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LaTeX can create citations in [number] format. However, I would need the format (author, year). And with the same cross-reference function as in the original numeric format. At the same time, I would like to use only the \begin{thebibliography} command, i.e. not use BibTeX and some other .bib file, but just write the literature manually via \bibitem. Is that possible?

Thanks for the answers and sorry if the question is already answered somewhere, I didn't find it.

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    yes, natbib can also handle manual bibliographies, you only need to use the optional argument of \bibitem correctly. See the natbib documentation. Mar 31, 2022 at 10:02

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Short answer: Yes.

Longer answer: Absolutely.

Even longer answer: Give the following minimalist document a try. Observe the use of the optional argument of \bibitem to store the citation call-out string and publication year. Compile the document twice to generate the citation call-outs correctly.

You will, of course, still be in charge of everything related to bibliography: The ordering of the bib entries, the formatting of the bib entries, etc., etc.

\documentclass{article}
\usepackage[authoryear,round]{natbib}

\begin{document}
\citet{jms:3001}, \citep{abcd:2005}

\begin{thebibliography}{2}

% Important: leave no space between citation callout 
% string and (year) in the optional argument.
\bibitem[Andersen et~al.(2005)]{abcd:2005} Andersen, Torben G., 
   Tim Bollerslev, Peter Christoffersen, and Francis X. Diebold 
   (2005), ``Volatility Forecasting,'' Manuscript.

\bibitem[Jones et~al.(3001)]{jms:3001} Jones, Hillary, Maria 
   Miller, and Sandra Smith, ``Thoughts,'' \emph{Circularity 
   Today}, 3(2) 456--789.
    
\end{thebibliography}

\end{document}

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