  We study the autocovariance functions of moving average random fields over
the integer lattice $\mathbb{Z}^d$ from an algebraic perspective. These
autocovariances are parametrized polynomially by the moving average
coefficients, hence tracing out algebraic varieties. We derive dimension and
degree of these varieties and we use their algebraic properties to obtain
statistical consequences such as identifiability of model parameters. We
connect the problem of parameter estimation to the algebraic invariants known
as euclidean distance degree and maximum likelihood degree. Throughout, we
illustrate the results with concrete examples. In our computations we use tools
from commutative algebra and numerical algebraic geometry.
