The government yesterday estimated that the economy has grown at a 2.8 percent annual rate between July and September, a growth rate well below Reagan administration forecasts.

The Commerce Department estimate of the third-quarter gross national product -- the nation's output of goods and services -- provided the first glimpse of just how strong the long-awaited economic rebound might be for the second half of the year.

The administration has been counting on a strong second-half performance to keep the unemployment rate from rising and to prevent the federal budget deficit from growing larger. The White House has predicted that GNP will grow at a 5 percent annual rate in the second half, bringing overall growth for the year to 3 percent.

Several private economists said yesterday the economy will have to grow at a 7.1 percent rate in the last three months of the year for the Reagan administration to achieve its 3 percent forecast.

The "flash" number released yesterday by the Commerce Department uses only one or two months of economic data plus Commerce Department projections. The GNP estimate is revised as more data become available in the next few months. So far this year, the flash number has been considerably higher than the final figures.

The Commerce Department reported yesterday that the economy grew at a 1.9 percent rate in the second quarter, a downward revision from 2 percent estimated last month. Growth in the first quarter was at a 0.3 percent rate.

Despite the lower than expected GNP report yesterday, the White House issued a statement in which it said it expects the economic recovery to continue. "The president remains convinced that the nation's strongest postwar recovery -- now in its 34th month -- can continue so long as the Congress holds spending down at least to the levels agreed to in the budget compromise," White House spokesman Larry Speakes said in the statement. "We expect real final sales to be up sharply for the third quarter and we expect a higher level of growth for the fourth quarter."

Commerce Secretary Malcolm Baldrige said that the "flash" estimate released yesterday "if realized, means that our economy is emerging from the doldrums of the past year." Baldrige said the sharp increase in assemblies of automobiles contributed about 1 percentage point to the third-quarter growth rate

"While stepped-up growth in output should bolster gains in employment and income during the months ahead, we still face serious economic challenges," Baldrige said. He said the nation's manufacturing sector has "yet to regain the vigor it displayed earlier in the expansion," and farmers are also having difficulties.

However, the government's "flash" report of GNP disappointed many economists who said they expected a sharper improvement from the first half of the year. The report, although forecasting higher growth than in the second quarter, failed to strongly clarify whether the economy was headed up or down, economists said.

The "flash" estimate "is below expectations," said Edward Yardeni, chief economist for Prudential Bache securities dealers. "People have been waiting for the economy to reaccelerate. . . . I don't see anything that suggests the economy is doing anything different. We are growing. We're not in a recession. But growth is sub-par."

Economist Alan Greenspan of Townsend-Greenspan & Co. said that if the third-quarter flash is revised, it is more likely that growth will be revised downward rather than up. Greenspan said that Commerce assumed that the trade picture had stabilized, which may not be the case.

The widening gap between imports and exports has been blamed for the economy's poor performance in more than a year, particularly what many economists call a recession in the manufacturing sector which competes with imports. The Commerce report assumed little change in the trade picture.

"The crucial issue is they assumed very little change in net exports, which may be the Achilles heel of the forecast," Greenspan said.

The "flash" estimate "is very close to expectations and is not terribly useful in giving us the answer to our very important question: Has the economy started to accelerate yet?" Greenspan said. "It's hard to imagine a number which is more neutral relative to the crucial elements we need to know."

The Commerce Department said it expects that consumer and government spending to account for "The crucial issue is they assumed very little change in net exports, which may be the Achilles heel of the forecast." -- Economist Alan Greenspan most of the increase in third-quarter growth. Business fixed investment, residential investment and net exports are expected to change little, Commerce said.

Business inventory investment, which fell in the second quarter, is also expected to decline in the third quarter, Commerce said.

Greenspan said that his analysis showed that significant gains in output were in passenger cars and consumer durables, services and federal, state and local government purchases.

Inflation, measured by the GNP fixed weighted price index that measures changes in prices and not composition of output, is expected to increase 2.9 percent in the third quarter compared with 3.9 percent in the second, Commerce said.

Commerce also reported that after-tax corporate profits increased 0.3 percent in the second quarter compared with an estimate last month that profits declined 0.4 percent. These profits dropped 2.8 percent in the first quarter.

Profits before taxes fell 0.6 percent, compared with last month's estimate of a 0.5 percent drop, Commerce said. These profits also fell 2.8 percent in the first quarter.