I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 A. Time series forecasting.
 B. Updating a linear forecast.
 C. Kalman filter I.
 D. Kalman filter II.
 E. Simultaneous equations.
 a. Simple linear reduction.
 b. Simultaneous equations bias.
 c. Two stage least squares procedure for simultaneous equations.
 d. General note of applicability.
 2 Classical statistics.
 3 Bayesian statistics.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Simple linear reduction.

uppose the observed real valued processes of discrete time are connected by the relationship where the is a Gaussian random variable with zero mean. Assume farther that we are given samples and . We would like to compute a maximal likelihood estimate of . The log-likelihood function is proportional to the sum where we use the notation . Hence, the maximal likelihood estimate has to satisfy Therefore, the value is given by the projection of on :

 Notation. Index. Contents.
 Copyright 2007