AbstractSynthetic ground motions (GMs) play a fundamental role in both deterministic and probabilistic seismic engineering assessments. This paper shows that the family of filtered and modulated white noise stochastic GM models overlooks a key parameter—the high‐pass filter's corner frequency, . In the simulated motions, this causes significant distortions in the long‐period range of the linear‐response spectra and in the linear‐response spectral correlations. To address this, we incorporate as an explicitly fitted parameter in a site‐based stochastic model. We optimize by individually matching the long‐period linear‐response spectrum (i.e., for ) of synthetic GMs with that of each recorded GM. We show that by fitting the resulting stochastically simulated GMs can precisely capture the spectral amplitudes, variability (i.e., variances of ), and the correlation structure (i.e., correlation of between distinct periods and ) of recorded GMs. To quantify the impact of , a sensitivity analysis is conducted through linear regression. This regression relates the logarithmic linear‐response spectrum () to 7 GM parameters, including the optimized . The results indicate that the variance of observed in natural GMs, along with its correlation with the other GM parameters, accounts for 26% of the spectral variability in long periods. Neglecting either the variance or correlation typically results in an important overestimation of the linear‐response spectral correlation.