- file: exchangeable

- file: likelihood_bayes

- file: blackwell_kihlstrom

- file: information_market_equilibrium

- file: mix_model

- file: navy_captain

- file: merging_of_opinions

- file: harrison_kreps

- file: morris_learn

- file: affine_risk_prices

- file: ross_recovery

- file: misspecified_recovery

- caption: Data and Empirics

numbered: true

chapters:

With costs, the consumer chooses quality investment $\theta$ to maximize *net value*.

If quality investment translates into experiment accuracy with diminishing returns — say, accuracy $\phi(\theta) = 1 - e^{-a\theta}$ for a rate parameter $a$ — then the marginal value of information eventually decreases in $\theta$.

If quality investment translates into experiment accuracy with diminishing returns -- say, accuracy $\phi(\theta) = 1 - e^{-a\theta}$ for a rate parameter $a$ -- then the marginal value of information eventually decreases in $\theta$.

With a convex cost $c(\theta) = c \, \theta^2$, the increasing marginal cost eventually overtakes the declining marginal value, producing an interior optimum.

- The jump in $\tau_c$ depresses $\bar{R}$ below $1$, causing a *sharp drop in consumption*.

- After $T = 10$:

- The effects of anticipated distortion are over, and the economy gradually adjusts to the lower capital stock.

- Capital must now rise, requiring *austerity* —consumption plummets after $t = T$, indicated by lower levels of consumption.

- Capital must now rise, requiring *austerity* --consumption plummets after $t = T$, indicated by lower levels of consumption.

- The interest rate gradually declines, and consumption grows at a diminishing rate along the path to the terminal steady-state.

+++

We now explore the impact of an increase in capital taxation in the domestic economy $10$ periods after its announcement at $t = 1$.

Because the change is anticipated, households in both countries adjust immediately—even though the tax does not take effect until period $t = 11$.

Because the change is anticipated, households in both countries adjust immediately--even though the tax does not take effect until period $t = 11$.

```{code-cell} ipython3

shocks_global = {

> But in reality the cycles ... are generally not damped.

> How can the maintenance of the swings be explained?

> ... One way which I believe is particularly fruitful and promising is to study what would become of the solution of a determinate dynamic system if it were exposed to a stream of erratic shocks ...

> Thus, by connecting the two ideas: (1) the continuous solution of a determinate dynamic system and (2) the discontinuous shocks intervening and supplying the energy that may maintain the swings—we get a theoretical setup which seems to furnish a rational interpretation of those movements which we have been accustomed to see in our statistical time data.

> Thus, by connecting the two ideas: (1) the continuous solution of a determinate dynamic system and (2) the discontinuous shocks intervening and supplying the energy that may maintain the swings--we get a theoretical setup which seems to furnish a rational interpretation of those movements which we have been accustomed to see in our statistical time data.

>

> — Ragnar Frisch (1933) {cite}`frisch33`

> -- Ragnar Frisch (1933) {cite}`frisch33`

Chow's main insight is that oscillations in the deterministic system are *neither necessary nor sufficient* for producing "cycles" in the stochastic system.

### The Slutsky connection

Chow connects this result to Slutsky's {cite}`slutsky:1927` finding that moving averages of a random series have recurrent cycles.

Chow connects this result to Slutsky's {cite}`slutsky1937` finding that moving averages of a random series have recurrent cycles.

The VAR(1) model can be written as an infinite moving average:

As $v$ increases, eigenvalues approach the unit circle: oscillations become more persistent in the time domain (left), and the spectral peak becomes sharper in the frequency domain (right).

Complex roots produce a pronounced peak at interior frequencies—the spectral signature of business cycles.

Complex roots produce a pronounced peak at interior frequencies--the spectral signature of business cycles.

```{solution-end}

```

The conditional Euler equation $E_t[M_{t+1}R_{t+1}^i - 1] = 0$ says that the pricing error is unpredictable given *everything* in the agent's time-$t$ information set.

That is a very strong restriction — it says the pricing error is orthogonal to every time-$t$ measurable random variable.

That is a very strong restriction -- it says the pricing error is orthogonal to every time-$t$ measurable random variable.

We cannot use the entire information set in practice, but we can pick any finite collection of time-$t$ observable variables $z_t$ and the orthogonality must still hold.

> rational expectations econometrics. A rational expectations equilibrium is a

> likelihood function. Maximize it.

>

> — An Interview with Thomas J. Sargent {cite}`evans2005interview`

> -- An Interview with Thomas J. Sargent {cite}`evans2005interview`

## Overview

- *Low estimated risk aversion:* The estimated $\hat\alpha$ values (and thus risk aversion $-\hat\alpha$) from the table above are similar to those in {cite:t}`hansen1983stochastic`, who report $\hat\alpha$ between $-0.32$ and $-1.25$.

- *Tiny return predictability:* The unrestricted-VAR $R_R^2$ values are comparable to the 0.02 to 0.06 range in {cite:t}`hansen1983stochastic` — the predictable component of stock returns is small relative to the unpredictable component.

- *Tiny return predictability:* The unrestricted-VAR $R_R^2$ values are comparable to the 0.02 to 0.06 range in {cite:t}`hansen1983stochastic` -- the predictable component of stock returns is small relative to the unpredictable component.

- *Strong rejection for Treasury bills:* The Euler-equation restrictions are decisively rejected for the nominally risk-free Treasury bill return, just as in Table 4 of {cite:t}`hansen1983stochastic`.