LONDON/SYDNEY, June 29 (Reuters) - Global shares edged lower on Tuesday, as new coronavirus outbreaks in Asia vied with strong U.S. consumer confidence, and as investors speculated about whether the Federal Reserve would accelerate its timetable to end easy monetary policy. MSCI"s all country world index, which tracks shares across 50 countries, shed 0.14%, as declines in Asian equities undercut new highs in European and U.S markets. The S&P 500 hit a record high for the fourth straight session, helped by technology and banks stocks, and a government survey that showed U.S. consumer confidence in June hit its highest since the pandemic started. European stocks measured by the the pan-European STOXX 600 index rose 0.31%, helped by a jump in industrial, financial and mining stocks set to benefit from economic improvements. However, MSCI"s broadest index of Asia-Pacific shares outside Japan closed 0.55% lower, while Japan"s Nikkei lost 0.81%, as some countries re-imposed lockdowns to contain the spread of the Delta variant of COVID-19. Chinese stocks lost 0.92% as investors booked profits after a rally on the economy"s strong rebound from the pandemic. Fears over the highly infectious Delta variant are denting sentiment in markets already on edge after the Fed appeared to take a hawkish tilt this month. Indonesia is grappling with record-high cases, Malaysia is extending its lockdown and Thailand announced new restrictions. Spain and Portugal were imposing travel restrictions on unvaccinated British travelers. "The outlook for policy in general and the U.S. specifically, both fiscal and monetary, is the more relevant factor in the market"s mind right now rather than the spread of the Delta variant," said James Athey, investment director at Aberdeen Standard Investments. "That may well prove to be naive or complacent." On Friday, markets will watch the U.S. jobs report for June, with economists polled by Reuters expecting a gain of 690,000 jobs this month, up from 559,000 in May. Richmond Federal Reserve President Thomas Barkin said on Monday the U.S. central bank has made "substantial further progress" toward its inflation goal in order to begin tapering asset purchases. Those comments and the anticipation of a strong jobs report have investors on edge that the Fed will bring forward its timeline for interest rate increases. "Inflation is already much higher than the Fed was anticipating, so it is really the pace of improvement in the labor market that stands head and shoulders above every other indicator in terms of when the Fed will feel comfortable signaling the start of tapering," said Ray Attrill, Head of FX Strategy at National Australia Bank in Sydney. On Wall Street, the Dow Jones Industrial Average fell 6.68 points, or 0.02%, the S&P 500 lost 3.05 points, or 0.07%, and the Nasdaq Composite added 3.44 points, or 0.02%. Germany"s DAX added 1%, grinding higher after data showed an easing of Germany"s annual consumer price inflation in June. The reading of 2.1% was still above the European Central Bank"s target of close to but below 2%. Yields for benchmark 10-year U.S. Treasuries last fell 1/32 in price to yield 1.4799%, from 1.478%. Germany"s 10-year bond yield was up 1 basis point at -0.173%. The U.S. dollar rose to a one-week peak. The dollar index, which tracks the greenback versus a basket of six currencies, rose 0.192%, with the euro down 0.2% to $1.19. Sterling was last $1.3848, down 0.25%. The Australian dollar, seen as a liquid proxy for risk appetite, fell 0.70% versus the greenback at $0.751. Oil prices rose as hopes for a demand recovery persisted despite new outbreaks of the Delta variant. Brent crude futures settled up 8 cents, or 0.11%, and U.S. crude settled up 7 cents, or up 0.10%. Spot gold dropped 0.8% to $1,764.92 an ounce. U.S. gold futures fell 0.97% to $1,762.60 an ounce.