The celebration of the Nativity of our Lord Jesus Christ is dated by the consensus of scholars to to be around the 3rd and 4th centuries. The reason being is evidence which was brought to light in the 354 A.D. Chronograph, the De Solstitiis by German scholar Herman Usener, who argued that the feast of Christmas was a way to incorporate pagans, who celebrated the feast of Sola Invictus, into Christianity The scholars who propose such a development for the date of Christmas are proponents of what is called in historical circles as History of Religions Theory.
However recently, British scholar, Philipp Nothaft argues there is considerable evidence that suggest a Christian origin to the celebration of the birth of our Lord. Naturally, the argument follows that Luke and Matthew included the infancy narrative in their Gospels because some importance, as well as the the Protoevangelium of James, a non-canonical early Christian document. Professor Nothaft argues for what is known as Calculation Theory which asserts that calculation interest of early Christians for dates within Christian history, much of this relies on establishing early Christian interest in dating and a basic understanding of how typology works in the Gospels in relationship with the Old Testament.
It’s important for faithful Catholics when examining scripture and typology must keep in mind what the Church teaches in the Catechism on the matter:
110 In order to discover the sacred authors’ intention, the reader must take into account the conditions of their time and culture, the literary genres in use at that time, and the modes of feeling, speaking, and narrating then current. “For the fact is that truth is differently presented and expressed in the various types of historical writing, in prophetical and poetical texts, and in other forms of literary expression.”
112 1. Be especially attentive “to the content and unity of the whole Scripture.” Different as the books which comprise it may be, Scripture is a unity by reason of the unity of God’s plan, of which Christ Jesus is the center and heart, open since his Passover. (128; 368)
The unity of the Old and New Testaments
128 The Church, as early as apostolic times, and then constantly in her Tradition, has illuminated the unity of the divine plan in the two Testaments through typology, which discerns in God’s works of the Old Covenant prefigurations of what he accomplished in the fullness of time in the person of his incarnate Son. (1094; 489)
129 Christians therefore read the Old Testament in the light of Christ crucified and risen. Such typological reading discloses the inexhaustible content of the Old Testament; but it must not make us forget that the Old Testament retains its own intrinsic value as Revelation reaffirmed by our Lord himself. Besides, the New Testament has to be read in the light of the Old. Early Christian catechesis made constant use of the Old Testament. As an old saying put it, the New Testament lies hidden in the Old and the Old Testament is unveiled in the New. (681; 2055; 1968)
130 Typology indicates the dynamic movement toward the fulfillment of the divine plan when “God [will] be everything to everyone.” Nor do the calling of the patriarchs and the exodus from Egypt, for example, lose their own value in God’s plan, from the mere fact that they were intermediate stages.
Pope Benedict XVI writes, “Before we consider the content of the texts, a brief word about their particular literary character is necessary. In their different ways, both Matthew and Luke closely link the events of Jesus’ childhood with Old Testament passages. Matthew demonstrates the connections for the reader each time by quoting corresponding Old Testament text. Luke describes events using Old Testament language…In Luke, there seems to be an underlying Hebrew text; at any rate, the whole account is marked by Semitisims, which are not otherwise typical of Luke.”
 C. Philipp E. Nothaft, “Early Christian Chronology and the Origins of the Christmas Date,” Questions Liturgiques: Studies in Liturgy Vol. 94 (2013), 248.
 Catholic Church, Catechism of the Catholic Church, 2nd Ed. (Washington, DC: United States Catholic Conference, 2000), 32-36.
 Pope Benedict XVI, Jesus of Nazareth: The Infancy Narratives (New York: Image, 2012), 15.