  In this study, we propose a test for the coefficient randomness in
autoregressive models where the autoregressive coefficient is local to unity,
which is empirically relevant given the results of earlier studies. Under this
specification, we theoretically analyze the effect of the correlation between
the random coefficient and disturbance on tests' properties, which remains
largely unexplored in the literature. Our analysis reveals that the correlation
crucially affects the power of tests for coefficient randomness and that tests
proposed by earlier studies can perform poorly when the degree of the
correlation is moderate to large. The test we propose in this paper is designed
to have a power function robust to the correlation. Because the asymptotic null
distribution of our test statistic depends on the correlation $\psi$ between
the disturbance and its square as earlier tests do, we also propose a modified
version of the test statistic such that its asymptotic null distribution is
free from the nuisance parameter $\psi$. The modified test is shown to have
better power properties than existing ones in large and finite samples.
