2.1 Logit Model

(Hendry and Nielsen 2007) As for the example ?? part, in order to describe the variation in the data, we need a joint model for all the observations. Assuming (\(i\)) that the pairs of observations are independent, we need to describe and parametrize the joint density of \(X_{i}, Y_{i}\) denoted \(f\left(x_{i}, y_{i}\right)\). The definition of conditional densities implies: \[ f\left(x_{i}, y_{i}\right)=f\left(y_{i} | x_{i}\right) f\left(x_{i}\right) \] A description of the joint distribution therefore comes about by describing (\(ii\)) the conditional distribution of participation given schooling, which is of primary interest; and (\(iii\)) the marginal distribution of the length of schooling, which is of less interest in this example.

The marginal density of education is much harder to describe. It would certainly require considerable institutional knowledge about the US school system prior to 1980 to provide a good description. since our primary interest is the variation in participation, it is unappealing to undertake such a detailed study. We will, therefore, seek to avoid describing the marginal distribution of the conditioning, or explanatory, variables, and draw our inference exclusively from a model based on the conditional density.

The opposite factorization, \(f\left(x_{i} | y_{i}\right) f\left(y_{i}\right),\) is also possible, but of less interest, as schooling happened in the past relative to participation.

The detailed discussion of exogeneity is in section 1.2.1.

References

Hendry, David F, and Bent Nielsen. 2007. Econometric Modeling: A Likelihood Approach. Princeton University Press.