* U.S. dollar falls as Summers withdraw from Fed consideration * Summers was seen as more hawkish than Yellen * View that monetary policy to stay easy for longer drives moves * U.S. stocks advance; Apple limits Nasdaq's rally By Ryan Vlastelica NEW YORK, Sept 16 (Reuters) - Stocks and bonds on major markets rallied on Monday after former U.S. Treasury Secretary Lawrence Summers withdrew from consideration to be the next chairman of the Federal Reserve, leading investors to believe U.S. monetary policy might stay looser for longer. Signs of progress in reducing tensions in the Middle East, after a Russia-brokered deal on Syria's chemical weapons also helped to support stocks. Summers' surprise decision on Sunday came just ahead of the Fed's policy meeting on Tuesday and Wednesday, when the U.S. central bank is expected to begin to scale back its asset purchases from the current pace of $85 billion a month. With the withdrawal of Summers, who was considered more hawkish, investors wagered that U.S. monetary policy would stay easier for longer should the other leading candidate for Fed chair, Janet Yellen, the Fed's current vice chair, get the job. Ben Bernanke's term as Fed chairman expires in January. "His passing as a contender for the top role has left in its wake a significant risk-on rally," said Andrew Wilkinson, chief economic strategist at Miller Tabak & Co in New York. The Dow Jones industrial average was up 138.38 points, or 0.90 percent, at 15,514.44. The Standard & Poor's 500 index was up 12.92 points, or 0.77 percent, at 1,700.91. The Nasdaq Composite Index was up 6.72 points, or 0.18 percent, at 3,728.90. Gains in the Nasdaq were limited by a 2.1 percent decline in Apple Inc shares. European shares rose 0.7 percent while the MSCI all-country world equity index rose 1 percent. With Summers' withdrawal, it was even possible a first Fed interest rate rise could be pushed out to 2016, rather than 2015 as currently expected, said Chris Rupkey, chief financial economist at Bank of Tokyo-Mitsubishi UFJ. Going by Yellen's past speeches, Rupkey said she would probably prioritize reducing unemployment. "Yellen looks like the clear front-runner and seems to be the public's popular choice," Rupkey said. "The Fed will shoot to lower the unemployment rate to the full employment level, and this means the new target could be more (like) 5.5 percent, not 6.5 percent." The benchmark 10-year U.S. Treasury note was up 8/32 in price, with the yield, which moves inversely to price, at 2.8606 percent. German Bunds tracked the moves and were last at 1.884 percent, well down on last week's peak of 2 percent. The more distant Eurodollar contracts rallied as the market pared back expectations for how quickly the Fed might finally start to tighten policy, as opposed to just tapering its bond-purchase program. Contracts from late 2014 out to 2016 all made double-digit gains suggesting a hike was now considered more likely in 2015 rather than in late 2014. DOLLAR DIVE The U.S. dollar index slipped 0.3 percent to 81.244, near an intraday trough of 81.029, its lowest level since Aug. 21. It fell 0.4 percent against the yen while the euro rose 0.35 percent to $1.3339. Liquidity in the yen was lacking, with Japanese markets closed for a holiday on Monday. In the latest U.S. data, industrial output rose 0.4 percent in August, as expected, while manufacturing output rose 0.7 percent, a slightly faster rate than had been forecast. MSCI's broadest index of Asia-Pacific shares outside Japan gained 1.6 percent overnight as South Korean shares added 1 percent, Australia's rose 0.5 percent and Indonesian stocks jumped 3.4 percent. PUSHING OUT THE HIKE Sentiment was underpinned by Saturday's deal between Russia and the United States to demand that Syrian President Bashar al-Assad account for his chemical arsenal within a week and let international inspectors eliminate all the weapons by the middle of next year. Emerging market stocks were up 1.6 percent and most emerging Asian currencies were on the front foot, with India's rupee leading the charge. Investors have pumped much of the cheap money from the Fed into emerging markets. Gold fell 0.7 percent, while Brent crude lost 1.4 percent to $110.13 a barrel and U.S. crude futures sank 1.2 percent to $106.97 per barrel.