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\section{Introduction}  \subsection{The negative comovement problem} One implication of simple RBC-like models with steady state growth is that a persistent increase in the growth rate of productivity can trigger a decline in output. More generally, positive changes in productivity growth can have negative impact on hours worked, output or investment. For instance \cite{Viard1993}, \cite{Carroll1994} showed that a \textit{decline} in the productivity growth rate should elicit an immediate rise in the saving rate. rate,  \cite{Campbell1994} showed that in a real business cycle model, a persistent decline in the productivity growth rate yielded the "perverse" effect of a rise in employment and output. In \cite{Edge2007} and in \cite{Boz2011}, in response to a positive and persistent trend growth shock, a representative agent reduces her labor supply due to the wealth effect while increasing her investment. When the persistence of the shock is higher than a threshold (around 0.2 in \cite{Boz2011} calibration) the decline in labor supply leads to a fall in output even after that capital starts to accumulate.