Efficient maximum likelihood estimation of linear models with APL

Abstract

Maximum likelihood estimators used in statistics and econometrics have desirable properties, however, due to the complexity of their solution maximum likelihood techniques are not widely used. This paper examines an APL implementation of one of the most efficient algorithms used to estimate the parameters of the univariate ARMA process. The straight-forward use of APL is found to be unacceptable and so a systematic search for optimizations is made. This search results in an approximate solution for the ML estimator, the use of □NA and FORTRAN, and special matrix techniques to increase the efficiency of the algorithm. The matrix techniques used are implementations of sparse, banded, and block diagonal data structures. Additional matrix techniques involve incremental updating of matrices. The effect of these optimizations is to bring the computational cost from an O(N*3) problem to an O(N*1) problem. This translates into a significant reduction in computer requirements. For example, to estimate a three parameter ARMA model with 5000 observations, memory requirements fall from approximately 95 megabytes to 300 kilobytes. Computer time falls from about 6.4 years to under 1 minute.