7 In Example 10.4, we wrote the model that explicitly contains the long-run propensity, u0, as gfrt 5 a0 1 u0 pet 1 d1( pet21 2 pet ) 1 d2( pet22 2 pet ) 1 ut , where we omit the other explanatory variables for simplicity. As always with multiple regression analysis, u0 should have a ceteris paribus interpretation. Namely, if pet increases by one (dollar) holding ( pet21 2 pet ) and ( pet22 2 pet ) fixed, gfrt should change by u0. (i) If ( pet21 2 pet ) and ( pet22 2 pet ) are held fixed but pet is increasing, what must be true about changes in pet21 and pet22? (ii) How does your answer in part (i) help you to interpret u0 in the above equation as the LRP? 8 In the linear model given in equation (10.8), the explanatory variables xt 5 (xt1, ..., xtk) are said to be sequentially exogenous (sometimes called weakly exogenous) if E(ut uxt , xt21, …, x1) 5 0, t 5 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 (HIVrate) as a distributed lag of per capita condom usage (pccon) for a state, region, or province: E(HIVratet upccont , pccontt21, …,) 5 a0 1 d0 pccont 1 d1 pccont21 1 d2 pccont22 1 d3 pccont23. Explain why this model satisfies the sequential exogeneity assumption. Does it seem likely that strict exogeneity holds too?