This article extends and generalizes the variance‐ratio (VR) statistic by employing an estimator of the asymptotic covariance matrix of the sample autocorrelations. The estimator is consistent under the null for general classes of innovations exhibiting statistical dependence including exponential generalized autoregressive conditional heteroskedasticity and non‐martingale difference sequence processes. Monte Carlo experiments show that our generalized test statistics have good finite sample size and superior power properties to other recently developed VR versions. In an application to two major US stock indices, our new generalized VR tests provide stronger rejections of the null than do competing VR tests.