This study develops a KPSS (Kwiatkowski et al., 1992)-type cointegration test utilizing residuals from integrated and modified ordinary least squares (IMOLS) estimation. The test statistic, denoted by $KPSS^{Fb}$ has a pivotal null limit distribution under fixed-$b$ assumption. The proposed test demonstrates reasonable performance in terms of size and power when the Andrews' AR(1) plug-in data-dependent ($DD$) bandwidth is employed and fixed-$b$ critical values are used. Additionally, two modified IMOLS residuals are proposed to obtain alternative data-dependent bandwidths. In the simulation experiment, these bandwidths deliver improved power properties for the proposed test.