  Reference panel-based estimators have become widely used in genetic
prediction of complex traits due to their ability to address data privacy
concerns and reduce computational and communication costs. These estimators
estimate the covariance matrix of predictors using an external reference panel,
instead of relying solely on the original training data. In this paper, we
investigate the performance of reference panel-based $L_1$ and $L_2$
regularized estimators within a unified framework based on approximate message
passing (AMP). We uncover several key factors that influence the accuracy of
reference panel-based estimators, including the sample sizes of the training
data and reference panels, the signal-to-noise ratio, the underlying sparsity
of the signal, and the covariance matrix among predictors. Our findings reveal
that, even when the sample size of the reference panel matches that of the
training data, reference panel-based estimators tend to exhibit lower accuracy
compared to traditional regularized estimators. Furthermore, we observe that
this performance gap widens as the amount of training data increases,
highlighting the importance of constructing large-scale reference panels to
mitigate this issue. To support our theoretical analysis, we develop a novel
non-separable matrix AMP framework capable of handling the complexities
introduced by a general covariance matrix and the additional randomness
associated with a reference panel. We validate our theoretical results through
extensive simulation studies and real data analyses using the UK Biobank
database.
