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Many unit root tests have distorted sizes when the root of the error process is close to the unit circle. This paper analyses the properties of the Phillips-Perron tests and some of their variants in the problematic parameter space. We use local asymptotic analyses to explain why the Phillips-Perron tests suffer from severe size distortions regardless of the choice of the spectral density estimator but that the modified statistics show dramatic improvements in size when used in conjunction with a particular formulation an autoregressive spectral density estimator. We explain why kernel based spectral density estimators aggravate the size problem in the Phillips-Perron tests and yield no size improvement to the modified statistics. The local asymptotic power of the modified statistics are also evaluated. These modified statistics are recommended as being useful in empirical work since they are free of the size problems which have plagued many unit root tests, and they retain respectable power.