The Permanent-Transitory Confusion: Implications for Tests of Market Efficiency and for Expected Inflation During Turbulent and Tranquil Times

Abstract

Even when all past and present information is known, individuals usually remain uncertain about the permanence of observed variables. After reviewing the history and role of adaptive expectations and its statistical foundations in modeling this permanent-transitory confusion, the paper investigates the consequences of this confusion for tests of market efficiency in the treasury bill and foreign exchange markets. A central result is that the detection of serial correlation in efficiency tests based on finite samples does not necessarily imply that markets are inefficient. The second part of the paper utilizes data on Israeli inflation expectations from the capital market to estimate the implicit speed of learning about changes in inflation and to examine the performance of adaptive expectations in tracking the evolution of those expectations during the 1985 shock stabilization as well as during the stable inflation targeting period.

6 August 2018

Personal note on the history of the paper by Alex Cukierman

The origin of this paper goes back to an old unpublished manuscript by Cukierman and Meltzer (1982). A couple of years before that time Karl Brunner and Allan Meltzer became aware of the importance of the permanent-transitory confusion. I first discovered the universality of this confusion for the formation of expectations and for economic behavior when, as a visiting scholar at Carnegie-Mellon during the end of the seventies and beginning of the eighties, I started to interact with Allan and Karl on this topic. This interaction culminated in a number of joint published papers. The Cukierman and Meltzer (1982) paper was a later spinoff of this research effort and was never completed mainly because the research attention of both Allan and myself had turned to other topics and I had returned to Tel-Aviv.

But I always felt that the ideas in our unpublished manuscript are sufficiently important to justify bringing it up to date and amplifying its message with empirical work. This is particularly important for younger generations of economists who, due to the early criticism by the rational expectation school that adaptive expectations are not rational, might not be aware of the fact that Muth (1960) provided a statistical foundation for the permanent-transitory confusion in which adaptive expectations are rational.

When the organizers of the conference on “Expectations: Theory and applications in historical perspectives” suggested I write a paper for the conference I felt the time had come to do that. Unfortunately, on May 8, 2017, my long-time friend and collaborator Allan Meltzer passed away.

My young collaborator, Thomas Lustenberger, joined me in this effort. The current updated and expanded version of the paper integrates the analysis in the original manuscript with some newer empirical work. Beyond the personal loss I feel along with Allan’s family, the history of the paper is one illustration of the process through which the research torch is being passed across generations of economists. I view it as a personal token of appreciation to Allan Meltzer who had a major influence on my professional life.

Allan Meltzer passed away on May 8 2017.

This research project was conducted while Thomas Lustenberger was affiliated with the Swiss National Bank and the University of Basel.

Appendix

1.1 Derivation of Equation (15)

Inserting Eq. (3) into Eq. (14) for all t

$$ u_{t + 1} = y_{t + 1}^{p} + y_{t + 1}^{q} - \sum\limits_{i = 0}^{\infty } \theta (1 - \theta )^{i} \left( {y_{t - i}^{p} + y_{t - i}^{q} } \right). $$

Grouping all the transitory terms into one expression and all the first differences of the random walk component into another expression and rearranging

$$ u_{t + 1} = Q_{t + 1} + P_{t + 1} $$

where Q_t+1 and P_t+1 are given by Eq. (18) in the text. QED.

1.2 Proof that Eu_t+1u_t = 0

It is convenient to first prove the following Lemma

Lemma 1

$(1 - \theta )\sigma_{p}^{2} - \theta^{2} \sigma_{q}^{2} = 0$

• Proof Rearranging Eq. (4) in the text

$$ \theta + \frac{a}{2} = \sqrt {a + \frac{{a^{2} }}{4}} . $$

Raising both sides of this equation to second power, cancelling terms and noting that $a \equiv \frac{{\sigma_{p}^{2} }}{{\sigma_{q}^{2} }}$

$$ \theta^{2} = \frac{{\sigma_{p}^{2} }}{{\sigma_{q}^{2} }}(1 - \theta ). $$

The proof is completed by moving $\sigma_{q}^{2}$ to the left hand side of this equation.

Since all the terms in P_t+1 are statistically independent from the terms in Q_t+1

$$ Eu_{t + 1} u_{t} = EQ_{t + 1} Q_{t} + EP_{t + 1} P_{t} . $$

(26)

Using the definitions of Q_t+1 and of P_t+1 from Eq. (18) in the text, it can be shown after some tedious algebra that

$$ EQ_{t + 1} Q_{t} = - \frac{{\theta \sigma_{q}^{2} }}{2 - \theta }, $$

(27)

$$ EP_{t + 1} P_{t} = \frac{{\sigma_{p}^{2} (1 - \theta )}}{\theta (2 - \theta )}. $$

(28)

Substituting those expressions into Eq. (26)

$$ Eu_{t + 1} u_{t} = \frac{{(1 - \theta )\sigma_{p}^{2} - \theta^{2} \sigma_{q}^{2} }}{\theta (2 - \theta )}. $$

By Lemma 1, the numerator of this expression is zero. Since the denominator is positive Eu_t+1u_t = 0. QED

1.3 Derivation of ρ_j(∆Y^p_t) (Eq. 20)

From Eq. (15) in the text,

$$ E\left[ {u_{t + j + 1} u_{t + j} |\varDelta y_{t}^{p} } \right] = EQ_{t + 1} Q_{t} + E\left[ {P_{t + j + 1} P_{t + j} |\varDelta y_{t}^{p} } \right] $$

(29)

where

$$ \begin{aligned} E\left[ {P_{t + j + 1} P_{t + j} |\varDelta y_{t}^{p} } \right] = & E\left\{ {\varDelta y_{t + j + 1}^{p} + (1 - \theta )\varDelta y_{t + j}^{p} + ..} \right\} \\ & \left\{ {\varDelta y_{t + j}^{p} + (1 - \theta )\varDelta y_{t + j - 1}^{p} + ..} \right\} \\ & + (1 - \theta )^{2j + 1} \left\{ {\left( {\varDelta y_{t}^{p} } \right)^{2} - \sigma_{p}^{2} } \right\} \\ \end{aligned} $$

(30)

Taking the expected value of the product in Eq. (30), summing up the resulting infinite series and rearranging this equation reduces to

$$ E\left[ {P_{t + j + 1} P_{t + j} |\varDelta y_{t}^{p} } \right] = \frac{{(1 - \theta )\sigma_{p}^{2} }}{\theta (2 - \theta )} + (1 - \theta )^{2j + 1} \left\{ {\left( {\varDelta y_{t}^{p} } \right)^{2} - \sigma_{p}^{2} } \right\}. $$

(31)

Substituting Eqs. (28) and (30) into Eq. (29), rearranging and using Lemma 1

$$ E\left[ {u_{t + j + 1} u_{t + j} |\varDelta y_{t}^{p} } \right] = (1 - \theta )^{2j + 1} \left\{ {\left( {\varDelta y_{t}^{p} } \right)^{2} - \sigma_{p}^{2} } \right\}. $$

(32)

From Eq. (17) in the text,

$$ Eu_{t}^{2} = EQ_{t}^{2} + EP_{t}^{2} \,{\text{for}}\,{\text{all}}\,t. $$

(33)

Using the expressions for Q_t and P_t from Eq. (18) in Eq. (33), taking expectations of the resulting expressions, rearranging and using Lemma 1 yields

$$ Eu_{t}^{2} = \frac{{\sigma_{q}^{2} }}{1 - \theta } {\text{for}}\,{\text{all}}\,t. $$

Hence

$$ \sqrt {E\left( {u_{t + j + 1} } \right)^{2} E\left( {u_{t + j} } \right)^{2} } = \frac{{\sigma_{q}^{2} }}{1 - \theta }. $$

(34)

Equations (32) and Eq. (34) imply that

$$ \rho_{j} \left( {\varDelta y_{t}^{p} } \right) \equiv \frac{{E\left\{ {u_{t + j + 1}, u_{t + j} |\varDelta y_{t}^{p} } \right\}}}{{\sqrt {E\left( {u_{t + j + 1} } \right)^{2} E\left( {u_{t + j} } \right)^{2} } }} = (1 - \theta )^{2(j + 1)} \left\{ {\frac{{\left( {\varDelta y_{t}^{p} } \right)^{2} }}{{\sigma_{q}^{2} }} - \frac{{\sigma_{p}^{2} }}{{\sigma_{q}^{2} }}} \right\} $$

QED

1.4 Derivation of Equation (23)

The proof is an immediate consequence of Lemma 1. QED

1.5 Proof that Using Observations on y_t and y^e_t or on π_t and π^e_t Yield Identical Estimates of $\theta$ and of $\sigma_{{\varDelta y_{t}^{p} }}^{2}$

When the pair $\left\{ {\pi_{t} ,\pi_{t}^{e} } \right\}$ is used the estimate of θ is obtained by running the regression

$$ \pi_{t}^{e} - \pi_{t - 1}^{e} = \theta \left( {\pi_{t} - \pi_{t - 1}^{e} } \right). $$

When the pair $\left\{ {y_{t} ,y_{t}^{e} } \right\}$ is used the estimate of θ is obtained by running the regression

$$ y_{t}^{e} - y_{t - 1}^{e} = \theta \left( {y_{t} - y_{t - 1}^{e} } \right). $$

The definitions of $\left\{ {y_{t} ,y_{t}^{e} } \right\}$ in Eq. (25) in the text imply that the first and the second equations are identical so the estimate of θ obtained from either equation is the same.

When π_t is used to estimate σ²_∆yt the estimate is the sample variance of $\pi_{t} - \pi_{t - 1}$ and when y_t is used it is the sample variance of $y_{t} - y_{t - 1}$. Since the definitions in Eq. (25) imply

$$ y_{t} - y_{t - 1} = \pi_{t} - \pi_{t - 1} $$

the two estimates are identical. QED