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 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.

## Simultaneous equations bias.

uppose the quantities are connected by the relationships where the numbers are unknown, the are Gaussian variables with zero mean and unknown deterministic variances. Such relationships may represent a simple supply-demand equilibrium model with being the supply-demand at the equilibrium and each of the equations representing the influence of price on supply (for first equation) or demand (second equation). The variable is not correlated with Note that the simple linear regression should not be used to estimate each of the parameters because it is neither a maximal likelihood nor unbiased estimate in such situation. This simple fact is called the ''simultaneous equations bias''. Intuitively, one has to remove the influence of second equation to apply the regression to the first equation. This is the idea of the two stage least squares procedure.

 Notation. Index. Contents.