  We present an analysis of high-precision pulsar timing data taken as part of
the North American Nanohertz Observatory for Gravitational waves (NANOGrav)
project. We have observed 17 pulsars for a span of roughly five years using the
Green Bank and Arecibo radio telescopes. We analyze these data using standard
pulsar timing models, with the addition of time-variable dispersion measure and
frequency-variable pulse shape terms. Sub-microsecond timing residuals are
obtained in nearly all cases, and the best root-mean-square timing residuals in
this set are ~30-50 ns. We present methods for analyzing post-fit timing
residuals for the presence of a gravitational wave signal with a specified
spectral shape. These optimally take into account the timing fluctuation power
removed by the model fit, and can be applied to either data from a single
pulsar, or to a set of pulsars to detect a correlated signal. We apply these
methods to our dataset to set an upper limit on the strength of the
nHz-frequency stochastic supermassive black hole gravitational wave background
of h_c (1 yr^-1) < 7x10^-15 (95%). This result is dominated by the timing of
the two best pulsars in the set, PSRs J1713+0747 and J1909-3744.
