Nonlinear Estimation of the CES Production Parameters: A Monte Carlo StudyAuthor(s): T. Krishna Kumar and James H. GapinskiSource: The Review of Economics and Statistics, Vol. 56, No. 4 (Nov., 1974), pp. 563-567Published by: The MIT PressStable URL: http://www.jstor.org/stable/1924476 .

Accessed: 28/06/2014 09:10

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to The Review ofEconomics and Statistics.

http://www.jstor.org

This content downloaded from 91.220.202.155 on Sat, 28 Jun 2014 09:10:47 AMAll use subject to JSTOR Terms and Conditions

NOTES 563

in general the sign and the size of the bias of S2 as an estimate of 0-2 depend both on V and on the explanatory variables (Watson 1955).

In the case when 1 is one of the columns of X we can obtain a sufficient condition for S2 to underestimate 0-2, which is only a function of the autocorrelation coefficients of the disturbances. The sufficient condition is that Nk < Vij,.

Proof: Let M and J be defined as the following projection matrices M I - X(X'X) -'X', J - 1 (l'l)-I 1'. Since JM MJ O then I-M-J is a projection matrix with the rank N - k - 1 and E'(I-M-J) E , 0. By taking expected values we obtain:

o-2N - (N-k) E(s2) - 2 2(1' V1)N > 0, (N - k)E(s2) < o-2 [N (IT'V 1)/N].

Therefore for V satisfying 1' V 1 > Nk, we obtain E(s2) < 02. Thus for example when V,j pi-i, p > 0, the sufficient condition can be written as

k < (I + p)/( -p) + 2p(1 + p + .. . + pl-')I[N(p -1)]

which means that (depending on N) p must be quite high.

REFERENCE

Watson, G. S., Serial Correlation in Regression Analy- sis I, Biometrika 42 (1955).

NONLINEAR ESTIMATION OF THE CES PRODUCTION PARAMETERS: A MONTE CARLO STUDY

T. Krishna Kumar and James H. Gapinski*

Over the past few years, a number of researchers have estimated the parameters of the CES produc- tion function by applying nonlinear regression pro- cedures directly to the function itself. Notable in this regard are Bodkin and Klein (1967), Kumar and Asher (1974), and Weitzman (1970). Despite the popularity of this practice, little has been dis- covered about the econometric characteristics of the accompanying estimators. This note illuminates a particular case of the directly estimated CES; it examines the small-sample properties of the non- linear least-squares estimators of the CES param- eters. The setting is time-series.

I. The Production Models and the Simulations

The CES production function can be written in several equivalent forms. Consider the "gamma- delta" specification,

Xi -ye"l [8Kj-P + (1- 8)LI-P] -/P. (1) Xi is the total output produced at time i; K, and Li are, respectively, the total capital and labor in- puts at time i. Two stochastic counterparts of (1) are

Q - -yegi [8Ki-P + (1 8)Li-P] -fl/PN (2) and

Vi - yegi [8Ki-P + (1 8)L--P] -:'P + w,,

(3) where ui - exp (vi - av .5XA). vi and w* are random disturbances, with vi N(a1v, A,) and wi N(-aw, Xw). Note that Eu- 1 and that var ui - exp (Av) -1.

The production models (2) and (3) were used to generate the data on output, and to create this data a simulation apparatus was developed. How- ever, before this tool could be pressed into service, several details had to be ironed out.

How should the capital and labor series behave temporally? Several smooth patterns immediately suggested themselves, but these were dubious be- cause the resulting degree of multicollinearity might be markedly different from that observed in actual practice. In view of this possibility, it seemed ad- visable to use real-world data. Readily available were the annual capital and labor figures for the United States Private Domestic Economy used by David and van de Klundert (1965). From these figures the 20 most recent observations were chosen; namely, those for 1941-1960.

The o- and /3 values for the production functions were chosen, in part, to permit inquiries about whether the estimate of Cr is biased toward unity and about whether the value of /8 has any systematic effect on the cr bias. Compatibility with empirical evidence and manageability of the experimental design were also considered. These guideposts led

Received for publication May 2, 1973. Revision accepted for publication October 19, 1973.

*We thank Professors Duane A. Meeter and T. N. Krishnamurti for supplying us with some of the computer algorithms used in this study. We also thank the anonymous referee editor for his comments. This note was part of a larger work presented at the New York meeting of the American Statistical Association, December 1973.

This content downloaded from 91.220.202.155 on Sat, 28 Jun 2014 09:10:47 AMAll use subject to JSTOR Terms and Conditions

564 THE REVIEW OF ECONOMICS AND STATISTICS

to the assignments o- = .35 and 1.35 and 8 -.8 and 1.2. The remaining parameters of the determin- istic components of the production functions were chosen more casually: y 1, p - .03, and 8 = .4. From these assignments four parameter combina- tions were obtained for each production model.

The means of the wi were set at zero; those of the ui were unity by construction. Consequently, (2) and (3) threw points randomly about the sur- face described by (1). a,, was equated to zero.

The variances X, and XA were chosen by trial and error. It was recognized that X values could be assigned which were so large that the output series from (2) and (3) would be almost completely stochastic. On the other hand, they could be so small that the production models would be virtually deterministic. To avoid these perverse situations, an expedient rule was followed whereby the variances were chosen to yield R2 values for the nonlinear regressions which would be of the magnitude typically encountered in practice-approximately 0.98. Thus, the X assignments were X,- .0005 and Xw- 1.75.

Since the input data consisted of 20 observations, the generation of one output series in either the multiplicative or additive disturbance regimes re- quired 20 disturbance values. A total of 150 sets of 20 vi values were drawn randomly from the N(0, .0005) distribution along with 150 sets -of 20 wi values from the N(0, 1.75) distribution. By varying only the disturbance-value sets, 150 sets of output data were produced for each of the four parameter combinations in each disturbance regime. The same i, Ki, and Li series were used for all multiplicative and additive output series.

II. The Nonlinear Regressions and the Regression Results

The nonlinear regression models for (2) and (3) are respectively

ln Qi 01i + 05 [04/(04 - 1)] ln [O2Li(04-1)/04 + 93K(04-1)/04] + In ui

(4) and

Q,- e0l [02L.(04-1)/04

+ 03K(04-1)/04]O504/(04-1) + w*, (5) where 01 =,a, 02 (1 - 8)y-P/I8, 03 = 8-P 04 = 0-, and 05 = 3.1 The error sum of squares

(ESSQ) associated with (4) and (5) are, respec- tively, I(ln u,)2 and Xw12.

To estimate the 0 parameters of (4) and (5) from the simulation data, the nonlinear least- squares regression technique developed by Mar- quardt (1963) was used. Two points deserve men- tion here. First, the true parameter values were used as the initial guesses for t1, t2, . . . , t5, the estimators of 01, 02, . . ., 05, respectively. This choice of initial guesses was made in the hope that the minimization of the ESSQ function could be accomplished in the smallest number of iterations. Second, the Marquardt algorithm has a built-in ten- dency to choose a direction orthogonal to the gradient of the ESSQ function as the optimum is approached. This feature will be referenced again in the discussion of the ESSQ contours.

From the previous section it is evident that E(ln U,) - - .5X, and that E(w,) -a = 0. It follows that the least-squares procedure applied directly to the nonlinear regression model (4) estimates the regression surface

ln Qi 01i + 05 404/(04 -1) ] ln [02Li(4-1)/04 + 03Ki(O4-1)/04] - .5 Xv.

(4.1) The regression estimates obtained for (4) were thus subject to a specification bias. While some adjust- ments could have been made to correct for the misspecification, this was not done since the inten- tion of the study was to simulate real-world situa- tions - and in these such a misspecification would be ignored. In actual practice the econometrician, in strict analogy with the ordinary linear regression model, would assume that the disturbances of (4), viz. ln ui, have zero means; and thus he would un- consciously assume away, not correct for, the mis- specification. In the present study, however, the existence of this misspecification should bias the estimators only slightly since -.5X, is very small (-.00025).

The application of the nonlinear least-squares program to the simulation data produced, among other results, 150 values for each of t1, t2, . . ., t5

for each parameter combination of each regression model. The sample size, of course, was always 20. The y and 8 estimates for both models were ob- tained from the conversions g = (t2 + t3)c, C

_t44/0- t4), and d = t3/(t2 + t3), respec- tively. Table 1 presents the sample mean (m1), the sample per cent bias (pb), and the sample variance (M2) of the estimators in equation (4). The corre- sponding statistics for equation (5) are strikingly similar and hence withheld.

Table 1 reveals that the bias and variance are generally small, particularly so for t1 and d. An

1 Equations (4) and (5) depart from the parametric form used in (2) and (3). The parameter p is replaced by 04, and the parameters y and b by 02 and 03. These changes were made because the regression algorithm cannot force inequality constraints such as p > -1 and 8 < 1. It can only impose sign restrictions.

This content downloaded from 91.220.202.155 on Sat, 28 Jun 2014 09:10:47 AMAll use subject to JSTOR Terms and Conditions

NOTES 565

obvious exception to this rule is t4. The bias and variance of t4 are always enormous, this being due in large measure to the presence of at least several extremely large t4 values in each case. The existence of these outliers renders questionable further analy- sis of t4 except for an inquiry into their cause. This is discussed in the next section.

A closer examination of table 1 reveals that the estimator t1 is biased downward for all parameter combinations; its bias decreases when C- shifts from 0.35 to 1.35 and increases when /8 shifts from 0.8 to 1.2. The estimator t5 is biased upward for all param- eter combinations. Its bias decreases either when C changes from 0.35 to 1.35 or when /3 changes from 0.8 to 1.2. The estimator g is biased upward in all cases, its bias decreasing when C- moves from 0.35 to 1.35 and increasing when /3 moves from 0.8 to 1.2. The estimator d is biased downward when

- .35 and biased upward when o- - 1.35, and its bias increases as /3 shifts from 0.8 to 1.2. It should be emphasized, however, that the systematic patterns exhibited by the bias are in many in- stances not very pronounced and that further Monte Carlo investigations or theoretical explanations are needed tq support the patterns which have been detected.

III. The Estimates of the Elasticity of Substitution and the ESSQ Contours

The results of the previous section demonstrate that alternate random drawings of data from the same production structure give rise to outlying values of t4 in all cases while the estimates of the other production parameters exhibit only slight variation. This gives rise to the suspicion that the

ESSQ function has a high degree of flatness in the 04 direction near the point of the true parameter values. While flatness alone would not explain why a general nonlinear least-squares regression package which starts with the true parameter values as the initial guesses would generate large t4 values, flat- ness would for the Marquardt program because of the functioning of that program near the optimum.

To obtain some idea of the shape of the ESSQ surface, the sectional contours of the ESSQ func- tion were plotted with the aid of a computer algo- rithm. For each of the four parameter combinations, two data sets were drawn from the relevant 150 sets. One corresponded to t4 values close to the true 04 value for both regression models, and one corre- sponded to extremely large t4 values for both models. Since the algorithm could plot in only two dimen- sions, the contours were mapped by pairing 04

against each of the other four 0's. The 04 values used in the plotting procedure ranged from 0.05 to 33.0, the increments being 0.03 for 0.05 < 04 ? 3.02 and 0.3 for 04 > 3.02. The finer partitioning of the 04 interval for the lower 04 values seemed ap- propriate in view of the sensitivity of the CES specification to 04 values close to unity. The re- maining 0's ranged thus: 01 from 0.005 to 0.05 in increments of 0.005; 02 and 03 from 0.1 to 1.0 in increments of 0.1; and 05 from 0.5 to 1.4 in incre- ments of 0.1. For each two-dimensional contour map, it was necessary to assign a value to each of the remaining 0's. These assignments were the true values. The four two-dimensional contours there- fore described partially the shape of the six-dimen- sional ESSQ function in the neighborhood of the point defined by the true parameter values.

TABLE 1. - SELECTED RESULTS FOR THE ESTIMATORS OF THE PARAMETERS IN EQUATION (4)

tI tt to g d

-= .35, / = .80 ml .02965 .58 X 1011 .83854 1.12126 .39825 pb -1.15% .17 X 1014% 4.82% 12.13% -.44%

I2 .10 X 10-4 .51 X 1024 .02935 1.15433 .02568

o= 1.35, =.80 ml .02976 .31 X 10109 .80570 1.08676 .40534 pb -.79% .23 X 10111% .71% 8.68% 1.33% m2 .40 X l0O5 .14 X 10220 .00948 .48361 .00694

o, = .35, /8 = 1.20 ml .02949 .75 X 102 1.24590 1.17405 .39721 pb -1.71% .21 X 105% 3.83% 17.41% -.70% m2 .12 X 10-4 .78 X 106 .03685 1.53486 .01313

= 1.35, 3 = 1.20 ml .02963 .85 X 1010 1.20749 1.11130 .40538 pb -1.23% .63 X 1012% .62% 11.13% 1.34% m., .57 X 10-5 .11 X 1023 .01243 .58927 .00582

This content downloaded from 91.220.202.155 on Sat, 28 Jun 2014 09:10:47 AMAll use subject to JSTOR Terms and Conditions

566 THE REVIEW OF ECONOMICS AND STATISTICS

The contours which emerged were remarkably uniform, their shapes being essentially unchanged across parameter combinations, across data sets, and across regression models. The contours for model (4) with o- - .35 and 3 - .8 are illustrated in figures 1-4. The dot in each figure locates the true parameter values.2 The shapes of these con- tours in the 04 direction near the true parameter values indicate that the ESSQ function in that neighborhood is almost flat with respect to 04 as suspected.3

IV. Conclusion

This note presented Monte Carlo results regard- ing the direct estimation of the CES production function parameters from small samples by the method of nonlinear least squares. The bias for the estimators of the crucial parameters IL, 8, and ,3 was always small regardless of whether the true produc- tion surface was specified with multiplicative or ad- ditive disturbances. The other critical parameter C was always estimated very imprecisely.4 This was

REPRESENTATIVE CONTOURS OF THE ESSQ FUNCTION

FIGURE 1. FIGURE 2.

1.5

.050.

.030 -- - _6_

.025- _ _ _ _ _ _ __ _ _ _ _ _

6. 35 l 2 3 . 35 1 2 A O4-

FIGURE 3. FIGURE 4.

63 35

1.5 1.5

1.0 1.0 __ _ _ _ _ _ _ _ _ _ _ _

.8-

.5 .5 _ _ _ _ _ _ _ __ _ _ _ _ _ _

A.4

035 1 2 e403 4

2 Although each contour map is everywhere dense, the computer did not plot the contour which passed through the point of the true parameter values. Also, because ol rounding errors inherent in the operation of the algorithm, the contour representing the minimum ESSQ value was difficult to locate precisely. In all cases, however, the lowest- numbered ESSQ contour passed close to the true parametei point; and the value of the ESSQ always increased steadily for steady upward or downward departures from the lowest-numbered contour.

3Two reasons could be advanced for this flatness in the ESSQ function. First, the ESSQ function could be insensi-

tive to large changes in t4 values for most of the values commonly assumed by the explanatory variables of the regression model. Second, the flatness could be due to multicollinearity. It should be emphasized, however, that the input series chosen were actual time series that would typically be encountered in a time-series regression model.

The flatness which marks the contours in the 02 and 03 directions (figs. 2 and 3) is associated with extreme vari- ability in the t2 and t3 values.

4 This phenomenon is not uncharacteristic of studies in- volving nonlinear regression techniques. See Bodkin and Klein (1967), Gapinski and Kumar (1972), and Kumar and Asher (1974).

This content downloaded from 91.220.202.155 on Sat, 28 Jun 2014 09:10:47 AMAll use subject to JSTOR Terms and Conditions

NOTES 567

due to the flatness of the ESSQ function with re- spect to oc combined with the functioning of the regression program near the optimum.

Nonlinear least squares appears to be an impor- tant tool for estimating the parameters of the CES production function as it seems to exhibit a ten- dency for providing accurate estimates, except for o-. This finding should prove useful to the econo- metrician who plays in the world of nonlinear tech- niques.

REFERENCES

Bodkin, R., and L. R. Klein, "Nonlinear Estimation of Aggregate Production Functions," this REVIEW, 49 (Feb. 1967), 28-44.

David, P. A., and Th. van de Klundert, "Biased Ef- ficiency Growth and Capital-Labor Substitution in

the U.S., 1899-1960," American Economic Review, 55 (June 1965), 357-394.

Gapinski, J. H., and T. K. Kumar, "Putty-Clay Capital and Small-Sample Properties of Ordinary and Non- linear Least-Squares Estimators," paper presented at the Winter Meeting of the Econometric Society, Toronto, December 1972.

Kumar, T. K., and E. Asher, "Soviet Postwar Economic Growth and Capital-Labor Substitution: A Com- ment," forthcoming in American Economic Review, 64 (March 1974), 240-242.

Marquardt, D. W., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters," SIAM Journal on Applied Mathematics, 11 (June 1963), 431-441.

Weitzman, M. L., "Soviet Postwar Economic Growth and Capital-Labor Substitution," American Eco- nomic Review, 60 (Sept. 1970), 676-692.

This content downloaded from 91.220.202.155 on Sat, 28 Jun 2014 09:10:47 AMAll use subject to JSTOR Terms and Conditions