The Regrettable Loss of Mathematical Molding in Econometrics - Mechanism and Causality in Biology and Economics - page 69

Statistical versus Theoretical Relations in Economic Macrodynamics ,toassess
Tinbergen's work.
The present memorandum does not discuss details of the various equations which
Tinbergen has obtained and whose coefficients he has determined statistically. My main
concern has been to discuss what equations of this type really mean , and to what extent they
can be looked upon as 'A Statistical Test of Business Cycle Theories'. (Frisch 1995 , p. 407)
The memorandum discussed two problems. The first problem was the question
“what sorts of equations it is possible to determine from the knowledge of the time
shapes that are actually produced” (p. 416). The answer to this question was that
only the so-called coflux equations were discoverable. These coflux equations were
defined algebraically by the set of functions that forms the actual solution of the
complete system, including those determined by the initial conditions. The second
and deeper problem was that these coflux relations may not come near to resem-
bling the more “fundamental” equations that form the “essence of theory,” the so-
called autonomous equations (p. 417). Frisch argued that “it is only coflux relations
that are determined by Tinbergen, and the lack of agreement between these equations
and those of pure theory cannot be taken as a refutation of the latter” (p. 419).
Frisch's analysis of the first problem, here labeled as the “identification prob-
lem,” 4 stimulated various members of the Cowles Commission to work on identifi-
cation in the 1940s (Hendry and Morgan 1995 , p. 57). The second problem, the
problem of autonomy, was crucial in the development of the concept of structural
equations (see Aldrich 1989 ). Although both problems are closely related, solving
the first does not imply a solution to the second. While the identification problem is
a mathematical problem, the autonomy problem remains basically an empirical
problem.
To understand the nature of both problems, they will be introduced according to
their original treatment in Frisch's memorandum. We start with the identification
problem. This problem deals with the relation between the “form” of the equations
representing the assumed relations between the economic variables and the “time
shape” of these variables.
Frisch defined the form of a difference equation,
X

a iθ x i ð t θÞ¼ 0 (4.11)

iθ

as the i range of the summation that determines the terms involved in the equation.

The time shape of a variable is the sum of the exponentials that make up this
variable,

θ

4 This was not Frisch's terminology, but Koopmans'. Aldrich ( 1994 ) gives an account of the
development of the identification theory from Frisch to Koopmans by focusing on Haavelmo
( 1944 ), including a discussion of the change in terminology.