Open Journal of Statistics
Vol. 2  No. 5 (2012) , Article ID: 25546 , 6 pages DOI:10.4236/ojs.2012.25064

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

Pham Van Khanh

Military Technical Academy, Hanoi, Vietnam


Received October 14, 2012; revised November 14, 2012; accepted November 27, 2012

Keywords: Random Coefficient Autoregressive Model; Quasi-Maximum Likelihood; Consistency


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.

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


where are random vectors with independent identical distribution defined in a certain probability space(2)

Firstly, we consider the property of the stochastic variable



Lemma 1. Suppose that condition (2) satisfied,




Y determined by (3) will be absolute convergence with probability 1.


Assume, according to the law of great numbers, existing stochastic variable such that:


where. Then


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.



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



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)


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:


where is a certain optional appropriate area of


Then (8) can be written as



Now, the consistence of maximum livelihood estimates is said.

Theorem 2. Suppose (2), (4), (5), (8), (9) satisfied and. We have





We will prove be continuous on.


On the other hand,



is a continuous function in acordance with

. Next, we will prove:


In fact,


and if and only if


If or,


But is a stationary series

But, take conditional expectations in both sides, we have:


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:.


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:


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.

6. 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.


  1. T. Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, Vol. 31, No. 3, 1986, pp. 307-327. doi:10.1016/0304-4076(86)90063-1
  2. D. Nicholls and B. Quinn, “Random Coefficient Autoregressive Models: An Introduction,” Springer, New York, 1982. doi:10.1007/978-1-4684-6273-9
  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.x