  This paper presents a double AR model without intercept (DARWIN model) and
provides us a new way to study the non-stationary heteroskedastic time series.
It is shown that the DARWIN model is always non-stationary and heteroskedastic,
and its sample properties depends on the Lyapunov exponent. An
easy-to-implement estimator is proposed for the Lyapunov exponent, and it is
unbiased, strongly consistent and asymptotically normal. Based on this
estimator, a powerful test is constructed for testing the stability of the
model. Moreover, this paper proposes the quasi-maximum likelihood estimator
(QMLE) for the DARWIN model, which has an explicit form. The strong consistency
and asymptotical normality of the QMLE are established regardless of the sign
of the Lyapunov exponent. Simulation studies are conducted to assess the
performance of the estimation and testing and an empirical example is given for
illustrating the usefulness of the DARWIN model.
