  We analyze the frequency dependence of the dispersion measure (DM), the
column density of free electrons to a pulsar, caused by multipath scattering
from small scale electron-density fluctuations. The DM is slightly different
along each propagation path and the transverse spread of paths varies greatly
with frequency, yielding time-of-arrival (TOA) perturbations that scale
differently than the inverse square of the frequency, the expected dependence
for a cold, unmagnetized plasma. We quantify DM and TOA perturbations
analytically for thin phase screens and extended media and verify the results
with simulations of thin screens. The rms difference between DMs across an
octave band near 1.5~GHz $\sim 4\times10^{-5}\,{\rm pc\ cm^{-3}}$ for pulsars
at $\sim 1$~kpc distance. TOA errors from chromatic DMs are of order a few to
hundreds of nanoseconds for pulsars with DM $\lesssim 30$~pc~cm$^{-3}$ observed
across an octave band but increase rapidly to microseconds or larger for larger
DMs and wider frequency ranges. Frequency-dependent DMs introduce correlated
noise into timing residuals whose power spectrum is `low pass' in form. The
correlation time is of order the geometric mean of the refraction times for the
highest and lowest radio frequencies used and thus ranges from days to years,
depending on the pulsar. We discuss the implications for methodologies that use
large frequency separations or wide bandwidth receivers for timing
measurements. Chromatic DMs are partially mitigable by using an additional
chromatic term in arrival time models. Without mitigation, our results provide
an additional term in the noise model for pulsar timing; they also indicate
that in combination with measurement errors from radiometer noise, an arbitrary
increase in total frequency range (or bandwidth) will yield diminishing
benefits and may be detrimental to overall timing precision.
