The steady progress of Sen. Robert Dole's big tax bill demonstrates an extraordinary shift of initiative and influence in American politics. The White House is no longer running American economic policy. The Republican leaders and committee chairmen in the Senate have taken charge of it.

The senior Republicans are now moving forcefully to rescue a stalled economy, a drifting administration and--not incidentally--their party's prospects. This process began nearly a year ago, late last summer, as the congressional Republicans began to see evidence that the economy was not responding as President Reagan's tax and budget program had anticipated. Their success represents a victory of pragmatism over the administration's ideology.

The bill represents a striking reversal of conventional party positions. The Dole bill puts three-quarters of its burden on business taxpayers. It revokes the worst excesses of last year's tax cut, and it includes important reforms like withholding taxes on interest and dividends--a blow against one of the most common areas of tax evasion. The Democrats, in lieu of any better ideas, have been pushing for a repeal of the 1984 reduction in personal income taxes--the burden of which would fall almost entirely on middle-income taxpayers.

When the Reagan administration first came to office, it was full of sharp, clear ideas to transform the economy. Despite the inconsistencies in some of those ideas, Congress passed the first year's program obediently. But then doubts began to set in almost immediately. According to all of the administration's theories, interest rates should have begun dropping rapidly in August, as soon as the tax and budget bills were enacted. And, of course, they did not.

Over the autumn, a wide consensus developed within the administration that the 1981 tax cut had gone too far and an increase would be necessary. Reagan refused to hear of it. When the administration brought out its proposed budget for 1983, last February, the most scathing criticism came from the congressional Republicans, who objected vigorously to its huge deficits. The White House, in effect, shrugged and told Congress to write its own budget.

Confounding all predictions, that's what Congress has done this year--a genuinely astonishing feat that has no real precedent in American politics. As it now stands in the first budget resolution, it's chiefly the handiwork of Sen. Pete Domenici and the Senate Budget Committee. Its adjustments in the Reagan program are not minor. For example, the Domenici budget contains a substantial increase in defense spending--but the increase is hardly more than half the increase in the original Reagan proposal. It's a conservatives' budget, but it follows a rule of reason that was absent from the Reagan original.

The administration held that the budget was to be balanced by spending cuts. The senators, knowing a good deal about the difficulties in cutting spending, saw that it would take more than that to get the deficits under control. That's where the Dole bill came from.

Now that it has passed the Senate, what will the House Democrats do with it? They see more economic trouble ahead, and they don't want Reagan and the Republicans to be able to blame it on the recalcitrance of the Democratic House. They have been following the cardinal rule that they must not leave themselves open to the charge of obstructing the president. If they follow past form, they will make enough of a fuss to fix it in everyone's minds that the responsibility for the tax bill belongs to Mr. Reagan and the Senate Republicans--but, in the end, provide enough votes to pass it.

The idea that this country's economic policy is being made in hand-to-hand combat between President Reagan and House Speaker Tip O'Neill is a fiction that serves the interests of both men-- but it is nonetheless a fiction. The reality is that the hands on the wheel this summer are those of Dole, Domenici, Senate majority leader Howard Baker, Appropriations Committee Chairman Mark Hatfield, and perhaps a few other Republican senators.