The thebibliography environment inside the book class is set as a \chapter*. From book.cls (comments added):
\newenvironment{thebibliography}[1]
{\chapter*{\bibname}% <----------- How the bibliography is set
\@mkboth{\MakeUppercase\bibname}{\MakeUppercase\bibname}%
\list{\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\@openbib@code
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\sloppy
\clubpenalty4000
\@clubpenalty \clubpenalty
\widowpenalty4000%
\sfcode`\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
titlesec strongly discourages the use of starred versions for section. From the titlesec documentation (section 4.2 Starred versions):
Using sectioning commands in the starred version is strongly discouraged. Instead, you can use a set of markup oriented commands which are easy to define and modify, if necessary. Thus, you can test different layouts before choosing amongst them.
Firstly remember if you say
\setcounter{secnumdepth}{0}
sections will be not numbered but they will be included in both ToC and headers.
With this in mind, you can use the following setup:

\documentclass{book}
\usepackage{titlesec}
\titleformat{\chapter}[display]
{\large \bfseries}{\chaptertitlename\ \thechapter}{16pt}
{\normalsize \thechapter{.}\ }[]
\begin{document}
\mainmatter
\chapter{Introduction}
\chapter{Theory}
\clearpage
\backmatter
\setcounter{secnumdepth}{-1}% Number only up to \part
\renewcommand{\thechapter}[2]{}% Gobble {.} and {\ }
\renewcommand{\bibname}{REFERENCES}
\begin{thebibliography}{x}
\bibitem{Taylor}
Taylor M.\ \& Mankiw, N., \emph{Economics}, Cengage Learning EMEA, UK (1980)
\end{thebibliography}
\end{document}
The two additional commands before called the thebibliography environment removes the numbering within the regular \chapter sequence, yet still sets it as a \chapter*.
\chapter{Introduction}, you'll see Chapter 1 1. Introduction. This seems odd, yet this is what you want for the first chapters. Are you sure? That's also the reason behind the 2. being part of the references chapter. – Werner Jun 26 '18 at 03:07