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The Unscented Kalman Filter Code

Equation (1) is called the measurement equation. It relates the measured observable variables that provide information on αt. We use Zt ∈ M (pt ×m) to denote the matrix of factor loadings. The Ht ∈M (pt × pt) matrix is the variance-covariance matrix of the measurement noise vector, εt. Equation (2) is called the transition equation. We use Gt ∈ M (m×m) to denote the matrix of factor coefficients. The transition equation models how the state vector αt evolves over time. In Cunha and Heckman (2007), the technology of skill formation is modeled as a transition equation. It tells us how current child’s skills and parental investments today increase the child’s skills tomorrow. Finally, (3) is the initial condition of the system. The matrices Zt,Ht, Gt, Qt are called system matrices. The state space model is said to be time invariant when Zt = Z, Ht = H, Gt = G, and Qt = Q for t = 1, ..., T. In many practical situations