  The ratio of branching fractions ${\cal{R}}(D^{*-})\equiv {\cal{B}}(B^0 \to
D^{*-} \tau^+ \nu_{\tau})/{\cal{B}}(B^0 \to D^{*-} \mu^+\nu_{\mu})$ is measured
using a data sample of proton-proton collisions collected with the LHCb
detector at center-of-mass energies of 7 and 8 TeV, corresponding to an
integrated luminosity of 3$~$fb$^{-1}$. The $\tau$ lepton is reconstructed with
three charged pions in the final state. A novel method is used that exploits
the different vertex topologies of signal and backgrounds to isolate samples of
semitauonic decays of $b$ hadrons with high purity. Using the $B^0 \to
D^{*-}\pi^+\pi^-\pi^+$ decay as the normalization channel, the ratio
${\cal{B}}(B^0 \to D^{*-} \tau^+ \nu_{\tau})/{\cal{B}}(B^0 \to
D^{*-}\pi^+\pi^-\pi^+)$ is measured to be $1.97 \pm 0.13 \pm 0.18$, where the
first uncertainty is statistical and the second systematic. An average of
branching fraction measurements for the normalization channel is used to derive
${\cal{B}}(B^0 \to D^{*-} \tau^+ \nu_{\tau}) = (1.42 \pm 0.094 \pm 0.129 \pm
0.054) \%$, where the third uncertainty is due to the limited knowledge of
${\cal{B}}(B^0\to D^{*-}\pi^+\pi^-\pi^+)$. A test of lepton flavor universality
is performed using the well-measured branching fraction ${\cal{B}}(B^0 \to
D^{*-} \mu^+\nu_{\mu})$ to compute ${\cal{R}}(D^{*-}) = 0.291 \pm 0.019 \pm
0.026 \pm 0.013$, where the third uncertainty originates from the uncertainties
on ${\cal{B}}(B^0 \to D^{*-}\pi^+\pi^-\pi^+)$ and ${\cal{B}}(B^0 \to D^{*-}
\mu^+\nu_{\mu})$. This measurement is in agreement with the Standard Model
prediction and with previous measurements.
