Autoregressive Order Selection Criteria and their Performances at Different Sizes of Time Series

Accepted 08/Aug/2018, Online 30/Aug/2018 Abstract—In the present study, the most popular model selection criteria are used as the autoregressive order selection criteria and the performances of time series model selection criteria at different sizes of the same time series are observed. In time series analysis and forecasting, selecting the most suitable model for a given time series and size of available time series plays a vital role. We verified that Final Prediction Error Criterion and Akaike’s Information Criterion are asymptotically equivalent and Akaike’s Information Criterion and Bias-Corrected Akaike’s Information Criterion are asymptotically equivalent when the size of time series is large with respect to the dimension of the parameters of the autoregressive process using empirical study. All the time series model selection criteria presented in the paper are evaluated by log-likelihood function.


I. INTRODUCTION
Model section criteria play important role in the selection of most appropriate model or the best model among the candidate models for the given time series. Mainly, we are determining the optimal order of the autoregressive model by these model selection procedures. Selecting the order of autoregressive model is one of the critical issues in time series analysis. A review of literature on some time series model selection methods is presented in Section II. A detailed study of the derivation of the above model selection procedures for autoregressive models is presented in Section II given by the different authors. In Section III, the most useful time series model selection procedures by Information Criteria are tabulated and a brief overview of the present study is discussed. In Section IV, we generated the autoregressive process of order 2 and the time series model selection procedures are used for selecting the optimal order of autoregressive process. Mainly, we observed the performance of the time series model selection procedures at different sizes of the same autoregressive process. In the present study, the behaviour of FPE, AIC, AICc, BIC, HQC, and MDL have been studied under standard normal errors. For this study, we used the most popular and powerful Rsoftware. Section V contains the conclusions.

II. SOME TIME SERIES MODEL SELECTION CRITERIA
Most popular and widely used model selection procedures for time series are discussed below

A. Final Prediction Error (FPE) Criterion:
Originally, the FPE was designed for autoregressive time series models. The FPE Criterion was developed by Akaike (1969) to select the appropriate order of the autoregressive process to fit a time series data. The final prediction error (FPE) criterion has been used widely in time series model selection. Suppose that 12 ,, n y y y is an observed series from AR (p) process and 12 ,, n x x x is an observed series from the same process which is independent of   t y and pn  [1]. Thus the model is    where n is the number of values in the estimation data.

B. Akaike Information Criterion (AIC):
AIC is probably the most commonly used model selection criterion for time series data. The most fundamental model in time series analysis is autoregressive model [2]. In the autoregressive model, the present value of the time series is expressed as a linear combination of past values of the time series and the random component. The AR (p) model is  is the estimated error or innovation variance for the fitted p th order candidate model.

C. Bias-Corrected Akaike Information Criterion (AICc):
The true auto regressive model with true order * p is Twice of the dimension ofâ may be much smaller than the bias adjustment (4), so AIC is negatively biased estimator of  (5), but it has no impact on selection behavior of criterion, by including additive constant.

Derivation of AICc for Autoregressive Models
Expectation of (-2 log-likelihood) of the true model is Expectation of (-2 log-likelihood) of the fitted model is The Taylor expansion of the log-likelihood  Where   J  is Fisher's information matrix [5,10].
The Minimum Description length is defined as

III. METHODOLOGY
The true auto regressive model with true order * p is Autoregressive order selection criterion like FPE, AIC, AICc, BIC, HQC, and MDL is used to select the best model. To fit the models and to find the optimal order of autoregressive model, we used packages "fpp", "forecast", "lmtest", "zoo", "fma", "expsmooth", "tseries" in R-Software.
The model with the minimum FPE, AIC, AICc, BIC, HQC, and MDL is highlighted with gray colour in Table 2      Bayesian information criterion is not performing well for n =15 and 30.
HQC for each candidate model for different sizes of the same time series are obtained using R Software and these values are presented in table 6. Hannan-Quinn criterion is not performing well for n =15.
MDL for each candidate model for different sizes of the same time series are obtained using R Software and these values are presented in table 7. Minimum description length is not performing well for n =15, 30.The performance of minimum description length is similar as the performance of Bayesian information criterion in model selection.

V. CONCLUSION and Future Scope
In the present work, we have investigated the use of autoregressive order selection criteria for determining the order of autoregressive model. These autoregressive order selection procedures useful for obtaining best model among the candidate models in time series and forecasting. Final Prediction Error (FPE) Criterion, Akaike Information Criterion (AIC), Bias-Corrected Akaike Information Criterion (AICc), Bayesian Information Criterion (BIC), Hannan and Quinn Criterion (HQC) and Minimum Description Length (MDL) are also useful for determining best ARMA model among the candidate ARMA models.