Suppose that the equation
yt = a + (t + (1xt1 + ...+(kxtk + u,
satisfies the sequential exogeneity assumption in equation (11.40).
(i) Suppose you difference the equation to obtain
(yt = ( + (1 (xt1 + ... + (k (xtk + (ut.
How come applying OLS on the differenced equation does not generally result in consistent estimators of the (j?
(ii) What assumption on the explanatory variables in the original equation would ensure that OLS on the differences consistently estimates the (j?
(iii) Let zt1,...., ztk be a set of explanatory variables dated contemporaneously with yt. If we specify the static regression model yt = (0 + (1zt1 + ... + (kztk + ut, describe what we need to assume for xt = z, to be sequentially exogenous. Do you think the assumptions are likely to hold in economic applications?