We　consider　a　problem　of　checking　whether　the　coefficient　of　the　scale　and　location　function　is　a　constant.　Both　the　scale　and　location　functions　are　modeled　as　single-index　models.　Two　test　statistics　based　on　Kolmogorov-Smirnov　and　Cramer-von　Mises　type　functionals　of　the　difference　of　the　empirical　residual　processes　are　proposed.　The　asymptotic　distribution　of　the　estimator　for　single-index　parameter　is　derived,　and　the　empirical　distribution　function　of　residuals　is　shown　to　converge　to　a　Gaussian　process.　Moreover,　the　proposed　test　statistics　can　be　able　to　detect　local　alternatives　that　converge　to　zero　at　a　parametric　convergence　rate.　A　bootstrap　procedure　is　further　proposed　to　calculate　critical　values.　Simulation　studies　and　a　real　data　analysis　are　conducted　to　demonstrate　the　performance　of　the　proposed　methods.