We consider excited random walks (ERWs) on Z with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta > 4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta is an element of (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [2] for delta is an element of (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies.