Economic Models - Convex Optimization

46 Dipak R. Basu and Alexis Lazaridis
where u ∗ i (i = 0, 1,...,T − 1), the optimal control sequence and x∗ i+1 , the
corresponding state trajectory, which constitutes the solution to the stated
optimal control problem.
In the above equations, i is the matrix of feedback coefficients and
g i is the vector of intercepts. The notation E i denotes the conditional
expectations, given all information up to the period i. Expressions like
E i C ′ K i+1 C, E i C ′ K i+1 A, E i C ′ K i+1 D are evaluated, taking into account
the reduced form coefficients of the econometric model and their covariance
matrix, which are to be updated continuously, along with the implementation
of the control rules. These rules should be re-adjusted according
to “passive-learning” methods, where the joint densities of matrices A, C,
and D are assumed to remain constant over the control period.
2.1. Updating Method of Reduced-Form Coefficients and Their
Co-Variance Matrices
Suppose we have a simultaneous-equation system of the form
XB ′ + UƔ ′ = R (14)
where X is the matrix of endogenous variable defined on E N × E n and
B is the matrix of structural coefficients, which refer to the endogenous
variables and is defined on E n ×E n . U is the matrix of explanatory variables
defined on E N ×E g and Ɣ is the matrix of the structural coefficients, which
refer to the explanatory variables, defined on E N × E g . R is the matrix of
noises defined on E N ×E n . The reduced form coefficients matrix is then
defined from:
=−B −1 Ɣ.
(14a)
Goldberger et al. (1961) have shown that the asymptotic co-variance
matrix, say of the vector ˆπ, which consists of the g column of matrix ˆ
can be approximated by
˜ =
[[ ˆ
I g
]
⊗ ( ˆB ′ ) −1 ] ′
F
[[ ˆ
I g
]
⊗ ( ˆB ′ ) −1 ]
(15)
where ⊗ denotes the Kroneker product, ˆ and ˆB are the estimated coefficients
by standard econometric techniques and F denotes the asymptotic
co-variance matrix of the n + g columns of ( ˆB ˆƔ), which assumed to be
consistent and asymptotically unbiased estimate of (BƔ).