  We investigate the short-time critical dynamics of the Baxter-Wu (BW) and
$n=3$ Turban (3TU) models to estimate their global persistence exponent $\theta
_{g}$. We conclude that this new dynamical exponent can be useful in detecting
differences between the critical behavior of these models which are very
difficult to obtain in usual simulations. In addition, we estimate again the
dynamical exponents of the four-state Potts (FSP) model in order to compare
them with results previously obtained for the BW and 3TU models and to decide
between two sets of estimates presented in the current literature. We also
revisit the short-time dynamics of the 3TU model in order to check if, as
already found for the FSP model, the anomalous dimension of the initial
magnetization $x_{0}$ could be equal to zero.
