  We address the problem of estimating the parameters of a time-homogeneous
Markov chain given only noisy, aggregate data. This arises when a population of
individuals behave independently according to a Markov chain, but individual
sample paths cannot be observed due to limitations of the observation process
or the need to protect privacy. Instead, only population-level counts of the
number of individuals in each state at each time step are available. When these
counts are exact, a conditional least squares (CLS) estimator is known to be
consistent and asymptotically normal. We initiate the study of method of
moments estimators for this problem to handle the more realistic case when
observations are additionally corrupted by noise. We show that CLS can be
interpreted as a simple "plug-in" method of moments estimator. However, when
observations are noisy, it is not consistent because it fails to account for
additional variance introduced by the noise. We develop a new, simpler method
of moments estimator that bypasses this problem and is consistent under noisy
observations.
