The Bivariate Stochastic Functional Form

25 Pages Posted: 25 Sep 2011

See all articles by Sanne De Boer

Sanne De Boer

Voya Investment Management

Aart F. de Vos

Vrije Universiteit Amsterdam, School of Business and Economics

Date Written: August 31, 1999


This paper gives the derivation of the Bivariate Stochastic Functional Form (BSFF), which may be seen as the direct generalization of the linear regression model. The derivation does not involve complex mathematical tools such as stochastic calculus. It extends the derivation of the univariate stochastic functional form proposed by De Boer et al. (1999). Our model imposes relatively strong smoothness conditions. The Kalman filter can be used calculate maximum likelihood estimates of the parameters and the smoothed posterior estimate of the function, which has several computational advantages. Numerical tests of the BSFF on constructed examples and real data show very promising results.

Keywords: bivariate smoothing splines, Kalman filter, nonparametric regression

Suggested Citation

De Boer, Sanne and de Vos, Aart F., The Bivariate Stochastic Functional Form (August 31, 1999). Available at SSRN: or

Sanne De Boer (Contact Author)

Voya Investment Management ( email )

230 Park Avenue
13th Floor
New York, NY 10069
United States

Aart F. De Vos

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV

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