Non-causality in Bivariate Binary Panel Data

The assumption of symmetry is particularly well suited when there is no natural
order in the responses, i.e. when 1 and 0 are logically interchangeable.¹⁹ In
particular, when ε_i is a normal r.v.

ε_i ~ N (θ,σ²)

we recover the standard probit model:

Pr(K_i = 0 I x_i) = Φ (-_i3⅛d),σ²)

Pr(y_i = l∣_a⅛) = Φ(∕3⅛0,σ²)

or, compactly, for y ∈ {0,1}:

Pr (Y_i = y∖ x_i) = Φ ((2 ∙ y - 1) ∙ β'x_i-0, σ²)

However, in this model β and σ² are not separately identified and this can be
easily shown considering a new latent continuous random variable Y ° = c∙Y_i* =
c ■ (_tβxi + ɛi) = ^rγxi + ŋi, with c > 0. In fact, in this case:

Pr (Y_i = 1 I x_i) = Pr (Y_i° > 0) = Φ (₇aι_i; 0, (c ∙ σ)²}

^fXi

= f √⅛7^exp(-⅛⅛)^d≡'

— ∞

7Xi                                   βxi

= I √⅛ ^eχP (-⅛) ^d © = / √⅛^exp (-⅛) ^dz
— ∞                      —∞

= Pr(y_i*>0)

To avoid this problem σ² is customarily set at the arbitrarily value of 1, and β
is evaluated under this condition.

A multivariate extension. The latent regression approach can be generalized
to two or more binary decisions.²⁰ Dropping the i subscript for notational ease,
we can suppose that every decision j can be expressed as a function of a latent
continuous regression:

У/ = β'jX + Ξj = Vj + Ξj,               j = 1, ..., n

Yj ⁼ ^17>0}

¹⁹This happens in most cases. However, suppose a group of individuals in a controlled
trial is observed on a window [0,T]: assuming the DGP to be a Cox (1972) Proportional
Hazards Model, survival probability in T is the c.d.f. of an Extreme Value Type I or Gumbel
distribution. Clearly, it is a skewed distribution.

²⁰The first attempt in this direction is due to Ashford and Sowden (1970); several papers
have followed this one, expanding the subject and proposing estimation metods different
from maximum likelihood. We refer to Amemiya (1981) for a survey of multivariate binary
regression models.

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