  Compositional data are common in many fields, both as outcomes and predictor
variables. The inventory of models for the case when both the outcome and
predictor variables are compositional is limited and the existing models are
difficult to interpret, due to their use of complex log-ratio transformations.
We develop a transformation-free linear regression model where the expected
value of the compositional outcome is expressed as a single Markov transition
from the compositional predictor. Our approach is based on generalized method
of moments thereby not requiring complete specification of data likelihood and
is robust to different data generating mechanism. Our model is simple to
interpret, allows for 0s and 1s in both the compositional outcome and
covariates, and subsumes several interesting subcases of interest. We also
develop a permutation test for linear independence. Finally, we show that
despite its simplicity, our model accurately captures the relationship between
compositional data from education and medical research.
