June is also a pivotal time for the economy. Fed Chair Janet Yellen has said the central bank could raise rates this year depending on the economy. So each measure of the economy's strength has become very important, and Friday's employment report tops the list.
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"The bar has been raised to see a turn in the data, if we're going to dig out of the first-quarter pothole in any way that makes us feel like things are getting back on track," said Lupton. "If last year was the template, we should have seen things turning, and we haven't. While we're not looking for the strong bounce we saw last year, we need to see things moving back up."
For the U.S., the big events are the May employment report Friday, and the Fed's meeting later in the month. There are 225,000 nonfarm payrolls expected, barely higher than the 223,000 in April.
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Economists are watching to see if the U.S. data begin to improve after the first quarter's 0.7 percent contraction in GDP. Second-quarter forecasts have been lowered from growth of more than 3 percent, to about 2.5 percent as the economic reports continue to disappoint.
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Stocks enter June on a positive footing, after May's 1 percent gain in the S&P 500. June historically is more often a weak month for stocks. The S&P 500 was up 4 at 2,111 on Monday, and the Dow was up 29 to 18,040. For the Dow, June has been the second-worst performer over the past 20 years, and the Dow has declined in eight of the past 10 Junes, for an average loss of 1.6 percent.
Some strategists expect to see a pullback in the stock market in the void ahead of the next earnings season.
"I'm not a big buyer of seasonal factors," said Savita Subramanian, chief U.S. equity and quantitative strategist at Bank of America Merrill Lynch.
But she said there are factors that give her concern, including the fact that second-quarter earnings could be negative. The S&P 500 was trading in a range between 2,102 and 2,120 on Monday, "and our target is 2,200. I think we could hit a soft patch here," she said.
But Subramanian expects stocks to do well this year, thanks in part to the easing of central banks. "It's hard to be short global growth when central banks are behind this effort to reinvigorate their economies," she said. That should help the stocks of multinationals.
The Fed is getting very close to set an opposite course, tightening as the other central banks continue to ease. While not likely, she said it would be very negative if the U.S. central bank were to find the economy too weak to raise interest rates. It would be even worse if it moved back to extraordinary easing programs. "That would be the death knell for risk assets," she said.
Subramanian said she's not concerned about the spillover on U.S. stocks from a negative event in Greece, and in fact it might drive funds to the U.S. Analysts give a Greek exit from the euro zone higher odds than in the past, but the markets have mostly expected a resolution.
"Our house view is the likelihood of an exit is low," she said. "I don't think it's going to undermine the entire global economy." Greece is due to make a 300 million euro ($327 million) repayment to the International Monetary Fund on Friday and another 1.2 billion euros this month.
As negotiations drag on without an agreement between Greece and its lenders, speculation has grown that Greece will have to impose capital controls. Lupton said that could buy more time. "There are uncertainties here. How much of a liquidity shock is this going to represent? The other uncertainty is what (Prime Minister Alexis) Tsipras wants to do with the time that is created by capital controls. Tsipras wants to have Greece stay in the euro area, but the road they've gone down is pushing them toward an outcome where that is more unlikely."
Lupton said while the ECB is not expected to take action at its Wednesday meeting, the bank's news conference will likely generate headlines about Greece.
The same could be true of the G-7 meeting. "I think probably the G-7 meeting continues to get consumed by the whole Greek thing," said Stephen Stanley, chief economist at Amherst Pierpont Securities. The summit will be held in Germany.
"It feels like the Europeans have done everything they could to inoculate the rest of the currency bloc in the event Greece defaults. Even so, we continue to hang on every word and act as if it's a big deal if something happens. Frankly, for as trivial as Greece's economy is in the world economic context, here we are five years later and we're still talking about it," Stanley said.
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Traders also are watching the Organization of the Petroleum Exporting Countries meeting Friday. While OPEC is expected to stick to its 30 million barrels a day production quota, headlines from the event could influence the oil market. The cartel has actually increased output, pumping 31.2 million barrels a day in May, according to Reuters.
OPEC, in November, decided to keep production unchanged and said it would let the markets determine oil prices. Since then oil collapsed, but has rebounded recently. West Texas Intermediate was trading near $60 a barrel Monday.
Markets are also watching for a June 9 decision on whether mainland China shares are going to be used in the MSCI index. That ruling, if affirmative, means that China shares would be included in its Emerging Markets Index, which is the index used for the largest Emerging Market ETF.
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Marc Chandler, chief currency strategist at Brown Brothers Harriman, said other events he's watching in June include the Turkish election June 7 and the EU summit June 25 and 26. "On June 15, Saudi Arabia is going to launch a QFII," he said. The qualified foreign institutional investors program will allow foreign investors access to the Saudi market.