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HumphreyYangthomassargent30Copilot
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Add lectures on Kihlstrom's paper, Ross Recovery Theorem and update lectures in probability theory (#859)
* Tom's edits of a new Kihlstrom lecture * Tom's April 20 edits of several lectures * updates * updates * updates * update * Tom's April 21 edits of new lecture * updates * Tom's April 22 edit of misspecified recovery lecture * Add Ross Recovery Theorem lecture - Add lectures/ross_recovery.md: new QuantEcon lecture on Ross (2015) covering the Recovery Theorem, Perron-Frobenius approach, natural vs risk-neutral distributions, tail risk, and efficient markets tests - Add ross_recovery to _toc.yml before misspecified_recovery in asset pricing section - Add 7 new BibTeX entries to quant-econ.bib: BreedenLitzenberger1978, CarrYu2012, BlackScholes1973, Merton1973, CoxRossRubinstein1979, JackwerthRubinstein1996, Weitzman2007 * Add Economic Networks citation and reference PF theorem to section 1.2.3 * updates * updates * update * updates * updates * updates * updates * updates Co-authored-by: Copilot <copilot@github.com> * updates Co-authored-by: Copilot <copilot@github.com> * updates * updates Co-authored-by: Copilot <copilot@github.com> * updates * updates * updates Co-authored-by: Copilot <copilot@github.com> * update Co-authored-by: Copilot <copilot@github.com> --------- Co-authored-by: thomassargent30 <ts43@nyu.edu> Co-authored-by: Copilot <copilot@github.com>
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‎lectures/_static/quant-econ.bib‎

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‎lectures/_toc.yml‎

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- file: exchangeable
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- file: likelihood_bayes
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- file: blackwell_kihlstrom
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- file: information_market_equilibrium
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- file: mix_model
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- file: navy_captain
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- file: merging_of_opinions
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- file: harrison_kreps
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- file: morris_learn
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- file: affine_risk_prices
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- file: ross_recovery
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- file: misspecified_recovery
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- caption: Data and Empirics
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numbered: true
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chapters:

‎lectures/blackwell_kihlstrom.md‎

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With costs, the consumer chooses quality investment $\theta$ to maximize *net value*.
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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 $\theta$.
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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 $\theta$.
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With a convex cost $c(\theta) = c \, \theta^2$, the increasing marginal cost eventually overtakes the declining marginal value, producing an interior optimum.
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‎lectures/cass_fiscal.md‎

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- The jump in $\tau_c$ depresses $\bar{R}$ below $1$, causing a *sharp drop in consumption*.
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- After $T = 10$:
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- The effects of anticipated distortion are over, and the economy gradually adjusts to the lower capital stock.
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- Capital must now rise, requiring *austerity* —consumption plummets after $t = T$, indicated by lower levels of consumption.
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- Capital must now rise, requiring *austerity* --consumption plummets after $t = T$, indicated by lower levels of consumption.
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- The interest rate gradually declines, and consumption grows at a diminishing rate along the path to the terminal steady-state.
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‎lectures/cass_fiscal_2.md‎

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We now explore the impact of an increase in capital taxation in the domestic economy $10$ periods after its announcement at $t = 1$.
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Because the change is anticipated, households in both countries adjust immediately—even though the tax does not take effect until period $t = 11$.
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Because the change is anticipated, households in both countries adjust immediately--even though the tax does not take effect until period $t = 11$.
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```{code-cell} ipython3
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shocks_global = {

‎lectures/chow_business_cycles.md‎

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> But in reality the cycles ... are generally not damped.
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> How can the maintenance of the swings be explained?
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> ... 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 ...
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> 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.
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> 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.
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> — Ragnar Frisch (1933) {cite}`frisch33`
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> -- Ragnar Frisch (1933) {cite}`frisch33`
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Chow's main insight is that oscillations in the deterministic system are *neither necessary nor sufficient* for producing "cycles" in the stochastic system.
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### The Slutsky connection
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Chow connects this result to Slutsky's {cite}`slutsky:1927` finding that moving averages of a random series have recurrent cycles.
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Chow connects this result to Slutsky's {cite}`slutsky1937` finding that moving averages of a random series have recurrent cycles.
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The VAR(1) model can be written as an infinite moving average:
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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).
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Complex roots produce a pronounced peak at interior frequencies—the spectral signature of business cycles.
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Complex roots produce a pronounced peak at interior frequencies--the spectral signature of business cycles.
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```{solution-end}
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```

‎lectures/hansen_singleton_1982.md‎

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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.
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That is a very strong restriction — it says the pricing error is orthogonal to every time-$t$ measurable random variable.
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That is a very strong restriction -- it says the pricing error is orthogonal to every time-$t$ measurable random variable.
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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.
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‎lectures/hansen_singleton_1983.md‎

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> rational expectations econometrics. A rational expectations equilibrium is a
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> likelihood function. Maximize it.
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> — An Interview with Thomas J. Sargent {cite}`evans2005interview`
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> -- An Interview with Thomas J. Sargent {cite}`evans2005interview`
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## Overview
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- *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 $-1.25$.
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- *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.
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- *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.
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- *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`.
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