This paper proposes a model of the US unemployment rate which accounts for both its asymmetry and its long memory. Our approach, based on the tests of Robinson (1994), introduces fractional integration and nonlinearities simultaneously into the same framework (unlike earlier studies employing a sequential procedure), using a Lagrange Multiplier procedure with a standard limit distribution. The empirical results suggest that the US unemployment rate can be specified in terms of a fractionally integrated process, which interacts with some non-linear functions of the labour demand variables (real oil prices and real interest rates). We also find evidence of a long-memory component. Our results are consistent with a hysteresis model with path dependency rather than a NAIRU model with an underlying unemployment equilibrium rate, hence giving support to more activist stabilisation policies. However, any suitable model should also include business cycle asymmetries, with implications for both forecasting and policymaking.