Coefficient of variation and policy ineffectiveness

In this chapter we examine a non-self-averaging performance index, and question the role of the "mean" dynamics when the measure of approximation errors by quadratic expression does not convey useful information, either in policy multiplier context or in conducting or designing large-scale Monte Carlo studies. Models with large values of coefficients of variation have smaller policy multipliers than models with small coefficients of variation.

Binary choice: model 1

As an example, consider a binary choice model (Aoki, 2002, s. 6.3) in which each agent makes a binary choice. Here we give a simple example and leave more detailed analysis to Aoki and Yoshikawa (2007, p. 63).

Agents are faced with two choices. They have some idea of the mean return of each of the two choices and the associated uncertainty expressed by some variance expressions.

Let two choices have values V_{t} and V_{2}, but we observe them with error e_{;} as V* , i = 1,2

Define e = e_{1} — e_{2} . Assume that e is distributed as

for real number x, and where f > 0 is a parameter of this distribution.

McFadden models agents' discrete choices as the maximization of utilities U, j = 1,..., K , where Uj is associated with choice j and K is the total number of available choices. Let Uj = Vj + e,, j = 1,2,..., K. We are really interested in picking the maximum of V's, not of U's.

McFadden's model (1973) may be used to illustrate a binary choice problem where under suitable conditions we obtain

where

where g (x) is the mean and a (x) is the variance in
and where

is the coefficient of variation.

As CV(x) tends to infinity, u approaches zero, and rj(x) approaches 1/2. This means that the choice between the two alternatives become equally likely, that is, n( x) ^ 1/2.

Galambos (1987, p. 11) discusses the following model. Suppose es are independent and identically distributed (i.i.d.) with the distribution function F. Writing F = 1 — (1— F), we have nlnF(x) = n[1 — F(a_{n} + b_{n}y]:= u(y) for some constant a_{n} and b_{n}. Then from

Choose F(x) = 1 — e ^{x}, a_{n} = ln(n), and b_{n}=. Let P = Pr(max^Uj = U_{{}).