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Add a lecture by @shizejin on Two Computations to Fund Social Security and Edits from Tom on BHS lecture (#810)
* updates
* Tom's Feb 18 edits of two-computations lecture
* Tom's Feb 19 edits of the bhs lecture, a little after noon
* updates
* Tom's second Feb 19 edits of bhs lecture
* updates
* update GPU note
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Co-authored-by: thomassargent30 <ts43@nyu.edu>
Copy file name to clipboardExpand all lines: lectures/doubts_or_variability.md
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## Overview
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{cite:t}`Tall2000` showed that a recursive preference specification could match the equity premium and the risk-free rate puzzle simultaneously.
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This lecture describes machinery that empirical macro-finance economists have used to evaluate the fits of structural statistical models that link asset prices to aggregate consumption.
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But matching required setting the risk-aversion coefficient $\gamma$ to around 50 for a random-walk consumption model and around 75 for a trend-stationary model, exactly the range that provoked the skepticism in the above quote from {cite:t}`Lucas_2003`.
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The Lucas asset pricing model {cite}`Lucas1978` functions as a benchmark that motivates much of this work.
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{cite:t}`BHS_2009` ask whether those large $\gamma$ values really measure aversion to atemporal risk, or whether they instead measure the agent's doubts about the underlying probability model.
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```{note}
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New Keynesians call the consumption Euler equation for a one-period risk-free bond in the Lucas {cite}`Lucas1978` model the **IS curve**.
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The distinguished **old Keynesian** disapproved of that name because the object it described was so remote from the investment function that was an important component of the IS curve of John R. Hicks {cite}`hicks1937mr` that Tobin used.
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See {cite}`tobin1992old`.
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```
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In two classic papers, Lars Peter Hansen and Kenneth Singleton used the method of maximum likelihood
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{cite}`hansen1983stochastic` and a generalized method of moments {cite}`hansen1982generalized` to investigate how well Lucas's model fit some post WWII data.
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The Hansen-Singleton papers systematically organized evidence about directions in which Lucas's model misfit the data that macroeconomists subsequently called
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- an **equity premium** puzzle
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- a **risk-free rate** puzzle
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```{note}
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{cite:t}`MehraPrescott1985` is widely credited for naming the **equity premium** puzzle.
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{cite:t}`Weil_1989` is widely credited for naming the **risk-free rate** puzzle.
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```
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These *puzzles* are just ways of summarizing particular dimensions along which a particular asset pricing model -- such as Lucas's -- fails empirically.
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They are thus special cases of specification failures detected by statistical diagnostics constructed earlier by {cite}`hansen1983stochastic` and {cite}`hansen1982generalized`.
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Their answer, and the theme of this lecture, is that much of what looks like "risk aversion" can be reinterpreted as **model uncertainty**.
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Macro-finance models that purport to resolve such puzzles all do so by changing features of the economic environment assumed by Lucas {cite}`Lucas1978`.
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The same recursion that defines Tallarini's risk-sensitive agent is observationally equivalent to a another recursion that expresses an agent's concern that the probability model governing consumption growth may be wrong.
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Many important papers have proceeded by altering the *preferences* that Lucas had imputed to a representative agent.
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Hansen-Jagannathan bounds are a key tool for evaluating how well such re-specifications do in
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correcting those misfits of Lucas's 1978 model.
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This lecture begins with a description of the {cite}`Hansen_Jagannathan_1991` machinery.
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After doing that, we proceed to describe a line of research that altered Lucas's preference specification in ways that we can think of as being designed with the Hansen-Jagannathan bounds in mind.
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We'll organize much of this lecture around parts of the paper by Thomas Tallarini {cite}`Tall2000`.
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His paper is particularly enlightening for macro-finance researchers because it showed that a recursive preference specification could fit both the equity premium and the risk-free rate, thus *resolving* both of the puzzles mentioned above.
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But like any good paper in applied economics, in answering some questions (i.e., resolving some puzzles), Tallarini's paper naturally posed new ones.
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Thus, Tallarini's puzzles-resolving required setting the risk-aversion coefficient $\gamma$ to around 50 for a random-walk consumption model and around 75 for a trend-stationary model, exactly the range that provoked the skepticism in the above quote from {cite:t}`Lucas_2003`.
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This brings us to the next parts of this lecture.
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Lucas's skeptical response to Tallarini's explanation of the two puzzles led
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{cite:t}`BHS_2009` to ask whether those large $\gamma$ values really measure aversion to atemporal risk, or whether they instead measure the agent's doubts about the underlying probability model.
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Their answer, and the theme of the remaining parts of this lecture, is that much of what looks like "risk aversion" can be reinterpreted as **model uncertainty**.
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The same recursion that defines Tallarini's risk-sensitive agent is observationally equivalent to another recursion that expresses an agent's concern that the probability model governing consumption growth may be wrong.
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Under this reading, a parameter value that indicates extreme risk aversion in one interpretation of the recursion indicates concerns about *misspecification* in another interpretation of the same recursion.
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{cite:t}`BHS_2009` show that modest amounts of model uncertainty can substitute for large amounts of risk aversion in terms of choices and effects on asset prices.
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This reinterpretation changes the welfare question that asset prices answer.
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Do large risk premia measure the benefits from reducing well-understood aggregate fluctuations, or do they measure benefits from reducing doubts about the model describing consumption growth?
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We begin with a {cite:t}`Hansen_Jagannathan_1991` bound, then specify the statistical environment, lay out four related preference specifications and the connections among them, and finally revisit Tallarini's calibration through the lens of detection-error probabilities.
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To proceed, we begin by describing {cite:t}`Hansen_Jagannathan_1991` bounds, then specify the statistical environment, lay out four related preference specifications and the connections among them, and finally revisit Tallarini's calibration through the lens of detection-error probabilities.
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Along the way, we draw on ideas and techniques from
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