  In this study, we propose a projection estimation method for
large-dimensional matrix factor models with cross-sectionally spiked
eigenvalues. By projecting the observation matrix onto the row or column factor
space, we simplify factor analysis for matrix series to that for a
lower-dimensional tensor. This method also reduces the magnitudes of the
idiosyncratic error components, thereby increasing the signal-to-noise ratio,
because the projection matrix linearly filters the idiosyncratic error matrix.
We theoretically prove that the projected estimators of the factor loading
matrices achieve faster convergence rates than existing estimators under
similar conditions. Asymptotic distributions of the projected estimators are
also presented. A novel iterative procedure is given to specify the pair of row
and column factor numbers. Extensive numerical studies verify the empirical
performance of the projection method. Two real examples in finance and
macroeconomics reveal factor patterns across rows and columns, which coincides
with financial, economic, or geographical interpretations.
