The half metallic and ferromagnetic properties of YX′CrZ (X′ = Fe, Co, Ni; Z = Al, Ga, In) quaternary Heusler alloys are studied using first principles calculations based on density functional theory. The most stable structure is analysed among α-, β- and γ-phase of LiMgPdSn crystal structure. The γ-phase is confirmed as the most stable phase for YFeCrAl, YFeCrIn, YCoCrAl, YCoCrGa, YCoCrIn, YNiCrAl and YNiCrIn, while β-phase is the lowest energy phase for YFeCrGa and YNiCrGa. At 5.2 GPa, 1.85 GPa and 6.38 GPa, structural phase transition is observed in YFeCrGa, YNiCrGa and YFeCrIn. The density of states shows that YFeCrAl, YCoCrAl, YCoCrGa, YCoCrIn, YNiCrAl and YNiCrIn are half metallic, whereas YFeCrGa, YFeCrIn and YNiCrGa are metallic. The band structure indicates that the band gap is indirect for these half-metallic compounds. Ferromagnetism is predicted in these alloys. The phonon dispersion curve shows that YCoCrAl, YCoCrIn, YNiCrAl and YNiCrIn are dynamically stable.