On geometric ergodicity of CHARME models
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Citations
Dilemmas of Robust Analysis of Economic Data Streams
Monitoring time series based on estimating functions
A uniform central limit theorem for neural network-based autoregressive processes with applications to change-point analysis
Change-Point Methods for Multivariate Autoregressive Models and Multiple Structural Breaks in the Mean
Volatility and correlation: Modeling and forecasting using Support Vector Machines
References
Maximum likelihood from incomplete data via the EM algorithm
Markov Chains and Stochastic Stability
Adaptive mixtures of local experts
On the convergence properties of the em algorithm
Related Papers (5)
Frequently Asked Questions (11)
Q2. What is the main purpose of CHARME?
CHARME is quite useful for modeling time series data which are piecewise stationary such that their dynamics switch sometimes from one state to another.
Q3. what is the t if e is a unit vector with the kth?
Then A1 contains at least one integer between 1 and K and it is enough to prove that there exists t such thatP (( St+1 Xt+1 ) ∈ {e} ×A2 |S1 = sl, X1 = x ) > 0with e a unit vector with the the kth component equal 1 and sl a unit vector with the lth component equal 1.
Q4. what is the l-th component of x?
Let x = (xt, · · · , xt−p+1)′ be a vector of real numbers and sl a K-dimensional unit vector with the l-th component equal 1 and considerE(g(ζt+1) | (Xt, · · · , Xt−p+1)′ = x, St = sl)= 1 + K∑k=1alk(m2k(x) + σ 2 k(x)) + bp−1x 2 t + · · ·+ b1x2t−p+2 (3.1)Now, let us focus onK∑ k=1 alk(m2k(x) + σ 2 k(x))=
Q5. What is the proof of the central limit theorem?
If the process {Xt} is also strictly stationary it is well known that this implies that {Xt} is absolutely regular with geometric decreasing rate, which gives a very useful condition for deriving limit theorems like the central limit theorem.
Q6. What is the simplest way to prove that t is irreducible?
0and doing so iteratively, the authors obtainP (( St+1 Xt+1 ) ∈ {e} ×A2 |S1 = sl, X1 = x ) =K∑ j,··· ,jt−1 alj1 · · · ajt−1kbj,··· ,jt−1(x)which is strictly greater than 0 for some t because of the irreducibility of {Qt} and the fact that bj,··· ,jt−1(x) > 0.• Analogously it can easily be seen that {ζt} is aperiodic.•
Q7. What is the main topic of this paper?
In this paper the authors are analyzing the so-called Conditional Heteroscedastic Autoregressive Mixture of Experts, henceforth CHARME, models.
Q8. What is the simplest way to explain the process Qt?
{St = (St1, · · · , StK)′} inherits the properties of {Qt}.A. 2 Let Gt−1 = σ{Xr, r ≤ t−1} be the σ-algebra generated by {Xr, r ≤ t−1} and Gt−1 any event in Gt−1.
Q9. What is the proof of the theorem 4?
Geometric ergodicity of ζt can clearly be obtained even if some of the underlying dynamics taken on their own are not geometric ergodic or even stationary, provided the probability to go from a stable dynamic to a non stable dynamic is low enough and the probability to move from a non stable dynamic to a stable dynamic is large enough.
Q10. what is the simplest way to prove the existence of a petite set?
In particular in their case where {φt} is a Markov chain, it is enough to prove the existence of a petite set C, a function g ≥ 1 and constants 0 < r < 1 and B > 0 such thatE(g(φt) |φt−1 = x) < rg(x), x 6∈ C E(g(φt) |φt−1 = x) < B, x ∈ C.
Q11. What is the main difference between the two types of CHARME models?
Independently of the type of estimates considered, a crucial condition for developing a theory for estimation and testing in the setting of CHARME is the existence of a stochastic process satisfying (1.2) which is geometric ergodic.