We compare estimates of the New Keynesian Phillips Curve (NKPC) when the curve is specified in two different ways. In the standard difference equation (DE) form, current inflation is a function of past inflation, expected future inflation, and real marginal costs. The alternative closed form (CF) specification explicitly solves the DE form to express inflation as a function of past inflation and a present-discounted value of current and expected future marginal costs. The CF specification places model-consistent constraints on expected future inflation that are not imposed in the DE form. In a Monte Carlo exercise, we show that estimating the CF version of the NKPC gives estimates that are much more efficient than the estimates obtained from the DE specification. We then compare DE and CF estimates of the NKPC with time-varying trend inflation on actual data. The data and estimation methodology are the same as in Cogley and Sbordone (2008). We show that DE and CF estimates differ substantially and have very different implications for inflation dynamics. As in Cogley and Sbordone, it is possible to estimate DE specifications of the NKPC where lagged inflation plays no role once trend inflation is taken into account. The CF estimates of the NKPC, however, typically imply as large a role for lagged inflation as for expected future inflation. These estimates thus suggest that trend inflation is not in itself sufficient to explain the persistent dynamics of inflation.