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We explore whether modelling parameter time variation improves the point, interval and density forecasts of nine major exchange rates vis-a-vis the US dollar over the period 1976-2015. We find that modelling parameter time variation is needed for an accurate calibration of forecast confidence intervals, and is better suited at long horizons and in high-volatility periods. The biggest forecast improvements are obtained by modelling time variation in the volatilities of the innovations, rather than in the slope parameters. Moreover, we do not find evidence that parameter time variation helps to unravel exchange rate predictability by macroeconomic fundamentals. Finally, an economic evaluation of the different forecast models reveals that controlling for parameter time variation leads to higher portfolios returns, and to higher utility values for investors.