Yoo and Cook (2007) developed an optimal sufficient dimension reduction methodology for the conditional mean in multivariate regression and it is known that their method is asymptotically optimal and its test statistic has a chi-squared distribution asymptotically under the null hypothesis. To check the effect of dimension used in estimation on regression coefficients and the explanatory power of the conditional mean model in multivariate regression, we applied their method to several simulated data sets with various dimensions. A small simulation study showed that it is quite helpful to search for an appropriate dimension for a given data set if we use the asymptotic test for the dimension as well as results from the estimation with several dimensions simultaneously.