Abstract

This study investigates alternative methods for estimating the parameters of state space models. Traditionally, maximum likelihood estimation is used for this purpose. However, this approach relies on assuming a specific probability distribution for the observations, which may not always be appropriate. This research explores distribution-free estimators, which can be advantageous when the underlying distribution of the observations is unknown or difficult to determine. The study focuses on leveraging the properties of the Kalman filter, a widely used technique for state estimation in linear systems. The core idea lies in analyzing the statistical characteristics of the Kalman filter's predictors and smoothers. Predictors provide estimates of the system's state at the current time based on past observations, while smoothers estimate the state for all time steps using the entire set of observations.
By examining these Kalman filter outputs, this study establishes relationships that allow for distribution-free estimation of the state space model parameters. The estimation is performed using two approaches: the Generalized Method of Moments (GMM) and Least Squares. The GMM approach involves matching the moments of certain functions of the observations and the corresponding moments of their theoretical counterparts based on the state space model and the Kalman filter. This matching process is achieved through an iterative procedure that refines the parameter estimates until a satisfactory convergence is reached. Least Squares, on the other hand, is considered to obtain parameters estimates by linear equations established by the Kalman filter stochastic properties. Both GMM and Least Squares methods, in this context, rely on stochastic relations and first-order Taylor series approximations to linearize the relationships between the observations, the Kalman filter outputs, and the state space model parameters. Additionally, the gradients of Kalman predictors are calculated iteratively to guide the parameter estimation process.
Overall, this research aims to contribute to the distribution-free framework for estimating the parameters of state space models. By utilizing the Kalman filter's dynamics and employing GMM or Least Squares techniques, the approach can be particularly useful in scenarios where the assumption of a specific observation distribution is not valid. This can potentially lead to more robust and reliable parameter estimates in various applications that rely on state space models.