Methods and systems for compensating for the absence or loss of a sensor measurement in a heading reference system such as an aircraft attitude and heading reference system, integrated standby unit, or vehicle inertial system, provides an estimate of the lost sensor measurement by estimating the bank angle after a detected vehicle turn. The estimate of the bank angle may also be used to estimate the vehicle's speed. Additionally, when the lost sensor measurement is a temperature measurement, the described methods and systems offer an improvement over estimating air temperature using a standard (e.g., ISA) model. The methods and systems also allow for the refinement of computed estimates using filtering techniques, such as low-pass or Kalman filtering. The methods may be iteratively repeated for each detected turn in order to maintain an accurate estimate of the lost sensor measurement or other estimates, such as vehicle speed.