The rate of major galaxy–galaxy merging is theoretically predicted to steadily increase with redshift during the peak epoch of massive galaxy development (1 ≤ z ≤ 3). We use close-pair statistics to objectively study the incidence of massive galaxies (stellar M1 > 2 × 1010 M⊙) hosting major companions (1 ≤ M1/M2 ≤ 4; i.e. <4:1) at six epochs spanning 0 < z < 3. We select companions from a nearly complete, mass-limited (≥5 × 109 M⊙) sample of 23 696 galaxies in the five Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey fields and the Sloan Digital Sky Survey. Using 5–50 kpc projected separation and close redshift proximity criteria, we find that the major companion fraction fmc(z) based on stellar mass-ratio (MR) selection increases from 6 per cent (z ∼ 0) to 16 per cent (z ∼ 0.8), then turns over at z ∼ 1 and decreases to 7 per cent (z ∼ 3). Instead, if we use a major F160W flux-ratio (FR) selection, we find that fmc(z) increases steadily until z = 3 owing to increasing contamination from minor (MR > 4:1) companions at z > 1. We show that these evolutionary trends are statistically robust to changes in companion proximity. We find disagreements between published results are resolved when selection criteria are closely matched. If we compute merger rates using constant fraction-to-rate conversion factors (Cmerg,pair = 0.6 and Tobs,pair = 0.65 Gyr), we find that MR rates disagree with theoretical predictions at z > 1.5. Instead, if we use an evolving Tobs,pair(z) ∝ (1 + z)−2 from Snyder et al., our MR-based rates agree with theory at 0 < z < 3. Our analysis underscores the need for detailed calibration of Cmerg,pair and Tobs,pair as a function of redshift, mass, and companion selection criteria to better constrain the empirical major merger history.