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

• Author: Aizcorbe, Ana M.
• Position: Communications

1. 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.

2. 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...