Improved GMM Estimation of AR(1) Time Series with a Root near 1

Received 16/Jan/2018, Revised 30/Jan/2018, Accepted 22/Feb/2018, Online 28/Feb/2018 Abstract—In this paper, to estimate AR(1) time series model First-difference GMM and Level GMM estimation methods have been considered, which have already performed well for estimation of AR(1) panel data model. A Monte Carlo simulation is carried out in order to study the performances of the above mentioned estimators and OLS estimator. Further, comparison among these estimators have been done in terms of bias and RMSE. Study reveals that, in many cases the OLS and First difference GMM estimators behave same in terms of Bias and RMSE. For all the negative values of autoregressive parameter the RMSE and bias of Level GMM estimator is larger than the remaining estimators. But in the case of positive values of autoregressive parameter Level GMM estimator performs better than First-difference GMM and OLS estimators especially, when sample size is small and autoregressive parameter is close to one.


I. INTRODUCTION
There are numerous time series models available in the literature. The most widely used models are the Autoregressive (AR) models, the Integrated (I) models and the Moving Average (MA) models. A common approach for modelling time series is AR model. The first order autoregression (AR(1)) is simple time series model, which can be analyzed through various standard methods. One of them is Ordinary Least Squares(OLS). The pioneers who worked in the area of OLS estimation of AR(1) time series model are [1] to [4] and more recent contributions include [5] to [7]. Their findings show that, the bias of the OLS estimator becomes large when sample size is small and an autoregressive parameter is near to unity. The OLS method requires the assumption of orthogonality between the error term and regressor, which is often not satisfied in various applications. In such cases, the OLS estimator becomes inconsistent. To overcome this problem many estimation methods emerged in the study of estimation of AR(1) model. GMM being one of them relaxes the assumption of orthogonality and is used for estimation of AR(1) model, see [8] to [17]. Recent studies show an estimation and inference of a panel AR(1) model with small T. In the context of panel data, to estimate AR(1) model many estimation methods are proposed. Two consistent estimation methods among them are First-difference GMM (Dif) Proposed by [18] and Level GMM (Lev) introduced by [19]. In this paper, the above mentioned two estimation methods to estimate AR(1) time series model have been considered. In First-difference GMM method, the constant is removed from the AR(1) model and then instruments from the differenced AR(1) model are considered, where as in Level GMM estimation method the constant is removed directly from the instruments and GMM estimation is performed. The bias and RMSE of the above two estimators are compared along with OLS estimator through simulation results.
The paper is organized as follows; Section 2 provides model, assumptions and model estimators. Section 3 presents Monte Carlo simulation to investigate the performances of the considered estimators. Section 4 contains results and discussion. Finally, section 5 concludes the paper.

II. THE MODEL AND ESTIMATORS
First order autoregressive time series model Following are the assumptions: The autoregressive process t y is initialized at some random

First-Difference GMM Estimation:
In the model (1), the constant  causes a correlation between the lagged endogeneous variable  [20] and H in the estimator proposed by [18].

Level GMM Estimation:
On the basis of Arellano and Bover (1995) Level GMM estimator is proposed for AR(1) time series model. Here the constant  is wiped out from the instrumental variable. The one-step Level GMM estimator is based on the moment conditions

III. MONTE CARLO SIMULATION STUDY
In this Monte Carlo simulation study, the data is generated from the following AR(1) model to investigate the finite sample performance of the above mentioned estimators.
which is similar to the assumption in [21]. For the parameters,

IV. RESULTS AND DISCUSSION
The results of the study are discussed through the tables and The graphs are plotted in figures 1-4 for the simulation results. Figure 1 depicts the comparison of means of  (dif),  (lev) and  (ols) with reference to true line over the entire range of  . From all graphs in figure 1, it is observed that, in terms of bias,  (dif) and  (ols) perform almost same in all the cases except at T=5. When 0 <  ,  (lev) has greater bias than other two estimators, but when  is near to unity  (lev) has small bias than other two. To understand clearly, in figure 2, the means of above three estimators have been plotted only for 0.5 >  .  (lev) has small bias when T is not so large. As T increases, bias of  (lev) also increases, although  (lev) has smallest bias for 0.80 >  for all values of T. Figure 3 shows the distinction of RMSE of  (lev),  (dif) and  (ols) for entire range of  . From figure 3, it is noticed that  (dif) and  (ols) behave almost same in all the cases except at T=5. RMSE of  (lev) is largest in negative range of  . Next, it starts decreasing from -0.5 and performs better than other two estimators as  approaches unity. For more clarity, the RMSE of above considered estimators is plotted in figure 4 for 0.5 >  . From figure 4, it is observed that,  (lev) has less RMSE, especially when T is too small. As T increases  (dif) and  (ols) perform better than  (lev) but as  approaches unity  (lev) performs excellent than remaining two estimators.

V. CONCLUSION
In this study, an estimation of AR(1) time series model is done by using First-difference GMM (Arellano and Bond (1991))and Level GMM (Arellano and Bover (1995)) estimation methods. Monte Carlo simulation is carried out to investigate the performances of the considered estimators. when  is close to unity, the Level GMM estimator has small bias and is more efficient than First-difference GMM and OLS estimators.