Parameter Estimation for ARMA(1,1) Model with MATLAB Implementation

The ARMA(1,1) model is a widely used time series model that combines the characteristics of both Autoregressive (AR) and Moving Average (MA) components. In the MATLAB environment, parameter estimation can be achieved through various methods, including Maximum Likelihood Estimation (MLE) and Least Squares Estimation (LSE).

The fundamental form of the ARMA(1,1) model can be expressed as: the current value of the time series is determined by both its previous value and white noise terms. The objective of parameter estimation is to identify optimal autoregressive and moving average coefficients that enable the model to best fit the observed data.

In MATLAB, parameter estimation can be performed using the `armax` or `arima` functions. These functions employ optimization algorithms based on Maximum Likelihood Estimation or Least Squares methods to automatically compute optimal parameters. After inputting the time series data, the functions return estimated parameter values along with their statistical significance metrics.

Furthermore, MATLAB provides diagnostic tools such as residual analysis to validate model adequacy. If the residuals exhibit white noise properties, it indicates good model fit; otherwise, model order adjustment or alternative optimization approaches may be necessary.

For advanced requirements, users can customize objective functions and leverage MATLAB's Optimization Toolbox to further refine the estimation process and enhance parameter accuracy. This allows for implementation of specialized estimation algorithms and constrained optimization techniques tailored to specific research needs.