Trying to figure out whether the dollar is headed up or down over the next six months?

That’s not exactly a purely academic exercise since a rising dollar has been linked lately to falling stock and commodity prices and a falling dollar has corresponded to rallies in stock and commodity prices.

Right now it looks like direction of the dollar—in that six month time period anyway—hangs on the relative performances of the U.S. and Japanese economies.

For the dollar to rally the U.S. economy doesn’t have to be the strongest in the world. It just needs to be stronger than the Japanese economy. And, frankly, that looks like a bet you can take to the bank for the first half of 2010.

The U.S. economy is projected to grow by 2.6% in 2010, according to a Bloomberg survey of 82 economists. That’s roughly twice as fast as this group predicts for the Japanese economy. (And in my opinion, these economists are being very optimistic if they think Japan is going to hit 1.3% growth in 2010. The Bank of Japan and the Japanese government are both launching new stimulus programs designed to keep the country from falling back into deflation.)

The difference between U.S. and Japanese growth rates is so important because it controls which currency funds the carry trade.

Recently traders have borrowed in U.S. dollars because U.S. interest rates were extremely low and the dollar was falling in price. That let them borrow at almost no cost and then feel confident that they could repay their loans with cheaper dollars. Those dollar loans then went into trades in commodities, emerging market stocks, real estate—anything that promised a return greater than the miniscule interest rate on the dollar-denominated loan.

The effect of all this borrowing in the dollar was to drive down the price of the dollar. Every loan put more dollars into circulation and that depressed the value of the dollar.

Before the dollar took the lead in this carry trade, the yen had been the currency of choice. Japanese interest rates were extremely low and with the Japanese economy stuck in the doldrums there seemed little risk that the yen would appreciate. That preference for the yen ended when the Federal Reserve promised to keep U.S. interest rates near 0% for an extended period and when the U.S. economy showed signs that it would underperform the Japanese economy. The yen has been a stronger currency than the dollar for much of the last half of 2009.

But now it looks like the tide is ready to turn. Investors are starting to bet that with the Japanese economy sinking again LIBOR (London Interbank Offered Rate), the interest rates that the most credit-worthy international banks charge each other for large loans, in Japan will sink below LIBOR in the United States by June.

From 1993 until August 2009 Japanese LIBOR was higher than it was in the United States. During most of that period the yen was in a downtrend versus the U.D. dollar.

The last time that Japanese loans become cheaper than U.S. loans, in August 1993, the yen fell by almost 8% in the following five months.

Since U.S. LIBOR rate began dropping in March, the dollar has tumbled 9.4% against the yen.

If traders begin to buy dollars to repay their dollar loans, and switch to yen loans, the U.S. dollar will strengthened against the yen in particular and to a lesser degree against currencies such as the euro.

The key to this shift will be the speed of the recoveries in the two countries. Right now the futures market is betting that the Bank of Japan will keep Japanese interest rates at a 0.1% benchmark through all of 2010, but that the U.S. Federal Reserve will start to raise rates by June at the earliest and the third quarter of 2010 by the latest.

If you want to follow this race to the bottom, keep your eye on the Bank of Japan and the U.S. Fed. Whichever raises rates first will set the direction for global cash flows. At least in the short term.