Unemployment and the incumbent president: look at the last 16 months

Binyamin Appelbaum recently warned in the New York Times:

No American president since Franklin Delano Roosevelt has won a second term in office when the unemployment rate on Election Day topped 7.2 percent. 

In response, political scientist Seth Masket plotted out presidential elections since 1948 and concluded, "the unemployment rate does not predict presidential elections at all." Carlisle Rainey dug deeper into the data and found that there does seem to be some relationship when incumbent presidents are up for re-election. Analyzing a massive series of data subsets, Rainey concludes more broadly, "the data are consistent with both very small and moderately large effects."  Nate Silver, similarly, dives deep and comes up agnostic, concluding that the unemployment rate does reflect (if not drive) the incumbent's fortunes,but that the relationship between the unemployment rate and electoral fortunes is too entwined with a host of other factors to map out with any confidence.

Silver also looks at the direction unemployment moves over an incumbent's term. Here I think he is getting warm. But as he looks over the whole four years, he finds little correlation.

I have for some time been (nervously) tracking Obama's political fortunes against Reagan's, since it's often pointed out that both inherited a vicious recession; at Obama's low moments I've taken some comfort in the fact that Reagan's approval rating reached a nadir of 35% in January 1983, shortly after the unemployment rate peaked at 10.8%.  The impression I've soaked up is that the unemployment rate really matters in the 16-odd months before a president stands for re-election, when the electorate starts to tune in and measure him against the opposing party's candidates.  For Reagan, I've noted, a decisive turn came in July 1983, when the unemployment rate dipped to 9.4% from 10.1% the prior month. From there he had the wind at his back to October 1984, when the rate was 7.4%. Hence Reagan's landslide.  He asked us to stay the course, and the course started running downhill run when it counted.

Might it be postulated that the direction in which employment moves in the 16 months prior to an incumbent president's reelection bid is a fair predictor of his political fortunes?  The postulate works for six of the seven incumbents since Nixon, the earliest president for whom I have monthly unemployment numbers. Based on yearly unemployment figures, it seems also to work for Roosevelt and Eisenhower; and while the data for Truman is neutral, the rate was extremely low to begin with, and recent research indicates that economic growth was very strong.

Here are the figures for each incumbent, with the one exception to the 'rule' highlighted:

Nixon:    July 1971 - 6.0%; Oct 1972 - 5.3%

Ford:      July 1975 - 8.6%; Oct 1976 - 7.7%

Carter;   July 1979 - 5.7%; Oct 1980 - 7.5%
Reagan:  July 1983 - 9.4%; Oct 1984 - 7.4%
Bush Sr: July 1991 - 6.8%; Oct 1992 - 7.3%
Clinton:  July 1995 - 5.7%; Oct 1996 - 5.2%
Bush Jr: July  2003 - 6.2%; Oct 2004 - 5.5%

Here too is some yearly data for presidents prior to 1960:

Roosevelt:   1933 - 19.8%; 1936 - 16.6%; 1940 - 14.6%
Truman:       1947  - 3.9%;  1948 - 3.8%
Eisenhower: 1955 - 4.4%;   1956 - 4.1%

I realize, as Masket and Rainey caution, that this is a small data set, so conclusions are tentative. But what say the math whizzes? Is there nothing here?