In this paper a modified identification algorithm for linear systems with noisy input-output data is developed. The main idea is to introduce extra equations so that the variances of the input noise and the output noise, which determine the noise-induced bias in the least-squares parameter estimator, can be estimated in a more accurate and quicker way. Due to this, such performance as estimation accuracy and convergence speed of the bias-correction based algorithm can be significantly boosted, at the expense of a fractional increase in the numerical cost. Computer simulations are presented, which leads to some useful conclusions.