NOVELTY - A catalyst for oxidative dehydrogenation of a lower hydrocarbon comprises oxycarbonate (I), hydroxycarbonate and/or carbonate of specific rare earth element, having a disordered and/or defect structure. The catalyst has a selectivity of 40% or more to higher hydrocarbon(s) and/or lower olefin when used for oxidative dehydrogenation of lower hydrocarbon at a pressure of more than 100 psig.    USE - Catalyst for oxidative dehydrogenation and oxidative coupling of a lower hydrocarbon to form higher hydrocarbon(s) and/or lower olefin (claimed), particularly for coupling methane to form ethane, ethylene and higher hydrocarbons, and for oxidative dehydrogenation of ethane to form ethylene.    ADVANTAGE - The catalyst has higher selectivity in oxidative dehydrogenation of lower hydrocarbons to produce higher hydrocarbons and/or lower olefins. During oxidative dehydrogenation, desired olefins such as ethylene and propylene are produced. Byproduct alkanes such as ethane and propane can be recycled and converted to olefins. Undesirable impurities such as poison for the catalyst are present at low levels that permit economical operation of the oxidative dehydrogenation reaction.    DETAILED DESCRIPTION - The catalyst for oxidative dehydrogenation of lower hydrocarbon comprises oxycarbonate of formula MXCYOZ (I), hydroxycarbonate and/or carbonate of rare earth element chosen from lanthanum, praseodymium, neodymium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium and/or thulium, having disordered and/or defect structure.    M = rare earth element;    X = 2;    Z = 3+AY;    A = less than 1.8;    Y = number of carbon atoms in the oxycarbonate;    Z/X = 1.5-4.5, preferably less than 3.75;    Z/Y = 1-6, preferably 1.5-4.5.    The catalyst has selectivity of 40% or more to higher hydrocarbon(s) and/or lower olefin when used for oxidative dehydrogenation of lower hydrocarbon at a pressure more than 100 psig.    INDEPENDENT CLAIMS are also included for the following:    (i) Preparation of a non-stoichiometric rare earth oxycarbonate catalyst by forming a catalyst precursor from rare earth compound(s) by treating them with water and/or organic compound containing a hydroxyl group, drying the treated compound and calcining it at 300-1000 degreesC in an atmosphere containing oxygen. The catalyst is formed by pressurizing the precursor to a pressure of 100 psig or more with a flowing gas containing hydrocarbon and/or oxygen, and heating the precursor and holding it for 20 minutes or more at 300-600 degreesC and oxygen conversion of less than 70%.    (ii) Oxidative dehydrogenation of lower hydrocarbon to form higher hydrocarbon(s) and/or lower olefin which involves contacting the lower hydrocarbon with oxygen and catalyst.    (iii) Oxidative coupling of lower hydrocarbon to form higher hydrocarbon(s) and/or lower olefin which involves contacting the lower hydrocarbon with oxygen and catalyst comprising an oxide of rare earth element and cocatalyst including metal chosen from vanadium, niobium, tantalum, chromium, molybdenum, tungsten, manganese, rhenium, iron, cobalt, nickel, copper, zinc, tin, lead, antimony and/or bismuth.    DESCRIPTION OF DRAWING(S) - The figure shows the plot of elemental mole ratios ((oxygen/rare earth element) and (oxygen/carbon)) of the non-stoichiometric oxycarbonate catalyst.