The Kalman Filter in Finance

As a beginning student of economics, I learned that economic coefficients were constants. A good example is provided by a passage in a introductory textbook on macro economics where Prof. Ackley “demonstrated” how he could use a “sex multiplier” to calculate changes in the enrollment in his economics class. (See Ackley (1961), pp. 309–312.) He would simply multiply the change in number of female students by his multiplier to get the change in total enrollment. As there were fewer women than men, this model saved him time. Of course, different years had different multipliers, but in the end it always worked. The point that Ackley was making was the difference between meaningful forecasts using ex ante models and tautologies derived from ex post calculations. “If we merely had some reason ... to assert the probable constancy or stability of the [sex multiplier]... then our theory would be ... potentially useful”, writes Ackley. The message is that theory should provide stable parameters that our analysis of the economy should be based on.¹

The second topic covered in this book will be the Kalman filter. As an analytical tool, it has been around since the early 1960’s when Kalman introduced the method as a different approach to statistical prediction and filtering. The problem addressed is that of estimating the state of a noisy system. For example, consider monitoring the position and the speed of an orbiting vehicle. Rather well known physical laws may be used to describe the system, but at any give moment of time, the exact position and speed of the vehicle may vary from that predicted by the model as there are always external forces at work that result in random impulses to the system. Again, consider the optimal control of an economic system which is described by estimated equations rather than exact physical laws. There will of course be a difference between the theoretical or estimated output from the model and the actual value of the economic variable due to the errors inherent in the model. To be more precise, consider estimating the marginal propensity to save (mps) out of disposable income. While it is possible to find a trend or steady state value for the mps, its actual value any given year may well deviate from this trend value. In both instances, a well defined concept may not be able to be observed exactly due to the random inputs into the system. Finally, consider the rather simple problem of estimating the time—varying mean of a stochastic system. The mean is assumed to follow a random walk which may or may not contain a deterministic parts. From historical data, an estimate of the mean at the current time is readily available. This estimate may be improved as additional observations of the process become available. Common to all four situations is the that the exact value of the variable studied is not observable and must be estimated.

The Kalman filter is has long been a standard tool for control engineers. However, the initial introduction of the filter to an audience of economists, while emphasizing the relevance of the filter in modeling economic systems, also pointed out the need to assume known covariance matrices for the various noise processes in the model. In an article in the Annals of Economic and Social Measurement, Athens (1972), after pointing out that the noise parameter in the system equation could represent “input uncertainties and deterministic modeling errors”, continues by saying: “Thus the covariance [of the error term in the system equation that is] selected by the designer should incorporate his judgment on the importance of the higher order terms in the validity of the linearized model. Thus, the ‘more nonlinear’ the system dynamics, the ‘larger’ the [covariance] should be. The white noise ... in the observation equation plays a similar role. Not only should it reflect the inherent uncertainty of the measurements due to sensor inaccuracies, but it should also be used to model the implications of neglecting [higher order terms] to obtain a linear equation.” (Athens (1972), p. 472, emphasis added)

In this chapter we reconsider some of the estimates. The first order of business is to address the question of those stocks for which the diagnostics were poor. We concentrate on 9 of the stocks, but what is said could just as well have been applied to the others with less than satisfactory estimation statistics.

Economic literature abounds with fixed coefficient models that could just as easily be estimated with the techniques outlined in chapters 4 and 5. There is no particular reason why variable coefficient models should be the exception rather than the rule. Indeed, that coefficients remain constant over a period of 15–20 years seems more unbelievable than the opposite. However, as the statistical methodology for fixed parameter models is more familiar to economists than those required for estimating variable coefficient models, it is models with constant parameters that abound in the literature. This chapter will present some examples of models that have been presented as ones with constant coefficients that just as easily could have been estimated using the techniques of this book.