Abstract Suppose the expected value of a response variable Y may be written h(Xβ +γ(T)) where X and T are covariates, each of which may be vector-valued, β is an unknown parameter vector, γ is an unknown smooth function, and h is a known function. In this article, we outline a method for estimating the parameter β, γ of this type of semiparametric model using a quasi-likelihood function. Algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed. The generalization of this approach to the case in which Y is a multivariate response is also considered. The methodology is illustrated on two data sets and the results of a small Monte Carlo study are presented.