By Jean-Pierre Florens, Velayoudom Marimoutou, Anne Peguin-Feissolle, Josef Perktold, Marine Carrasco
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We can see, therefore, that numerous rigorous definitions of noncausality can be introduced, and one of the natural questions that arise lies in their relationship and possibly their equivalence. Another type of question consists of the transformation of these general concepts into properties that can be expressed as restrictions on parameters, which in turn can be tested. At this moment, we are required to stay with a stationary specification in order to guarantee the limiting distribution of estimators and of tests.
4. 5. 6. to extract stationary linear combinations z i , whose distributions have a certain temporal stability. The conditional modeling obeys a rationale of limited modeling. We generally consider a large set of variables xi whose generating process may be complex, and we focus on the distribution of a variable given the others, assuming that the conditional distribution is sufficiently stable (across individuals and through time) to be studied statistically. For example, a model of individual consumption considers besides this variable, household income, prices, as well as socio-demographic characteristics (size of the family, type and localization of the housing, socioprofessional category).
And with = σ yy σ yz σ yz σzz . Then, θ ∈ = R2 × C2 . d. and satisfies n ⊥⊥i=1 z i |θ and z i |θ ∼ N (ζ, σzz ) and a conditional model characterized by yi |z i , θ ∼ N (α + βz i , σ 2 ) with β= σ yz , σzz α =η− σ yz ζ σzz and σ 2 = σ yy − 2 σ yz σzz . This example has the property that the parameter θ of the original model can be decomposed into two functions of θ, θmarg = (ζ, σzz ) and θcond = (α, β, σ 2 ), 13:56 P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO Ecoeok CUFX117-Florens 0521876407 April 22, 2007 Econometric Modeling and Inference 12 whose values can be assigned independently from each other because the mapping θ → (θmarg , θcond ) + from R2 × C2 into (R × R+ ∗ ) × (R ×R∗ ) is bijective.