A note on empirical tests of separability and the "approximation" view of functional forms.

Aizcorbe, Ana M.

I. Introduction

An influential paper by Denny and Fuss |6~ provided two important contributions to the literature dealing with empirical tests of separability. First, they found an important limitation in empirical tests of separability when the translog is viewed as an "exact" representation of the underlying function (hereafter referred to as the "exact" view of functional forms). Second, they demonstrated that the translog functional form may, in principle, be viewed as a Taylor series expansion to an arbitrary function. Appealing to this view, they then derive testable restrictions for the existence of an aggregate, something empirical tests under the "exact" view are incapable of providing. Their proposed test has become the norm in empirical work.(1)

The applicability of their method for empirical work hinges on viewing estimated functional forms, such as the translog, as second-order Taylor series expansions (also referred to as second-order approximations). Although they demonstrated that, in principle, one may view translog coefficients as Taylor series coefficients to an arbitrary function, their paper did not address the relevant question from an empirical point of view: Can estimated translog coefficients be viewed as Taylor series coefficients to an arbitrary function? This distinction between properties of the translog in principle and properties of the translog once estimated is clearly important because testing the separability restrictions requires that the translog be estimated.

This note examines the approximation properties of the estimated translog functional form. In particular, conditions under which approximations derived from estimated translog functional forms may be viewed as a Taylor series expansion are examined. Another way to state the issue is: Under what conditions can coefficient estimates obtained using a translog functional form be viewed as estimates for a Taylor series coefficients? This issue was rigorously addressed by White |13~ for linear approximations (i.e., first-order approximations), where he demonstrated that OLS estimates of first-order approximations do not provide consistent estimates of Taylor series coefficients. The continuing interpretation of the translog and other functional forms as second-order approximations suggests that applied researchers are not convinced that White's argument for first-order approximations generalizes to higher-order approximations |4~. Therefore, this note may be viewed as an examination of these issues as they relate to the translog functional form. Although the focus of this note is on least-squares techniques (since it is the commonly used estimation method) and the widely-used translog, it is readily demonstrated that the arguments made here apply to other functional forms which take the form of quadratic expansions (such as the quadratic and generalized Leontief).

The note is organized as follows. Section II reviews the Denny and Fuss (DF) argument. The applicability of their argument for the estimated translog is examined in section III. Here, it is demonstrated that the estimated version of a translog functional form provides unbiased estimates of Taylor series coefficients only if the underlying function holds exactly. Therefore, assuming that the estimated translog yields estimates of a second-order "approximation" involves the maintained assumption that it holds "exactly". Section IV points out the unfortunate implications for empirical tests of separability. Specifically, conventional methods for estimating the theoretical separability restrictions do not yield an empirical test of the existence of an aggregate.

II. The Denny and Fuss Argument

The general point made by DF is that the empirical restrictions for separability and their interpretation depends on how one interprets the functional form used to conduct the test. Thus, they defined two alternative views of functional forms:

DEFINITION 1 (Denny and Fuss). A second-order approximation, A(z), to the production function Q = Q(z), where z = ||Z.sub.1~...|Z.sub.N~~ represents the inputs, is the Taylor-series quadratic expansion

|Mathematical Expression Omitted~

where |z.sup.*~ = |Mathematical Expression Omitted~ is the point of expansion.

A function satisfying this definition may be viewed as a second order-approximation. Alternatively, one may assume that the functional form holds exactly:

DEFINITION 2. An approximation A(z) is exact for Q(z) iff A(z) = Q(z) for all z.

DF then (1) demonstrated a problem with separability tests conducted under the exact view of functional forms, and (2) proposed an alternative method using the approximation view. Each of these issues is examined in turn.

Problems with Separability Tests under the Exact View.

An important contribution of the DF paper was in pointing out a problem with conducting separability tests under the exact view. To see this, note that, in theory, a production function

y = f(|X.sub.1~, |X.sub.2~, |X.sub.3~) (1)

is separable in |X.sub.1~ and |X.sub.2~ if (1) may be restated as:

y = f(g(|X.sub.1~, |X.sub.2~), |X.sub.3~). (2)

In that case, it is said that an aggregate, |Mathematical Expression Omitted~, exists. The well-known theoretical condition that allows one to move from (1) to (2) is that the marginal rate of substitution between |X.sub.1~ and |X.sub.2~ be invariant to changes in the level of |X.sub.3~:

d/d|X.sub.3~|(dy/d|X.sub.1~)/(dy/d|X.sub.2~)~ = 0. (3)

The problem with testing the restrictions in (3) in the exact view is that choosing a functional form for |Mathematical Expression Omitted~ implicitly places restrictions on the functional form of the aggregate |Mathematical Expression Omitted~. DF demonstrated this for the translog functional form:

PROPOSITION 1 (Denny and Fuss). The separable form of a translog function interpreted as an exact production function must be either a Cobb-Douglas function of translog subaggregates or a translog function of Cobb-Douglas subaggregates.

This is demonstrated by considering the well-known translog:

|Mathematical Expression Omitted~

and the empirical counterpart to the separability conditions in (3) for the translog |2~:

|Mathematical Expression Omitted~

So, for example, a sufficient condition for (5) to hold is that ||alpha~.sub.13~ = ||alpha~.sub.23~ = 0, which, once imposed on (4) yields:

ln y = ||alpha~.sub.0~ + ||theta~.sub.g~ ln g + ||theta~.sub.h~ ln h, (6)

where |Mathematical Expression Omitted~ is a translog function of |X.sub.1~ and |X.sub.2~, and |Mathematical Expression Omitted~ is a translog function of |X.sub.3~. This presents a problem for empirical testing since the theoretical conditions for separability in (3) represent a test for the existence of some aggregate |Mathematical Expression Omitted~ while the empirical test (using the translog) implies functional form restrictions on the particular form that |Mathematical Expression Omitted~ may take: In this case, |Mathematical Expression Omitted~ must be translog. Therefore, the empirical test is not just a test for separability, but a test of separability, conditional on given functional forms for |Mathematical Expression Omitted~. They also demonstrated that other conditions that guarantee condition (5) also place restrictions on the particular form that |Mathematical Expression Omitted~ can take. Therefore, empirical tests of separability under the exact view of functional forms do not represent a test for existence.

Empirical Tests of Separability under the Approximation View.

To circumvent this problem, DF proposed the now-conventional "approximate" tests for separability where one views the translog as an approximation rather than as an exact representation of the underlying function. To justify this view, they provided conditions under which the translog may be viewed as an approximation: In Proposition 2 of their paper, DF demonstrated that the translog functional form in (4), with symmetry imposed, is a second order approximation if the following holds:

|Z.sub.i~ = ln |X.sub.i~

||Z.sup.*~.sub.i~

||alpha~.sub.0~ = Q(z)/|z.sup.*~

||alpha~.sub.i~ = dQ(z)/d|Z.sub.i~/|z.sup.*~

||alpha~.sub.ij~ = |d.sup.2~Q(z)/d|Z.sub.i~d|Z.sub.j~/|z.sup.*~ = |d.sup.2~Q(z)/d|Z.sub.j~d|Z.sub.i~/|z.sup.*~ = ||alpha~.sub.ji~. (7)

The key conditions in (7) are the last three, which state that the coefficients of the approximation must equal the derivatives of the underlying function at |z.sup.*~. Their proof involves imposing these restrictions on the translog in (4), to obtain an expression that satisfies Definition 1. Therefore, their proposition demonstrated that in principle one may view the translog as a second-order approximation.

The usefulness of this for separability testing is that under the approximation view, one does not impose functional form restrictions on the underlying function and thus, one does not restrict the functional form of the aggregate |Mathematical Expression Omitted~. Specifically, Proposition 4 in DF showed that under the approximation view, the separable version of the translog still satisfies the definition for a second-order approximation. Therefore, the test of the empirical separability restrictions, under the approximation view, represent a test for the existence of an aggregate.

III. Problems with Separability Tests under the Approximation View

The difficulty with the DF solution, from an empirical perspective, is that there is no guarantee that a translog estimating equation, once estimated, will yield parameters that are in some sense equal to the derivatives of the underlying function, as (7) requires. This raises the issue of potential biases in estimation and suggests the following criteria for determining whether an estimated translog is a second-order approximation:

DEFINITION 1'. The translog estimating equation:

|Mathematical Expression Omitted~

yields an approximation:

|Mathematical Expression Omitted~

which may be viewed as a second-order quadratic approximation to an arbitrary function, Q(x), at the point |Mathematical Expression Omitted~ if

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the expectations operator.

Alternatively, one may view the translog functional form as an exact representation of the underlying function, in which case the resulting approximation is called exact:

DEFINITION 2'. An estimated translog approximation |Mathematical Expression Omitted~ is exact for |Mathematical Expression Omitted~ for all x.

Definition 1' states that an approximation obtained from an estimated translog may be viewed as a second-order approximation if the expected value of the estimated translog coefficients equal the derivatives of the underlying function. Another way to state this is that it may be viewed as a second-order approximation if the DF conditions in (7) hold in expectation. Similarly, Definition 2' states that the approximation obtained from an estimated translog is exact if it provides unbiased predictions of the underlying function Q(x) at all points (not just the point of approximation).

These definitions are used below to explore conditions under which an approximation obtained using an estimated translog function will satisfy Definition 1'. At this point it is useful to give an intuitive overview of the argument to be made. First, the relationship between two approximations to the same function, but using different points of approximations is examined and it is demonstrated that changing the point of approximation does not alter the predicted values of the approximation. This means that all translog approximations from an estimated translog function will yield the same predicted values, regardless of the point of approximation used in the estimation. Using this result, it is demonstrated that if one assumes that an approximation yields unbiased estimates at the point of approximation, then it must yield unbiased estimates at all points, which cannot occur unless the approximation holds exactly. Therefore, assuming that Definition 1' holds for the translog approximation involves the implicit assumption that Definition 2' holds as well.

Each of these arguments are now examined in turn.

PROPOSITION 1. When estimating a translog equation, changing the point of approximation does not affect the predicted values of the approximation at any point.

Proof. Consider two translog approximations. The first

|Mathematical Expression Omitted~

is obtained using estimates from the following translog estimating equation which uses ||X.sup.*~.sub.i~ = 1 as the point of approximation (without loss of generality):

|Mathematical Expression Omitted~

The second approximation:

|Mathematical Expression Omitted~

is obtained using estimates from the following translog estimating equation which uses some arbitrary point ||X.sup.k~.sub.i~ as the point of approximation:

|Mathematical Expression Omitted~

Since both estimating equations involve the same dependent variable (ln y), set the right-hand sides of (12) and (14) equal to obtain the following equalities(2):

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

||alpha~.sub.ij~ = ||beta~.sub.ij~

||epsilon~.sub.1~ = ||epsilon~.sub.2~. (15)

Note, then, that these equalities, once imposed on (12) yield an estimating equation identical to that in (14): Both regression equations have identical independent variables (ln y); dependent variables and residuals. Therefore, the estimated coefficients obtained from (12) and (14) will satisfy the following conditions:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Finally, substituting (16) into (11) and comparing the result to (13) yields:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Equation (17) says that |Mathematical Expression Omitted~ evaluated at any point will take on the same value as |Mathematical Expression Omitted~. Taking expectations over (17) yields equation (18). Therefore, changing the point of approximation does not affect the prediction given by the approximation.(3) Q.E.D.

A numerical example is helpful in examining this issue. Consider two second-order quadratic TABULAR DATA OMITTED approximations to an arbitrary function in one variable: y = f(X). Table I provides estimated coefficients and predicted values for the dependent variable and its derivatives for two regressions, both using a translog functional form. Each approximation is performed at a different point of approximation: The first is centered at X = 1 while the second is centered at the mean. Note that although most of the coefficient estimates differ, the estimated values for y and its derivatives do not vary as the point of approximation changes. That is, OLS provides the same estimated function regardless of the "point of approximation" chosen. Another way to state the point is that knowing the equalities in (16) makes one of the regressions superfluous. That is, if one estimates (12), one can use the equalities in (16) to obtain the estimates of the regression in (14).

We now make the central point.

PROPOSITION 2. A translog approximation obtained by estimating the translog functional form satisfies the definition for second order approximations (Definition 1') only if it is exact (satisfies Definition 2').

Proof. Suppose Definition 1' holds for the approximation |Mathematical Expression Omitted~, above. That is, evaluate (11) at the point of approximation x = 1 and impose the first equality in (10):

|Mathematical Expression Omitted~

Further, suppose |Mathematical Expression Omitted~ does not satisfy Definition 2'. That is, suppose there exists a point |x.sup.k~ such that

|Mathematical Expression Omitted~

This condition will be used to raise a contradiction. In particular, it is demonstrated that given (19), a translog approximation will not satisfy Definition 1' unless it violates (20).

To do this, consider another approximation |Mathematical Expression Omitted~, also satisfying definition 1' at this arbitrary point |x.sup.k~:

|Mathematical Expression Omitted~

The relationship between the expectations of the two approximations |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ is given in (18), which when applied to the point of approximation is stated as:

|Mathematical Expression Omitted~

This, combined with (21) yields:

|Mathematical Expression Omitted~

which contradicts (20). Q.E.D.

This implies that if Definition 1' holds (i.e., the translog approximation yields unbiased estimates at the point of approximation), then it must be the case that one obtains unbiased estimates everywhere, which cannot occur unless the functional form holds exactly (i.e., Definition 2' holds).

IV. Implications for the Conventional Separability Test

Having demonstrated that approximations obtained from translog estimating equations must be viewed as "exact" approximations, the implications for separability testing are fairly obvious. Since these approximations must be viewed as "exact", Proposition 1 of DF applies to the translog approximation and the approximation view of the translog fails to circumvent the problems in the exact view.

This means it is inappropriate to interpret the conventional empirical test for separability as a test for the existence of an aggregate. Since applying the definition for second-order approximations to an estimated functional form requires that the functional form of the estimated function match that of the true, underlying function, one is actually testing for separability, conditional on the functional form chosen. That is, testing the separability conditions still involves a maintained functional form assumption which affects the possible functional forms of aggregates being tested. Thus, one can not test for existence of an aggregate, and can only test for aggregates of a particular functional form (i.e., those allowed by the functional form chosen for the estimated function).

This argument motivates the need for new methods for testing separability restrictions. Promising techniques which do not impose functional form assumptions on the underlying function are the non-parametric tests discussed in Varian |11~. Another approach would focus on testing the validity of aggregates with a particular functional form |1~. Finally, the development of new functional forms which provide unbiased estimates of underlying elasticities |7~ could provide a useful way to test for the existence of an aggregate.

1. Among the numerous studies applying the Denny and Fuss separability tests are Norsworthy and Malmquist |10~, Hazilla and Kopp |8~, Wang-Chang and Friedlaender |12~, McMillan and Amoako-Tuffour |9~, and Chung |5~.

2. If one multiplies out the bracketed terms in (14) and simplifies, one obtains:

|Mathematical Expression Omitted~

Setting this equal to the right-hand side of (12) yields the equalities.

3. This point may also be examined by using tedious algebra to demonstrate that the predictions from (12) and (14) are identical:

|Mathematical Expression Omitted~

where |X.sup.1~ and |X.sup.k~ are the matrices of explanatory variables for equations (12) and (14) respectively, and the batted vectors represent parameter estimates.

References

1. Aizcorbe, Ana M., "Testing the Validity of Aggregates." Journal of Business and Economics Statistics, 8, 1990, 373-83.

2. Berndt, Ernst and Laurits Christensen, "Testing for the Existence of a Consistent Aggregate Index of Labor Inputs." American Economic Review, June 1974, 391-403.

3. Blackorby, Charles, Daniel Primont, and R. R. Russell. Duality, Separability, and Functional Structure: Theory and Economic Applications. New York: North-Holland, 1978.

4. Byron, Randall P. and Anil K. Bera, "Least Squares Approximations to Unknown Regression Functions: A Comment." International Economic Review, February 1983, 255-60.

5. Chung, Jae W., "On the Estimation of Factor Substitution in the Translog Model." Review of Economics and Statistics, 1987, 409-17.

6. Denny, M. and M. Fuss, "The Use of Approximation Analysis to Test for Separability and the Existence of Consistent Aggregates." American Economic Review June 1977, 404-18.

7. Gallant, A. Ronald, "An Elasticity Can Be Estimated Consistently Without A Priori Knowledge of Functional Form." Econometrica, November 1983, 1731-51.

8. Hazilla, Michael and Raymond Kopp, "Testing for Separable Functional Structure using Partial Equilibrium Models." Journal of Econometrics, 1986, 119-41.

9. McMillan, Melville L. and J. Amoako-Tuffour, "An Examination of Preferences for Local Public Sector Outputs." Review of Economics and Statistics, 1988, 70(1):45-54.

10. Norsworthy, J. R. and David Malmquist, "Input Measurement and Productivity Growth in Japanese and U.S. Manufacturing." American Economic Review, December 1983, 947-67.

11. Varian, Hal, "The Nonparametric Approach to Demand Analysis." Econometrica, July 1982, 945-73.

12. Wang-Chang, S. Judy and Ann F. Friedlaender, "Output Aggregation, Network Effects, and the Measurement of Trucking Technology." Review of Economics and Statistics, 1985, 267-76.

13. White, Halbert, "Using Least Squares to Approximate Unknown Regression Functions." International Economic Review, February 1980, 149-70.