Comment: added figures for Section 2
corrected typosIn 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.