Fitting differential equation models to observed economic data—I. quasilinearization

Fitting Differential Equation Observed Economic Data-I. Jay Helms, Harriet

Kagiwada,

Models to Quasilinearization

Robert

Kalaba,

and John Niedercorn

Department of Economics University of Southern California Los Angeles, California 90007

ABSTRACT This paper discusses the fitting of differential equation models to economic data. In particular, it treats the problem of describing the growth of capital in terms of a differential equation containing several parameters. The parameters are to be estimated on the basis of data. This estimation problem is formulated as a nonlinear boundary value problem. The rapidly convergent successive approximation method of quasilinearization is described and applied. Representative results of numerical experiments are presented, showing the effectiveness of the approach. Suggestions for additional studies are made.

I.

INTRODUCTION

Classical regression analysis concerns itself with the fitting of polynomial, exponential, and other functions to observed data [l], and the theory is highly developed. On the other hand, there is frequently little theoretical justification for the functional form of the assumed relation among the variables. Sometimes, though, as in growth theory [2, 31, general considerations lead us to postulate linear or nonlinear relations among variables and their rates of change. The problem now is to select arbitrary constants in the resulting differential equation or equations, and a complete set of initial conditions, so that the solution of the differential equations agrees as closely as possible with the observations. We shall show how this may be done using the method of quasilinearization and modern computing machines. A detailed presentation of the method is given in [4, 5, 61. APPLIED MATHEMATICS AND COMPUTATION 4, 13%145 (1978)

0 Ekevier North-Holland, Inc., 1978

fX%3OO3/78/0004-0139/~1.75

139

JAY HELMS

140 This

paper

is the first in a series.

We

ET AL.

show how to fit a differential

equation describing the growth of capital to artificially generated data. In the second paper we shall consider real data concerning the American economy in the post-World War II era. Subsequent studies will deal with more realistic assumptions concerning growth of the labor force, technological improvement,

II.

THE

and other matters.

MODEL

It is hypothesized that capital growth in an economy can be represented by a nonlinear ordinary differential equation of the form

o< t< tNN,

$=uk(+c(t)-yk(t),

k(0) = 6, where

k(t) = the accumulated

real capital stock per capita at time t, and

c(t) = the rate of real consumption This

is a simple

conservation

per capita at time t.

equation

stating

that

the

rate

of real

net

investment

per capita is equal to the rate of total production per capita using and a Cobb-Douglas production function, less the rates of consumption depreciation per capita. Of course, it is not expected that any single equation model can accurately depict the growth of capital in an economy; however, such a model (a) may be able to capture the most crucial aspects of observed investment data, and (b) will serve to illustrate the use of quasilinearization to estimate selected parameters in a nonlinear ordinary differential equation, which is of relevance in the study of economic processes. In our numerical experiments for the consumption function c we use

c = 1.67836 + 0.051962,

III.

QUASILINEARIZATION

o< t< thi.

[5]

Suppose there are N observations on the dependent variable k(t) at times t = t,, t,, . . . , tN, and that the observed values are hi, h,, . . . , h,. For any values of the parameters (Y, /?, y, and 6 the nonlinear ordinary differential equation

141

Fitting Differential Equation Models

(1) can be integrated numerically to yield k(t,),k(t,), . . . , k(t,). We seek values of these parameters yielding values for k( ti) so that the fit is optimal in the sense that

Q=

-$[k(ti)-

h,]’

i=-1

is a minimum. One method of solving this minimization problem is to employ the iterative procedure of quasilinearization [4, 51. To illustrate this process and to avoid complications in generating numerical solutions, it will be assumed in this first article that values for the parameters 1y and y are given (whether correctly or incorrectly), and that Q in Eq. (2) is to be minimized by the choice of /? and the initial condition 6 only. With (Yand y thus fixed, Eq. (1) may be rewritten as

f

=f(WP),

o< tf

tN,

(3)

k(O)=& Given initial estimates /3’ and 6’ for the unknown parameters, Eq. (3) can be integrated numerically [7] over the indicated domain to provide an initial estimate k” of the function k(t), 0 < t < tN. Applying Taylor’s theorem for functions of two variables then yields

af +(pl-po)ap(ko(t),~O)+..., where k’ and p ’ are the as yet unknown revised estimates for the function k and the parameter fi, respectively. Dropping the higher order terms, suppressing the arguments of the functions and partial derivatives, and rearranging gives dk’ af -=kk’ak+f-k dt

0 af ;ii;-pO$++-,

af

(5)

JAY HELMS

142

which is the associated linear differential general solution of (5) can be written

equation

for the function

dpl

of the initial condition,

af

0

dt- --pl+f-k ak af

dp2

af

q-P

af

o<

dt=xp2+ap>

0

t<

and

af ap,

af zh3

dt

04

tN,

WI

P,(O) = P2 (0) = 07

dh -=

k’. The

(6)

k’(t)=p,(t)+P’~,(t)+6’h(t), where 6 r is the revised estimate

ET AL.

o<

t< tN’

h(O)=l.

(74 (7d)

The differential equations and initial conditions (7a-d) can then be integrated numerically to provide values for p,(t), pz( t), and h(t) on the interval 0 < t < N. The minimization criterion, initially provided by Eq. (2) for the nonlinear ordinary differential equation, can now be adapted to the linearized equation. We now seek values for the revised parameter estimates P1 and a1 which minimize S=

$j

[k’(4)-bi12

i=l

=

5 [pl(ti)+P1p2(ti)+6112(ti)-hi]2.

(8)

i=l

Taking partial derivatives of S with respect equal to zero, and regrouping yields

to /3’ and 6 ‘, setting

plitlp2(t,)h(t~)+S1i~l’(ti)2= I$[‘ih(ti)-PI(ti i=l

the results

(9b)

Fitting Differential Equation Mod&s

143

Equations (9a,b) can readily be solved analytically or numerically. With values thus obtained for p ’ and 6 ‘, Eq. (6) p rovides the revised estimate k1 of the function k. At this point, the entire process may be repeated to provide still better estimates p2, S2, and k2. Th e quasilinearization procedure may be applied iteratively until the parameter estimates converge or diverge, as determined by an appropriate stopping rule. In the typical case convergence is rapid, being quadratic in the limit. IV.

NUMERICAL

RESULTS

Numerical experiments were performed on a computer using a FORTRAN program to instrument the quasilinearization process and to investigate its convergence properties in this application. For this paper, synthetic data for k were generated by numerically integrating Eq. (1) with (Y= 2.0, p = 0.3, y =0.07, and S=5.8 and setting bi= k(tJ,

i = 1,2,. . . ,26,

where

t z-1 2

t,=

i=l,2

’

,..., 26,

thus yielding N = 26 data with a known solution for which the minimized Q in Eq. (2) is zero. Taking the correct values of 2.0 and 0.07 for a and y, respectively, the quasilinearization procedure was employed iteratively for various initial parameter estimates p” and 6’. Convergence was in each case rapid, as the results in Table 1 indicate. In a second series of experiments the assumed values of (Y and y were different from those values which produced the data. Again the estimates of

TABLE 1 EXAMPLE

OF CONVERGENCE GIVEN

(Y =

IN ESTIMATION

2.0

AND

y =

OF p

AND

Estimate Approximation: P s

6,

0.07 True Value

1

2

3

4

0.4000 6.0000

0.3304 5.0050

0.3000 5.7082

0.3000 5.8000

0.3000 5.8000

JAY HELMS ET AL.

144 TABLE 2 EXAMPLES WHEN

OF CONVERGENCE ASSUMED

VALUES

IN ESTIMATION OF a AND

y ARE

OF ,8 AND

8

INCORRECT

(a) Given a = 2.4, y = 0.06 Estimate Approximation:

P s

0

1

2

3

4

0.4000 6.0000

0.3178 3.2134

0.2397 4.2823

0.2271 4.9737

0.2268 5.0100

0.2852 6.8219

0.3852 6.8219

(b) Given a = 1.6, y = 0.08 0.4000 6.7717

P 6

0.3858 6.7717

0.3852 6.8223

P and 6 converged, although the values to which they converged significantly upon the modified values of (Y and y (see Table 2). Additional

experiments

revealed

that, given certain

any reasonable initial estimates of ,8 and values. In a final series of computer runs the of normally distributed random error standard deviation of the errors increased values of ,l3 and S and the computed

depended

fixed values for (Y and y,

6 lead to convergence

to the same

data were modified by the addition terms. In general, increasing the the differences between the “true”

solutions (see Table 3).

TABLE 3 EXAMPLES WITH

OF CONVERGENCE

DATA

MODIFIED

IN ESTIMATION

WITH

GAUSSIAN

OF fl AND

RANDOM

6

ERRORSa

(a) Mean of errors = 0, standard deviation = 0.1 Estimate Approximation:

Initial

1

P s

0.4000 0.3365 6.CKKKl 5.0809

True Value

2

3

4

0.3023 5.6861

0.3002 5.7742

0.3002 5.7748

0.3000 5.8000

(b) Mean of errors = 0, standard deviation = 1.O

P s

0.4000 6.oooO

0.3372 4.9565

0.3040 5.4915

Given a = 2.0 and y = 0.07

0.3022 5.5513

0.3022 5.5504

0.3ooo 5.8000

Fitting Differential V.

Equation Models

145

DISCUSSION This paper takes a first step toward fitting a nonlinear

ordinary

tial equation as an investment model. It has formulated problem, outlined a method of solution and demonstrated numerical characteristics of the model. In the model there are four parameters,

differen-

an estimation some of the

two of which are held fixed while

the remaining two are being estimated. When the non-estimated parameters are properly specified, convergence is rapid and accurate regardless of the initial specifications of the parameters being estimated. However, the results depend heavily upon the specification of those parameters which are not being estimated. This points to the desirability of numerically estimating all four parameters using an extension of this procedure, be the subject of a subsequent paper.

a procedure

that will

Further experimentation indicates that the model is not oversensitive to random errors in the data, a fact which encourages work using actual economic data with this model. Modificiations of the model itself, such as a revision of the minimization criterion or the inclusion of an exponential factor

to account

for the increase

in productivity

of capital,

should also be

explored.

REFERENCES 1 2 3 4 5 6 7

R. Wonnacott and T. Wonnacott, Econometrics, Wiley, New York, 1970. E. Burmeister and A. Dobell, Mathematical Theories of Economic Growth, Macmillan, New York, 1970. A. Takayama, Mathematical Economics, Dryden, Hinsdale, 1974. R. Bellman and R. Kalaba, Quasilineariultion and Nonlinear Bounday-Value Problems, American Elsevier, New York, 1965. H. Kagiwada, System identification Methods and Applications, Addison-Wesley, Reading, 1974. R. Kalaba, On nonlinear differential equations, the maximum operation and monotone convergence, J. Math. and Mech., 8 (1959), 519-574. L. Shampire and M. Gordon, Computer Solutions of Ordinary J3ifferential Equations, Freeman, San Francisco, 1975.