p[R] = m(T+l) (2.72)
where R is of course equal to the matrix of (2.70). Furthermore, from
II
0 1 T+l T+S
P = p[R] = m(T+l) (2.73)
0 TT
S
The fulfilment of the sufficient condition for dynamic policy existence
allows a solution to the problem (2.71) to be obtained. The solution can
be found by using the dynamic instrument multipliers in the controllability
matrix and using the dynamic multipliers for the exogenous variables which
are related to the state space realisation in the same general way as the
instruments, to give an expression for the dynamic values of the instrument
(2.74) is conceptually a very simple problem to solve, certainly more
simple than the linear/quadratic approach to dynamic stabilisation.
However, there are some very real problems associated with solution of
(2.76) in an applied framework as we shall see below and in a later chapter.
The sufficient condition if satisfied, enables the policy-maker to
exactly achieve his targets over the period T+l even in a non strongly-
Tinbergen world where k < m so long as the policy-maker is willing to
anticipate his target by s periods. The dimensions of the controllability
matrix R and the rank criterion for sufficiency lead to a dynamic
counterpart of Tinbergen's counting rule. Thus, a necessary condition
for dynamic target path achievement is that the number of time indexed
instruments must be greater than or equal to the total number of targets
over the target path. That is,
In man y instances the column dimension of R, k ( T+s+1)f will be greater than .-1. .-1.. ,-l~
u = R y - R D z - R Px
- S
(2.74)
the row dimension m(T+l) and the matrix will not be square hence
precluding the existence of the regular inverse. That is, if the matrix
has full rank then we will have more instruments than are necessary to
achieve the desired target configuration. To overcome this problem, time
indexed instruments can be dropped from R until R has dimension
(m(T+l)xm(T+l)). The excluded "instruments" can then be assigned pre
determined values and treated as exogenous variables although in practice
it would be difficult to ascertain at which level to fix the instruments.
The numerical values chosen for the "slack" instruments will of course
affect the values obtained for u in the solution. While there may be
some justifiable reasons for government spending at a pre-determined level,
for example social and political reasons, there would appear to be little
justification for setting monetary instruments at a particular level for
say one period. This would be especially so when the time indexed monetary
instrument was dropped from the anticipation period. The sensitivity of
fixed target solutions to the dropping of slack instruments and the values
assigned to those instruments would be of prime concern in an applied
framework and will be a matter for investigation in this study. As with
the target point problem, an upper bound on the amount of target path
anticipation can be derived from the Cayley-Hamilton theorem. The upper
bound for the target path problem is s = n, that is, once the level of
anticipation equals the number of state variables no further linearly
independent time indexed instruments can be found.
Tentative comparisons between the fixed target framework and the
linear/quadratic framework have already been made. The most striking
difference between the two techniques which has already been alluded to in
Chapter One is that in many cases the policy-maker will not be able to
instruments. Implicit in the linear/quadratic case is a zero policy lead
implying that the policy-maker expresses excessive impatience in trying
to achieve his targets. The end result is that his targets will be
compromised. On the other hand, the fixed target approach allows a policy
maker to exactly hit his target within the bounds of the planning horizon
as long as he is prepared to wait. Which technique to choose? The linear/
quadratic solution for a particular problem may result in all targets
being compromised but the degree of compromise may be small in all periods.
The fixed target solution for an identical problem may entail being
considerably off target for two thirds of the planning period and only
exactly on target for the remaining third of the planning period. The great
divergence from targets in the initial stages of the planning period may
result in structural shifts as the expectations of the private sector change.
The maintenance of a constant structure is crucial for a fixed target
approach and while a constant structure has been assumed for the linear/
quadratic case, recent developments in that area have allowed for learning
techniques to be incorporated in the stabilisation procedure. In this
respect the fixed target approach is more restrictive.
There are also some similarities in the existence criteria of
both techniques. Specifically, if the targets chosen are independent
and the number of instruments equals the number of targets then the
existence of policy in both frameworks is guaranteed. However, in such
a strongly-Tinbergen world the linear/quadratic solution becomes redundant
and the problem can be solved by the computationally simpler fixed target
technique. Both stabilisation techniques allow for policy existence
instruments- However, the dynamic Tinbergen solution still requires
equality betweeen "targets" and "instruments" in a time indexed sense
and hence the policy planner is constrained in relation to the number of
targets he can choose given his available instruments and the length
■of the planning period. In this respect, the optimisation approach is
more flexible as the number of time indexed instruments and targets has
no part in establishing the existence of policy and the policy planner is
free to choose any number of targets relative to instruments and will
not be constrained by the length of the planning period. While the
optimising policy planner is free to choose many more targets than
instruments than may be possible in a fixed target framework, the choice
of targets still must be consistent with the computation of (C*H^_C + R^_)
While it m ay not be possible to ascertain if the sufficient conditions for
policy existence are satisfied in the linear/quadratic case prior to
computation, it is possible in the target point situation when we only
need to establish the existence of the inverse of a matrix of order m.
If the number of targets is about four then this is a relatively easy
exercise - not so however, for the target path problem. Indeed, the
size of the matrix to be inverted in the target path situation is one
of the drawbacks of the technique. Consider a planning horizon of
twenty periods with four targets and two instruments. Assuming existence,
the linear/quadratic solution would require twenty (2x2) matrix inversions,
the target point solution would require one (4x4) inversion while the
target path solution would require one (40x40) matrix inversion and
in addition, the controllability matrix will contain a large number of
zero elements, from (2.74). Typically, the non-zero elements will be
very small which could make inversion difficult with the possibility
rank deficiencies and linear dependence between instruments when in fact
none exist. Further insight into this problem can be gained from
Chapter Seven.
The relative adjustment of both techniques to uncertainty is also
important. As presented here, neither technique is able to adjust to
parameter uncertainty although it must be recognised that the optimisation
literature has begun to advance techniques to handle fully stochastic
systems for example Chow (1975b) and Kendrick and Majors (1974) to name
just two of an ever expanding literature. The inclusion of parameter
uncertainty in the fixed target framework presents some problems. The
nature of the solution procedure makes learning techniques questionable
in effectiveness. One technique would be to replace the coefficients by
expected values and run monte carlo simulations to gain a knowledge of
the variance in policy and target performance. The problem here would
be that existence of policy could not be obtained in many cases whereas
existence could be obtained in a purely deterministic framework.
Optimisation techniques are clearly ahead of the fixed target approach in
terms of parameter uncertainty but as this study constitutes the first
attempt to employ fixed targets techniques, it will not be concerned
with parameter uncertainty.Nonetheless it does suggest a very interesting
area for future research. Of more interest is the reaction of the
alternative solutions to additive uncertainty. The feedback nature of
optimisation allows the controls to adjust past shocks and in this
sense the technique is more flexible than the fixed target technique
which in general cannot adjust to additive uncertainty as the past
state of the system only enters into the solution once in the form of
initial conditions prevailing at the beginning of the planning period.
approach is able to adjust to past shocks as policies are computed
for each time period without anticipation and it is only under these
conditions that such adjustment can take place- The removal of past
state behaviour from the computation of controls for target path
achievement removes the problem of obtaining accurate information
about the past State vector - a difficult problem for any real world
application of optimisation techniques and one which has not been
adequately explored in the literature. Both techniques suffer from
the perhaps unpalatible assumption of perfect foresight about the future
behaviour of the exogenous variables although filtering and adaptive
techniques can be employed in optimisation solutions. The advantage of
optimisation is that because it reflects a situation of zero
anticipation and considerable compromise, it would, in the event of new
information becoming available about future exogenous variables, be
feasible to abandon the current plan and re-compute the optimal controls
for the remaining of the planning period. This procedure would not be
feasible in a fixed target framework unless the policy-maker was dealing
with a strongly-Tinbergen situation as the need to re-plan would
constitute hitting the targets outside of the current planning period
due to the need to adequately anticipate the exact achievements of the
targets. The answer to the question of which technique to use cannot be
answered from theory or armchair speculation. Each stabilisation problem
has its own characteristics which perhaps may favour one or the other
technique. The applied results which are presented in the remainder of
In this chapter a small open model of the Australian economy will
be developed and estimated. The philosophy behind the construction of
the model is that only the "minimal" size model required to illustrate the
important linkages between the open, monetary and income sectors will be
discussed. As is the practice with most econometric studies, a wide variety
of differing structures were tested but because of the limited space
available the alternative specifications will not in general be discussed.
As we have indicated in the previous chapter, non-linear techniques are
available for the approximate optimal control of non-linear models. However,
as the dynamic Tinbergen approach requires a strictly linear model, we
shall proceed directly to the construction of a linear model and disregard
non-linear versions which may be linearised. The need to employ a linear
model does place some restriction on the structure of the model and can cause
difficulties through the mixture of real and nominal values of variables in
equations. If the inherent non-linearities are not too severe however,
then it could be expected that the choice of a linear version will not
result in excessive information loss. The term "minimal" model needs some
clarification at this point. The aim of the applied investigation is to
observe the behaviour of the major broad aggregates rather than a multitude
of subsectors. This can be effectively carried out by aggregating the
system into a smaller system which captures all desired structural
characteristics. Secondly, the use of very large models does not guarantee
any additional information so the model presented here was originally
formulated as a much larger version of the final structure presented and
influence on the general results were either simplified or deleted with
the requirement that the remaining structure gave similar results to any
preceding larger versions. On a more practical level, a large model would
be very difficult to handle in the dynamic Tinbergen fixed target framework
and as this study is the first to contain any applied analysis of dynamic
fixed target problems, it is of some benefit to keep the analysis as
simple as possible. Even with a small simple model, the solution of the
fixed target problem presents a formidable computational problem as will
be seen in Chapter Seven. The requirements of linearity, simplicity and
smallness leave stabilisation studies of the kind presented here open to
charges of "bending the problem to fit the technique" but the use of a
small simple model does not necessarily imply any significant information
loss at an aggregated level and as we have already pointed out a linearity
requirement need not be too restrictive in all cases.
Before proceeding to a discussion of the model, a number of issues
need to be considered. Firstly, the proliferation of optimal stabilisation
studies has given rise to a debate concerning the endogeneity of
government controls in estimated models. Clearly, if governments do have
some kind of cost function in mind and controls are continually being
adjusted to offset current stocks in order to minimise accumulated welfare
costs^then controls are endogenous and should be treated as such in the
estimation of models to avoid serious specification errors. Recent work
by Crotty (1973) (1976), Boddy and Crotty (1975) and Blinder and Goldfeld
(1972) has focussed on a theoretical assessment of this problem. The
possibility of such a feature raises some interesting econometric problems
in regard to the degree of coefficient bias that may be present if the
controls are not treated as endogenous. However, the adjustment of current
decision period for policy is shorter than the observation period for the
data and that policy authorities are able to react to current shocks and
observations. Clearly the authorities would not know the size and
direction of shocks until after the data for the current period becomes
available. In this case, it could be reasonably argued, depending on the
time unit chosen, that policy must be formulated on the basis of past
performance and as such controls become a function of predetermined
variables and can legitimately be treated as exogenous. The use of feed
back control laws also reinforces this as controls are always adjusted
in response to past target deviations. The choice of unit time period is
important in this case as there could be an argument for treating the
controls as endogenous if annual data were used when the correct model
is quarterly, as controls would most likely have been adjusted within
the current time period. To avoid this specification problem, the time
interval chosen was quarterly although as we shall see below this can
also lead to problems in a stabilisation framework. The fixed target
approach presents some conceptual problems with respect to endogenous
instruments as the current controls are a function of current desired
target levels. It is not clear whether or not controls should be treated
as endogenous for estimation purposes in this situation. However, as
fixed target techniques have not yet been employed for stabilisation
purposes, especially by present or past Australian governments, no
specification errors are likely to occur in the estimated model from this
source. With the level of knowledge of stabilisation techniques now
making control of national economies more feasible, it is of some
importance to consider the possibility of specification error when
The linear/quadratic controls laws set out in Chapter Two
utilised the reduced from of an economic system. Similarly, the fixed
target solution can be derived directly from the reduced form. If the
stabilisation techniques are amenable to the direct use of reduced forms
then why bother carrying out a structural form estimation when the reduced
form can be estimated directly? Monetarists argue that it is exceptionally
difficult to model the transmission of policy, particularly monetary policy,
and even a complicated model of considerable size will not be able to capture
the underlying structure. As such, reduced forms should be estimated.
Blinder and Goldfeld (1972) have shown that the direct estimation of
reduced forms can seriously bias the estimated impacts of monetary and
fiscal policy. Extending this to a stabilisation framework it is clear that
given an incorrect assessment of the effectiveness of policy, an
inappropriate mix of policy can result. Even when a structural form is
specifically outlined, the direct estimation of a reduced form can be
inconsistent with the underlying structure. The work of Blinder and
Goldfeld places serious doubt on the acceptability of using reduced form
models and in particular questions the findings of perhaps the most
widely known monetarist model - the St Louis model (Andersen and Jordan
(1970)). To avoid problems of reduced form estimation, the model will be
estimated in its structural form and then the "true" reduced form will
be obtained from the structure. While criticism of the structure can be
made, the reduced form will be at least consistent with the structure and
provided the structural form can adequately capture the general relative
importance of monetary and fiscal policy, the reduced form instrument
multipliers will display a similar tendency.
The model itself is essentially Keynesian in nature and assumes a
into and interrelate with each other. The monetary sector is closely
related to the open sector through the concept of a money base and
endogenous money supply. The inclusion of an endogenous money supply
raises the interesting proposition that the planning authorities are