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In recent years there has been increasing interest in a good technique for identifying the order of the autoregression. In spite of the fact that several testing procedures have been proposed for testing the adequacy of an estimated time series model, the available tests are not, however, satisfactory. They are subjective, and the power of many of the tests to discriminate between different models is not good. Many testing procedures for identifying the order of an autoregressive model are based on the use of the information theoretic criteria introduced by Akaike and by Schwartz and Rissanen. In this paper, a completely different technique is presented to recognize the order of an autoregressive model. It avoids the setting of a penalty function, the choice of which depends on the criterion used. The approach which is taken here is to apply the theory of generalized likelihood ratio testing for composite hypothesis testing. A procedure is used for calculating the exact likelihood function for a Gaussian zero-mean autoregressive process, and then we show that the true maximum likelihood parameter estimates for this process can be obtained. A generalized likelihood ratio test, coupled with the proposed estimation scheme, is used to solve the order recognition problem. The decision rule about the model order for data observed from a Gaussian zero-mean autoregressive process is based on the generalized long-likelihood ratio statistics, which are computed and combined by Fisher's method to form a new test for comparing the model orders. In the new technique, the autoregressive order is estimated by increasing the order of autoregression until the corresponding null hypothesis is accepted. The first autoregressive model, i.e., the autoregressive model with the lowest order transforming the observed time series into test for this hypothesis, gives the estimated order. In essence, the proposed technique represents an iterative procedure which consists of repeated use of the above decision rule. The results of computer simulation are presented as an evidnce of the validity of the theoretical predictions of performance.
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