We present an analysis using the MOSFIRE Deep Evolution Field (MOSDEF) survey on the nature of MIR-excess galaxies, which have star formation rates (SFR) inferred from mid-infrared (MIR) data that is substantially elevated relative to that estimated from dust-corrected UV data. We use a sample of $sim$200 galaxies and AGN at $1.40<z<2.61$ with 24 $mu$m detections (rest-frame 8$mu$m) from MIPS/textit{Spitzer}. We find that the identification of MIR-excess galaxies strongly depends on the methodologies used to estimate IR luminosity ($rm L_{IR}$) and to correct the UV light for dust attenuation. We find that extrapolations of the SFR from the observed 24 $mu$m flux, using luminosity-dependent templates based on local galaxies, substantially overestimate $rm L_{IR}$ in $zsim2$ galaxies. By including textit{Herschel} observations and using a stellar mass-dependent, luminosity-independent $rm L_{IR}$, we obtain more reliable estimates of the SFR and a lower fraction of MIR-excess galaxies. Once stellar mass selection biases are taken into account, we identify $sim24%$ of our galaxies as MIR-excess. However, $rm SFR_{Halpha}$ is not elevated in MIR-excess galaxies compared to MIR-normal galaxies, indicating that the intrinsic fraction of MIR-excess may be lower. Using X-ray, IR, and optically-selected AGN in MOSDEF, we do not find a higher prevalence for AGN in MIR-excess galaxies relative to MIR-normal galaxies. A stacking analysis of X-ray undetected galaxies does not reveal a harder spectrum in MIR-excess galaxies relative to MIR-normal galaxies. Our analysis indicates that AGN activity does not contribute substantially to the MIR excess and instead implies that it is likely due to the enhanced PAH emission.
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