The Regrettable Loss of Mathematical Molding in Econometrics - Mechanism and Causality in Biology and Economics - page 70

X

n

C ik e γ i t x i ð t Þ¼ (4.12)

k ¼ 1

In the memorandum, the identification problem was phrased in terms of “reduc-
ibility” and “irreducibility” and was linked to the time shapes of the variables.
It is clear that the property of irreducibility must be important when we are studying the
nature of those equations that can be determined from the knowledge of the time shapes of
the functions that are to satisfy the equations. (Frisch [1938] 1995 , p. 413)

The (ir)reducibility of an equation was defined with respect to a set of functions.
An irreducible equation of the form (p. 11) is “one whose coefficients are uniquely
determined and allow of no degree of freedom if the equation is to be satisfied by
this set of functions (apart from the arbitrary factor of proportionality which is
always present in the case of a homogeneous equation)” (p. 413).
By inserting the function x i ( t ) defined in formula ( 4.12 ) into Eq. ( 4.11 ), one can
derive algebraically the following rule 5 :

Rule about reducibility: If the functions with respect to which reducibility is defined are
made up of n exponential components ... , the equation is certainly reducible - and hence its
coefficients are affected in a more or less arbitrary manner - if it contains more than n +1
terms. And it may be reducible even if it contains n + 1 terms or less . (Frisch 1995 , p. 414)
In other words, only equations that contain at most n + 1 terms may be irreduc-
ible - uniquely identified - with respect to the time shape of a variable. For example,
if the time shape has the form of a (dampened, undampened, or antidampened) sine
function, then it is equivalent to a combination of two exponential components and
therefore cannot identify an equation with more than three terms.
However, the time shapes of the variables do not satisfy just one equation but
form the actual solution of the complete system, including those determined by the
initial conditions. Frisch called an equation that is identified by the time shape of
this actual solution a “coflux” equation. The other equations were called
“superflux” equations. The word “flux” suggested that both kinds of equations
were defined with respect to the time shape actually possessed by the phenomena.
Thus, only “coflux equations and no other equations are discoverable from the
knowledge of the time shapes of the functions that form the actual solution” (p. 416).
This is the nature of passive observations , where the investigator is restricted to observing
what happens when all equations in a large determinate system are actually fulfilled
simultaneously . The very fact that these equations are fulfilled prevents the observer from
being able to discover them, unless they happen to be coflux equations. (Frisch 1995 ,p.416)
Should one bother about these other equations that are not discoverable through
passive observations? Frisch's answer was yes; the other equations, the superflux
equations, are well worth knowing because they have a higher degree of

5 Boumans ( 1995 ), which discusses the more technical details of Frisch's memorandum, also
provides the derivation of this rule.