Non-causality in Bivariate Binary Panel Data

Similarly, K_tL₁ does not strongly cause Yf one step ahead, given УД_Х if

Pr{₂∕,² I ₂∕_i.₁} = Pr{y_i² ∣₂∕Li} ∀t ∈ {1,... ,7¹}          (5)

Definition 4 - Strong simultaneous independence for a Markov chain with sta-
tionary transition probabilities: Yf and Yf are strongly simultaneously indepen-
dent given y_t_i if

P⅛ I yt-ι} = Pr{2∕_i¹ I y_t-ι}Pr{yt I yt-ι} Vtel (6)
or equivalently

Pr{‰¹ I y⅛,yt-ι} = Pr{⅜¹ I yt-ι} VteI

or equivalently

Pr{‰² I y},yt-ι} = Pr{⅜² I yt-ι} VteI

The appropriate statistical model where these conditions may be tested is
the joint distribution of Yt given Yt-ι ■ Granger поп-causality conditions involve
only the marginal distributions of Yf and Yf (conditional on Yt-ι), whereas
testing for simultaneous independence requires the joint distribution to be fully
specified, and compared to the product of the marginal distributions. Notice
that, since Yt-ι, as well as Yt, may belong to a finite set of four states, the most
general model representing Pr{t∕_i ∣ yt-ι} involves 16 parameters, correspond-
ing to the transition probabilities from each of the states in (t — 1) to each of
the states in t (or some one to one transformation of the transition probabili-
ties). More precisely, since the sum of the transition probabilities for transitions
starting from each of the states is equal to 1, just 12 parameters are enough to
describe the conditional distribution completely.

The paper is organized as follows. Under the maintained assumption that
{Tj} is a Markov chain with stationary transition probabilities, and that the
information set is restricted to (½-ι, Section 2 shows how Pr{t∕_i ∣ yt-ι} may
be represented with no loss of generality by a dynamic bivariate probit model.
Within this framework, the restrictions on the parameters implying (4), (5)
or (6) are illustrated. Section 3 extends the simple dynamic probit model il-
lustrated in Section 2 in two directions. First, the assumption of stationary
transition probabilities is dropped, allowing the transition probabilities to de-
pend on covariates. Then, the Markov assumption is also relaxed, allowing for
more complex dynamic structures. Section 4 shows under which conditions the
Maximum Likelihood estimates of the parameters of the proposed models, as
well as the Likelihood Ratio tests for hypotheses on such parameters, display

which in turn implies (4), being

The same argument holds for the other definitions.

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