The index of leading economic indicators shot up 3.6 percent in January, the largest increase since 1950, unmistakably signaling that a recovery from the recession is under way.

President Reagan said the indicators, which usually foreshadow changes in the economy, "flashed a bright green light for recovery."

Most economic forecasters saw the large gain as validating their increasingly optimistic predictions of how fast the recovery's pace will be this year. With the prospect of further large declines in oil prices, many forecasters have raised their estimates of real growth this year by about a percentage point while lowering their inflation predictions by a similar amount.

But there were some words of caution.

Martin S. Feldstein, chairman of the president's Council of Economic Advisers, said recent economic statistics have convinced him that the recession hit bottom in December and that January was the first month of the recovery.

Nevertheless, he noted that part of the big jump in the leading indicators was due to a large increase in the nation's money supply in January, which was related more to the advent of the popular money market deposit accounts than to any change in monetary policy.

Feldstein, whose forecasting caution has been criticized by some other administration officials speaking on a not-for-attribution basis, said the February figures on employment and unemployment, due out Friday, will show whether January's strong gains continued. "We are making a lot of January," he said.

At the Commerce Department, Secretary Malcolm Baldrige spoke in the same vein. "There has never been a gain in the leading index this large in the past without an economic recovery." However, he added, the increase "should not be taken as a sign of a coming economic boom" since a variety of special factors, including unseasonally mild weather, reinforced the climb.

And Alexander B. Trowbridge, president of the National Association of Manufacturers, also expressed considerable reservation about the speed of the recovery. "On the basis of the January numbers, we think we can revise upward our estimate of real growth in the economy this year to a level of 3 1/2 to 4 percent, which is higher than our previous estimate of 2 1/2 to 3 percent," he said.

But Trowbridge added, "We need a little more time and evidence to know how strong the recovery will be because some manufacturers are seeing a substantial improvement in their order books while others are not. And we are hoping that some of our basic industries will come along stronger than they look now."

CEA chairman Feldstein acknowledged that the administration's official forecast of a 3.1 percent increase in the gross national product, adjusted for inflation, between the fourth quarter of 1982 and the fourth quarter of 1983, is probably too low. The figure could be as much as 5 percent, he said.

That will depend in part on what happens to oil prices. Official OPEC prices per barrel could well drop into the mid-$20 range, conceivably lower, Feldstein agreed. Then he added, "The good news on oil that is so much fun to talk about hasn't happened yet."

Part of the unexpectedly swift beginning for the recovery is due to a sharp decline in business inventories in the fourth quarter of last year. That decline occurred when sales of goods outpaced production, which was severely cut back. Now, just to halt the depletion of their stocks of goods, businesses must increase production to bring it more in line with sales.

After that, further production increases are not needed unless sales rise or businesses decide to increase their inventories. That is one reason some economists are still cautious about the outlook.

Production clearly is rising this quarter, but the picture for sales is still cloudy, particularly for autos. Some of the larger production increases have occurred on auto assembly lines, and output is once again running well ahead of sales of domestic cars, which fell to a seasonally adjusted annual rate of 5.6 million units in the second 10 days of February from a 6.2 million rate in the first 10 days. If sales don't pick up again, the ambitious production plans of the automakers may have to be scaled back.

The economic consulting firm of Townsend-Greenspan & Co., headed by Alan Greenspan, recently warned its clients, "Despite the very strong tone in the economy at the moment, it should be emphasized that it is essentially an inventory reversal that is currently in place coupled with considerable strength from residential construction which reflects, in part, the surprisingly warm winter throughout the country. The extent of follow-through in retail and capital goods markets is, as yet, an unproved element."

At Wharton Econometric Forecasting Associates, Donald H. Straszheim was more ebullient. He said his group's forecast was already for a strong 4.6 increase in real GNP over the four quarters of 1983 before oil prices began to fall. "We have just done a new forecast knocking the oil price down from $34 to $27 a barrel. It does wonders. The oil price drop should add half a percentage point or more" to real growth this year, he said.

Of the dozen individual indicators comprising the leading index, 10 are available for January and nine of them rose that month, the Commerce Department reported.

About one-fourth of the total increase was due to the jump in the money supply, while large increases in the length of the average workweek in manufacturing and in new orders for manufactured consumer goods, adjusted for inflation, also contributed heavily. The only indicator falling was the level of contracts and orders for business plants and equipment, adjusted for inflation.