Stochastic processes

Question 1

Stochastic process

Answer 1

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

Question 2

First order markov process

Answer 2

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)

Question 3

Strictly stationary process

Answer 3

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

Question 4

Covariance stationarity (weakly stationary)

Answer 4

Mean, and covariances are stationary.

• A covariance stationary normal process is stricly strationary

Question 5

White noise process

Answer 5

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.

Question 6

Ergodic process

Answer 6

• 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

Question 7

Sufficient condition for ergodicity

Answer 7

• Covariance stationarity
• Series of covariances if absolutely convergent

Question 8

LLN

Answer 8

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

Question 9

CLT

Answer 9

• Covariance stationarity
• Ergodicity

Question 10

Normality in AR(1)

Answer 10

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

Question 11

Variance of sum of covariance stationary variables

Answer 11

Question 12

Martigale difference sequence

Answer 12

Question 13

Independent white noise

Answer 13

White noise +

Question 14

Condition for applicability of CLT

Answer 14

w_t is an infinite-order moving average process

Question 15

AR(1) definition

Answer 15

Question 16

MA process

Answer 16

Question 17

ARMA(1,1)

Answer 17

Question 18

Super consistent estimator

Answer 18

Its rate of convergence is greater then sqrt(N)