  This paper is the first in a set that analyses the covariance matrices of
clustering statistics obtained from several approximate methods for
gravitational structure formation. We focus here on the covariance matrices of
anisotropic two-point correlation function measurements. Our comparison
includes seven approximate methods, which can be divided into three categories:
predictive methods that follow the evolution of the linear density field
deterministically (ICE-COLA, Peak Patch, and Pinocchio), methods that require a
calibration with N-body simulations (Patchy and Halogen), and simpler recipes
based on assumptions regarding the shape of the probability distribution
function (PDF) of density fluctuations (log-normal and Gaussian density
fields). We analyse the impact of using covariance estimates obtained from
these approximate methods on cosmological analyses of galaxy clustering
measurements, using as a reference the covariances inferred from a set of full
N-body simulations. We find that all approximate methods can accurately recover
the mean parameter values inferred using the N-body covariances. The obtained
parameter uncertainties typically agree with the corresponding N-body results
within 5% for our lower mass threshold, and 10% for our higher mass threshold.
Furthermore, we find that the constraints for some methods can differ by up to
20% depending on whether the halo samples used to define the covariance
matrices are defined by matching the mass, number density, or clustering
amplitude of the parent N-body samples. The results of our configuration-space
analysis indicate that most approximate methods provide similar results, with
no single method clearly outperforming the others.
