* Dollar up versus yen, dips against the euro * Fed's Dudley, Powell played down stimulus tapering plan * Investors square positions on last trading day of quarter NEW YORK, June 28 (Reuters) - The dollar rose against the yen but fell against the euro on Friday, as investors squared positions on the last trading day of the quarter and the month. For the month, the dollar fell 1.4 percent against the yen, snapping eight straight months of gains against the Japanese currency, while the euro rose 0.5 percent against the dollar. But the medium-term outlook remained in favor of the dollar, with investors increasingly pricing in the chance that the U.S. Federal Reserve will begin to downsize its asset purchase program, perhaps as early as September. Federal Reserve Governor Jeremy Stein said on Friday the Fed must consider the overall economic improvements since it launched the stimulus and not give undue weight to the most recent economic data as it considers scaling back asset purchases.. That prompted a minor pull-back in the dollar but did not alter the day's trading trend. "For the most part, there is overall improvement in risk appetite as the general direction of U.S. monetary policy is keeping demand for the dollar intact," said Kathy Lien, managing director at BK Asset Management. Against the yen the dollar was up 0.6 percent on Friday at 98.94 yen. The dollar index, which tracks the greenback's performance against other major currencies, was down marginally at 82.797 , though not far off Thursday's 83.171, a peak last reached on June 3. The index was on track for its second straight week of gains but lost 0.4 percent on the month. The dollar lost ground against the euro, after Federal Reserve officials on Thursday sought to play down expectations that the central bank was about to cut back its stimulus programme. The euro was up 0.5 percent at $1.3095, with stop-loss sell orders cited at $1.3020. "Today is dominated by quarter-end flows and one should not be reading too much into the price moves," said Chris Walker, currency strategist at Barclays. Large funds rebalance their investment portfolios at the end of the month and the quarter, and their flows and requirements to square positions often dominate trade on the last trading day. "The dollar's uptrend remains intact going into next quarter as we are expecting Fed tapering to start in September. We forecast euro/dollar to fall to $1.26 in six months and the dollar to rise to 103 yen in the next three months," Walker added. Deutsche Bank said in a note that a recent drop in euro zone government yields was likely to weigh on the euro. Yields on euro zone bonds, like those on German Bunds, had risen with U.S. Treasuries, but they have started to fall this week after several European Central Bank officials said monetary policy will remain accommodative. "This (rise in yields) is now reversing, and so rate differentials which were providing support for the euro are longer doing so," Deutsche strategists said. Analysts said growing worries about the euro zone's faltering economy, in contrast to the relative optimism around the U.S. economy, could hurt the single currency. This week European Central Bank President Mario Draghi pointed out downside risks to growth and said the ECB was nowhere near exiting its accommodative monetary policy.