The problem of quantifying errors due to nonlinear undermodeling is addressed. It is assumed that the system consists of a linear dynamic block in cascade with a static nonlinearity and that the objective is to identify the linear part using a purely linear model. The stochastic embedding approach is applied to capture the on-average properties of the undermodeling. As compared to previous methods, the priors on the covariance matrix of the embedding parameters are reduced. As a result an expression for the amplitude error bounds of the estimated transfer function, that does not require knowledge of the true system parameters, is obtained. The quality of this measure depends on the degree of accuracy with which the unknown nonlinearity can be represented using a set of known basis functions. The proposed method simultaneously delivers error bounds on the estimated transfer function and an indirect estimate of the size of the nonlinearity. The importance of obtaining error bounds on transfer functions and estimates of static nonlinearities for controller design is well established in the literature.