  For data segmentation in high-dimensional linear regression settings, the
regression parameters are often assumed to be sparse segment-wise, which
enables many existing methods to estimate the parameters locally via
$\ell_1$-regularised maximum likelihood-type estimation and then contrast them
for change point detection. Contrary to this common practice, we show that the
sparsity of neither regression parameters nor their differences, a.k.a.
differential parameters, is necessary for consistency in multiple change point
detection. In fact, both statistically and computationally, better efficiency
is attained by a simple strategy that scans for large discrepancies in local
covariance between the regressors and the response. We go a step further and
propose a suite of tools for directly inferring about the differential
parameters post-segmentation, which are applicable even when the regression
parameters themselves are non-sparse. Theoretical investigations are conducted
under general conditions permitting non-Gaussianity, temporal dependence and
ultra-high dimensionality. Numerical results from simulated and macroeconomic
datasets demonstrate the competitiveness and efficacy of the proposed methods.
