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Theory of convolutional neural networks suggests the property of shift equivariance, i.e., that a shifted input causes an equally shifted output. In practice, however, this is not always the case. This poses a great problem for scene text detection for which a consistent spatial response is crucial, irrespective of the position of the text in the scene. Using a simple synthetic experiment, we demonstrate the inherent shift variance of a state-of-the-art fully convolutional text detector. Furthermore, using the same experimental setting, we show how small architectural changes can lead to an improved shift equivariance and less variation of the detector output. We validate the synthetic results using a real-world training schedule on the text detection network. To quantify the amount of shift variability, we propose a metric based on well-established text detection benchmarks. While the proposed architectural changes are not able to fully recover shift equivariance, adding smoothing filters can substantially improve shift consistency on common text datasets. Considering the potentially large impact of small shifts, we propose to extend the commonly used text detection metrics by the metric described in this work, in order to be able to quantify the consistency of text detectors.