## Overview

{cite:t}`Tall2000` showed that a recursive preference specification could match the equity premium and the risk-free rate puzzle simultaneously.

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.

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`.

The Lucas asset pricing model {cite}`Lucas1978` functions as a benchmark that motivates much of this work.

{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.

```{note}

New Keynesians call the consumption Euler equation for a one-period risk-free bond in the Lucas {cite}`Lucas1978` model the **IS curve**.

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.

See {cite}`tobin1992old`.

```

In two classic papers, Lars Peter Hansen and Kenneth Singleton used the method of maximum likelihood

{cite}`hansen1983stochastic` and a generalized method of moments {cite}`hansen1982generalized` to investigate how well Lucas's model fit some post WWII data.

The Hansen-Singleton papers systematically organized evidence about directions in which Lucas's model misfit the data that macroeconomists subsequently called

- an **equity premium** puzzle

- a **risk-free rate** puzzle

```{note}

{cite:t}`MehraPrescott1985` is widely credited for naming the **equity premium** puzzle.

{cite:t}`Weil_1989` is widely credited for naming the **risk-free rate** puzzle.

```

These *puzzles* are just ways of summarizing particular dimensions along which a particular asset pricing model -- such as Lucas's -- fails empirically.

They are thus special cases of specification failures detected by statistical diagnostics constructed earlier by {cite}`hansen1983stochastic` and {cite}`hansen1982generalized`.

Their answer, and the theme of this lecture, is that much of what looks like "risk aversion" can be reinterpreted as **model uncertainty**.

Macro-finance models that purport to resolve such puzzles all do so by changing features of the economic environment assumed by Lucas {cite}`Lucas1978`.

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.

Many important papers have proceeded by altering the *preferences* that Lucas had imputed to a representative agent.

Hansen-Jagannathan bounds are a key tool for evaluating how well such re-specifications do in

correcting those misfits of Lucas's 1978 model.

This lecture begins with a description of the {cite}`Hansen_Jagannathan_1991` machinery.

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.

We'll organize much of this lecture around parts of the paper by Thomas Tallarini {cite}`Tall2000`.

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.

But like any good paper in applied economics, in answering some questions (i.e., resolving some puzzles), Tallarini's paper naturally posed new ones.

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`.

This brings us to the next parts of this lecture.

Lucas's skeptical response to Tallarini's explanation of the two puzzles led

{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.

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**.

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.

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.

{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.

This reinterpretation changes the welfare question that asset prices answer.

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?

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.

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.

Along the way, we draw on ideas and techniques from

Σ_R_inv = np.linalg.inv(Σ_R)

```

## The equity premium and risk-free rate puzzles

## Asset pricing 101

### Pricing kernel and the risk-free rate

\frac{1}{R_t^f} = E_t[m_{t+1}] = E_t \left[\beta\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}\right].

```

### The Hansen--Jagannathan bound

### Hansen--Jagannathan bounds

Let $R_{t+1}^e$ denote the gross return on a risky asset (e.g., the market portfolio) and $R_{t+1}^f$ the gross return on a one-period risk-free bond.

The figure below reproduces Tallarini's key diagnostic.

Because it motivates much of what follow, we show Tallarini's figure before developing the underlying theory.

Because it motivates much of what follows, we show Tallarini's figure before developing the underlying theory.

Closed-form expressions for the Epstein--Zin SDF moments used in the plot are derived in {ref}`Exercise 2 <dov_ex2>`.

plt.show()

```

The next figure makes the "doubts or variability?" question by decomposing the log SDF into two additive components.

The next figure poses the "doubts or variability?" question by decomposing the log SDF into two additive components.

Taking logs of {eq}`bhs_sdf` gives