends on their distance in the sequence
4
Q

Covariance stationarity (weakly stationary)

A

Mean, and covariances are stationary.

  • A covariance stationary normal process is stricly strationary
5
Q

White noise process

A

Covariance stationary process with: - zero unconditional mean - constant unconditional variance - zero across-tome covariances If we add f(w_t|w_t-1,…,w1)=f(w_t) we have an independent white noise process.

6
Q

Ergodic process

A
  • A stationary process is ergodic if any two variables positioned far apart in the sequence are almost independently distributed.
    • An ergodic process satisfies lim gamma_j=0 as j goes to infty
  • Notice covariance stationarity do not imply ergodicity
7
Q

Sufficient condition for ergodicity

A
  • Covariance stationarity
  • Series of covariances if absolutely convergent
8
Q

LLN

A

We need:

  • Constant mean
  • Covariances depending only in differences of time indexes
    • i.e. covariance stationarity
  • Series of covariances converging absolutely
    • With this two combined we have ergodicity
9
Q

CLT

A
  • Covariance stationarity
  • Ergodicity
10
Q

Normality in AR(1)

A

If we have:

  • Conditional normality w_t|w_t-1 is normal
  • Weak stationarity
  • Normality of w_0

Then the whole process is jointly normal

11
Q

Variance of sum of covariance stationary variables

A
12
Q

Martigale difference sequence

A
13
Q

Independent white noise

A

White noise +

14
Q

Condition for applicability of CLT

A

w_t is an infinite-order moving average process

15
Q

AR(1) definition

A
16
Q

MA process

A
17
Q

ARMA(1,1)

A
18
Q

Super consistent estimator

A

Its rate of convergence is greater then sqrt(N)