A Simplified Approach to Understanding the Kalman Filter Technique

The Kalman Filter is a time series estimation algorithm that is applied extensively in the field of engineering and recently (relative to engineering) in the field of finance and economics. However, presentations of the technique are somewhat intimidating despite the relative ease of generating the algorithm. This paper presents the Kalman Filter in a simplified manner and produces an example of an application of the algorithm in Excel. This scaled down version of the Kalman filter can be introduced in the (advanced) undergraduate classroom as well as the graduate classroom. 2 INTRODUCTION: Many models in economics and finance depend on data that are not observable. These unobserved data are usually in a context in which it is desirable for a model to predict future events. The Kalman Filter has been used to estimate an unobservable source of jumps in stock returns, unobservable noise in equity index levels, unobservable parameters and state variables in commodity futures prices, unobservable inflation expectations, unobservable stock betas, and unobservable hedge ratios across interest rate contracts 1. In the field of engineering a Kalman Filter (Kalman, 1960) is employed for similar problems involving physical phenomena. The technique is appearing more frequently in the fields of finance and economics. However, understanding the technique can be very difficult given the available resource material. When viewing chapter thirteen of Hamilton's Times Series Analysis text (1994), one can understand why the topic of Kalman Filters is generally reserved for the graduate classroom. However, as we will demonstrate, the technique is not quite as difficult as one may perceive initially and has similarities to standard linear regression analysis. Consequently, if placed in the correct context, it is accessible to the undergraduate student. In order to make the Kalman Filter more accessible, an Excel application is developed in this paper to work the student through the mechanics of the process. In the first section, a derivation of the Kalman Filter algorithm is presented in a univariate context and a connection is made between the algorithm and linear regression. In the second section, the Kalman Filter is combined with Maximum Likelihood Estimation (MLE) to create an iterative process for parameter estimation. In the third 3 section, an Excel application/example of using the Kalman Filter/MLE iterative routine is performed. There are two basic building blocks of a Kalman Filter, the measurement equation and the transition equation. The measurement equation relates an unobserved variable (X …