The empirical migration literature often identifies the labor market effects of immigration using exogenous variation in migration concentration across sectors or regions. However, this approach differences out macroeconomic effects which occur in all sectors. In this paper we apply macroeconomic time series methods to UK data from 2001-2019 for 35 different sectors, to model, for the first time, immigration, native wages and hours worked, as responding to demand, supply and immigration shocks at both aggregate and sectoral levels. The labor market is thereby modeled as being subject to multiple shocks at any one time. Using a VAR approach, we find that the share of migrant labor is `Granger caused' by other labor market variables which suggests that immigration is, in part, endogenously determined by aggregate demand and supply. However, it also retains a component which has a negative association between immigration and native wages, which may be thought of as a `migration shock'. Using historical decompositions which decompose both the error terms and, novelly, the constant terms into their structural parts, we show that the `migration shock' accounts for most of the change in migration share over the sample period and plays a significant negative role in the determination of native wage growth, particularly in unskilled sectors. However other contemporaneous shocks have offsetting positive associations between immigration and native wages, whose effects differ substantially across sectors.