I have underlined by green the things that I don\'t know how to get 2. Setup 2.1
ID: 3040919 • Letter: I
Question
I have underlined by green the things that I don't know how to get 2. Setup 2.1. Model and assumptions The RegARMA model we consider is specified as ..Xal is a d-dimensional vector of regression covariates where Y is the response variable at time t, X K ( 1- d) is the corresponding vector of regression coefficients, and lwa is an ARMA (pq) process satisfying recursions houn or in short (BW (BZ , where Bis the backward shift operator, 1 'z Oze is the AR polynomial, and variance . and finite fourth moment. Also, we assumie that the polynomials biz, and ror, have no common roots and that V . a(z) = 1 +0,2+ . . . +ee" is the MA polynomia.we assume that·Z) is a sequence of iid. tandom vanables with mean t, their roots lie outside the unit circle in the complex plane, Then, there exists aunique causal-invertible stationary solution W. Moreover, we assume that X:) is a strictly stationary and ergodic process with finite second moment, namely Eix, 12Explanation / Answer
Ergodic process: A random process is said to be ergodic if the time averages of the process tend to appropriate ensemble averages. This definition implies that with probability 1, any ensemble average of {X(t)} can be determined from a single sample function of {X(t)}. Clearly, for a process to be ergodic, it has to necessarily be stationary.
Lasso regression: In the orthonormal case, i.e. That is, the lasso estimate is related to the OLS estimate via the so-called soft threshold function. (depicted here for =1)