The First Order Autoregressive Model with Coefficient Contains Non-Negative Random Elements: Simulation and Esimation ()
1. Introduction
It is well-known that many time series in finance such as stock returns exhibit leptokurtosis, time-varying volatility and volatility clusters. The generalized autoregressive conditional heteroscedasticity (GARCH) and the random coefficient autoregressive (RCA) model have been caturing three characteristics of financial returns.
The RCA models have been studied by several authors [1-3]. Most of their theoreic properties are well-known, including conditions for the existence and the uniqueness of a stationary solution, or for the existence of moments for the stationary distribution. In this paper, we address the stationary conditions for the RCA model, the existence and the uniqueness of a stationary solution and parameter estimation problem for the RCA model with the coefficient have a non-negative random elements.
2. Stationary Conditions of the Series
Consider time series 
satisfying
(1)
where 
are random vectors with independent identical distribution defined in a certain 
probability space(2)
Firstly, we consider the property of the stochastic variable
(3)
Let
.
Lemma 1. Suppose that condition (2) satisfied,
(4)
If
(5)
Y determined by (3) will be absolute convergence with probability 1.
Proof.
Assume
, according to the law of great numbers, existing stochastic variable 
such that:
(6)
where
. Then
(7)
We will prove
. Indeed, due to
and in accordance with lemma Borel-Cantelli, sufficient condition here means proving
with 
We have:
From (7), we have
.
If
, (6) can always correct with some
. Therefore, (7) is always true.
Lemma 2. Suppose that (2) and (5) meet
with some
. Then, existing 
such that
.
Proof.
Suppose 
We have
, and owing to
, 
is a decreasing function in the neighborhood of 0. Hence, existing 
such that
. Generally less, suppose that 
Due to the convex, we have 
với
.
Use condition (2) and
, we obtain:
Lemma 3. Assume (2) and (5) are satisfied with
. Then
.
Proof.
Due to condition (2) and inequality Minkowski
. Hence,
.
Theorem 1: Suppose that (1), (4) and (5) satisfied with the almost sure convergence of
and process 
is the stationary solution of (1)
Proof.
is convergent absolutely, acording to Lemma 1 We have:
. Therefore:
is the single solution of (1)
Obviously, 
is a stationary series and 
is independent of
.
3. Estimation of Model Parameters
Suppose that
In this section, we care about estimating vectors of 
based on Quasi-Maximum Likelihood method.
With
, we have:
but
, so
Therefore, we have following likelihood function
Maximum likelihood estimators determined by:
(8)
where 
is a certain optional appropriate area of 
Let
Then (8) can be written as 
Assume
(9)
Now, the consistence of maximum livelihood estimates 
is said.
Theorem 2. Suppose (2), (4), (5), (8), (9) satisfied and
. We have
.
Proof.
Let
and 
We will prove 
be continuous on
.
Indeed,
On the other hand,
But
Then
is a continuous function in acordance with
. Next, we will prove:
.
In fact,
where
and 
if and only if
If
If 
or
, 
But 
But 
is a stationary series
But
, take conditional expectations 
in both sides, we have:
But 
Return to theorem
, so
But series 
is stationary and ergodic with
, according to Ergodic theorem, we have:
With each positive integer 
is a continuous function in compact set G, so
Let 
-compact set in 
with positive distance to
. Owing to g1(u) being continuous in
, existing an open sphere U(u) with center u with 
such that:
.
Sets 
are open covers of C, so C holds such finite open covers, are called 
of C. In accordance with Ergodic thoerem, with every
, we have:
See that
In out of events 
with 
with 
satisfying:
.
Therefore, 
But 
is continuous and 
is singly minimum of 
Let U is a open sphere with center 
and enough small radius and
. If
, existing a random subseries 
such that with
, we have:
But 
hence, with each above
, existing random variable 
such that 
This completes the proof. 
4. Simulation
In this section, we simulate series (1) with different values of
. These simulations show stationary and non-stationary series cases.
We simulate series (1) with different values of 
and in each case we can check the stationary conditions of the series (1) by Lemma 1. In Figure 1, we see that the series is not stationary with the negagtive slope 
and in Figures 2 and 3 we simulate the not stationary series with positive slope 
and
. Figure 4 presents a stationary but clustering series, Figures 5-7 present stationary series with parameters are
, 
and
.
Figure 1. Simulation for series Yt defined by (1) with
.
Figure 2. Simulation for series Yt defined by (1) with
.
Figure 3. Simulation for series Yt defined by (1) with
.
Figure 4. Simulation for series Yt defined by (1) with
.
Figure 5. Simulation for series Yt defined by (1) with
.
Figure 6. Simulation for series Yt defined by (1) with
.
Figure 7. Simulation for series Yt defined by (1) with
.
5. Application for Real-Time Series
In this section, we use model (1) for the model of return series of the price of gold on the free market in Hanoi, Vietnam. Figure 8 show the Return series of Gold price
.
From the data series we estimate for vector
is
. So, we can use the following model to forecast the future value of gold price:
Figure 8. Return series of Gold price rt.
Figure 9. Simulation for series Yt defined by (1) with
.
Figure 9 below is a simulation of the process (1) with parameters
.