Consider the geometric distributed model in equation (18.8), written in estimating equation form as in equation (18.11):

y_t = α₀ + γz_t + ry_t–1 = v_t,

where v_t = u_t – ρu_t–1.

(i) Suppose that you are only willing to assume the sequential exogeneity assumption in (18.6). Why is zt generally correlated with v_t?

(ii) Explain why estimating (18.11) by IV, using instruments (z_t, z_t–1), is generally inconsistent under (18.6). Using the IV estimator, can you test whether z_t and v_t are correlated?

(iii) Evaluate the following proposal when only (18.6) holds: Estimate (18.11) by IV using instruments 1zt21, zt22 2.

(iv) Explain what you gain by estimating (18.11) by 2SLS using instruments (z_t, z_t–1, z_t–2).

(v) In equation (18.16), the estimating equation for a rational distributed lag model, how would you estimate the parameters under (18.6) only? Might there be some practical problems with your approach?