Non-causality in Bivariate Binary Panel Data

A Multivariate Discrete Choice Models

The binary case. Choice among discrete alternatives is a deeply studied topic
in modern econometrics. Consider an individual facing a binary choice. This
situation may be modelled by assuming the existence of a latent continuous
variable, supposed to be an index of the propensity to undertake this decision.
A threshold model is a model in which the choice of the individual is supposed
to be caused by this index crossing a deterministic value.¹⁸ In the following we
will consider a threshold of 0, since no restriction is imposed in this setting.

If we indicate with Y* the latent continuous variable, we have:

_Y=( 1, ify_i*>o
^г ɪ 0, otherwise

or

5^zi = l{γ,*>o}

where l{.j is the indicator function. If Y* has a cumulative distribution function
F_γ* (∙), we can express the probability of Yi taking a value 1 as:

Pr (Y_i = 1) = E(y_i) = E (l_{y?>0}) = Pr (y_i* > 0) = 1 - F_γ∙ (0)

Similarly the probability of having a value of 0 is:

Pr (Y_i = 0) = 1 - Pr (Y_i = 1) = Pr (У/ ≤ 0) = F_γ∙ (0)

A particular case of this general framework is the latent regression model, that
can be obtained when Y* is a (usually linear) function of observed and unob-
served characteristics of the individual and of the environment:

E(y_i* I x_i) = β'x_i

Y*=β'x_i+ε_i

Here Xi is a vector of observed covariates, while e⅛ represents the unobserved
ones. Assuming that e⅛ has a c.d.f. F_e (∙), the probability of the discrete binary
variable Yi is:

Pr (Y_i = 0 I x_i) = Pr (У/ ≤ 0) = Pr (β'x_i + ε_i ≤ 0)
= Pr (ɛi ≤ -β'xi) = F_e (-β'x_i)

Pr (Y_i = 1 I x_i) = 1 - Pr (Y_i = 0) = 1 - F_e (-β^,x_i)

If the density function f_e (∙) is symmetric around 0, then F (-y) = 1 — F (y)
and:

Pr (Y_i = 1 I x_i) = 1 - F_e (-β'x_i) = F_e (β'x_i)

¹⁸Stochastic threshold are commonly advocated in biometrical response study: here an
individual is treated with a deterministic quantity of a reagent; the reaction (death or survival)
is caused by the crossing of a stochastic threshold, function of observed (age, etc.) and
unobserved (frailty or proneness, etc.) characteristics of the individual. In economics, it
seems more natural to suppose the threshold deterministic.

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