  We study the phase transition behavior of the ferroelectric
BaTi$_{0.8}$Zr$_{0.2}$O$_3$ in the paraelectric region. The temperature
dependencies of thermal, polar, elastic and dielectric properties indicate the
presence of local structures above the paraelectric-ferroelectric transition
temperature Tc = 292 K. The non-zero remnant polarization is measured up to a
characteristic temperature T* ~350 K, which coincides with the temperature
where the dielectric constant deviates from Curie-Weiss law. Resonant
Piezoelectric Spectroscopy shows that DC field-cooling above Tc using fields
smaller than the coercive field leads to an elastic response and remnant
piezoelectricity below T*, which likely corresponds to the coherence
temperature associated with polar nanostructures in ferroelectrics. The
observed remnant effect is attributed to the reorientation of polar
nanostructures above Tc.
