Non-Linear Programming for AR(2) Parameter Estimation: Gradient Projection and Convex Quadratic Programming Methods

Assessing parameters in autoregressive (AR) models is a fundamental aspect of time-series analysis and is widely used in fields such as finance, climate science, and engineering. Ordinary least squares and maximum likelihood estimation are commonly used for this purpose, but their performance can be affected when the estimation process is subject to complex constraints. In this study, two nonlinear programming techniques—the Gradient Projection (GP) method and Convex Quadratic Programming (CQP) based on the complementary pivot algorithm—were examined for estimating the parameters of an AR(2) model. The two nonlinear methods were compared with traditional estimation techniques using the Mean Squared Error (MSE) as the primary criterion. The findings indicated that both nonlinear methods achieved lower MSE values than the conventional methods. The GP method gave the smallest MSE (25.8959), while CQP produced a value very similar to this (25.8960), showing that both achieved comparable estimation accuracy while meeting the stationary constraints of the AR(2) model. The GP method also required fewer computational resources and converged quickly, whereas the CQP approach converged to the global optimum under the assumed convexity conditions. In general, the results show the promise of nonlinear programming for achieving accurate and computationally efficient parameter estimation in AR(2) models and offer a foundation for applying optimization-based estimation to higher-order autoregressive models.