  In this work we investigate the holographic dark energy models with slowly
time-varying model parameter defined based on the current Hubble horizon length
scale. While the previous studies on the three popular holographic dark energy
models defined based on the future event horizon, Ricci scale and
Granda-Oliveros IR cutoffs showed that these models cannot fit the
observational data [1], in this work we show that the holographic dark energy
models with time-varying model parameter defined on the current Hubble radius
are well favored by observations. Using the standard $\chi^2$ minimization in
the context of Markov Chain Monte Carlo method, we compare the ability of
holographic dark energy models with time-varying $c^2$ parameter constructed on
the current Hubble length scale against different sets of observational data
namely expansion data, growth rate data and expansion+growth rate data
respectively. Based on the values of Akaike and Bayesian information criteria,
we find that these types of holographic dark energy models are well fitted to
both expansion and growth rate observations as equal to $\Lambda$CDM cosmology.
We also put constraints on the cosmological parameters and show that the
transition epoch form early decelerated to current accelerated expansion
calculated in holographic dark energy models with time-varying model parameter
defined on the Hubble length is consistent with observations.
