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GLOBAL MARKETS-Dollar falls, stocks stay higher after Fed statement

* Dollar dips after Fed statement

* U.S. companies report more upbeat earnings

* U.S. Treasury yields fall after statement (Updates to after Fed statement)

NEW YORK, July 26 (Reuters) - The dollar turned lower while U.S. stocks stayed higher on Wednesday after the Federal Reserve signaled it could begin to cut its massive bond portfolio "relatively soon."

U.S. Treasury yields fell to a session low after briefly rising following the statement.

The U.S. central bank, which also kept its benchmark lending rate in a target range of 1.00 to 1.25 percent, said it was continuing the slow path of monetary tightening that has lifted benchmark U.S. interest rates by a percentage point since 2015.

It said it expected to start winding down its holdings of bonds "relatively soon" in a sign of confidence in the U.S. economy.

The dollar index, which measures the greenback against six world currencies, dropped 0.4 percent and was hovering just above a 13-month low touched on Tuesday. The dollar also hit session lows versus the yen after the statement.

"All the talk in the lead-up to the decision was about some kind of hawkish surprise. It's clear the market was positioned that way, it didn't happen, so bets on the dollar cleared out. Expect the dollar to stabilize quickly and for the market to refocus on data," said Adam Button, chief currency analyst at Forexlive in Montreal.

The Dow Jones Industrial Average was up 109.03 points, or 0.5 percent, to 21,722.46, the S&P 500 had gained 2.88 points, or 0.12 percent, to 2,480.01 and the Nasdaq Composite had added 9.86 points, or 0.15 percent, to 6,422.03.

All three major U.S. stock indexes hit record highs early in the session as Boeing and AT&T led another set of strong earnings reports from U.S. companies.

MSCI's index of stock markets across the world was up 0.3 percent, while European shares ended up 0.5 percent.

Benchmark 10-year U.S. Treasury notes were up 10/32 in price to yield 2.29 percent, down from 2.33 percent Tuesday.

Oil prices settled higher, with Brent futures rising 1.5 percent to $50.97 a barrel, after data showing a fall in U.S. inventories.

For Reuters Live Markets blog on European and UK stock markets see reuters://realtime/verb=Open/url=http://emea1.apps.cp.extranet.thomsonreuters.bi z / c m s / ? p a g e I d = l i v e m a r k e t s

(Additional reporting by Jonathan Spicer and Karen Brettell in New York and Patrick Graham in London; Editing by Mark Trevelyan and James Dalgleish)