The Einstein relation which is known as the Fluctuation-Dissipation (FD) theorem in physics is applied to financial time series data in this paper. The fundamental equation for the time series is the generalized Langevin equation with colored noise. The autocorrelation function and the probability density function are derived from this equation and the Einstein relation. Those functions are then fitted to the sample autocorrelation and the histogram by means of nonlinear regression and all the parameters in this formalism are estimated consistently. This paper shows that the Einstein relation is useful in the stochastic time series analysis.

The data in financial markets are usually analyzed by the stochastic method for time series analysis (Gardiner, 2003). The essential concept of this method was formulated in a study by Einstein (1906) on the Brownian motion. In this article, Einstein studied the motion of a Brownian particle in liquid by the thermodynamical method. Later, Langevin (1908) formulated this phenomenon by the method of the Newtonian differential equation (Coffey et al., 2004). It is common for economists to understand the Einstein theory of the Brownian motion in the framework of Langevin.

To apply the Einstein’s method to financial data, we have to generalize Equation (1) to the Generalized Langevin (GL) equation, which was introduced by Kubo (1966), and the white noise (Equation 2) to colored noise (Hanggi and Jung, 1995). Corresponding to these generalizations, we define a new Einstein relation. By the use of this relation, we can solve the GL equation, calculate the autocorrelation function, and decide the probability density function (pdf) of x( t). These functions are then fitted to the data, and we estimate the values of all the parameters introduced in this model.

Applied Economics Journal, Einstein Relation, Financial Data, Financial Markets, Brownian Motion, Fiuctuation Dissipation Theorem, Gaussian Distributions, Weak Stationary Process, Einstein-Langevin Formalism, Fokker-Planck Equation.