This paper studies the generalized method of moments (GMM) in the presence of nonstationary time series with a unit root. We investigate asymptotic properties of the GMM estimator in such a situation. It is shown that the GMM estimator is a consistent estimator regardless of whether or not the model under study contains nonstationarity. On the other hand, the asymptotic distribution of the GMM estimator is nonstandard in the presence of nonstationarity. Such nonstandard limiting behavior of the GMM estimator in the presence of unit roots causes difficulty in statistical inference. However, under a reasonable condition implied in an equilibrium relation the asymptotic distribution of the GMM estimator is described by a mixed normal distribution. The mixed normality of the estimator itself is, in general, not of direct use for statistical inference. However, the mixed normality of the estimator sometimes enables us to derive results that are useful for statistical inference. We show that, under some reasonable conditions, the asymptotic behavior of the $J$-statistic for testing the validity of moment restrictions is characterized by a chi-square distribution.