  The Siegert relation connects the first- and second-order coherence
properties of light. While strictly valid only in the thermal regime and in the
absence of correlations, this relation is routinely extended to the partially
coherent regime in the study of high-$\beta$ nanolasers, where it aids in the
identification of the lasing threshold. We re-evaluate the use of a generalized
Siegert relation in different device regimes. A full two-time quantum-optical
theory is derived as a reference for obtaining first- and second-order
correlation functions $g^{(1)}(\tau)$ and $g^{(2)}(\tau)$ in the steady state.
We find that even in the partially coherent regime the generalized Siegert
relation is well suited as an approximation to $g^{(2)}(\tau)$ as long as
emitter correlation are negligible, but does not apply well in the quantum
regime of few-emitter nanolasers, or to devices featuring sub- and superradiant
emission.
