Abstract A global methodology of identification, estimation and forecasting of transfer function (Box-Jenkins) models with deterministically varying parameters is provided. First, properties of stability and forecasting algorithms are investigated by means of Markovian representations and methods of solution of nonstationary difference equations. Next, the degree of the polynomials of the system is specified with typical off-line methods, and the shape of the coefficients (parameter functions) is identified by means of recursive (on-line) algorithms. Finally, the identified parameter functions are inserted in the model and their coefficients are estimated (off-line) on the original data by means of pseudolinear regression techniques.