Errors of Measurement and Least Squares Estimation in a Simple Recursive Model of Dynamic Equilibrium - Aigner, Dennis J. (Econometrica, 1966)
where Yt and Xt are subject to random errors of measurement ut and vt, respectively, with E(u) = E(vt) = 0, E(u2) = u, E(v2) = 2, E(utvt) = , and E(utu')E(vtsv) = 0 for t # t', is applied to a simple system of difference equations. The paper treats two such models, which differ in that one system is specified as deterministic except for measurement error, while the second model includes random in the equations as well. The investigation focuses on the possible consistency of least squares estimates of structural parameters in both cases. Mann and Wald [3] have demonstrated the consistency property of least squares estimates in stochastic difference equations which contain a shock term, and T. W. Anderson [1] has more recently shown such estimates to possess asymptotic normal distributions. One way argue, and perhaps quite legitimately, against the inclusion of measurement errors and shocks as separate entities in systems such as those under consideration. This separation, however, does provide a useful contrast with regard to the consistency property of LS estimates as compared to the case when only shocks (which subsume measurement error) are present in the specification of the system. When measurement errors are separated from shocks, LS yields consistent estimates for the explosive case of cobweb equilibrium and inconsistent estimates under convergence. The above phenomenon rests on the perhaps more interesting results for a recursive model where only measurement errors are present. This change in how the random terms enter the system, as contrasted to the Mann and Wald or Anderson formulations, causes zero correlation between observed variables and inconsistent LS estimates in the equations under convergence. Again, under explosion, LS provides consistent estimators of structural parameters.