A process for the oxidative dehydrogenation of a lower hydrocarbon to form at least one higher hydrocarbon and/or lower olefin, which comprises contacting the lower hydrocarbon with oxygen and a catalyst comprising an oxycarbonate, hydroxycarbonate, and/or carbonate of at least one rare earth element selected from the group consisting of La, Pr, Nd, Sm, Eu, Gd, Th, Dy, Ho, Er, and Tm, wherein said catalyst, when used for the oxidative dehydrogenation of said lower hydrocarbon exhibits higher selectivity to at least one higher hydrocarbon and/or lower olefin at a pressure above about 100 psig than said catalyst or a precursor of said catalyst exhibits at a pressure in the range of about atmospheric pressure to about 25 psig, and wherein at a pressure above about 100 psig said catalyst has a selectivity to said at least one higher hydrocarbon and/or lower olefin of at least 40%.