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This paper deals with the problem of local sensitivity analysis in regression, i.e., how sensitive the results of a regression model (objective function, parameters, and dual variables) are to changes in the data. We use a general formula for local sensitivities in optimization problems to calculate the sensitivities in three standard regression problems (least squares, minimax, and least absolute values). Closed formulas for all sensitivities are derived. Sensitivity contours are presented to help in assessing the sensitivity of each observation in the sample. The dual problems of the minimax and least absolute values are obtained and interpreted. The proposed sensitivity measures are shown to deal more effectively with the masking problem than the existing methods. The methods are illustrated by their application to some examples and graphical illustrations are given.