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I am fitting a multinomial logit model with ridge penalty and in turn estimating the value of time (VOT) or availability to pay (WTP). I want to work with real and simulated data. For the real data I have 3 modes of transport: car, bus and train; where each individual or observation has its travel time and its cost for each mode of transport, that is, i want estimate $\beta_{0}^{auto},\beta_{time}^{auto},\beta_{cost}^{auto},\beta_{0}^{bus},\beta_{time}^{bus},\beta_{cost}^{bus}$.

My question is, how do I choose the lambda penalty parameter? if I want to verify if there is indeed a reduction in the variance of the VOT estimators for the "car" ($var(\hat{VOT}_{auto})$) mode and the VOT for the "bus" ($var(\hat{VOT}_{bus}$) mode, if in the model I consider the train as a reference category? the variance of the VOT estimators I want to determine through the Delta method.

I choose the parameter according to the MSE (mean square error) of the $\beta$ estimators ($MSE(\hat{\beta})=variance+bias^2$)? or by the prediction error using the same data with which I adjusted the multinomial logit model with ridge penalty? or is there a better method or idea than what I propose?

I will read them carefully, thank you very much!

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