Economic Models - Convex Optimization

44 Dipak R. Basu and Alexis Lazaridis
time. For this purpose, adaptive control systems with time-varying parameters
(Astrom and Wittenmark, 1995; Basu and Lazaridis, 1986; Brannas
and Westlund, 1980; Nicolao, 1992; Radenkovic and Michel, 1992) provide
a fruitful approach. There are alternative estimation methods, e.g., rational
expectation modeling, VAR approaches etc., to estimate time-varying systems.
These methods, however, are mainly descriptive, i.e., cannot have
immediate application in policy planning. The purpose of this paper is to
explore this possibility in terms of a planning model where adaptive control
rules will be implemented so that we can derive time-varying reduced form
coefficients from the original structural model to demonstrate the dynamics
of monetary-fiscal policies.
The model follows the basic theoretical ideas of monetary approach to
balance of payments and structural adjustments (Berdell, 1995; Humphrey,
1981; 1993; Khan and Montiel, 1989). In Section 2, the method of adaptive
optimization and the updating method of the time-varying model are analyzed.
In Section 3, the model and its estimates are described. This includes
some necessary adjustments for a developing country. In Section 4, the
results of the time-varying impacts of monetary-fiscal policies are analyzed.
2. The Method of Adaptive Optimization
Suppose a dynamic econometric model can be converted to an equivalent
first order dynamic system of the form
˜x i = Øx i−1 + ˜Cũ i + ˜D˜z i +ẽ i (1)
where ˜x i is the vector of endogenous variables, ũ i is the vector of control
variables, ˜z i is the vector of exogenous variables, and ẽ i is the vector
of noises, which are assumed to be white Gaussian and Ã, ˜C, and ˜D are
coefficient matrices of proper dimensions. It should be noted that a certain
element of ˜z i is 1 and corresponds to the constant terms. The parameters of
the above system are assumed to be random.
Shifting to period i + 1, we can write
˜x i+1 = Øx i + ˜Cũ i+1 + ˜D˜z i+1 +ẽ i+1 . (2)
Now, we define the following augmented vectors and matrices.
[ ] [ ] [ ]
˜xi
˜xi+1
ẽi+1
x i = , x
ũ i+1 = , e
i ũ i+1 = ,
i+1 0
A =
[ ]
à 0
, C =
0 0
[ ] ˜C
, D =
I
[ ] ˜D
.
0