Add lectures on Kihlstrom's paper, Ross Recovery Theorem and update lectures in probability theory

lectures/_toc.yml

@@ -43,6 +43,7 @@ parts:
   - file: exchangeable
   - file: likelihood_bayes
   - file: blackwell_kihlstrom
+  - file: information_market_equilibrium
   - file: mix_model
   - file: navy_captain
   - file: merging_of_opinions
@@ -141,6 +142,8 @@ parts:
   - file: harrison_kreps
   - file: morris_learn
   - file: affine_risk_prices
+  - file: ross_recovery
+  - file: misspecified_recovery
 - caption: Data and Empirics
   numbered: true
   chapters:

lectures/blackwell_kihlstrom.md

@@ -962,7 +962,7 @@ The Blackwell order says that, absent costs, more information is always better f
 
 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.
 

lectures/cass_fiscal.md

@@ -1133,7 +1133,7 @@ and capital stock across time:
     - 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.
 
 +++

lectures/cass_fiscal_2.md

@@ -498,7 +498,7 @@ This means that foreign households begin repaying part of their external debt by
 
 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 = {

lectures/chow_business_cycles.md

@@ -351,9 +351,9 @@ The second equation is the discrete Lyapunov equation for $\Gamma_0$.
 > 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.
 
@@ -844,7 +844,7 @@ The peak appears at $\omega/\pi \approx 0.10$, which corresponds to a cycle leng
 
 ### 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:
 
@@ -1408,7 +1408,7 @@ plt.show()
 
 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}
 ```

lectures/hansen_singleton_1982.md

@@ -225,7 +225,7 @@ The vector $z_t$ plays the role of **instruments**.
 
 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.
 

lectures/hansen_singleton_1983.md

@@ -36,7 +36,7 @@ kernelspec:
 > 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
 
@@ -1869,7 +1869,7 @@ Our estimates reproduce the pattern that {cite:t}`MehraPrescott1985` later calle
 
 - *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`.