Mode Choice - Econometric Estimation

Econometric Estimation

Turning now to some technical matters, how do we estimate v(x)? Utility (v(x)) isn’t observable. All we can observe are choices (say, measured as 0 or 1), and we want to talk about probabilities of choices that range from 0 to 1. (If we do a regression on 0s and 1s we might measure for j a probability of 1.4 or −0.2 of taking an auto.) Further, the distribution of the error terms wouldn’t have appropriate statistical characteristics.

The MNL approach is to make a maximum likelihood estimate of this functional form. The likelihood function is:

we solve for the estimated parameters

that max L*. This happens when:

The log-likelihood is easier to work with, as the products turn to sums:

Consider an example adopted from John Bitzan’s Transportation Economics Notes. Let X be a binary variable that is γ and 0 with probability (1 − gamma). Then f(0) = (1 − γ) and f(1) = γ. Suppose that we have 5 observations of X, giving the sample {1,1,1,0,1}. To find the maximum likelihood estimator of γ examine various values of γ, and for these values determine the probability of drawing the sample {1,1,1,0,1} If γ takes the value 0, the probability of drawing our sample is 0. If γ is 0.1, then the probability of getting our sample is: f(1,1,1,0,1) = f(1)f(1)f(1)f(0)f(1) = 0.1×0.1×0.1×0.9×0.1 = 0.00009 We can compute the probability of obtaining our sample over a range of γ – this is our likelihood function. The likelihood function for n independent observations in a logit model is

where: Y_i = 1 or 0 (choosing e.g. auto or not-auto) and Pi = the probability of observing Y_i = 1

The log likelihood is thus:

In the binomial (two alternative) logit model,

, so

The log-likelihood function is maximized setting the partial derivatives to zero:

The above gives the essence of modern MNL choice modeling.