NEW YORK, Nov 5 (Reuters) - U.S. stocks closed sharply higher in a broad rally on Tuesday after data signaled a solid economy, but investors braced for volatile trading this week as voting was underway in an extremely tight U.S. presidential election.

The Institute for Supply Management said its non-manufacturing purchasing managers index, a gauge of the services sector, accelerated to 56.0 last month, its highest since August 2022, from 54.9 the prior month and above the 53.8 expected by economists polled by Reuters.

The election outcome could take days to be finalized as the latest polls showed the race between Republican Donald Trump and Democrat Kamala Harris, which has impacted markets in recent months, was too close to call.

The former president's odds improved on Tuesday in betting markets that many investors see as election indicators.

"The market continues to try and price for what is the outcome of this election," said Rob Haworth, senior investment strategist at U.S. Bank Wealth Management in Seattle.

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"It's been so tight and ...we've been in a tight price range, and so what's really moving us is marginal positioning for one result or the other."

More:How Trump and Harris differ on taxes, spending, inflation, tariffs

"Both the bond market and the equity market are looking at Congress as important as well," he added. "Most base cases are for divided government, but this election is so close we could get any outcome. That's the challenge."

The Dow Jones Industrial Average .DJI rose 427.28 points, or 1.02%, to 42,221.88, the S&P 500 .SPX gained 70.07 points, or 1.23%, to 5,782.76 and the Nasdaq Composite .IXIC gained 259.19 points, or 1.43%, to 18,439.17.

Volatility was more pronounced in government debt and currency markets. The benchmark 10-year U.S. Treasury note yield US10YT=RR rose more than 10 basis points to a high of 4.366% before paring gains on a solid auction, and was last down 2 basis points on the day.

Equity markets avoided Monday's volatility on expectations of a soft landing for the economy, bolstered by corporate earnings, lower interest rates and a resilient labor market.

Other economic data on Tuesday showed the trade deficit hit a 2-1/2 year high in September, as domestic demand draws in imports while concerns about higher tariffs under a Trump presidency have led to a front loading of imports by businesses.

Still, the CBOE Volatility Index.VIX, also known as Wall Street's "Fear Gauge," closed at 20.49, above its long-term average of 19.46, although it had eased from a near-two month high hit last week of 23.42.

Industrials .SPLRCI, up 1.67%,and consumer discretionary .SPLRCD, up 1.83%,led S&P 500 sectors higher and were among five to gain at least 1.3% on the session.

Investors are also keeping an eye on Congressional elections to determine the balance of power in Washington. Many analysts predict a split government, which would limit the ability of the president to enact significant policy changes.

More:US yields dip as 'Trump trades' unwind in tight presidential race

Stocks viewed as proxies on a win for the former president experienced large swings, with Trump Media & Technology Group DJT.O climbing as much as 18.64% and dropping as much as 8.42%, while also being halted for volatility multiple times. Its shares eventually closed down 1.16% on the session.

Crypto stocks tracked bitcoin BTC=higher, with the cryptocurrency up roughly 3%, as Trump has positioned himself as an ally to the sector.

Palantir PLTR.Nsurged 23.47% to close at a record $51.13 after the data analytics firm raised its annual revenue forecast for the third time.

The Federal Reserve will announce its latest policy statement on Thursday. Markets have almost completely priced in a 25-basis point interest rate cut, but the outlook for the path of future easing is less certain given the U.S. economy's strength.

Advancing issues outnumbered decliners by a 4.44-to-1 ratio on the NYSE and by a 2.67-to-1 ratio on the Nasdaq.

The S&P 500 posted 23 new 52-week highs and seven new lows while the Nasdaq Composite recorded 110 new highs and 104 new lows.

Volume on U.S. exchanges was 12.64 billion shares, compared with the 11.77 billion average for the full session over the last 20 trading days.

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Reporting by Chuck Mikolajczak; Editing by Richard Chang