  In this contribution we are concerned with tight a posteriori error
estimation for projection based model order reduction of $\inf$-$\sup$ stable
parameterized variational problems. In particular, we consider the Reduced
Basis Method in a Petrov-Galerkin framework, where the reduced approximation
spaces are constructed by the (weak) Greedy algorithm. We propose and analyze a
hierarchical a posteriori error estimator which evaluates the difference of two
reduced approximations of different accuracy. Based on the a priori error
analysis of the (weak) Greedy algorithm, it is expected that the hierarchical
error estimator is sharp with efficiency index close to one, if the Kolmogorov
N-with decays fast for the underlying problem and if a suitable saturation
assumption for the reduced approximation is satisfied. We investigate the
tightness of the hierarchical a posteriori estimator both from a theoretical
and numerical perspective. For the respective approximation with higher
accuracy we study and compare basis enrichment of Lagrange- and Taylor-type
reduced bases. Numerical experiments indicate the efficiency for both, the
construction of a reduced basis using the hierarchical error estimator in a
weak Greedy algorithm, and for tight online certification of reduced
approximations. This is particularly relevant in cases where the $\inf$-$\sup$
constant may become small depending on the parameter. In such cases a standard
residual-based error estimator -- complemented by the successive constrained
method to compute a lower bound of the parameter dependent $\inf$-$\sup$
constant -- may become infeasible.
