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Title | Bivariate probit with partial observability and a single dependent variable | |

Author | Vince Wiggins and Brian Poi, StataCorp |

I’m trying to estimate a bivariate probit with partial observability
following Abowd and Farber (1982),
Maddala (1983), and Poirier (1980). The problem is that we have only one
dependent variable (the product of the two latent dependent variables), and
the **biprobit** command in Stata requires two different dependent
variables!

The bivariate probit (**biprobit**) model has two binary dependent
variables that we assume are correlated. Partial observability occurs when
we can observe a positive outcome for only one of the dependent variables
when the other is also positive. For example, assume **y1** and
**y2** are our two dependent variables, and we have the following
cross-tabulation of the outcomes:

. tabulate y1 y2

y2 | ||||

y1 | 0 1 | Total | ||

0 | 26 26 | 52 | ||

1 | 8 14 | 22 | ||

Total | 34 40 | 74 |

With partial observability, we know only 14 outcomes are positive for
both **y1** and **y2**. We could think of this as a single dependent
variable, say **y**, that is the product of **y1** and **y2**.

The user who raises this question says he does not have two dependent variables; his
single dependent variable already reflects the partially observed data. He
has a single dependent variable **y** with 14 positive outcomes and 60
zero outcomes.

The syntax for **biprobit** is designed so that we can fit a partial
observability model whether we have complete data, such as **y1** and
**y2** above, or the product of the two, such as **y** above. The
partial observability model uses only the information from the product of
the two dependent variables. So, if we already have that product, we can
use any pair of dependent variables that, when multiplied together,
produce the same set of positive outcomes observed in the product dependent
variable, **y**.

Many other pairs of variables will do this, and any pair when
multiplied to produce the pattern in **y** will imply the same
partial observability model. **biprobit** will not, however, let us
specify a dependent variable that is always 1. To duplicate **y** would be the easiest way to produce
two binary variables that when multiplied together have the same pattern of
0s and 1s as our product variable **y**.

Taking the easy way and assuming the single product dependent variable
is **y**, we can type

. generate y_copy = y . biprobit (y x1 x2 x3)(y_copy x1 x2 x4), partial

to estimate a bivariate probit model with partial observability. **x1**, **x2**,
**x3** are the covariates for the first dependent variable **y1**, and
**x1**, **x2**, **x4** are the covariates for the second dependent variable **y2**.

We use the syntax for a seemingly unrelated bivariate probit model, so
we can specify different regressors for the equations for **y1** and
**y2**. With the partially observable variant of the model, we only
observe the product of **y1** and **y2**. The partially observable
model is particularly difficult to estimate when the same set of regressors
is used for both equations, and the parameters may not even be identified.
Poirier (1980) discusses in detail identification for this model.

- Abowd, J. M., and H. S. Farber. 1982.
- Job queues and the union status of workers.
*Industrial and Labor Relations Review*35: 354–367.

- Maddala, G. S. 1983.
*Limited-Dependent and Qualitative Variables in Econometrics*. Cambridge: Cambridge University Press.

- Poirier, D. J. 1980.
- Partial observability in bivariate probit models.
*Journal of Econometrics*12: 209–217.