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Whats the benefit of transforming the explanatory variables to have mean 0 and standard deviation 1 in the lasso method, if we didn't do the standardisation what would change?

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If you have multiple features and do regularization(lasso) without normalization, they will be penalized in amounts proportional to their values. You would not want that, and would want them to be penalized on an equal scale relative to each other. Hence you should do that

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