Notice that the assumption that εt is independently distributed matches per-
fectly the Markov assumption, since failure of this condition means that there is
some information left in the history of the process after conditioning on Yitt-ι∙
The conditions for strong one step ahead поп-causality and strong simulta-
neous independence are easily stated as restriction on the parameter space of
(7):
where
|
H1^2 H1^2 -H1<⅛>2 |
(Y1 * (У1 O |
1-У2 >y2 |
) : β1 = H1φ1 ) : β2 = H2φ2 ): 7 = O |
(11) (12) (13) |
|
rɪ |
0 |
1 1 Ol | ||
|
0 |
1 |
0 0 | ||
|
H1 = |
= 0 |
0 |
’^=01 |
(14) |
|
0 |
0 |
0 0 |
Under H1^2, y"i-ι and are excluded from (9), so that Pr {y'it ∣ yi,t-ι} =
Pr{y)i I 2∕lj-1}. Similarly, under H1→t2, yj-ι and yj-iVt-i are excluded from
(10), so that Pr{y?t I yi,t-1} = Pr{y?t ∣ ylt-ι}∙ Finally, under Hlφ2, Pi,t is
equal to zero, and hence the joint distribution (7) factors out in the product of
the marginal distributions (9) and (10).
3 Introducing Covariates and Relaxing the Markov
Hypothesis
In this Section, the model presented in Section 2 will be extended in two direc-
tions. First we will relax the assumption of stationary transition probabilities
by introducing covariates, in order to account for individual and/or time het-
erogeneity. This will be done under the Markov assumption. Then we will drop
the Markov assumption to allow for more complex dynamics. This will be done
in the absence of covariates. Relaxing both hypotheses is straightforward and
left to the reader.
Extending the information set. The information set available to predict Yt
is now enlarged to Ft-ι = ʃt-i V Xt-1∙β Notice that replacing Xt~↑ by Xt
is completely irrelevant for the following discussion. Let us first maintain the
Markov assumption, and for notational simplicity let us also assume, without
loss of generality, that all the information in Xt-1 which is relevant for the
transition probabilities in t is given by Xt-1.
Extending model (7) so that the transition probabilities depend on xt-1 may
be easily done by replacing Sjιt-ι by
_________________≤u = κ'-'∙<-r (15)
6Let Λ4ι and M be σ-fields. Mi VM2 denotes the σ-field generated by Mi UM2. Hence
{77t} = {Tt V A⅛} corresponds to the canonical filtration associated to {(Y⅛, A7)}.