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2017 | OriginalPaper | Buchkapitel

13. Persistent Pessimism and Optimism in Forecasts: Implicit Means and Law of Iterated Integrals

verfasst von : Kiyohiko G. Nishimura, Hiroyuki Ozaki

Erschienen in: Economics of Pessimism and Optimism

Verlag: Springer Japan

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Abstract

Economic activities have become increasingly complicated in recent years. For example, financial innovation, past and present, has increased the complexity of financial information. Securitization is one manifestation of this trend, and it generates an extraordinary degree of complexity.

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Vorheriges Kapitel A Simple Characterization of Pessimism and Optimism: -Contamination Versus -Exuberance

Nächstes Kapitel Learning Under Knightian Uncertainty

We say that $f = f'$ $\mu $-a.e. by definition if there exists a set $N \in \mathcal {F}$ such that $\{ \, \omega \in \Omega \, | \, f(\omega ) \ne f'(\omega ) \, \} \subseteq N$ and $\mu (N) = 0$.

The indicator function is defined in Sect. 2.2.3.

An assumption of skew-symmetry is introduced later. For Fishburn’s axiomatization, see the next footnote. In related works, Chew (1983, 1989) and Dekel (1986) axiomatized preferences that are represented as the unique solutions to functional equations similar to (13.1) in the framework of risk. Grant et al. (2000) extended it to the framework of uncertainty.

Fishburn (1986) axiomatizes the skew-symmetric implicit mean. To be more precise, he considers seven axioms that may be imposed on a real-valued function defined on a convex set of probability measures on a real interval, and then shows that such a function is the skew-symmetric implicit mean if and only if it satisfies all the seven axioms, one of which is his cancellation axiom that is responsible for skew-symmetry. Note that the definition of the skew-symmetric implicit mean requires only an existence of some skew-symmetric betweenness function. It is clear in light of Proposition 13.2.1 that there does exist (many) non-skew-symmetric betweenness functions that generate the same skew-symmetric implicit mean.

Let $\varphi $ and $\varphi '$ be such betweenness functions. Then, by Proposition 13.2.1 and the skew-symmetry, there exists a function $a: \mathbb {R} \rightarrow \mathbb {R}$ that satisfies $(\forall x, z) \; a(z)\varphi (x,z) = \varphi '(x,z) = -\varphi '(z,x) = -a(x)\varphi (z,x) = a(x)\varphi (x,z)$. This shows that $(\forall x, z) \; a(z) = a(x)$; that is, the function a is constant.

A subset E of $\Omega $ is a $\mu $ -null set by definition if $E \in \mathcal {F}$ and $\mu (E) = 0$.

See Footnote 1.

To see that $F(\mathcal {G}) \ne L^{\infty }(\mathcal {G})$ in general, consider a simple example: $\Omega := \{ \omega _{1}, \omega _{2}, \omega _{3} \}$; $\mathcal {F} := 2^{\Omega }$; $\mathcal {G} := \{ \phi , \{ \omega _{1} \}, \{ \omega _{2}, \omega _{3} \}, \Omega \}$; $\mu (\{ \omega _{1} \}) = \mu (\{ \omega _{2} \}) := 1/2 \text { and } \mu (\{ \omega _{3} \}) := 0$; $f \in F$ is such that $f(\omega _{1}) = f(\omega _{2}) := 1 \text { and } f(\omega _{3}) := 2$. Then, $f \in F(\mathcal {G})$, but $f \notin L^{\infty }(\mathcal {G})$. Note that $\mathcal {G} \notin \mathcal {F}^{\circ }$.

Equation (13.4) is typically applied to $L^{2}$ spaces, as a definition of conditional expectation, in which case the minimum is always uniquely attained by the orthogonal projection because $L^{2}(\mathcal {G})$ is a closed subspace of $L^{2}(\mathcal {F})$. Because $L^{\infty } \subseteq L^{2}$, this definition can be directly applied to the current context. Furthermore, if it is extended to the $L^{1}$ space by the standard approximation argument, it coincides with a more common definition of the conditional expectation: $E[f|\mathcal {G}]$ is defined as a $\mathcal {G}$-measurable integrable function that satisfies

For the definitions of $L^{1}$ and $L^{2}$ spaces and other details, see Billingsley (1986, in particular, p. 477, 34.15).

Equation (13.5) holds true even if we understand as usual that $\{ \, \phi , \Omega \, \}$ represents the $\sigma $-algebra generated by $\phi $ and $\Omega $. The text definition of $\{ \, \phi , \Omega \, \}$ is to ensure that $\{ \, \phi , \Omega \, \} \in \mathcal {F}^{\circ }$.

When $p_{1} = p_{2}$, any real number between $x_{1}$ and $x_{2}$ can be a solution of the minimization problem defining (13.2), and hence, $M^{q}$ is not well defined. This is why we excluded the case where $q = 1$ at the outset. Thus, the argument in the text stands only heuristically.

When $q = +\infty $, we think of the minimization problem defining (13.2) as

$$ \min _{z\in \mathbb {R}}\lim _{q\rightarrow +\infty }\left( \int _{\Omega } \left| f(\omega ) - z \right| ^{q}d\mu (\omega ) \right) ^{1/q} \, . $$

Again, the argument remains only heuristic.

In fact, any betweenness function $\varphi '$ that is “equivalent” to $\varphi $ in the sense that $(\exists a)(\forall x,z) \; \varphi '(x,z) = a(z)\varphi (x,z)$ is not separable. See Proposition 13.2.1.

The function $\hat{u}$ may be regarded as a summary statistic of all relevant factors that could affect the agent’s overall utility.

This is simply because we like to show an aversion to the information. Introducing the cost merely facilitates our job.

For the negative-surprise aversion, see Sect. 13.4.4.

Billingsley, P. 1986. Probability and Measure (2nd ed.), Wiley-Interscience.

Chew, S.H. 1983. A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox. Econometrica 51: 1065–1092.CrossRef

Chew, S.H. 1989. Axiomatic utility theories with the betweenness property. Annals of Operations Research 19: 273–298.CrossRef

Dekel, E. 1986. An axiomatic characterization of preferences under uncertainty. Journal of Economic Theory 40: 304–318.CrossRef

Ellsberg, D. 1961. Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics 75: 643–669.CrossRef

Epstein, L.G. 1986. Implicitly additive utility and the nature of optimal economic growth. Journal of Mathematical Economics 15: 111–128.CrossRef

Epstein, L.G., and S.E. Zin. 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57: 937–970.CrossRef

Farmer, R.E.A. 1990. RINCE preferences. Quarterly Journal of Economics, February, 43–60.

Fishburn, P.C. 1986. Implicit mean value and certainty equivalence. Econometrica 54: 1197–1205.CrossRef

Grant, S., A. Kajii, and B. Polak. 2000. Decomposable choice under uncertainty. Journal of Economic Theory 92: 169–197.CrossRef

Hardy, G., Littlewood, J.E and Pólya, G. 1952: Inequalities (2nd ed.), Cambridge University Press.

Kolmogorov, A. 1930. Sur la notion de la moyenne. Rendiconti Accademia dei Lincei 6 (12): 388–391.

Kreps, D.M., and E.L. Porteus. 1978. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46: 185–200.CrossRef

Nagumo, M. 1930. Uber eine Klasse der Mittelwerte. Japan Journal of Mathematics 7: 71–79.CrossRef

Ozaki, H. 2009. Conditional implicit mean and the law of iterated integrals. Journal of Mathematical Economics 45: 1–15.CrossRef

Ozaki, H. 2013. Subjective error measure, Keio University, mimeo. Presented at Risk, Uncertainty and Decision 2013 (Paris).

Weil, P. 1990. Nonexpected utility in macroeconomics. Quarterly Journal of Economics, February, 29–42.

Titel: Persistent Pessimism and Optimism in Forecasts: Implicit Means and Law of Iterated Integrals
verfasst von: Kiyohiko G. Nishimura
Hiroyuki Ozaki
Verlag: Springer Japan
Buch: Economics of Pessimism and Optimism
Print ISBN: 978-4-431-55901-6

Electronic ISBN: 978-4-431-55903-0

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-4-431-55903-0_13

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