For the attitude estimation problem, the quaternion is preferred to describe the rotation of spacecraft, which obeys a unit-norm constraint. However, this constraint is not considered in the most of the traditional estimation algorithms. This work will develop a new estimator named norm-constrained predictive filter to solve this problem based on the predictive filter frame. The optimization theory of Lagrange multiplier is adopted by minimizing a constrained cost function to improve the traditional predictive filter. In addition, the obtained estimator is further analyzed and applied to the spacecraft attitude estimation problem by numerical simulations.