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Stock prices show interesting behavior over time. For example, the price variations are often modeled with a deterministic part and a random part where the deterministic part has usually a slow growing (or decaying) pattern which shows the long term movements of the value of the asset. Long-term investors are usually interested in the deterministic part, as it is relatively less risk prone. On the other hand, a naive model of the stochastic part is that it is a “random noise” with a bit of colouring. The “colouring” sets the random variables apart from the white noise. These stochastic components (identified and denoted by the stochastic variables in the process or observation models) mostly govern the risks associated with the stock prices. Determining the patterns or the “colours of the coloured noise”, is a fascinating field of study called technical analysis for the short term market players. Modeling the said stochastic components remains challenging for the researchers since these components associated with the major risk or uncertainty underlying the time series.
Also directly observed returns over time from the prices of the financial instruments may be a stochastic variable composed of market independent part and a market dependent part in Capital Asset Pricing Model (CAPM). This market dependent part is assumed to be composed of a sensitivity term (denoted by identifying the market risk of the considered instruments) multiplied by market returns. Even this sensitivity itself behaves like stochastic variables. Estimation of such sensitivities is a challenging task for the financial engineers to understand the effects or risks of the market as whole on that particular instrument.
Discovering the state (or parameters) from the noisy observations of the system identified by a mathematical (or analytical) processes (or model) remained an area of interest for the researcher in recent five decades at least. Such technique of discovering is known as estimation in statistics literatures in broad sense and filtering is more inclined to the dynamic (or time varying) engineering systems. The term estimation is used in different context with different shades of meaning. In statistics, one may like to “estimate” the mean and standard deviation of a random variable from a sample of realizations as a measure of central tendency and dispersion respectively. The procedure which produces such estimate is called an estimator. For linear signal models the so named Kalman Filter (KF) produces optimal estimation. The state estimate using KF can be obtained recursively if the problem is formulated in discrete domain. Optimality of KF relies on various assumptions such as (i) linearity of the model, (ii) whiteness and Gaussianity of the noise sequence and (iii) absence of correlation between process and observation noise. In real life application most of the systems do not follow the above assumptions. Deviations in these assumptions forced researchers to think beyond KF and hence various post KF techniques evolved. During the last couple of decades remarkable contributions have been noted in the field of state estimation, KF, EKF (Extended KF), AKF (Adaptive Kalman Filter), UKF (Unscented KF or Sigma Point Filters), broadly in the field of Linear Regression Kalman Filter and its family to obtain the optimal estimates for a state-space model with Gaussian noise constraints. GMF (Gaussian Mixture Filter) and PF (Particle Filter) provide optimal estimates for non-Gaussian noise and for non linear signal models.
In the present work, variants of AKF would be employed to filter financial time series data. In such applications covariance matrices Q (process noise covariance) and/or R (output or measurement noise covariance) are usually unknown. The AKF would be used in such restricted situations and the nominal (noise-free) process and observation models would be assumed to be known. The models being a first order in this case. Present work explores some alternative formulations (some new and some proposed in other domain) of first order filters for empirical beta estimation specially to take care of “inadequate noise filtering” problem observed in the Das et al. (2010). This work proposed and characterized four novel alternative formulations of Q adaptive AKF or QAKF (higher root, moving average, scaled and rate limited (RLF)). This work also characterized an alternative QAKF formulation (Mohamed-QAKF) for empirical smooth β estimation proposed by Mohamed et al (1999). Scaled-MQAKF, RLF-MQAKF (with suitable tuning parameters) and Mohamed-QAKF provided time varying β estimates comparable to that by benchmark KF. This work also characterizes two alternative R adaptive AKF or RAKF methods proposed by Almagbile et al. (2010) for empirical β estimation.