This is a guest post by political scientist Nicholas Goedert.

In the wake of the government shutdown and declining Republican popularity, Democrats are growing more optimistic about winning a majority in the House of Representatives in 2014.   Several recent polls show Democrats leading the “generic ballot” for Congress by eight or nine points.  Yet they still face an uphill battle; in 2012, Democrats won a majority of the congressional popular vote but a minority of seats, due to district maps biased against them through both partisan gerrymandering and asymmetric population distributions.   But just how steep is this hill?  How big of a national majority would Democrats need to win that 218th seat?

David Wasserman of the Cook Political Report estimates that it would take a 6.8 percent lead in the national vote for Democrats to win control.  And if ranked by vote share, the Republican candidate won the 218th seat by a similar amount (6.3 percent) in 2012.  But the 2014 electoral landscape might not mirror the 2012 congressional results, when Democrats concentrated so much more of their campaign efforts on retaining the Presidency and Senate.  If the House appears more competitive at a national level in 2014, it is likely that Democrats will field higher-quality candidates and devote more resources to these new swing seats.

Under two alternate methods, I estimate that Democrats could win the House with an even smaller lead in the generic ballot — a lead closer to 5 percent.  But there is still much uncertainty as to how much the bias from a previous cycle will carry over into future elections.

In 2012, Barack Obama was reelected with a 3.9 percent margin of the national popular vote, yet won only 209 of the nation’s 435 congressional districts. To have won a majority of districts, Obama would have needed to win nine more.  So one way to quantify any GOP bias in the congressional landscape is to ask: “How much more would Obama need to have won by in order to win nine more districts?”

One way to do this is to rank all the districts by Obama’s margin of victory, and see how much he lost by in the 218th district.  In 2012, Obama lost this district by 1.7 percent.  If we assume a uniform swing toward Obama, he would have needed to win by 5.6 percent to win the median district (3.9 percent+1.7 percent=5.6 percent).  So based on the 2012 results, a Democrat would need to win the national vote by about 5.6 percent to win a majority of congressional seats.  On the other hand, we can also examine Obama’s 2008 vote share under the 2012 district lines.  In 2008, Obama won the popular vote by 7.3 percent and the re-ranked 218th district by 2.9 percent, implying that in 2012 he would have needed to win by 4.4 percent overall to win a majority of districts.  Splitting the difference between these two estimates suggests that Democrats would need to win the popular vote by about 5 percent to win a majority of seats.

Another method of estimating the vote margin the Democrats would need to retake the House is to compare their actual performance in 2012 to historical levels of responsiveness in congressional elections.  That is, based on historical averages, how many more votes would it take to win the 17 additional seats Democrats need on top of what they won in 2012?

In 1973, Edward Tufte estimated that from 1900-1970, a 1 percent increase in a party’s national congressional vote generally led to a 2 percent increase in their seat share, a seats-to-vote ratio has also remained strikingly stable since then. We can apply this 2:1 ratio to estimate what the Democrats need to win the majority.  In 2012, Democrats won 50.6 percent of the two-party vote share, and 46.2 percent of the seats.  Thus, in order to win the additional 3.8 percent of seats necessary for a majority, Democrats would need 1.9 percent more of the vote, or about 52.5 percent of the major-party vote share.  This would imply winning the national popular vote by 5 percent, the same threshold we arrived at by the “median presidential district” method above.

Yet even though these methods generate similar estimates, there is still considerable uncertainty in these estimates, as bias in congressional elections has varied widely between cycles.  The graph below plots the difference between median and mean Democratic vote share in congressional elections over 50 years, with positive figures indicating bias in favor of the Democrats. (These data are from Eric McGhee at the Public Policy Institute of California.)  There have been several dramatic changes in bias, even in the middle of decades.  For example, the bias favoring Republicans shrank from 5.4 percent in 2004 to essentially 0 percent in 2008.  So it would not be unheard of for the bias in 2012 to disappear in a subsequent election.