In the linear model given in equation (10.8), the explanatory variables xt = (xt1, ...., xtk) are said to be sequentially exogenous (sometimes called weakly exogenous) if
E(ut|xt, xt-1,_,, ...,x,) = 0, t = 1,2, .....,
So that the errors are unpredictable given current and all past values of the explanatory variables.
(i) Explain why sequential exogeneity is implied by strict exogeneity.
(ii) Explain why contemporaneous exogeneity is implied by sequential exogeneity.
(iii) Are the OLS estimators generally unbiased under the sequential exogeneity assumption? Explain.
(iv) Consider a model to explain the annual rate of HIV infections as a distributed lag of per capita condom usage for a state, region, or province:
E(HIVratet|pccont, pccontt-1,....,) = a0 + (0 pccnot + (1 pccon t-1.
+ (2 pccon t-2 + (3 pccon t-3.
Explain why this model satisfies the sequential exogeneity assumption. Does it seem likely that strict exogeneity holds too?