This paper shows that econometric models that include categorical variables are not invariant to choice of ‘base’ category when random parameters are estimated, unless they are allowed to be correlated. We show that the invariance can lead to significant increases in Type I errors, and distortions in the implied behaviour of respondents. We hypothesise that these biases may influence the economic policy implications of published models that contain this error, but it’s impossible to be sure without re-estimating the model correctly.