Prior work variously ascribes the forward puzzle -the low slope in the Fama (1984) regression of the exchange rate change on the forward premium- to various model misspecications or statistical problems with non-stationary forward premia, but no single theory fully succeeds in explaining the puzzle. In this paper we simultaneously address the model-misspecification problem and the non-stationarity issue. On the basis of competing hypotheses about the risk premium we consider the nonlinear models that specify the Fama beta as approximately quadratic or spline functions of the forward premium. We estimate these relations using overlapping one-month observations for ERM-member exchange and forward rates against the DEM. The standard deviations are calculated under the Monte Carlo Method for overlapping observations. Wald test confirms the presence of such nonlinearities, and the models outperform the Fama in terms of various in the goodness-of-fit measures, but the spline adds little relative to the simple quadratic. To handle the non-stationary forward premium problem, we decompose the forward premium into a long-memory co-movement component and a short-term filtered forward premium. In regressions that link exchange-rate changes to the long-memory co-movement component the forward puzzle worsens, while it is substantially reduced when, instead, the filtered component isused as the regressor, suggesting that the filtered component loads relatively heavily on expectations and the slow-moving trend on the missing variable. Beta appears to be an inverse-U-shaped function of the forward premium. This contradicts the Bansal risk premium and the transaction-cost/limit-to-arbitrage hypotheses, but is consistent with a Fallen-angel effect, where traders or portfolio managers shun long positions in assets with danger signals like forward discounts.