Feature scaling changes the penalty

A ridge model uses one feature in dollars and another in millions of dollars. Why can the units change the fitted model?

A rescaled feature requires a correspondingly rescaled coefficient to produce the same prediction. Ridge penalises coefficients, so equivalent predictions can receive different penalties under different units. Fit scaling statistics on the training window and apply them unchanged to validation data. Leave the intercept unpenalised. Standardisation improves comparability, although intentional economic penalty weights may be appropriate.