Ronnyreagan, when Bush left office, the stimulus plan was just a thin outline. It wasn't until the latter half of the year when the plan had ballooned into the $800+ billion monster that Obama signed (after personally lobbying for it).
This isn't like any regular appropriations bill that was introduced during the tenure of one president, then signed by the next.
Obama's crew also puts the debt for the various business loans into GW's debt column, but conveniently uses the repayment of those loans a couple of years later to reduce his own deficits.