We suggest a new approach to the detection of gravitational waves using observations of a group of millisecond pulsars. In contrast to the usual method, based on increasing the accuracy of the arrival times of pulses by excluding possible distorting factors, our method supposes that the additive phase noise that is inevitably present even in the most accurate observational data has various spectral components, which have characteristic amplitudes and begin to appear on different time scales. We use the “Caterpillar” (Singular Spectral Analysis, SSA) method to decompose the signal into its components. Our initial data are the residuals of the pulse arrival times for six millisecond pulsars. We constructed the angular correlation function for components of the decomposition of a given number, whose theoretical form for the case of an isotropic and homogeneous gravitational-wave background is known. The individual decomposition components show a statistically significant agreement with the theoretical expectations (correlation coefficient ρ = 0.92 ± 0.10).