Stocks faded in the last half-hour of trading Tuesday after staying positive most of the day, closing mostly lower for a fourth consecutive session, with Washington politics the biggest threat to the 2013 stock rally, investors said. The Dow industrials and broader Standard & Poor’s indexes fell on speculation that the U.S. government may shut down on October 1. Even if a shutdown is avoided, the larger concern is whether Congress will raise the debt limit before the government risks defaulting on its debt. U.S. economic news was mixed. Home prices in August were up sharply from a year ago, but consumer confidence fell in September. The U.S. dollar rose versus the euro and the yen. Light sweet crude for November delivery fell slightly to near $103 a barrel on the New York Mercantile Exchange. Gold futures were little changed at around $1,320 an ounce. The Dow Jones industrial average fell 66.79, or 0.4 percent, to 15,334.59. The S&P 500 index fell 4.42, or 0.3 percent, to 1,697.42. The technology-heavy Nasdaq composite index was little changed, rising 2.96, or less than 0.1 percent, to 3,768.25. Shares of Facebook rose almost 3 percent to a record high, as an analyst upgraded the social-networking giant’s stock and there were reports that China may soon loosen some of its internet restrictions, which could allow the company to enter the world’s most populous market.