Solved – Multinomial logit with aggregate data

Question

I am asking a general question here. Can multi-nomial model be applied to aggregate data. If so , can you give me a reference list.

Answer 1

Yes, it can. Suppose we have a dataset on choices $j=1 \ldots J$. Each of the choices, let's say products, has observable characteristics $X_j$ and unobservable characteristics $\xi_j$. Using the typical formulation of a conditional logit, say that the utility of product $j$ to consumer $i$ is: \begin{align} U_{ij} = X_j\beta + \xi_j + \epsilon_{ij} \end{align} With the additional assumptions that $\epsilon_{ij}$ is iid Type I Extreme Value and that consumers each pick the choice with highest utility, you get: \begin{align} P_{ij} = \frac{exp(X_{j}\beta+\xi_j)}{\sum_{k}exp(X_{k}\beta+\xi_{k})} \end{align}

As for data, assume that, in addition to the $X_j$, we have the share of consumers, say, who choose each of these $s_1,s_2,\ldots,s_J$. Let's say that the number of consumers from which these shares were calculated was large enough that it's reasonable to view them as probabilities (i.e. to ignore sampling variation). Choose one of the shares/goods/choices to be a contrast (I'll pick 1), and write: \begin{align} ln(P_{ij})-ln(P_{i1}) &= (X_j-X_1)\beta + \xi_j - \xi_1\\~\\ ln(s_{j})-ln(s_{1}) &= \xi_1 + (X_j-X_1)\beta + \xi_j \end{align} The second of these two is the estimating equation. You construct the left-hand-side variable by taking logs of all the shares and then subtracting the log-share of choice 1 from the log-shares of each of the other goods. You calculate the RHS variables by taking each of the characteristics for each of the choices and subtracting off the value of the corresponding characteristic for choice 1. This gives you $J-1$ observations (you drop the contrast share's observation). The intercept in the equation is $\xi_1$. The error terms are the $\xi_j$. You can estimate via OLS.

The simple procedure I outline above has been known more-or-less since the 1960s.
This kind of model has been an active area of research in empirical Industrial Organization for about twenty years now. Much, much richer and more complicated models similar to this are now relatively easy to estimate.

To repeat, don't be put off by how simple and inflexible the model I write above is. You can have individual characteristics in the model. You can have interactions between individual characteristics and product/choice characteristics in the model. You can have endogenous variables in the model. You can have random coefficients. You can dispense with the iid Type I Extreme Value assumption. You can have a dynamic model. Etc. This kind of modelling has been a hot area in empirical IO for twenty years, now. There is a tool to do almost anything you want. And Matlab code available online for a lot of it.

You asked for cites. Read Berry (1994, Rand J) and Berry, Levinsohn, and Pakes (1995, Econometrica) first (called "bee ell pee" by approximately all empirical IO guys). Then use SSCI to go backwards and forwards in the published literature from this. If you want a textbook treatment, read Train's free online books/courses: One and Two. Also, Aviv Nevo has a guidebook here.

I see from your profile that you are at McGill. Normally, I would tell you to go talk to the empirical IO guy on your faculty, but I don't see an empirical IO guy on your faculty on a quick look.