  In this manuscript, we discuss a class of difference-based estimators of the
autocovariance structure in a semiparametric regression model where the signal
is discontinuous and the errors are serially correlated. The signal in this
model consists of a sum of the function with jumps and an identifiable smooth
function. A simpler form of this model has been considered earlier under the
name of Nonparametric Jump Regression (NJRM). The estimators proposed allow us
to bypass a complicated problem of prior estimation of the mean signal in such
a model. We provide finite-sample expressions for biases and variance of the
proposed estimators when the errors are Gaussian. Gaussianity in the above is
only needed to provide explicit closed form expressions for biases and
variances of our estimators. Moreover, we observe that the mean squared error
of the proposed variance estimator does not depend on either the unknown smooth
function that is a part of the mean signal nor on the values of difference
sequence coefficients. Our approach also suggests sufficient conditions for
$\sqrt{n}-$ consistency of the proposed estimators.
