With measurements taken on subjects over time, on matched pairs of subjects or on clusters of subjects, the data often contain pairs of correlated dichotomous responses. McNemar's test is perhaps the best known test to compare two correlated binomial proportions. The salient feature of McNemar's test is that we compute the variance of the contrast estimator under the restriction that the null hypothesis is true. Wald's test, on the other hand, does not require that restriction. As a consequence, Wald's statistic is always greater in magnitude than McNemar's statistic when the marginal proportions are unequal, but there is a problem with the validity of both McNemar's test and Wald's test with small to moderate samples. There have been various modifications suggested for McNemar's test to improve its performance. We propose a modified Wald's test that is valid in small to moderate samples and maintains good power. We also evaluate the performance of McNemar's test and Wald's test with and without modifications to enhance validity as well as the performance of the large sample likelihood ratio test and an exact test of the equality of correlated binomial proportions. In a smaller study, we compare the behaviour of a test based on the James-Stein estimator of the common odds ratio proposed by Liang and Zeger to McNemar's test and Wald's test.