The foundation of the New Keynesian Phillips curve (NKPC) is a model of price setting with nominal rigidities that implies that the dynamics of inflation are well explained by the evolution of real marginal costs. In this paper, we analyze whether this is a structurally invariant relationship. We first estimate an unrestricted time-series model for inflation, unit labor costs, and other variables, and present evidence that their joint dynamics are well represented by a vector autoregression (VAR) with drifting coefficients and volatilities. We then apply a two-step minimum distance estimator to estimate deep parameters of the NKPC. Given estimates of the unrestricted VAR, we estimate parameters of the NKPC by minimizing a quadratic function of the restrictions that this theoretical model imposes on the reduced form. Our results suggest that it is possible to reconcile a constant-parameter NKPC with the drifting-parameter VAR; therefore, we argue that the price-setting model is structurally invariant.