Estimation of Econometric Models with Measurement Errors

Abstract

Having formulated the econometric model with errors in the variables in state-space, according to the following:

$$ {{x}_{t+1}}=\Phi {{x}_{t}}+\Gamma {{u}_{t}}+E{{w}_{t}} $$

(4.1)

$$ {{z}_{t}}=H{{x}_{t}}+D{{u}_{t}}+C{{v}_{t}} $$

(4.2)

with

$$ E\left[ {{w}_{t}} \right]=0,\,E\left[ {{v}_{t}} \right]=0 $$

$$E\left[ {\left[ {\begin{array}{*{20}c} {w_{t_1 } } \\ {v_{t_1 } } \\ \end{array} } \right]\left[ {w'_{t_2 } \,v'_{t_2 } } \right]} \right] = \left[ {\begin{array}{*{20}c} {Q\,} & S \\ {S'} & {\,R} \\ \end{array} } \right]\delta _{t_1 t_2 }$$

we shall go on to estimate the vector θ, containing p parameters, in which we shall include all the unknown elements within the following matrices: Φ, Γ, E, H, D, C, Q, R and S. Recall that vectors x _t, u _t and z _t have dimensions n, r, and m, respectively.