We study a minimal string model possessing the same massless spectra as the MSSM on $Z_N\times Z_M$ orbifolds. Threshold corrections of the gauge coupling constants of SU(3), SU(2) and U(1)$_Y$ are investigated in a case of an overall modulus. Using computer analyses, we search ranges of levels of U(1)$_Y$ allowed by the LEP experiments. It is found that $Z_3\times Z_3$ can not derive the minimal string model for a $M_Z$ SUSY breaking scale. The minimum values of the overall moduli are estimated within the ranges of the levels.