The child, Josiah, is attributed by Farlow (1911) to both Joseph BALLARD, Sr., (p. 71) and to his son, Joseph BALLARD, Jr. (p. 74).  Given the birth date of Josiah (22 Jun 1699) and the marriage date of Joseph BALLARD, Jr., & Rebecca JOHNSON (17 Aug 1698, ten months earlier), it is far more reasonable (to me) that Josiah was the child of Joseph & Rebecca (JOHNSON) BALLARD, Jr., and not that of Joseph & Rebeckah (REA) BALLARD, Sr., and that the mixup was the result of both couples being:  "Joseph & Rebecca BALLARD."

1.  Charles Frederic Farlow.  1911.  Ballard Genealogy. Charles H. Pope, Boston, MA.

2.  Lucius Manlius Boltwood.  1849.  "Marriages in the town of Andover, Ms., from 1647 to 1700." New England Historical and Genealogical Register 3(Jan): 65-68.

p. 65	Joseph Ballerd & Elizabeth Philps married Feb 18 1665
p. 68	Joseph Ballard Sen & Rebeccah Horn (widdow) married Nov 15 1692

3.  Thomas Hutchinson.  1870. "The Witchcraft Delusion of 1692."  New England Historical and Genealogical Register 24: 381-414.  In a written statement signed by six of those accused of witchcraft who survived the trials:

p. 401	We whose names are underwriten, inhabitants of Andover; when as that horrible and tremendous judgment beginning at Salem village in the year 1692, by some called witchcraft, first breaking forth at Mr. Paris's house, several young persons, being seemingly afflicted, did accuse several persons for afflicting them, and many there believing it so to be, we being inform-
p. 402	ed that, if a person was sick, the afflicted persons could tell what or who was the cause of that sickness: Joseph Ballard, of Andover, his wife being sick at the same time, he either from himself or by the advice of others, fetched two of the persons, called the afflicted persons, from Salem village to Andover, which was the beginning of that dreadful calamity that befel us in Andover...

4.  Joshua Coffin.  1852.  "Early Settlers of Essex and Old Norfolk." New England Historical and Genealogical Register 6(Apr): 205-214.  Consists of isolated genealogical facts gleaned from the public records.  On p. 207:

BALLARD... William, ae. 44 in 1662. Joseph, ae. 26 in 1671.

William is obviously the father of our subject, and the document is doubtless the deposition he gave in 1662 (see Farlow 1911: 63), though Farlow says he's age 45, not 44.  The question is, is this Joseph the son of William?  Seems highly likely; his age is just right.

5.  Lucius Manluis Boltwood.  1848.  "Deaths in the town of Andover, Ms., from 1650 to 1700." New England Historical and Genealogical Register 2: 377-381.

p. 378	Tabitha Ballard dau of Joseph & Elizabeth died Mch 30 1687
p. 379	Tabitha Ballard dau of Joseph & Elizabeth died Feb 24 1690-1