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.