The government's index of leading indicators, a guide to the future health of the economy, turned down in August after four months of consecutive increases, while initial claims for unemployment benefits soared to a record high in mid-September, official reports said yesterday.

The Commerce Department's index of leading indicators sank by 0.9 percent in August, while the number of new claims for state unemployment benefits climbed to 703,000 in the week ending Sept. 18, according to the Bureau of Labor Statistics.

The new figures add to fears that unemployment, already at a post-World War II peak, is likely to climb past 10 percent and stay high for several months to come, analysts said yesterday.

Commerce Secretary Malcolm Baldrige said, however, he was "confident" the economy would recover in the final three months of this year, despite the August dip in the leading indicators.

President Reagan's new chairman of the Council of Economic Advisers, Martin Feldstein, who was confirmed by the Senate yesterday, said, "It's a number that bounces around." The August decline in the index, after four months of increases, does not change his basic view that the "economy is bottoming out and ready to recover," Feldstein said.

Baldrige warned reporters that the jobless rate would likely not come down much until "three, four or five" months after a recovery began. He later added that unemployment would begin to decline "sometime around the end of the year or beginning of next year."

"We feel this month's dip in the leading indicators is a temporary interruption," the Commerce Secretary said, echoing Reagan's comments in a news conference on Wednesday that the foreshadowed decline in the index -- the government's main guide to the future course of the economy -- was no more than a "glitch" or "blip." The leading indicator index will probably rise in September, Baldrige said. The dramatic rally in the stock market that began in August will show up in the September figures.

But other analysts are more pessimistic. The earlier increases in the leading indicators were "less than a third" as large as normal for a recovery period, economist Otto Eckstein of Data Resources Inc. said yesterday. He added, "even if the index is up . . . in September, it wouldn't mean anything. That's not recovery." Alan Greenspan, chairman of the CEA under President Ford, commented that "what we are looking at is an economy which clearly has exhibited no signs yet of recovery," although he added that "we must assume the turn is near at hand."

The administration has been predicting an economic recovery since the beginning of the year, but its forecasts have so far been confounded. Baldrige said that the recent sharp decline in interest rates had laid the groundwork for recovery, together with a rise in real, after-tax incomes, and slowing inflation.

Democrats seized on yesterday's bad economic news, with House Speaker Thomas P. O'Neill Jr. (D-Mass.) describing Reagan as "Hoover with a smile." Reagan earlier this week blamed the nation's economic troubles on previous Democratic administrations and claimed to have brought America back from the brink of disaster.

"The fact is he [Reagan] brought it to the brink of disaster," O'Neill said, adding that the president has broken his promises to create 13 million new jobs, to balance the budget by 1984 and to boost business investment and expansion.

A separate report published by the Commerce Department and the Department of Housing and Urban Development yesterday showed a 2 percent increase in sales of new single-family homes in August. New houses were sold at an annual rate of 359,000 in August, up from July's total but down a bit from the June rate. Meanwhile, resales of existing family homes dropped 6.3 percent to an annual rate of 1.79 million, the lowest level of the recession so far, the National Association of Realtors said yesterday.

The decline in the leading indicators had been anticipated by the administration last week, softening the blow from yesterday's announcement. Earlier this summer officials had pointed to the four months of increases as evidence of recovery.

Five out of the 10 indicators in the gauge contributed to the August decline, yesterday's Commerce Department report said. A drop in building permits played the biggest part. The initial claims for unemployment in August, which averaged 597,000 a week, were another factor in the overall decline, along with average workweek, manufacturers' new orders for consumer and intermediate goods after allowing for inflation, and changes in sensitive prices.

The record level of initial claims in the latest week was partly because of a spillover of claims from the preceding short week, which included Labor Day, analysts said. However, when the two weeks are averaged together they still show claims at a very high level of 657,000, well up on the August average.