We investigate the lagged correlation between a selection of geomagnetic indices and solar wind parameters for a complete solar cycle, from 2000 to 2011. We first discuss the mathematical assumptions required for such a correlation analysis. The solar wind parameters and geomagnetic indices have inherent timescales that smooth the variations of the correlation coefficients with time lag. Furthermore, the solar wind structure associated with corotating interaction regions and coronal mass ejections, and the compression regions ahead of them, strongly impacts the lagged correlation analysis results. This work shows that such bias must be taken into account in a correct interpretation of correlations. We then evidence that the magnetospheric response time to solar wind parameters involves multiple timescales. The simultaneous and quick response of the PC and AE indices to solar wind dynamic pressure with a delay of ~5 min suggests that magnetospheric compression by solar wind can trigger substorm activity. We find that the PC and AE indices respond to interplanetary magnetic field (IMF) BZ with a response time of respectively ~20 and ~35 min. The response of the SYM-H index takes longer (~80 min) and is less sharp, SYM-H being statistically significantly correlated to the IMF BZ observed up to more than ~10 h before. Our results suggest that the solar wind velocity's dominant impact on geomagnetic activity is caused by the compression regions at the interface of fast/slow solar wind regimes, which are very geo-effective as they are associated with high solar wind pressure and strong interplanetary magnetic field.