# Multinomial logit model with alternative-independent coefficients

Having a multinomial logit model $U_{ij}=x'_{i}\theta_{j} + \epsilon_{ij}$ for 3 alternatives, I checked the efficiency of the estimators of this model through 1000 Monte-Carlo iterations, and compared it with the same procedure, but only for simple logit model. Those estimators from MNL are obviously more efficient.

However, what will happen if I change the model to $U_{ij}=x'_{ij}\theta + \epsilon_{ij}$ what will happen with efficiency of the estimators from MNL and Logit? Will logit be more efficient?

• I don't understand your question. The two models you list look the same to me. Are you trying to compare the asymptotic relative efficiency of the maximum likelihood estimator for a multinomial logit w/ a 3-level response to the binomial logit? Why? That seems like apples & oranges to me. – gung - Reinstate Monica Mar 21 '17 at 14:50
• Yes, exactly. That is the question. I just need an intuition for it. Thank you for quick answer. – Kakalukia Mar 21 '17 at 15:46
• I don't quite understand the question as well. – SmallChess Mar 21 '17 at 15:51
• The models are different, because in the first one $x'_{i}$ denotes individual specific regressors and $\beta_{j}$ choice specific coefficients, but in the second model $x'_{ij}$ are alternative-specific, but $\theta$ is not. I just need to find an intuition - how this affects the efficiency of the estimator of $\theta$ in logit and MNL. – Kakalukia Mar 21 '17 at 15:54
• This is a relevant question that needed to be asked. The first thing to understand is, as asked previously, that both multinomial logit as well as conditional logit respect the IIA. For the efficiency part, you only have two hours to figure it out but I strongly advice you to check this paper: chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/eml.berkeley.edu/~train/revelttrain.pdf – user154133 Mar 22 '17 at 13:47