  In this paper we investigate panel regression models with interactive fixed
effects. We propose two new estimation methods that are based on minimizing
convex objective functions. The first method minimizes the sum of squared
residuals with a nuclear (trace) norm regularization. The second method
minimizes the nuclear norm of the residuals. We establish the consistency of
the two resulting estimators. Those estimators have a very important
computational advantage compared to the existing least squares (LS) estimator,
in that they are defined as minimizers of a convex objective function. In
addition, the nuclear norm penalization helps to resolve a potential
identification problem for interactive fixed effect models, in particular when
the regressors are low-rank and the number of the factors is unknown. We also
show how to construct estimators that are asymptotically equivalent to the
least squares (LS) estimator in Bai (2009) and Moon and Weidner (2017) by using
our nuclear norm regularized or minimized estimators as initial values for a
finite number of LS minimizing iteration steps. This iteration avoids any
non-convex minimization, while the original LS estimation problem is generally
non-convex, and can have multiple local minima.
