Merge pull request #75 from QuantEcon/ci_1

Fix link checker

Author: Smit-create

lectures/_static/quant-econ.bib

@@ -2518,8 +2518,6 @@ @article{benhabib_wealth_2019
 	volume = {109},
 	issn = {0002-8282},
 	shorttitle = {Wealth {Distribution} and {Social} {Mobility} in the {US}},
-	url = {https://www.aeaweb.org/articles?id=10.1257/aer.20151684},
-	doi = {10.1257/aer.20151684},
 	abstract = {We quantitatively identify the factors that drive wealth dynamics in the United States and are consistent with its skewed cross-sectional distribution and with social mobility. We concentrate on three critical factors: (i) skewed earnings, (ii) differential saving rates across wealth levels, and (iii) stochastic idiosyncratic returns to wealth. All of these are fundamental for matching both distribution and mobility. The stochastic process for returns which best fits the cross-sectional distribution of wealth and social mobility in the United States shares several statistical properties with those of the returns to wealth uncovered by Fagereng et al. (2017) from tax records in Norway.},
 	language = {en},
 	number = {5},
@@ -2535,7 +2533,6 @@ @article{benhabib_wealth_2019
 
 @article{cobweb_model,
  ISSN = {10711031},
- URL = {http://www.jstor.org/stable/1236509},
  abstract = {In recent years, economists have become much interested in recursive models. This interest stems from a growing need for long-term economic projections and for forecasting the probable effects of economic programs and policies. In a dynamic world, past and present conditions help shape future conditions. Perhaps the simplest recursive model is the two-dimensional "cobweb diagram," discussed by Ezekiel in 1938. The present paper attempts to generalize the simple cobweb model somewhat. It considers some effects of price supports. It discusses multidimensional cobwebs to describe simultaneous adjustments in prices and outputs of a number of commodities. And it allows for time trends in the variables.},
  author = {Frederick V. Waugh},
  journal = {Journal of Farm Economics},
@@ -2556,8 +2553,6 @@ @article{hog_cycle
 number = {4},
 pages = {842-853},
 doi = {https://doi.org/10.2307/1235116},
-url = {https://onlinelibrary.wiley.com/doi/abs/10.2307/1235116},
-eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.2307/1235116},
 abstract = {Abstract A surprisingly regular four year cycle in hogs has become apparent in the past ten years. This regularity presents an unusual opportunity to study the mechanism of the cycle because it suggests the cycle may be inherent within the industry rather than the result of lagged responses to outside influences. The cobweb theorem is often mentioned as a theoretical tool for explaining the hog cycle, although a two year cycle is usually predicted. When the nature of the hog industry is examined, certain factors become apparent which enable the cobweb theorem to serve as a theoretical basis for the present four year cycle.},
 year = {1960}
 }

lectures/markov_chains.md

@@ -9,7 +9,7 @@ kernelspec:
   name: python3
 ---
 
-# Markov Chains 
+# Markov Chains
 
 In addition to what's in Anaconda, this lecture will need the following libraries:
 
@@ -24,7 +24,7 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 Markov chains are a standard way to model time series with some
 dependence between observations.
 
-For example, 
+For example,
 
 * inflation next year depends on inflation this year
 * unemployment next month depends on unemployment this month
@@ -34,7 +34,7 @@ Markov chains are one of the workhorse models of economics and finance.
 The theory of Markov chains is beautiful and insightful, which is another
 excellent reason to study them.
 
-In this introductory lecture, we will 
+In this introductory lecture, we will
 
 * review some of the key ideas from the theory of Markov chains and
 * show how Markov chains appear in some economic applications.
@@ -53,7 +53,7 @@ import numpy as np
 In this section we provide the basic definitions and some elementary examples.
 
 (finite_dp_stoch_mat)=
-### Stochastic Matrices 
+### Stochastic Matrices
 
 Recall that a **probability mass function** over $n$ possible outcomes is a
 nonnegative $n$-vector $p$ that sums to one.
@@ -93,7 +93,7 @@ Therefore $P^{k+1} \mathbf 1 = P P^k \mathbf 1 = P \mathbf 1 = \mathbf 1$
 The proof is done.
 
 
-### Markov Chains 
+### Markov Chains
 
 Now we can introduce Markov chains.
 
@@ -126,7 +126,7 @@ dot.edge("mr", "ng", label="0.145")
 dot.edge("mr", "mr", label="0.778")
 dot.edge("mr", "sr", label="0.077")
 dot.edge("sr", "mr", label="0.508")
-    
+
 dot.edge("sr", "sr", label="0.492")
 dot
 ```
@@ -199,7 +199,7 @@ More generally, for any $i,j$ between 0 and 2, we have
 $$
 \begin{aligned}
     P(i,j)
-    & = \mathbb P\{X_{t+1} = j \,|\, X_t = i\} 
+    & = \mathbb P\{X_{t+1} = j \,|\, X_t = i\}
     \\
     & = \text{ probability of transitioning from state $i$ to state $j$ in one month}
 \end{aligned}
@@ -234,11 +234,11 @@ For example,
 
 $$
 \begin{aligned}
-    P(0,1) 
-        & = 
+    P(0,1)
+        & =
         \text{ probability of transitioning from state $0$ to state $1$ in one month}
         \\
-        & = 
+        & =
         \text{ probability finding a job next month}
         \\
         & = \alpha
@@ -303,7 +303,7 @@ $$
 Going the other way, if we take a stochastic matrix $P$, we can generate a Markov
 chain $\{X_t\}$ as follows:
 
-* draw $X_0$ from a marginal distribution $\psi$ 
+* draw $X_0$ from a marginal distribution $\psi$
 * for each $t = 0, 1, \ldots$, draw $X_{t+1}$ from $P(X_t,\cdot)$
 
 By construction, the resulting process satisfies {eq}`mpp`.
@@ -458,7 +458,7 @@ mc.simulate_indices(ts_length=4)
 ```
 
 (mc_md)=
-## Marginal Distributions 
+## Marginal Distributions
 
 Suppose that
 
@@ -827,7 +827,7 @@ mc.stationary_distributions  # Show all stationary distributions
 ```
 
 (ergodicity)=
-## Ergodicity 
+## Ergodicity
 
 Under irreducibility, yet another important result obtains:
 
@@ -900,7 +900,7 @@ Another example is Hamilton {cite}`Hamilton2005` dynamics {ref}`discussed above
 
 The diagram of the Markov chain shows that it is **irreducible**.
 
-Therefore, we can see the sample path averages for each state (the fraction of time spent in each state) converges to the stationary distribution regardless of the starting state 
+Therefore, we can see the sample path averages for each state (the fraction of time spent in each state) converges to the stationary distribution regardless of the starting state
 
 ```{code-cell} ipython3
 P = np.array([[0.971, 0.029, 0.000],
@@ -915,7 +915,7 @@ plt.subplots_adjust(wspace=0.35)
 for i in range(n_state):
     axes[i].grid()
     axes[i].set_ylim(ψ_star[i]-0.2, ψ_star[i]+0.2)
-    axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black', 
+    axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black',
                     label = fr'$\psi^*(X={i})$')
     axes[i].set_xlabel('t')
     axes[i].set_ylabel(fr'average time spent at X={i}')
@@ -962,7 +962,7 @@ dot
 
 As you might notice, unlike other Markov chains we have seen before, it has a periodic cycle.
 
-This is formally called [periodicity](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/16:_Markov_Processes/16.05:_Periodicity_of_Discrete-Time_Chains#:~:text=A%20state%20in%20a%20discrete,limiting%20behavior%20of%20the%20chain.). 
+This is formally called [periodicity](https://www.randomservices.org/random/markov/Periodicity.html).
 
 We will not go into the details of periodicity.
 
@@ -979,7 +979,7 @@ fig, axes = plt.subplots(nrows=1, ncols=n_state)
 for i in range(n_state):
     axes[i].grid()
     axes[i].set_ylim(0.45, 0.55)
-    axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black', 
+    axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black',
                     label = fr'$\psi^*(X={i})$')
     axes[i].set_xlabel('t')
     axes[i].set_ylabel(fr'average time spent at X={i}')
@@ -1016,7 +1016,7 @@ strictly positive, then $P$ has only one stationary distribution $\psi^*$ and
 $$
     \psi_0 P^t \to \psi
     \quad \text{as } t \to \infty
-$$    
+$$
 
 
 (See, for example, {cite}`haggstrom2002finite`. Our assumptions imply that $P$
@@ -1090,24 +1090,24 @@ plt.subplots_adjust(wspace=0.35)
 x0s = np.ones((n, n_state))
 for i in range(n):
     draws = np.random.randint(1, 10_000_000, size=n_state)
-    
+
     # Scale them so that they add up into 1
     x0s[i,:] = np.array(draws/sum(draws))
 
 # Loop through many initial values
 for x0 in x0s:
     x = x0
     X = np.zeros((n,n_state))
-    
+
     # Obtain and plot distributions at each state
     for t in range(0, n):
-        x =  x @ P 
+        x =  x @ P
         X[t] = x
     for i in range(n_state):
         axes[i].plot(range(0, n), X[:,i], alpha=0.3)
-    
+
 for i in range(n_state):
-    axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black', 
+    axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black',
                     label = fr'$\psi^*(X={i})$')
     axes[i].set_xlabel('t')
     axes[i].set_ylabel(fr'$\psi(X={i})$')
@@ -1139,13 +1139,13 @@ for i in range(n):
 for x0 in x0s:
     x = x0
     X = np.zeros((n,n_state))
-    
+
     for t in range(0, n):
         x = x @ P
         X[t] = x
     for i in range(n_state):
         axes[i].plot(range(20, n), X[20:,i], alpha=0.3)
-    
+
 for i in range(n_state):
     axes[i].axhline(ψ_star[i], linestyle='dashed', lw=2, color = 'black', label = fr'$\psi^* (X={i})$')
     axes[i].set_xlabel('t')
@@ -1260,7 +1260,7 @@ TODO -- connect to the Neumann series lemma (Maanasee)
 
 ## Exercises
 
-````{exercise} 
+````{exercise}
 :label: mc_ex1
 
 Benhabib el al. {cite}`benhabib_wealth_2019` estimated that the transition matrix for social mobility as the following
@@ -1291,7 +1291,7 @@ P_B = np.array(P_B)
 codes_B =  ( '1','2','3','4','5','6','7','8')
 ```
 
-In this exercise, 
+In this exercise,
 
 1. show this process is asymptotically stationary and calculate the stationary distribution using simulations.
 
@@ -1323,7 +1323,7 @@ codes_B =  ( '1','2','3','4','5','6','7','8')
 np.linalg.matrix_power(P_B, 10)
 ```
 
-We find rows transition matrix converge to the stationary distribution 
+We find rows transition matrix converge to the stationary distribution
 
 ```{code-cell} ipython3
 mc = qe.MarkovChain(P_B)
@@ -1360,7 +1360,7 @@ We can see that the time spent at each state quickly converges to the stationary
 ```
 
 
-```{exercise} 
+```{exercise}
 :label: mc_ex2
 
 According to the discussion {ref}`above <mc_eg1-2>`, if a worker's employment dynamics obey the stochastic matrix
@@ -1443,7 +1443,7 @@ plt.show()
 ```{solution-end}
 ```
 
-```{exercise} 
+```{exercise}
 :label: mc_ex3
 
 In `quantecon` library, irreducibility is tested by checking whether the chain forms a [strongly connected component](https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.components.is_strongly_connected.html).