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‎lectures/additive_functionals.md‎

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@@ -1265,9 +1265,11 @@ These probability density functions help us understand mechanics underlying the
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### Multiplicative Martingale as Likelihood Ratio Process
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{doc}`This lecture <likelihood_ratio_process>` studies **likelihood processes** and **likelihood ratio processes**.
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[This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies **likelihood processes**
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and **likelihood ratio processes**.
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A **likelihood ratio process** is a multiplicative martingale with mean unity.
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Likelihood ratio processes exhibit the peculiar property that naturally also appears in {doc}`this lecture <likelihood_ratio_process>`.
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Likelihood ratio processes exhibit the peculiar property that naturally also appears
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[here](https://python.quantecon.org/likelihood_ratio_process.html).
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‎lectures/amss.md‎

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:depth: 2
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```
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**Software Requirement:**
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This lecture requires the use of some older software versions to run. If
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you would like to execute this lecture please download the following
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<a href=_static/downloads/amss_environment.yml download>amss_environment.yml</a>
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file. This specifies the software required and an environment can be
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created using [conda](https://docs.conda.io/en/latest/):
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Open a terminal:
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```{code-block} bash
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conda env create --file amss_environment.yml
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conda activate amss
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```
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In addition to what's in Anaconda, this lecture will need the following libraries:
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```{code-cell} ipython

‎lectures/amss2.md‎

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:depth: 2
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```
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**Software Requirement:**
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This lecture requires the use of some older software versions to run. If
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you would like to execute this lecture please download the following
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<a href=_static/downloads/amss_environment.yml download>amss_environment.yml</a>
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file. This specifies the software required and an environment can be
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created using [conda](https://docs.conda.io/en/latest/):
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Open a terminal:
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```{code-block} bash
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conda env create --file amss_environment.yml
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conda activate amss
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```
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In addition to what's in Anaconda, this lecture will need the following libraries:
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```{code-cell} ipython

‎lectures/amss3.md‎

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:depth: 2
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```
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**Software Requirement:**
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This lecture requires the use of some older software versions to run. If
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you would like to execute this lecture please download the following
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<a href=_static/downloads/amss_environment.yml download>amss_environment.yml</a>
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file. This specifies the software required and an environment can be
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created using [conda](https://docs.conda.io/en/latest/):
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Open a terminal:
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```{code-block} bash
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conda env create --file amss_environment.yml
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conda activate amss
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```
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In addition to what's in Anaconda, this lecture will need the following libraries:
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```{code-cell} ipython

‎lectures/arellano.md‎

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@@ -79,8 +79,8 @@ import numpy as np
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import quantecon as qe
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import random
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from numba import jit, jitclass, int64, float64
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from numba import jit, int64, float64
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from numba.experimental import jitclass
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%matplotlib inline
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```
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‎lectures/black_litterman.md‎

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@@ -197,8 +197,7 @@ sample = excess_return.rvs(T)
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w = np.linalg.solve(δ * Σ_est, μ_est)
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fig, ax = plt.subplots(figsize=(8, 5))
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ax.set_title('Mean-variance portfolio weights recommendation \
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and the market portfolio')
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ax.set_title('Mean-variance portfolio weights recommendation and the market portfolio')
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ax.plot(np.arange(N)+1, w, 'o', c='k', label='$w$ (mean-variance)')
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ax.plot(np.arange(N)+1, w_m, 'o', c='r', label='$w_m$ (market portfolio)')
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ax.vlines(np.arange(N)+1, 0, w, lw=1)
@@ -219,7 +218,7 @@ Black and Litterman's responded to this situation in the following way:
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- They want to continue to allow the customer to express his or her
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risk tolerance by setting $\delta$.
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- Leaving $\Sigma$ at its maximum-likelihood value, they push
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$\mu$ away from its maximum value in a way designed to make
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$\mu$ away from its maximum-likelihood value in a way designed to make
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portfolio choices that are more plausible in terms of conforming to
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what most people actually do.
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@@ -314,8 +313,7 @@ d_m = r_m / σ_m
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μ_m = (d_m * Σ_est @ w_m).reshape(N, 1)
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fig, ax = plt.subplots(figsize=(8, 5))
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ax.set_title(r'Difference between $\hat{\mu}$ (estimate) and \
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$\mu_{BL}$ (market implied)')
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ax.set_title(r'Difference between $\hat{\mu}$ (estimate) and $\mu_{BL}$ (market implied)')
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ax.plot(np.arange(N)+1, μ_est, 'o', c='k', label='$\hat{\mu}$')
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ax.plot(np.arange(N)+1, μ_m, 'o', c='r', label='$\mu_{BL}$')
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ax.vlines(np.arange(N) + 1, μ_m, μ_est, lw=1)
@@ -418,8 +416,7 @@ def BL_plot(τ):
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ax[0].vlines(np.arange(N)+1, μ_m, μ_est, lw=1)
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ax[0].axhline(0, c='k', ls='--')
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ax[0].set(xlim=(0, N+1), xlabel='Assets',
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title=r'Relationship between $\hat{\mu}$, \
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$\mu_{BL}$and$\tilde{\mu}$')
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title=r'Relationship between $\hat{\mu}$, $\mu_{BL}$, and $ \tilde{\mu}$')
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ax[0].xaxis.set_ticks(np.arange(1, N+1, 1))
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ax[0].legend(numpoints=1)
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‎lectures/cattle_cycles.md‎

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@@ -357,15 +357,15 @@ econ2.irf(ts_length=25, shock=shock_demand)
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econ3.irf(ts_length=25, shock=shock_demand)
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fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
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ax1.plot(econ1.c_irf, label='$\\rho=0.6$')
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ax1.plot(econ2.c_irf, label='$\\rho=1$')
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ax1.plot(econ3.c_irf, label='$\\rho=0$')
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ax1.plot(econ1.c_irf, label=r'$\rho=0.6$')
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ax1.plot(econ2.c_irf, label=r'$\rho=1$')
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ax1.plot(econ3.c_irf, label=r'$\rho=0$')
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ax1.set_title('Consumption response to demand shock')
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ax1.legend()
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ax2.plot(econ1.k_irf[:, 0], label='$\\rho=0.6$')
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ax2.plot(econ2.k_irf[:, 0], label='$\\rho=1$')
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ax2.plot(econ3.k_irf[:, 0], label='$\\rho=0$')
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ax2.plot(econ1.k_irf[:, 0], label=r'$\rho=0.6$')
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ax2.plot(econ2.k_irf[:, 0], label=r'$\rho=1$')
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ax2.plot(econ3.k_irf[:, 0], label=r'$\rho=0$')
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ax2.set_title('Breeding stock response to demand shock')
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ax2.legend()
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plt.show()
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fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
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ax1.plot(econ1.k_irf[:, 0], label='Breeding Stock')
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ax1.plot(total1_irf, label='Total Stock')
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ax1.set_title('$\\rho=0.6$')
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ax1.set_title(r'$\rho=0.6$')
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ax2.plot(econ3.k_irf[:, 0], label='Breeding Stock')
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ax2.plot(total3_irf, label='Total Stock')
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ax2.set_title('$\\rho=0$')
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ax2.set_title(r'$\rho=0$')
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plt.show()
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```
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‎lectures/classical_filtering.md‎

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### Implementation
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Here's the code that computes solutions to LQ control and filtering problems using the methods described here and in :doc: lu_tricks.
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Here's the code that computes solutions to LQ control and filtering problems using the methods described here and in {doc}`lu_tricks <lu_tricks>`.
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```{literalinclude} _static/lecture_specific/lu_tricks/control_and_filter.py
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```
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example = LQFilter(d, h, y_m, r=d)
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```
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The Wold representation is computed by example.coefficients_of_c().
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The Wold representation is computed by `example.coeffs_of_c()`.
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Let's check that it "flips roots" as required
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‎lectures/cons_news.md‎

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@@ -123,7 +123,7 @@ y_{t+1} - y_t = a_{t+1} - \beta a_t \quad
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```
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where $\{a_t\}$ is another i.i.d. normally distributed scalar
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process, with means of zero and now variances $\sigma_a^2$.
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process, with means of zero and now variances $\sigma_a^2 > \sigma_\epsilon^2$.
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The two i.i.d. shock variances are related by
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@@ -175,7 +175,7 @@ Using calculations in the {doc}`quantecon lecture <classical_filtering>`, where
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$z \in C$ is a complex variable, the covariance generating
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function $g (z) =
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\sum_{j=-\infty}^\infty g_j z^j$
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of the $\{(y_t - y_{t-1})\}$ process equals
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of the $\{y_t - y_{t-1}\}$ process equals
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$$
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g(z) = \sigma_\epsilon^2 h(z) h(z^{-1}) = \beta^{-2} \sigma_\epsilon^2 > \sigma_\epsilon^2 ,
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The consumer with information associated with representation {eq}`eqn_1`
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responds to each shock $\epsilon_{t+1}$ by leaving his consumption
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unaltered and **saving** all of $a_{t+1}$ in anticipation of the
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permanently increased taxes that he will bear to pay for the addition
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$a_{t+1}$ to his time $t+1$ nonfinancial income.
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unaltered and **saving** all of $\epsilon_{t+1}$ in anticipation of the
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permanently increased taxes that he will bear in order to service the permanent interest payments on the risk-free
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bonds that the government has
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presumably issued to pay for the one-time addition
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$\epsilon_{t+1}$ to his time $t+1$ nonfinancial income.
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The consumer with information associated with representation {eq}`eqn_2`
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responds to a shock $a_{t+1}$ by increasing his consumption by

‎lectures/hs_recursive_models.md‎

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$$
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$$
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mu_t= - \beta^t [\Pi^\prime \Pi\, c_t - \Pi^\prime\, b_t]
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\mu_t= - \beta^t [\Pi^\prime \Pi\, c_t - \Pi^\prime\, b_t]
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$$
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723723
$$
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c_t = - (\Pi^\prime \Pi)^{-1} \beta^{-t} mu_t + (\Pi^\prime \Pi)^{-1}
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c_t = - (\Pi^\prime \Pi)^{-1} \beta^{-t} \mu_t + (\Pi^\prime \Pi)^{-1}
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\Pi^\prime b_t
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$$
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This is called the **Frisch demand function** for consumption.
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We can think of the vector $mu_t$ as playing the role of prices,
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We can think of the vector $\mu_t$ as playing the role of prices,
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up to a common factor, for all dates and states.
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The scale factor is
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b_t) \cdot ( s_t - b_t) + \ell_t^2 ] \bigl| J_0 , \ 0 < \beta < 1
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$$
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*Next steps:** we move on to discuss two closely connected concepts
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**Next steps:** we move on to discuss two closely connected concepts
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- A Planning Problem or Optimal Resource Allocation Problem
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- Competitive Equilibrium
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17541754
$$
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\begin{aligned}
1756-
[\Pi &+ \beta^{1/2} L^{-1} \Lambda (I - \beta^{1/2} L^{-1}
1756+
&[\Pi + \beta^{1/2} L^{-1} \Lambda (I - \beta^{1/2} L^{-1}
17571757
\Delta_h)^{-1} \Theta_h]^\prime [\Pi + \beta^{1/2} L
1758-
\Lambda (I - \beta^{1/2} L \Delta_h)^{-1} \Theta_h]\\
1759-
&= [\hat\Pi + \beta^{1/2} L^{-1} \hat\Lambda
1758+
\Lambda (I - \beta^{1/2} L \Delta_h)^{-1} \Theta_h] \\
1759+
&\quad = [\hat\Pi + \beta^{1/2} L^{-1} \hat\Lambda
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(I - \beta^{1/2} L^{-1} \Delta_h)^{-1} \Theta_h]^\prime
17611761
[\hat\Pi + \beta^{1/2} L \hat\Lambda
1762-
(I - \beta^{1/2} L \Delta_h)^{-1} \Theta_h]\end{aligned}
1762+
(I - \beta^{1/2} L \Delta_h)^{-1} \Theta_h]
1763+
\end{aligned}
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$$
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The factorization identity guarantees that the

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