Summary: This article studies identification of the threshold-crossing model of binary response. Most research on binary response has considered specific estimators and tests. The study of identification exposes the foundations of binary response analysis by making explicit the assumptions needed to justify different methods. It also clarifies the connections between reduced-form and structural analyses of binary response data.
Assume that the binary outcome z is determined by an observable random vector x and by an unobservable scalar u through a model $z=1[x\beta +u\ge 0]$. Also assume that $F\sb{u\vert x}$, the probability distribution of u conditional on x, is continuous and strictly increasing. Given these maintained assumptions, we investigate the identifiability of $\beta$ given the following restrictions on the distributions $(F\sb{u\vert x},x\in X):$ mean independence, quantile independence, index sufficiency, statistical independence, and statistical independence with the distribution known.
We find that mean independence has no identifying power. On the other hand, quantile independence implies that $\beta$ is identified up to scale, provided that the distribution of x has sufficiently rich support. Index sufficiency can identify the slope components of $\beta$ up to scale and sign, again provided that the distribution of x has a rich support. Statistical independence subsumes both quantile independence and index sufficiency and so implies all of the positive findings previously reported. If u is statistically independent of x with the distribution known, identification requires only that the distribution of x has full rank.

##### MSC:

62J99 | Linear statistical inference |

62G99 | Nonparametric inference |