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PETER GXRDENFORS

D Y N A M I C M O D E L S AS T O O L S F O R F O R E C A S T I N G

AND P L A N N I N G : A P R E S E N T A T I O N A N D SOME

M E T H O D O L O G I C A L A S P E C T S *

I. INTRODUCTION

Many of the mathematical models which have proved to be most useful as tools for forecasting and planning can be described as dynamic models. A

dynamic model consists essentially of a set of equations where the variables

included in the equations can be divided into input variables (or exogenous variables) and state variables (or endogenous variables). Dynamic models can

be used to combine, in a strictly mathematical manner, an analysis of causes and effects, with or without time-lags, and various feed-back relationships.

The abstract definition of a dynamic model has grown out from various applications within control theory. But also within other areas dynamic models have been used to model processes. Above all, many macroeconomic models are of this kind, but applications can be found also in other social and biological sciences.

As regards the classical applications within control theory, the equfitions

which describe the connections between the variables in the model can be obtained from some well-developed theory, viz. some branch of physics.

However, when the applications within the 'soft' social sciences are considered, there is no such generally accepted theory from which the mathe-

matical formalism can be acquired. When a dynamic model is used to produce forecasts within an area of this kind, the forecasting process is not the methodologically most difficult part of the work, but it is rather the model

construction, i.e., the description of the dynamic model, that requires the greatest caution and the soundest judgments. There is always some portion

of fortuitousness in the choice of equations in, for instance, macroeconomic

models. A central problem, when judging a forecast obtained from such a model, is thus how much the uncertainty about the form of the equations influences the accuracy of the forecast.

It would be an exaggeration to maintain that the dynamic models which

Theory and Decision 14 (1982) 237-273. 0040-5833/82/0143-0237503.70. Copyright �9 1982 by D. ReideI Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

238 PETER G.~RDENFORS

are used within the social sciences are applications of the mathematical control theory and that the models can be compared to steam engine governors, autopilots etc. The reason is that the methodological problems connected with forecasting models are quite different from the problems connected with physical applications of control theory. Much of the mathematical apparatus which belongs to the engineer's tool kit is simply irrelevant to applications within the 'soft' sciences. On the other hand, problems of equation choice and parameter estimation do not bother the engineer to the same extent as when constructing forecasting models.

This argument does not prevent that some of the conceptual framework which has developed within control theory may be useful also when non- classical applications are considered. For instance, in connection with the evaluation of forecasting models I will emphasize the methodological importance of the concepts of stability and sensitivity of a model. I believe that, when social science applications are considered, these aspects have been swept under the carpet in favour of statistical concepts of 'goodness of fit', etc.

One of the main reasons why dynamic models have become so frequently used as tools for forecasting and planning is that much of the heavy work in connection with model construction and model analysis can be done by a computer. The computer can be used when estimating equation parameters, when analysing the dynamic properties of the model by simulation, when forecasting and when planning with the aid of the model. A number of user directed programs (and even programming languages) have been specially constructed for simulating and analysing dynamic models.

When studying the use of dynamic models in forecasting and planning it is easy to be swamped with mathematical analysis and statistical hair-splitting. Here I will try to keep my head above the surface, but still look for what is going on in the depth. The paper will mainly be a surveyal presentation of dynamic models and their use in forecasting and planning. This will be complemented with some methodological remarks and recommendations in connection with model construction and model evaluation.

TOOLS FOR FORECASTING AND PLANNING 239

II. DYNAMIC MODELS

1. Definitions o f Deterministic Dynamic Models

The abstract definition of a dynamic model has developed from various ways of analysing and synthesising control systems. The methods used when con-

structing centrifugal governors for steam engines, electronic amplifiers, auto-

pilots, etc. have resulted in an abstract mathematical theory where the basic

'objects' of study are models of dynamic systems. In this section I will intro-

duce some basic concepts from control theory and give a definition of a

discrete deterministic dynamic modelJ Some examples of dynamic models will also be outlined.

A. Basic Concepts A fundamental postulate is that the system or process which is to be

modelled can be described by a set of state variables taking real numbers as values. These variables are functions of time and they will be denoted by

xl(t), x2(t) . . . . . The state of a system at a time to is identified with the set

of values xl(to), x2(to) , . . . . In this paper only models with finitely many

state variables will be discussed. The state of a model can then be described

by a vector 2(t) = (xl( t) , x2(t) . . . . . Xn(t)). The set of all possible states of a dynamic model is called the state space. The first state during the time

interval which is considered is called the initial state. Normally, a system can be controlled by external factors. It is assumed

that the control can be characterised by a set of real-valued input variables (or

exogenous variables). These variables are also functions of time and will be

denoted ul(t), u2(t) . . . . . Here, only finitely many input variables will be considered, which then can be described by a vector ti(t) = (ul(t), u 2 ( t ) , . . . , urn(t)). The set of all possible input vectors is called the input space.~

In a deterministic model the state of the model is uniquely determined by

the input vector together with earlier and present values of the state variables.

If the earlier and present values of the state variables have no influence on the

state of the model, but this is determined from the present values of the input variables only, then the model is said to be static. In a stochastic model the input vector and the earlier and present states of the model determine a probability distribution over the state space.

The time variable is either characterised by a continuous line or by a

240 PETER G)~RDENFORS

discrete set o f points o f time. In the first case, the system is said to be continuous, and in the second, discrete.

The central part in the definition of a dynamic model is to describe how

the state o f the model depends on the time variable and the input and state

variables. In this paper I shall concentrate on the case when the state variables

are differentiable functions of time. The change of the state of the model can

then be defined as a function of the previous state 2 ( t -- 1), the present state

~(t), the present input vector fi(t) and the time t. For continuous models the

changes can be characterised by first order differential equations and for

discrete models first order difference equations can be used.

B. Discrete dynamic models Next, a more systematic definition of a discrete dynamic model will be given.

DEFINITION 1. A discrete deterministic dynamic model consists of:

(i) a set T which is a set of consecutive natural numbers to, tl, t2, �9 �9 �9

representing the points of time;

(ii) a set X which is a continuous subset of R n, representing states of the

system. The number of state variables is n and an element in X, a state, is

written Yc = (xl, x2 . . . . . Xn); (iii) a set U which is a subset of R m, representing the input space. The

number of input variables is m and an element in U, an input vector is written

17 = ( u l . . . . ,Urn); (iv) a set Y which is a subset o f R k, representing observation vectors. The

number of observation variables is k and an element in Y, an observation

vector, is written)7 = (Yl . . . . , Yk);

(v) for each state variable xi(t), a function fi from X x U x T to R which

is continuously differentiable with respect to time such that

xi(t) = k ( ~ ( t - 1),~(t), • t) , t). []

A special case o f this definition is when xi(t ) can be described as a function

of 2 ( t -- 1) and t~(t) only. This case is of interest in many respects, and if the

equations of a model are written in this form one says that the model is given

a state-space description or that the model is given in reduced form. A definition of a continuous dynamic model would be quite similar

except that the points of time in (i) would be replaced by a time interval and


that the difference equations in (v) would be replaced with differential

equations. The rather abstract definition of a discrete deterministic dynamic model

which has been given here will now be illustrated by two examples.

Example 1. A very simple macroeconomic model will be outlined and it wiP be shown how it can be given a state-space description. The model contains four variables: aggregate consumption C(t), gross investment I(t), gross national product Y(t), and government spending G(t). These variables are related by the following three equations:

(1) C(t) = al + a2Y(t -- 1);

(2 ) I ( t ) = b~ + b 2 ( r ( t - 1) - - Y ( t - - 2));

(3) r ( t ) = C(t) + I(t) + G(t).

In these equations as, a2, bl and b2 are parameters. 2 A good starting point when reformulating these equations as a state-space

description is to sort up the variables. According to economic terminology C, I and Y are endogenous variables while G is an exogenous variable. Exogenous variables correspond to input variables in a dynamic model, and G is thus the only input variable in the desired dynamic model.

The Equations (2) and (3) do not quite fit the pattern of a state-space description since they do not only contain values of state variables at t - 1, but also at t and t - - 2. We can get away from this deviation by some mani- pulations of the Equations (1)-(3). If we put in (1) and (2) in (3), we obtain

(4) Y(t) = a 1 + a 2 Y ( t - - 1 ) + b l + b 2 Y ( t - - 1 ) - - b 2 Y ( t - - 2 ) + G ( t ) .

From (1) it follows that

(5) Y ( t - - 2 ) = 1 C ( t _ 1) al a2 a2"

If we now put (5) in (2) and (4) we get finally

(6) I(t) = bl + b2ala2 + b2Y( t - - 1) --~2 2 C(t - - 1); and

242 PETER G~ .RDENFORS

(7) Y(t) = a i + b l + b 2 a l +(a2+b2)Y( t - -1) - -b-~2C(t - -1) a2 a2

+ G(t).

The form of the Equations (1), (6) and (7) now accords with the require- ments of a state-space descdption. []

Example 2. The 'world models' which have been constructed by, among others, Forrester [11] and Meadows et al. [23] can easily be rewritten as

discrete dynamic models. These models do not explicitly contain any input

variables, but the value of a state variable at t is determined solely from the

state of the model at t - 1. On the other hand, some of the parameters that appear in the model equations can be interpreted as input variables and

with the aid of this it is possible to perform simple policy experiments. In Forrester's model the state of the model is described by five state vari-

ables, viz., population, natural resources, pollution, capital investment, and capital-investment-in-agriculture fraction. Besides these state variables the

model contains a number of auxiliary variables which are introduced by static

equations. The model constructed by Meadows et al. contains eleven state variables and also here a large number of auxiliary variables. All state variables are assumed to be measurable, but in many cases the high level of aggregation,,

makes this assumption seem very artificial. For example, Forrester measures natural resources in 'natural resource units' and pollution in 'pollution units'.

The equations of Forrester's model are well documented in [ 11 ]. In Limits to Growth, the equations which have been used when simulating the model are not explicitly stated, but these have been presented in a technical report ([24]) which has been published only afterwards. []

C. Classifying Dynamic Systems A model with input variables is said to be autonomous (or closed). The world models presented in Example 2 are autonomous models. A model is said to be time invariant, if the functions f l . . . . , fn and gl . . . . . gh do not depend on t. In this paper only time invariant models will be considered. A model is said to be linear, if the functions f l . . . . , fn and g~ . . . . . g~ are linear in the state variables, the control variables and t. The model in Example 1 is linear, while both Forrester's and Meadows's world models contain non-linear equations.

T O O L S F O R F O R E C A S T I N G A N D P L A N N I N G 243

2. Some Comments on the Definitions o f Dynamic Models

A. Classifying Variables

Although some variable may be classified as an input variable in a dynamic

model it is not always possible for human agents to have any influence to 'control' the values of the variable; or sometimes the 'control' may be very

restricted. For example, in a dynamic model which is intended to model

agricultural production, meteorological variables as temperature and precipitation may occur as input variables. For this reason, we can divide the input variables of a model used for forecasting and planning into policy

variables (or control variables), which are controlled by those who use the

model when forecasting and planning, and external variables, which are not

controlled. The borderline between these kinds of variables is not sharp since

some variables may be controlled only to a limited degree. In econometric models a distinction is made between endogenous and

exogenous variables. Exogenous variables correspond to input variables in the definition of a dynamic model. Endogenous variables correspond to state variables and other auxiliary variables which may be introduced by static

equations or identities.

B. Higher Order Equations

The equations used in (v) of Definition 1 are first order difference equations. It is now easy to give examples from models in physics or econometrics where higher order difference (or differential) equations are exploited. For

this reason, the requirement that when formulating dynamic models only first order equations may be used seems, a priori, as a serious restriction.

It is, however, easy to show that this requirement does not involve any real restriction. By introducing an appropriate number of new state variables it is possible to reduce a higher order equation to a set of first order equations. Suppose, for example, that we are given a third order difference equation of the form

x ( t ) = f ( x ( t -- 1), x ( t -- 2), x ( t -- 3), t~, t).

If we now introduce two variablesy(t) and z(t) which represent x ( t - 1) and x ( t -- 2) respectively, then the equation above can be replaced by the following set of first order equations:

x ( t ) = f ( x ( t - - 1 ) , y ( t - - 1 ) ,z ( t - - 1),K, t);

244 PETER G~RDENFORS

y ( t ) = x ( t - - 1);

z( t ) = y ( t - - 1).

If all equations are linear (and in many other cases), then it is, in this way,

possible to present all equations in accordance with (v) of Definition 1.

C. Stochastic Models

A generalisation of Definition 1 is to relax the assumption that the model e q~Ja~.ms are deterministic and introduce stochastic equations which will

then give us stochastic dynamic models. In continuous stochastic models the equations are so-called stochastic differential equations. It would take me too far to present here this kind of equations and their analysis. The interested reader is referred to Astr6m [4].

The equations used in discrete stochastic systems are called stochastic difference equations. These equations have the following form:

xi( t) = f i ( x ( t -- 1), x( t ) , u(t), t) + v i ( x ( t - 1), x( t ) , u(t) , t).

Here v i is supposed to be a random variable with fixed mean. Sometimes it is assumed to have certain additional statistical properties as being normally distributed, having some fixed standard deviation, being serially uncorrelated,

etc. Another form of stochastic equations is obtained if the input variables are

regarded as stochastic variables. For the solution of stochastic difference equations and further analysis of

stochastic models the reader is referred to Astr6m [4]. I will return to stochastic dynamic models in connection with simulation and when analysing forecasts made with the aid of dynamic models.

3. Analysing Dynamic Models

There are two main ways of studying the properties of a dynamic model. The first way is to search for analytic solutions to the set of equations given by the model. An analytic solution consists of l~mding, for each state variable, an equation which expresses the state variable as a function of the time and of the input variables (and with the initial value as a parameter), but which is independent of all state variables, including earlier values of the variable itself. From these equations the interesting properties of the model can then be derived.


The second way to study a dynamic model is simulation, By providing

initial values to the state variables and values to the input variables during

some subsequent time interval, one can use the equations in a deterministic model to compute the values of the state variables at all points of time in the

interval. By varying the initial values, the values for the input variables and

the equation parameters one can then obtain some information about the

dynamic properties of the model. I will now turn to a more detailed

discussion of these two methods of analysing dynamic models.

A. Analytic Solutions The goal, when looking for the analytic solution to a set of model equations,

is to 'solve' these equations so that the references to the state variables in the

equations are eliminated. In deterministic models the equations are ordinary

difference (or differential) equations. When solving these kinds of equations

we can rely on standard classical techniques.

Example 3. Let us assume that the following difference equation is one of

the equations in a discrete dynamic model:

x(t) = 2 + 0 .9"x ( t - - 1).

This equation is a purely autoregressive equation since the new value of the

variable depends only on an earlier value of the same variable and neither on

time nor on any input variable.

If we put X(to) = Xo, the solution to this equation is given by

x(t) = (Xo-- 20)" 0.9(t -to) + 20.

If xo > 20, x(t) will describe a curve which decreases exponentially towards

the limit value 2 0 ; i f x o < 20, the curve will increase towards 20. []

If the model is linear and if the input variables are ignored in the sense that

their values during the time interval are assumed to be constantly zero, then

there is a general method of solving a differential or difference equation which essentially involves finding the roots of the so-called characteristic equation of the model equation. With the aid of these roots the transient part of the solution of the model equations is obtained. If now each input variable varies according to some simple function u(t), it may be possible to find some particular solution to the model equation. The complete solution


is then obtained by adding the transient part to the particular solution (and

fixing the arbitrary constants so that the initial conditions are satisfied). For many simple equations particular solutions are known and in these cases the

analytical solutions are easy to obtain. However, as soon as the course of the input variables becomes more complex, it is in general extremely difficult,

not to say impossible, to fred particular solutions, a For non-linear models

there is no such general method but the equations in such a model have to be

attacked by more or less ad hoc methods. A deterministic difference or differential equation for a state variable x(t)

is solved if we know the values of x(t) for all t. The solution of a stochastic equation will not produce determinate values but probability distributions

over the values of the state variable. Thus a stochastic equation for a state

variable x(t) is solved if we know the probability distribution ofx( t ) for all t. The solution of a stochastic difference equation will be a Markov process. It would take us too far to present here any solution methods for stochastic

equations. The interested reader is referred to Astr6m [4].

B. Simulation It does not take many variables and complicated functions for the input variables before it becomes overwhelmingly difficult to find the analytic solution

even for a linear deterministic model; and as soon as non-linearities appear in the equations or stochastic equations are considered, the search for an analytic solution is, from a practical point of view, hopeless. In these situations other means for analysing dynamic models must be exploited; and

here computer simulation techniques offer great possibilities. A simulation of a dynamic model during a time interval T consists of

supplying initial values for the state variables and values for the input variables during T and then using the model equations to compute the values of

the state variables for a number of points of time during T. As regards discrete deterministic models, the equations of which are given

in reduced form, this process is computationally simple. From the initial values of the state variables, the values for the next period can be obtained immediately from the system equations. These new values can then be put back into the system equations and the values for the subsequent period can

be computed, and so on. If the equations cannot be put in reduced form, then the computational


process may be more complicated. In this case one may have to rely on

algorithms for solutions of equation systems where an approximative solution

is computed by an iterative procedure. Among the most common are algorithms based on the Gauss-Seidel, Newton-Raphson and Runge-Kutta procedures.

There is a danger in using approximate solutions. During a simulation, computed values for the state variables, which contain some approximation

errors, are' used to compute new values for the state variables at a later point

of time. Although the error at each step may be small, it may happen that

these errors concur and magnify each other so that the simulated value of a

state variable soon deviates considerably from the analytic value. For this reason some form of error analysis of the approximations used in a simulation is necessary.

When stochastic models are considered the advantages of using simulation

are even greater since analytic solutions are difficult to obtain and use for this

kind of models. The inclusion of random variables in the model equations involves no computational problems. There are, however, some methodo-

logical problems in connection with simulation of stochastic models as is pointed out by Howrey and Keleijan [17]. I will retum to this topic in connection with the evaluation of stochastic dynamic models.

With the aid of a computer the simulation methods can be programmed and the simulations performed rapidly and efficiently. A number of special programming languages have been developed with the particular purpose of simulating dynamic models. 4 As input to such a program one gives a model

description, essentially consisting of edited versions of the difference (or differential) equations, together with initial values for the state variables and values for the input variables during the simulation interval. The program then performs the simulation and produces as output either tables of values of the state variables during the simulation interval, or plotted curves

describing how the state variables vary with time.

Example 4. The curves which can be found in the books by Forrester and

Meadows et al. are the results of a number of computer simulations of their world models. Since these models are autonomous only initial values for the state variables are necessary as input when the system equations are described.

248 PETER GARDENFORS

The programming language which has been used for the simulation process is DYNAMO. s []

If one is able to work with an interactive simulation program, it is possible to

produce the state variable curves directly on a screen. One can then comfort-

ably see the effects on the dynamic behaviour of the model of changes in

initial values, parameters in the equations, and the values for the input vari-

ables. In this way one may, without knowing anything about the analytic

solution of the model, work out a 'Fingerspitzengefuhl' for the dynamic

properties of the studied model, even if it is quite complex.

C. Stability and Sensitivity A central problem within control theory is whether a system is stable or not.

In an unstable system a small disturbance may, for example, lead to self-

oscillations which reinforce and the system collapses. The disturbances may

originate from different sources. The initial conditions for a system may not

be set accurately, which will lead to a different course of the system than the

desired one. Another type of disturbance occurs if the equations in the model

used to represent the system are not exact because of error in the equation parameters or because some aspect of the modelled system is neglected or

simplified when the equation is formulated. A third type occurs if the control

variables cannot be perfectly controlled but are subject to some error. An

important question is now how these kinds of disturbances affect the course

of the state variables in general and the output variables in particular. For

some of the possible kinds of disturbances of a dynamic model criteria of

stability have been formulated.

The first definition of stability is connected with disturbances in the initial

values of a dynamic model. The basic idea is that small disturbances in the

initial values will only lead to small changes in the course of the state vari-

ables, if the model is stable.

DEFINITION 2. A dynamic model with given initial values at to for the state variables is said to be Lyapunov stable if and only if for every 8 > 0 there is

some e > 0 such that whenever an initial value is changed at to by an amount

the absolute value of which is smaller than 8, then the changes of the values of the state variables for all points of time t after to never exceed e. []


Example 5. I return to the simple macroeconometric model which was pre-

sented in Example 1. If the equation for Y(t) is solved as a function of lagged

values and the control variable G(t), one obtains the following equation:

Y ( t ) - (a2 + b~)" Y ( t -- l) + b2" Y ( t -- 2) = al + bl + G(t) .

Now suppose the initial values for this equation are Y(0) and Y(1). For simplicity, it will be assumed that G(t) is kept constant and equal to

- - (al + bl) during the relevant time interval. 6 Thus we wilt be studying the

behaviour of the homogenous equation

Y ( t ) -- (a2 + b2)" Y ( t -- 1) + b2" Y ( t -- 2) = 0

under disturbances of the initial conditions. The analytic solution to this

second order difference equation, under the given initial conditions, is (unless

as = 2x/b2 -- b2)

Y ( t ) = Ax" 3Ì + A2" 3 t2

where Aa = (3`2 Y(0) -- Y(1))/(3`2 -- 3`1), A2 = (Y(1) -- 3 a Y(0))/(3,2 -- 3à) and where 3`1 and 3 2 are the roots of the characteristic polynom

3`2_(a 2+b2 ) . 3 `+b2 = 0

of the homogenous equation.

(If a2 = 2x/b2 -- b2, then 3 1 = 3`2 and the solution is then Y ( t ) = Y(O)" 3`t,

which entails that Y(1) = 3,1" Y(0).) The dynamic behaviour of Y(t) depends on the values of 3 1 and 3`2 and

thereby on the values of a2 and b2. For some values of 3 1 and X2, Y(t) will converge towards 0 without oscillations (independently of the values of Y(0)

and Y(1)) and the model described by the equation above will be Lyapunov stable. For other values, Y ( t ) will converge towards 0 with damped oscil-

lations and also be Lyapunov stable. For still other values of 3Ì and X2, Y(O will grow exponentially, with or without oscillations, and in these cases the model is not Lyapunov stable] []

It can be shown that if a linear time invariant dynamic model is Lyapunov stable for some initial values of the state variables, then it is stable for all initial values. The stability of a linear time invariant dynamic model is thus a property of the model, and not only a property referring to a particular set of initial conditions.

250 PETER G)kRDENFORS

Another concept of stability is introduced in the following definition:

DEFINITION 3. A dynamic model is input-output stable if and only if

limited values for the input variables result in limited values for the output variables. []

It can be shown that if a linear time invariant dynamic model where all output variables are state variables is Lyapunov stable, then it is also input-

output stable. Input-output stability covers disturbances in the input variables, since such

will only have a limited effect if the system is input-ouput stable. In a sense,

this criterion also covers disturbances in the equation specifications since these can be seen as changes in the values of the input parameters. As regards linear time invariant models the result mentioned above shows that if a model is stable under disturbances of initial values, then it is stable under other kinds of disturbances as well.

When investigating the dynamic properties of a linear economic model, the

method most frequently used is that of examining the multipliers associated with the model's exogenous variables. If the model is stable then a one-period increase in the value for one of the input variables would result in limited

changes in the following values of the state variables. These changes are called the dynamic multipliers of the input variable. The first-period change of a

state variable is called the impact multiplier of the variable.

The long-run multiplier of a state variable is the sum of all the dynamic multipliers over time. The multipliers of a model often give a good picture

of the dynamic properties of the model, but it should be noted that as regards

non-linear models all questions concerning the stability and sensitivity of such

a model are not settled by multiplier analysis. Another important part of the study of a dynamic model is the sensitivity

analysis where the goal is to find out how sensitive the model is to different kinds of changes. This kind of analysis is, of course, closely related to the

analysis of the stability of the model. The most important aspects are to analyse the model with respect to (i) changes in initial values for the state variables, (ii)changes in the parameters in the model equations, and 0ii) changes in the values of the input variables. The sensitivity of a factor in a model is a measure of how much a change in the factor is magnified when


comparing the behaviour of the state variables before and after the change. If

it is possible to compute an analytic solution to the equations of a dynamic

model, then it is possible to answer all questions concerning sensitivity

from this solution. If simulation is used, the sensitivity of a model can be

determined by running through various combinations of changes in initial

values, equation parameters, and input variables. I will return to this kind of

analysis in the sequel since the sensitivity of a dynamic model is an important

feature when evaluating the adequacy of using the model as a tool for fore-

casting.

III . FORECASTING AND PLANNING WITH DYNAMIC MODELS

1. Model Construction

When one wishes to use a dynamic model as a tool for forecasting and plan-

ning the most difficult part, from a methodological point of view, is not how

to do the forecasting or planning with the model, but how to construct the

model. In this section I will dis.cuss some methodological issues connected

with the choice of variables in the model, the choice of the form of the model equations, and the estimation of the equation parameters.

A. Choice of Variables The 'normal' procedure when constructing a dynamic model as a model of

a real process consists roughly of the following main steps: (i) choose the set

of variables to be included in the model and divide them into input and state

variables; (ii) formulate the system equations (at least with respect to 'form'

and which variables should be included in each equation); and (iii) use avail-

able data for the state and input variables to estimate the parameters in the equations. I do not intend to say that one of these steps must be finished

off before the next may be commenced. In practice one often starts with a

rough work on step (i) and formulates some preliminary equation in step (ii)

to obtain some first estimation results and then returns to step (i) or (ii) to search for less approximate solutions.

It is hardly possible to formulate any general rules for which variables should be included in a dynamic model as state variables or as input variables.

For this problem any form of theory of the process one wishes to construct

a model for may be useful. Theoretical considerations may result in anything

252 PETER G ~ R D E N F O R S

from well corroborated mathematical formulas to loose speculations about causal connections.

A variable, for which one intends to obtain a forecast, is never an isolated

phenomenon. It is part of some causal structure and is influenced by other

variables and the variable also influences other factors itself. It may be a

part of causal cycles, i.e., feed-back loops. A first delimitation when con-

structing a model is to find some causal network, which includes the variables

to be forecasted, but which is relatively isolated in the sense that variables

'within' the network, i.e., those which are potential state variables, do not

influence the variables 'outside' the network, i.e., potential input variables,

in any essential way. This causal network can be chosen more or less

inclusive. To make the demarcation one has to weigh the completeness and

richness of detail against perspicuity and manageability.

Another problem when selecting the set of variables for a dynamic model

is the level of aggregation. For the construction of a macroeconometric model

we may have available sufficient data for a large number of variables as individual consumers, individual entrepreneurs etc. It might be possible to

write out various kinds of relationships between these magnitudes as model equations and thus obtain an extremely detailed model. Such a dynamic

model, however, would have a rather limited practical interest. In order to

explain and predict the behaviour of macroeconometric variables, it is

necessary to deliberately disregard a considerable amount of details. Here,

again, completeness and manageability must be weighed against each other.

The advantage of a model which contains a large number of variables is

that it may be possible to reach a closer representation of the 'real' process

than with a model containing fewer variables. The main disadvantage is that

things get complicated and much more work is needed; more model equations

have to be specified, more equation parameters have to be estimated, and it

may be difficult to collect sufficient data for all variables. Another, more fundamental, disadvantage is that if the theory for the process to be modelled is 'weak', i.e., not articulated in an exact way or not so well corroborated,

then it is difficult to motivate the choice of model equations; and the more equations one has to construct, the more uncertain will thermodel as a whole be.

Even if some variable has been chosen to be included in the model, one has to decide how the variable is to be represented in the model equations.


Sometimes the derivative of a variable may seem more appropriate than the

variable itself or some functional transformation of the values of a variable

may yield a representation which is easier to include in the equations. Even if

the resulting models are, in a sense, equivalent, the choice of the represent-

ation of a variable may have devastating consequences for the outcomes of

parameter estimation procedures.

When choosing the control variables to be included in the model one is

faced with the same problems as when choosing state variables. Since a

dynamic model is often used as a tool for planning, the choice of control

variables may even have political consequences. A dynamic model can be

used to give recommendations of which values of the control variables to

aim at, if a certain goal is to be attained. Since different control variables

may be governed by different political institutions, the mere choice of

control variables may reflect or influence a political power structure.

When choosing input variables which are not policy variables, i.e., what

was earlier called external variables, the accessibility of actual and future

data for these variables is an important aspect. The theory of the process to

be modelled may prescribe an input variable which, because of measurement

problems or other data collecting problems, may be useless when the model is

to be used for forecasting or planning. Actual and past data of input variables

are necessary for the estimation of equation parameters and forecasted" future

data are necesary when using the model to forecast future values of the state

variables. If the forecast for an external variable is very unreliable, then this

usually affects the reliability of the forecast for the state variables of the model; and it may then be preferable to replace the external variable with

another for which there exists a more reliable forecast.

B. Choice of Equations Within areas where precisely formulated and well corroborated theories are

available, as e.g. the classical theories of physics, the choice of model

equations does not normally present any fundamental problems, since the

equations are, on the whole, provided by the theory. On the other hand,

within the 'soft ' sciences, where the theoretical foundations are more shaky and less informative, the choice of model equations often is the most difficult part when constructing dynamic models.

Even if an incomplete theory does not provide us with the exact model


equation for a state variable, it may nevertheless give some guidance as to which state and input variables are causally connected with the state variable

under consideration and thus determine, at least, which variables ought to be

essentially included in the equation. The theory may also indicate the 'form'

of the equation, e.g., whether it should be linear, exponential or logarithmic in the variables. But frequently the information provided by the theory is

scanty and one will then be forced to experiment with different types of equations.

Within economic theory it is often assumed that all equations are linear in the variables. The main motivations for this assumption seem to be

practical: Having linear equations allows you to, more or less mechanically,

solve the system of equations analytically and thereby you will be able to

study the dynamic properties of the model in a careful way; and, further-

more, the statistical techniques available for estimating linear equations are

more thoroughly investigated, better developed and have more appealing

properties than the corresponding methods for non-linear equations. It is

also argued that the restriction to linear equations is not a serious short-

coming since linear equations (the linear terms of a Taylor expansion) are

good approximations of the 'real' equations as long as the variations in the

variables are relatively small. This argument is, naturally, of limited value.

For many processes stochastic models seem more appropriate than deter-

ministic ones. When formulating the equations of a stochastic model the

choice of the error term involves another source of problems. In most appli-

cations the error term is simply added to the rest of the model equation. This

error term is then furnished with various statistical properties (normally dis-

tributed, zero mean, no serial correlation with earlier errors, homoscedastic,

etc.) depending on what the theory of the process says about the error and what data show.

An important problem is how the error term in an equation is to be inter-

preted. One way is that the model equation for a state variable is not perfectly specified but, due to some idealisations or some unknown features

of the modelled process, there will be a small residual difference between the true value of the state variable at a certain point of time and the value obtained from the equation at the same point of time. The residuals can be seen as the unexplained part of the value of the variable.

Another way of interpreting the error term, which can be combined with

TOOLS F O R F O R E C A S T I N G AND P L A N N I N G 255

the first, is that some or all of the variables used in the equations are

measured with error. For this kind of error, assumptions o f normal distribu-

tion, zero mean and no serial correlation between different error terms seem

quite natural. I f the stochastic parts of the model equations consist only o f

measurement errors, then the underlying process is essentially deterministic

and the stochastic dynamic model may, i f correctly specified, be replaced by

a deterministic one. If, on the other hand, the error terms are principally

residuals, then there is no reason, a priori, why they should have the statistical

properties ment ioned above.

In many cases the theory of the process to be modelled does not give

much guidance as to which form of the model equations is the appropriate

one. In these cases one may acquire some information by studying the avail-

able data for the state variables. The curves may suggest a linear or an

exponential trend or there may be some cyclical movements; and this

information can be helpful when formulating the model equations.

I t is dangerous, however, to let the apparent form of the curves for the

state variables dictate the choice of the model equations. This danger was

pointed out by Haavelmo [15] so long ago that much of his argument needs

be repeated:

The degree of conformity between. . , theoretical solution and the corresponding observed time series is used as a test of the validity of the model. In particular, since most economic time series show cyclical movements, one is led to consider only mathematical models the solutions of which are cycles corresponding approximately to those appearing in the data. This means that one restricts the class of admissible hypotheses by inspecting the apparent form of the observed time series.

This condit ion. . , may not even be a necessary condition, and its application may result in a dangerous and misleading discrimination between theories. (p. 312)

The reason why inspection of data series may lead to prejudiced conclusions

about the form of the model equation is, according to Haavelmo, that

the whole question is connected with the type of errors we have to"lntroduce as a bridge between pure theory and actual observations. (p. 312)

Haavelmo then discusses two kinds of errors corresponding to the two inter-

pretat ions of stochastic equations, i.e., measurement error and unexplained

residuals, mentioned above.

256 PETER GXRDENFORS

The conclusion to be drawn from this discussion is that the form of a

model equation should not be chosen only because the data series shows a

certain pattern - there should be some theoretical reason, even if it is weak, for making the choice.

This ends my discussion of the choice of model equations. The results of

the search for a set of appropriate equations may not be univocal but one is

often left with a number of equation forms which all are reasonable candi- dates for being included in the final model. The final choice among these

equations is postponed until their ability to 'explain' data and their dynamic properties are investigated.

C. Estimation o f Equation Parameters

Apart from state and input variables the model equations contain a number

of parameters. Even if the mathematical form of the equation can be fully

inferred from the theory of the process to be modelled, the actual value of

the parameters are not derivable from theory alone. In very particular cases

only it is possible to obtain the 'correct' values from data in combination

with theory - in general one must be content with estimates.

The theory of parameter estimation in multi-equation models is complex.

There are a great number of statistical procedures available for the estimation

of the parameters in a system of equations. Some of the more well-known

methods are ordinary least squares, two-stage least squares, and full-

information maximum likelihood. These methods are constructed to be

applicable under certain assumptions concerning the error terms in the

equations. It would, however, take us too far to present any of these methods here.

There are no general rules for determining the choice of estimation pro-

cedure for a particular system of equations. Partly the choice depends on

what is known about the statistical properties of the errors in the equations, partly it depends on the purpose for which the estimated system of equations is to be used. For example, the choice of estimation technique can have sub-

stantial effects on the dynamic properties of the estimated model.

Example 6. Pindyck and Rubinfeld [29], pp. 329-330, present a simple

macroeconomic dynamic model. The equation for the consumption function C(t) is estimated as a linear function of C(t -- 1), G(t -- 1), Y ( t -- 2) and

TOOLS FOR F O R E C A S T I N G AND P L A N N I N G 2 5 7

R ( t -- 4), where G is government spending, Y is GNP, and R is a short-term

interest rate. Using quarterly data during the period 1955-1970, the

equation was estimated by two-stage least squares combined with an auto-

regressive transformation to correct for serial correlations. The result was

C(t) = 2.097 + 0.966C(t -- 1) + O.187G(t -- 1) --

-- 0.009Y(t -- 2) -- 0.609R(t -- 4).

I f the equation is estimated using ordinary least squares instead of the pro-

cedure above, then the result is:

C(t) = 5 .842+ 0 . 9 2 6 C ( t - 1)+ 0 . 4 3 7 G ( t - 1 ) -

- O . l O O r ( t - 2) - 1.687R(t - 4).

The dominating part of both of these equations is the autoregressive term

C(t -- 1). The behaviour of the variable C is very sensitive to changes in the

parameter for this term. So even if there seems to be little difference between

the two values 0.966 and 0.926, the dynamic properties of the two equations

are quite different. Which equation is 'best ' cannot be determined by

statistical techniques only, but the dynamic behaviour of the model as a

whole must be considered when choosing the final estimates of the

parameters. []

2. Model Evaluation

When a dynamic model has been constructed its performance must be evalu-

ated. This means that the values for the state variables produced by the sys-

tem is to be compared to the 'real' values, past, present, or future, of the state

variables. The better the values produced by the model fit the real values, the

better is the model. But how is one to estimate the degree of fit between

model output and actual values? Frijda [13] notes that

There is hardly any methodology existing here. As much ingenuity as has been invested in the making of computer programs, as little has been spent on the assessment of their value. Next to high precision there always seem to be spots of rough approximations which undercut this very precision. We are left largely to our subjective impressions of what we consider good or bad correspondence. (p. 65)

There exists a large number of criteria which can be used to evaluate dynamic

258 PETER G.g, RDENFORS

models. Here, I will not attempt at a systematic presentation but confine my-

self to a discussion of some of the most important criteria.

A. Goodness o f Fit The quantitative measure of how closely individual state variables track their

corresponding data series that is most often used is called the root-mean- square (RMS) error. In order to define this measure for a discrete state vari-

able x( t ) , we distinguish between xs(t), the value o fx ( t ) at t predicted by the

given model under the given initial conditions, and Xa(t), the actual value of

x( t ) at t.

DEFINITION 4. Let xs(to), Xs(tl ) . . . . . xs(tn) be a series of values obtained from a discrete dynamic model and let xa(to ), Xa(tl ), . . . , xa(tn) be the

corresponding data series. Then the RMS error for this time interval is

x/Zi(xs(t i) -- xa(ti))2/(n + 1). []

The square root of the mean square error is taken in order to obtain a

measure which has the same dimension as the predicted and actual values themselves. If a dimensionless measure is desired, one can use either the RMS

percent error which is defined as

x/Zi(xs(t i) -- xa(tt)/xa(ti))2/(n + 1)

or Theil's inequality coefficient as it is defined in [31], which is just a way of

normalising the RMS error.

All these measures can be decomposed into three parts. The first part, the

error in central tendency, is zero if and only if the average predicted change

from the mean of the predicted values coincides with the average actual

change. The second part, the error due to unequal variation, is zero if and

only if the standard deviations of predicted and actual changes are equal. The third term, the error due to incomplete covariation, is zero if and only if the

covariance of predicted and actual values takes its maximum value. This kind of decomposition is often useful since each term refers to a particular kind of

prediction error, s

B. The R 2 Measure Another measure which is often used as an estimate of the goodness of fit is

TOOLS FOR F O R E C A S T I N G AND P L A N N I N G 259

the R 2 measure. It measures the proportion of the variation of a discrete state

variable x( t ) which is 'explained' by the system equation. R 2 is often

informally used to compare the validity of alternative specifications of system

equations for a state variable.

To define this measure we begin by defining the total variation of x(t) about its mean (or total sum of squares) as

TSS = ~i(xa(ti) -- ~a) 2.

Here, Xa(ti) are the actual values, as before, and xa is the mean of these values

during the time interval t o , . . . , tn. The residual variation o f x ( t ) ( o r error sum o f squares) is

ESS = "ti(xa(ti)-- xs(ti)) 2.

As before, xs(ti) denotes the value of x(t) at t i produced by the model.

Finally, the explained variation o f x ( t ) (or regression sum o f squares) is

RSS = ~i(xs(ti)-- ~a) 2.

Now, if the mean error of the predicted values is zero and the errors are

perfectly uncorrelated with the variables in the equation, it can be shown that

TSS = ESS + RSS.

DEFINITION 5. The R 2 measure for a given discrete equation during the

time interval to . . . . . t,, isR 2 = RSS/TSS. []

When the above mentioned equality holds, this is equivalent to R 2 = 1 -- ESS/

TSS. Thus R 2 is the proportion of the total variation in x( t ) explained by the

model equation. The R 2 ranges in value between 0 and 1, and is higher the

better the xs(t) values correspond with the Xa(t ) values. An R ~ value of 1 can

occur only in case o f a perfect fit.

A high R 2 value cannot be the only criterion to determine whether an

equation describes some data 'correctly', since if further variables are added

to the model equation, the R 2 value is never lowered and it is likely to be

raised (the addition of a new explanatory variable does not alter TSS, but is likely to increase RSS). There must be additional reasons for including a variable in an equation - theoretical reasons or, if the theory is 'weak', tests for statistical significance.


In Gardenfors [14] the R 2 measure is further criticised as a measure of the explanatory value of a model equation. In that paper it is argued that the explanatory value of a forecasting equation is to be measured by the improve- ment of fit to the actual values obtained from the forecasting model in com- parison to the values obtained from a 'naive' model. Here, 'naive' means that

the forecast of a variable is determined without the aid of any other explanatory variables, as e.g. in a time series model.

C. The t Statistic

A mistake that is difficult to discover when evaluating a system of dynamic

equations is that some variable is missing in an equation. This problem occurs

mainly when the theory of the process to be modelled does not give much guidance as to which variables have a direct influence on a given variable. For

this reason it is easy to include too many variables in the first attempts at

formulating the model equations and then later exclude some variables when

the statistical analysis shows that they have no explanatory value. When the model is linear, one of the main tools in this statistical anaysis is the so called t statistic which, for each variable, determines whether the estimated parameter coefficient for the variable is significantly different from 0, i.e.,

that the null hypothesis that the coefficient equals 0 can be rejected with a

certain degree of confidence. It is often demanded, for each estimated coefficient in a linear equation, that the t statistic should show that the null

hypothesis can be rejected at the 5 percent level of significance. However, if this demand is followed too strictly, then it may very well

happen that some important variables will be excluded from the model

equations. If, for example, an input variable (say, oil price), which has a strong influence on the variable one wishes to explain; is constant or almost

constant during the time interval which is used for the parameter estimation, then it is likely that the t statistic for the coefficient for this input variable is not large enough to warrant the rejection of the null hypothesis. If there are theoretical reasons for including a certain variable in an equation, my recom- mendation is that it should be included even if the t statistic for the corresponding parameter is low. It should be noted, however, that, in these cases, the estimated value of the parameter may not be reliable.

Example 7. The following quotation from Pindyck and Rubinfeld [29], p. 379, illustrates the argument above:


Despite the fact that the estimated coefficient of the long-term interest rate is not significant at the 5 percent level, we do not drop the variable from the equation. Because we believe a priori that the interest rate variable provides the link between monetary policy and investment, it is necessary that we include an interest rate as one of the deter- minants of interest. []

The conclusion to be drawn is that one should not rely blindly on the

statistical tools when choosing and specifying model equations. The

parameter values, acquired by some estimation procedure, must be inter-

preted and their reasonableness checked against whatever theory is available.

D. Dynamic Properties When studying a ' real ' process one acquires some intuitive knowledge about

the dynamic propert ies of the process. One may know that it is stable under

quite general condit ions and one may have a fairly detailed knowledge of how

sensitive the process is to different changes in the input variables. When trying

to represent such a process by a dynamic model it is important that the

model has roughly the same dynamic properties.

By using the definitions of stability presented earlier, it may be possible to

express some of the properties of the model in a precise manner. But even if

the available knowledge about the stability and sensitivity properties of a pro-

cess cannot be expressed formally, it is nevertheless an important comple-

ment to the data series for the state and control variables when the tentative

dynamic model is to be evaluated. Since a system of equations, by including

more and more variables, can be made to fit actual data series for the state

variables arbitrarily well, the goodness of fit cannot be the only critierion of

the validity o f the model, but the dynamic properties also play an important

part in the evaluation. Kmenta [21], p. 593, notes that

The stability condition is clearly important from the economic point of view. It should also be realised that the existence of stability (or, at worst, of a regular oscillatory pattern) is, in fact, assumed in the process of estimation. The assumption that the pre- determined variables of the system have finite variances as t-~ ~ applies also to the lagged endogenous variables, and this assumption would be violated ff the endogenous variables were to grow or decline without limit. If this assumption were not made, there would be difficulties in proving the asymptotic properties of estimators.

Example 8. Pindyck [28] aims at formulating an equation for inventory

investment, which is denoted by I(t). As explanatory variables he chooses

262 PETER G~ ,RDE NFORS

Y(t), disposable income, C(t), consumption, together with lagged values of

I( t) , Y(t) and C(t). The model is discrete and the equation is assumed to be

linear in these variables. The following two equation forms are then selected

for estimation:

( i ) I ( t ) = a l Y ( t ) + b l ( Y ( t ) - - Y ( t - - 2)) + c l ( C ( t ) - - C ( t - - 2)) +

+ d ~ I ( t - - i) + k~ ;

(i i) I ( t ) = a 2 ( Y ( t ) - - Y ( t - - 1)) + b 2 ( C ( t ) - - C ( t - - 1)) +

+ e 2 I ( t - - 1) + k2.

The results of the estimations are:

(i) I ( t ) = 0.0113Y(t) + 0.4647(Y(t) - Y ( t - - 2)) - 0.6002(C(t) --

- - C ( t - 2)) + 0.4219I( t -- 1 ) - 2.4615;

(ii) I ( t ) = 0.8407(Y(t) -- Y ( t - 1)) -- 1.0087(C(t) -- C ( t - - 1)) +

+ 0.7362I( t - 1) + 0.8726.

The R 2 value for (i) is 0.740 and the standard error is 2.228 while the R 2

value for (ii) is 0.905 and the standard error 1.349. Pindyck has the following

comment on these statistical results:

[Equation (ii)] gives better statistical fit along the regression line, which is not surprising since a large proportion of the high frequency change in Y can be accounted for by fluctuations in inventory investment. But ff [Equation (ii)] is used in simulation of the model as a whole, it results in extreme instability. This is not surprising - a sudden large increase results in a large I, which in turn results in a larger increase in Y and thus an even larger I, etc. This is an example of statistical fit that is somewhat artificial - we in effect are regressing a noisy variable against its own noise. For these reasons [Equation (i)] was found preferable for use in the final model. (p. 59)

There are no particular reasons within macroeconomic theory for choosing (i) instead of (ii). Pindyck's argument shows that the good statistical fit of

Equation (ii) is outweighed by its dynamic properties. The interpretation of

the equation coefficients shows that Equation (i) is preferable to (ii). []

E x a m p l e 9 . In connection with the world models constructed by Forrester

[11] and Meadows e t al. [23] some interesting sensitivity analyses have been


made. The authors are well aware of the problems connected with the esti-

mation of the equation parameters in their models. It is argued, however, that

accuracy is not necessary since the main variables of the models show the

same qualitative behaviour, which essentially means an impending population

catastrophe, even if the parameter values are allowed to vary within 'reasonable' limits. In support of this claim both Forrester and Meadows et al. pro-

vide the results of a number of simulations of their models where various

equation parameters and initial values are changed. For instance, Meadows

et al. show that even if the assumed initial amount of natural resources is

doubled, there is no essential change in the behaviour of the population vari-

able, but a catastrophe is also in this case predicted to occur in the middle of the 21st century.

It is, however, not sufficient to make sensitivity tests for some model

parameters only in order to evaluate the reliability of the models, but a more

systematic investigation is necessary. Zwicker [32] has pointed out a most

undesirable case of sensitivity in Meadows' model. He studies the effects of

changing three coefficients in some equations concerning capital. In Meadows'

model, the constants are called ALIC (average lifetime of industrial capital),

ICOR (industrial capital coefficient) and FIOAC (consumption fraction).

Zwicker increases the values for ALIC and ICOR by 10% and decreases the

value for FIOAC by 10%. These changes are clearly within the error limits of

the estimations of the coefficients. Zwicker shows that if these constants are

changed in the year 1900 in a simulation run, then all important state vari-

ables (population, natural resources, food per capita, industrial production,

and pollution) will remain at a constant or almost constant level for all foreseeable future. In another simulation it is shown that if these changes of

the equation coefficients are made in the year 1975, instead of 1900, then

the curves will have almost the same form as those considered desirable by Meadows et al.

Zwicker's investigation shows, contrary to what is maintained in Limits to

Growth, that Meadows' model is unwarrantably sensitive to changes in the

values of the equation parameters. His simulations are alone sufficient to

considerably lower the trustworthiness of the conclusions and recommendations presented in Limits to Growth. []


3. Forecasting

Once a dynamic model has been constructed, forecasting with the aid of the

model is, from a methodological point of view, comparatively simple. What is

needed when using a deterministic model is a set of initial values for the state

variables and a set of values for the input variables for the time interval of the

forecast. With the aid of these requisites the model is then run into the

future. If the model is stochastic it is necessary to rely on so-called stochastic

simulation. Such a simulation can be made by including various kinds of

errors in the description of the model - making the external variables and the

equation parameters random variables and adding error terms to the

equations. By running a large number of simulations with this kind of model

one gains a comprehension o f the most likely values of the forecasting vari-

ables and with the aid of the distribution of the simulation curves one also

obtains a rough measure of the uncertainty of the forecast. 9

The acquisition of future values of the input variables may be problematic.

As regards policy variables, the forecaster chooses their values himself or

relies on the decision makers' plans, but when it comes to external variables,

their values must be obtained from already given forecasts.

A. Conditional and Unconditional Forecasts

A distinction is often made between conditional and unconditional forecasts.

In an unconditional forecast, it is assumed that values for all the input vari-

ables are known with certainty. In a conditional forecast, values for one or

more of the input variables are not known with certainty, but the forecast is

made under the condition that certain assumed values of the input variables

will become the actual values.

If the dynamic model, which is used for making a forecast, is a closed

model, i.e., without input variables, then all forecasts will, by definition, be

unconditional. When models with control variables are used, which are the most common and most important cases, unconditional forecasts will occur only in extreme cases, for instance when the input variables influence the state variables only after some lag and the forecast interval is so short that only known values of the input variables are needed. In all other cases, the required values of the input variables have to be found in some other way.

For all external variables utilised when forecasting with a dynamic model


one needs forecasts of their future values. These forecasts are afflicted with

some uncertainty and this uncertainty is the source of a corresponding uncer-

tainty in the forecast made by the dynamic model. If the uncertainty of the

external variables can be described by some probability distribution around

an expected mean, then, with the aid of statistical analysis or stochastic simu-

lation, this distribution can be used to determine the sensitivity of the fore-

cast obtained by the dynamic model to errors in the forecasts of the external

variables.

As regards the policy variables, one cannot, in the same sense, forecast their future values, since they are dependent, at least to some extent, on

decisions which are not yet made. This means that a forecast which depends

on policy variables is made under the condition that certain decisions are

made. Thus, the distinction between forecasting and planning is dissolved in

these cases, since the forecasts are dependent on the planning for the future.

B. The Uncertainty of Forecasts A fundamental source of uncertainty is the model used when producing the

forecast. The choices of variables and model equations are, as pointed out

earlier, always dependent on the theory of the process which is to be

modelled; and within the fields where forecasts are most desirable, i.e., much of the social sciences, theories are often vaguely formulated and considered

to be first approximations only. Dynamic models which are constructed with the aid of such theories are, at best, rough simplifications of the process of

interest. This kind of uncertainty is unavoidable and it is extremely difficult

to judge how much the simplifications and idealisations influence a given

forecast.

When judging the uncertainty of forecast obtained from econometric models it is often assumed, simply, that the model is a correct description of

the real process to the extent that the right variables are included in the

model and the mathematical form, at least, of the model equations is correct.

Even if this strong assumption is made, there are several sources of uncer-

tainty and error in the forecast. I have already mentioned the uncertainty due

to the forecasts of the values of the external variables. Other uncertain factors are the parameters in the model equations. These are almost never the 'real'

coefficients but only estimated values. The process of estimating the parameters may give some clues to the degree of uncertainty associated with


these values. For instance, the standard error of the parameter can be estimated.

If a model equation contains a stochastic error term, this is another source

of uncertainty since the future values of this error term have to be predicted. Here it is possible to form a picture of the properties of the error term by studying the residuals of the estimation from the set of available data. For instance, one may estimate the mean of the error (in general zero), the standard error and the serial correlation of the errors. These estimated properties are then assumed to be valid also for the errors in the forecasting

period.

For each of these types of uncertainty it may be possible to use some kind of statistical analysis to compute a set of confidence intervals for the forecasts

of the state variables. However, this kind of analysis is complicated even if strong assumptions are made and as soon as several kinds of uncertainty are

to be combined one has to use other tools. Here stochastic simulation can be

helpful.

4. Policy Experiments

A. Trial and Error

One of the main advantages of using dynamic models is that they are very

suitable for policy experiments. If we assume that reliable forecasts for the external variables are available, then by studying the behaviour of the model

for various choices of the future values of the policy variables one may get a rough idea of which planning leads to the most desired combinations of values for the input and state variables. As was pointed out above, forecasting with the aid of a dynamic model in general involves a certain amount of plan-

ning and there is thus little difference between the two kinds of activities. Necessary prerequisites for this method of planning are ( i)a certain

knowledge of the dynamic properties of the model, (ii)enough energy to investigate a large number of choices of future values of the policy variables in a systematic manner, and (iii)a sufficient amount of computer time (analytic solutions are, in general, too complicated and therefore expensive to use). By repeated simulation trials and by learning from mistakes, it is possible to pick out some plans for the future which will result in acceptable consequences.


Example 10. Both Forrester and Meadows et al. use their dynamic models as

tools for policy recommendations. Their world models are, as pointed out

above, defined as closed models, but, in practice, some of the equation parameters function as policy variables. In both cases, the authors' aim is an

equilibrium state of the world with constant population and constant quality

of life. Forrester shows, by some examples, that such an equilibrium is

possible in his model, but he does not give any account of an attempt to show that the final equilibrium is the 'optimal' one (which may mean that it has as high quality of life as possible). The changes of model parameters which leads

to the equilibrium state in his final example can hardly be taken as recom-

mendations for future planning since one change is that food production is

reduced by 20% in 1970. The method used by Meadows et al. is similar. It is not argued that the

recommended changes are optimal, but it is claimed that if the changes are

postponed (to 2000 instead of 1975), then it is no longer possible to maintain the equilibrium state. []

B. Optimal Control Trial-and-error is no efficient method of finding the optimal planning for the future with a dynamic model and there is no guarantee that it will ever be

found. As a consequence, one has tried to use certain results from'control

theory by constructing the optimal planning problem as a mathematical optimisation problem.

Such an optimisation problem contains the following components:

(i) a dynamic model; (ii) a time interval for which the optimal planning is to be computed;

(ii 0 a set of initial values of the state variables, and, possibly, a set of final values which the state variables must reach in the final state;

(iv) restrictions on the state and input variables (these restrictions may consist of maximum or minimum limits which must not be overstepped);

(v) a function which measures the utility of the behaviour of the model for various choices of values for the input variables (this utility function is a quantitative aggregation of the planner's goals, values and costs).

When these five components are given, the optimisation problem is then to find the set of values for the control variables during the time interval such that the utility of the behaviour of the model is maximised.


For certain special cases there are general algorithms for solving this optimisation problem. The cases which are most extensively studied are linear time-invariant models. As regards the utility measure, it is constantly assumed that the utility of the behaviour of the model during the time interval is the sum of the utilities of the states of the model during that interval. The utility of a state of the model is then, normally, determined in the following way: for each state and input variable, first determine an 'optimal' value of the variable, and then let the utility of the actual value of the variable in the given state depend on its distance from this optimum. It is often assumed that the utility of a variable is inversely proportional to the square of the distance from the optimal value. The utility of a state, finally, is then computed as a weighted sum of the utilities of the state and input variables, where the weight of a variable depends on its relative importance.

For the kind of optimisation problem outlined above, there are numerical

methods for computing the optimal sequence of values for the input variables

(cf. e.g. Chow [6] and Pindyck [28]). The computations soon get complex and a computer is unavoidable. I f the dynamic model to be used corresponds

well to the process to be analysed and if the utility function realistically

describes actual goals, costs and values, then one may avoid intuitive planning

and instead rely on the computed solution.

Example 11. Pindyck [28] describes a small macroeconometric model for the American economy which he then uses as a basis for optimisation investi-

gations. He uses the same kind of utility function as described above. He com-

putes the optimal series of values for the input variables during a five year period and studies the effects of using different variants of the utility func-

tion (obtained by varying the weights of the variables). In this way it is possible to acquire a conception of how sensitive the optimal planning is to

the uncertainty of the utility valuations. Pindyck also studies how sensitive the optimal planning is to small vari-

ations of the parameters in the model equations. He shows, by an example, that a change in a parameter (for the autoregressive term of the price equation) which is definitely within the margin of error, will radically change the series of optimal values for the poficy variables. Pindyck confesses that

these results are rather disconcerting, since they say that a small change in a coefficient value can result in a large change in the optimal policy and in the resulting behaviour of


the economy. This throws some doubt on the usefulness of our particular econometric model for policy planning. In particular, the explanatory power of the model as indi- cated by the standard regression statistics is probably misleading because autoregressive terms were used so extensively. In any case, a sensitivity testing of the optimal policy solution is, as we have here seen, essential before that policy is used. (pp. 146, 148)

Somewhat later, in connection with a similar result for a small change in the

autoregressive term in the consumption function, he complements the argu-

ment above with the following methodological remark:

What this says is that the optimal policy is very sensitive to a change in a particular coefficient because the model itself is sensitive to such a change. In particular, the use of autoregressive terms in building models must be done with caution, since the apparent increase in explanatory power that results is just an extrapolation of past trends, the effects of changes in other coefficients are magnified, and a change in the autoregressive coefficient itself has a large effect on the behaviour of the model. (pp. 149, 151)

Pindyck's remark that the standard way of judging the explanatory value of a

model may be misleading accords well with the discussion in Ggrdenfors [14]

of the appropriateness of the R 2 value as a measure of the explanatory value. []

The discussion so far has been limited to the optimal control of deterhainistic

systems. When stochastic systems are considered one faces additional

problems when determining the optimal sequence of control values. The com-

putat ional techniques become even more complicated. But, apart from this

complication, there is a more fundamental problem which arises from the fact

that for a stochastic system we cannot 'know' the state of the system at a

future point of time, even if all values of the input variables are known up to

that point.

When considering deterministic systems the planner may specify the time

paths o f the policy variables at the beginning of the planning period. These

paths can then be followed without regard to future events. For stochastic

systems another type of pol icy making may be necessary. Here the policy

variables should be specified as functions of observations of the state of the

system yet to be made, so that the future values of the variables depend on

future observations. This type of control is called feedback control or a

closed loop policy, while the former type is called an open loop policy.


IV. CONCLUSION

Dynamic models are, no doubt, among the best tools for forecasting and plan-

ning. The conceptual framework is rich enough to cover most mathematically formulated models in which temporal aspects are central, and it allows precise formulations of causal relationships and mutual dependencies between vari-

ables. The equations in a model are formal representations of the structure of a complex situation. Our intuitions and theoretical education may be of good

help when formulating these equations. But when the pieces of the model

have been assembled, our intuitions are poor when estimating the dynamic interactions of the parts of the model. Here the computer may be extremely helpful. The mould of a dynamic model, as given by the definitions, is well

suited for computer representations. Much of the estimation and experiment- ing in connection with the forecasting and planning process can be efficiently

carried out on a computer. The fundamental problem when using dynamic models is how to find the

model which is the most suitable representation of the 'real' system or pro-

cess one wishes to forecast. Since dynamic models are such powerful tools for forecasting and planning it is easy to forget that the choice of model for a

particular process presents the greatest methodological problems. In my opinion, too much emphasis has been put on mathematical and statistical

methods for analysing given dynamic models and too little on how to choose among the potential models. Only when it comes to parameter estimation,

some detailed work has been done. However, the efforts put into the construction of efficient estimation procedures have tended to overshadow other, more fundamental, problems of modal construction. Also, the work with

estimation procedures has concentrated on the statistical fit of the model to given data and neglected the dynamic properties of the resulting model. Even though the problems connected with choice of variables and choice of model equations do not lend themselves to mathematical formalism as parameter estimation does, these problems should not be neglected since they are of fundamental importance for the success of the dynamic model to be

constructed. Throughout the paper I have illustrated my arguments with examples

taken from world models and macroeconomic models. I will now, finally, make some brief remarks on the methodological status of these models.

T O O L S F O R F O R E C A S T I N G A N D P L A N N I N G 271

The main advantage of a small world model is that it may present a general view of a complex structure. Even if such a small model must be based on

gross simplifications, it may have a basically correct dynamic structure, which can be studied and evaluated. Already in models of this small size our mental

powers are not sufficient to keep count of the dynamic changes in the model,

but a computer is necessary. I have already commented upon some of the drawbacks of Forrester's and

Meadows' world models. The variables in the models are difficult to measure, and this is mainly due to their high level of aggregation. This makes any

numerical relationships in the system equations hard to test. Even if simulations of the world models show a reasonable fit with actual data, the dynamic properties of the models must be considered highly unrealistic. They are, for instance, unwantedly sensitive to changes of some parameters, and thus the forecasts produced by the models are on the verge of nonsense.

The macroeconomic models fare somewhat better. Variables are only allowed in these models if they are objectively measureable. A great deal of effort is spent on estimating the parameters in the model equations. Some

analysis of the sensitivity of the models is often made, in particular by studying dynamic multipliers.

From a methodological point of view, the standard way of constructing and exploiting dynamic macroeconomic models nevertheless has some flaws. The class of permitted model equations is too often restricted to linear equations. This restriction is motivated by the applicability of the 'best'

estimation procedures. However, this argument points to another drawback: too much emphasis is put on the statistical fitting of the model to data and

too little on the dynamic properties of the model. Finally, among the many

measures used to evaluate a model, few have been supplied with a good motivation for being appropriate for forecasting and planning models.

Lund University, Sweden

N O T E S

* The work with this paper has been supported by a research project on future studies sponsored by the Planning Division o f the Research Inst i tute of Swedish National Defence (FOA P). I wish to thank Professor Eckhart Zwicker for extensive and construc- tive criticism of earlier versions o f this paper.

272 P E T E R G X R D E N F O R S

i I have befitted from the presentation in AstrSm [3]. 2 A note on terminology: Economists sometimes distinguish between models, where the parameters in the equations are undetermined variables, and structures, which are models where the parameters have been replaced by fixed numbers. a For an elementary presentation of the solutions of difference equations and their economic interpretations, the reader is referred to Baumol [5 ]. 4 For a presentation of simulation languages, cf. Kiviat [19]. A popular tool for econometric forecasts is the TROLL system. The use of the language DYNAMO is presented in Pugh [30], and the SIMNON language, which is a general programming language for simulating (linear and non-linear) dynamic systems, is presented in Elmqvist [9]. 5 Cf. Forrester [12] and [11] and Pugh [30]. 6 This assumption is not realistic, but it will simplify the mathematics.

For stability analyses of difference equations, cf. Baumol [5]. 8 This decomposition was first presented and discussed by Thei1 [ 31 ]. 9 In the case of non-linear stochastic models Howrey and Keleijan [17] argue that only stochastic simulations of models containing stochastic errors are appropriate since the non-stochastic portions of the model may yield inconsistent forecasts. Their arguments are questioned by Aigner [2]. Also cf. Haavelmo's paper which was mentioned earlier.

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