Non-causality in Bivariate Binary Panel Data

5 Non-Causality with Survival Data

Special cases of the model discussed in Sections 2 and 3 can be obtained when
some of the transition probabilities are set at O. In the following paragraph we
will deal with the case of survival models,¹² i.e. models in which the states
with Y_t = O are not accessible from the states with = 1, j = {1,2}; this
implies that every decision with respect to a variable У/ is in a certain sense
irreversible. This imposes on the model the following constraints:

Pr{y⅛ = o∣y_i‰ = l} = 0,
Pr{¾ = O∣¾-₁ = l}=O,

or, in a different form:

(0,0)' I y_i,_t-ι = (l,0)'} = O,
(0,0)' I y_i,_t-ι = (0,1)'} = O,
(0,0)' I y_i,_t-₁ = (1,1)'} = O,
(l,0)' I y_i,_t-₁ = (0,1)'} = O,
(l,0)' I y_i,_t-₁ = (1,1)'} = O,
(0,l)' I y_i,_t-₁ = (l,0)'} = O,
(0,l)' I y_i,_t-₁ = (1,1)'} = O.

Then the state-transition diagram of this kind of model can be represented as
in Figure 2.

Probably the simpler way to think about this model is to consider a bivariate
probit model starting from state O, that is from Y_t = (0,0)' with transition
probabilities:

Pr {y_i,t I Y_i,_t-1 = (0,0)'} = Φ₂ (¾,_t [ Jθ ] ; O, ¾,_t [ ɪ ^p₁ ] dQ ,

where we have eliminated the subscript from p since no other correlation coef-
ficient is present in the model.

Transitions from states 1 and 2 have respectively probabilities:

Pr ⅛¹, I y,_i.₁ = (0,1)'} = Φ₁ ((2₂∕¹_i, - 1) (∕3₁₀ + ft₂) ; 0,1) ,
Pr {yl_t I Y_i,_t~₁ = (1,0)'} = Φ₁ ((2_y²_i - 1) ‰ + ∕¾₁) ; 0,1) .

¹²A standard reference for survival models is Kalbfleisch and Prentice (1980). A counting
processes perspective on these models is in Andersen et al. (1993). A review of multivariate
survival models is Hougaard (1987). The model presented in this Section is discussed in greater
detail in Mosconi, Sartori and Seri (1998), where in particular the effect of time aggregation
on causality tests is discussed.

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