By John S. McClenahen To signal serious improvement in the U.S. jobs market, initial claims for unemployment insurance need to fall well below the 400,000 mark and keep moving down. Last week claims did fall, but at 425,000 they were still well above the point of serious improvement, U.S. Labor Department data show. Indeed, the department's four-week moving average of initial unemployment claims was 446,000 for the week ending May 3, an increase of 3,250 from the previous week's revised average of 442,750. Many economists consider the moving average to be a better indicator of job market conditions, because it tends to smooth out sometimes volatile week-to-week changes. The Labor Department's latest data also show that 3.665 million people were drawing benefits under unemployment insurance during the week ending April 26, an increase of 6,000 people from the previous week.