The First Order Autoregressive Model with Coefficient Contains Non-Negative Random Elements : Simulation and Esimation

This paper considered an autoregressive time series where the slope contains random components with non-negative values. The authors determine the stationary condition of the series to estimate its parameters by the quasi-maximum likelihood method. The authors also simulate and estimate the coefficients of the simulation chain. In this paper, we consider modeling and forecasting gold chain on the free market in Hanoi, Vietnam.


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][2][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., according to the law of great numbers, existing stochastic variable i such that:

Stationary Conditions of the Series
nd in accordance with lemma Borel-Cantelli, sufficient condition here means proving 6) can always correct with some   .Therefore, ( 7) is always true.Lemma 2. Suppose that (2) and ( 5) meet 0 0 with some and We have , and owing to

 
M t is a decreasing function in the neighborhood of 0. Hence, existing 0 , we obtain: 2) and ( 5) are satisfied with 0 0 1: , and Proof.
Due to condition (2) and inequality Minkowski Theorem 1: Suppose that (1), ( 4) and (5) satisfied with the almost sure convergence of and process   is the stationary solution of (1) Proof.
k is convergent absolutely, acording to Lemma 1 We have: is the single solution of ( 1)

Estimation of Model Parameters
Suppose that In this section, we care about estimating vectors of  , we have: With Therefore, we have following likelihood function Copyright © 2012 SciRes.OJS Maximum likelihood estimators determined by: where is a certain optional appropriate area of Let Now, the consistence of maximum livelihood estimates n  is said.Theorem 2. Suppose (2), (4), ( 5), ( 8), (9) satisfied and and Eg u   We will prove be con- On the other hand,

Then
Copyright © 2012 SciRes.OJS is a continuous function in acordance with .Next, we will prove: where

P Y c c b c e b eY as
, take conditional expectations 1 1 , e b in both sides, we have: is stationary and ergodic with   1 E g    , according to Ergodic theorem, we have: -compact set in  with positive distance to  .Owing to g 1 (u) being continuous in , existing an open sphere U(u) with center u with Copyright © 2012 SciRes.OJS such that: Sets are open covers of C, so C holds such finite open covers, are called 1 2 k of C. In accordance with Ergodic thoerem, with every , we have: In out of events with with satis-

Simulation
In this section, we simulate series ( 1

Conclusion
This paper has solved some problems relating to a kind of first order time series with coefficient regression affected by non-negative random elements.In subsequent studies, the author will consider the asymptotic estimates of the parameters.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   .0002,0.0069 is .So, we [3] A. Aue, L. Horvath and J. Steinbach, "Estimation in Random Coefficient Autoregressive Models," Journal of Time Series Analysis, Vol. 27, No. 1, 2006,  pp.61-76.doi:10.1111/j.1467-9892.2005.00453.xcan use the following model to forecast the future value of gold price: Firstly, we consider the property of the stochastic variable

Figure 4 .
Figure 4. Simulation for series Y t defined by (1) with