Merriam-Webster Collegiate Dictionary - bye-bye

 I. interjection  or by-by  Etymology: baby-talk reduplication of goodbye  Date: circa 1736 — used to express farewell  II. adverb  or by-by  Date: 1917 out especially for a walk or ride — used with the verb go  III. noun  or by-by  Date: 1867 bed, sleep lie down…and go to ~ — Rudyard Kipling  IV. adverb  or by-by  Date: 1920 to bed or sleep — used with the verb go