In this interview, CEA chair Kevin Hassett (around minute 4:00) dismisses the adverse impact of the government shutdown on real GDP. It seems to me that he is more wrong than right. Kevin appears to be assuming that government workers don't produce anything of value when they are at work, or that they will make up all the undone work when they return, so making them stay at home has no significant economic impact. If that were really the case, we should give them all shorter work weeks, so they can enjoy more leisure.
To me, that does not seem tenable. Take, for example, the national parks that are now closed because of the shutdown. Those families that would otherwise be enjoying them are suffering a true reduction in economic well-being that is forever lost.
My very rough calculation is that economic cost of the government shutdown is in the ballpark of about $100 million per day.