  A random coefficient autoregressive process is deeply investigated in which
the coefficients are correlated. First we look at the existence of a strictly
stationary causal solution, we give the second-order stationarity conditions
and the autocorrelation function of the process. Then we study some asymptotic
properties of the empirical mean and the usual estimators of the process, such
as convergence, asymptotic normality and rates of convergence, supplied with
the appropriate assumptions on the driving perturbations. Our objective is to
get an overview of the influence of correlated coefficients in the estimation
step, through a simple model. In particular, the lack of consistency is shown
for the estimation of the autoregressive parameter when the independence
hypothesis is violated in the random coefficients. Finally, a consistent
estimation is given together with a testing procedure for the existence of
correlation in the coefficients. While convergence properties rely on the
ergodicity, we use a martingale approach to reach most of the results.
