sd update

Author: HumphreyYang

lectures/stationary_densities.md

@@ -746,10 +746,13 @@ for the look-ahead estimator is very good.
 
 The first exercise helps illustrate this point.
 
+
 ## Exercises
 
 (statd_ex1)=
-### Exercise 1
+```{exercise-start}
+:label: sd_ex1
+```
 
 Consider the simple threshold autoregressive model
 
@@ -799,89 +802,12 @@ density estimator.
 
 If you repeat the simulation you will see that this is consistently the case.
 
-(statd_ex2)=
-### Exercise 2
-
-Replicate the figure on global convergence {ref}`shown above <statd_egs>`.
-
-The densities come from the stochastic growth model treated {ref}`at the start of the lecture <solow_swan>`.
-
-Begin with the code found {ref}`above <stoch_growth>`.
-
-Use the same parameters.
-
-For the four initial distributions, use the shifted beta distributions
-
-```{code-block} python3
-ψ_0 = beta(5, 5, scale=0.5, loc=i*2)
+```{exercise-end}
 ```
 
-(statd_ex3)=
-### Exercise 3
-
-A common way to compare distributions visually is with [boxplots](https://en.wikipedia.org/wiki/Box_plot).
-
-To illustrate, let's generate three artificial data sets and compare them with a boxplot.
-
-The three data sets we will use are:
-
-$$
-\{ X_1, \ldots, X_n \} \sim LN(0, 1), \;\;
-\{ Y_1, \ldots, Y_n \} \sim N(2, 1), \;\;
-\text{ and } \;
-\{ Z_1, \ldots, Z_n \} \sim N(4, 1), \;
-$$
-
-Here is the code and figure:
-
-```{code-cell} python3
-n = 500
-x = np.random.randn(n)        # N(0, 1)
-x = np.exp(x)                 # Map x to lognormal
-y = np.random.randn(n) + 2.0  # N(2, 1)
-z = np.random.randn(n) + 4.0  # N(4, 1)
-
-fig, ax = plt.subplots(figsize=(10, 6.6))
-ax.boxplot([x, y, z])
-ax.set_xticks((1, 2, 3))
-ax.set_ylim(-2, 14)
-ax.set_xticklabels(('$X$', '$Y$', '$Z$'), fontsize=16)
-plt.show()
-```
-
-Each data set is represented by a box, where the top and bottom of the box are the third and first quartiles of the data, and the red line in the center is the median.
-
-The boxes give some indication as to
-
-* the location of probability mass for each sample
-* whether the distribution is right-skewed (as is the lognormal distribution), etc
-
-Now let's put these ideas to use in a simulation.
-
-Consider the threshold autoregressive model in {eq}`statd_tar`.
-
-We know that the distribution of $X_t$ will converge to {eq}`statd_tar_ts` whenever $|\theta| < 1$.
-
-Let's observe this convergence from different initial conditions using
-boxplots.
-
-In particular, the exercise is to generate J boxplot figures, one for each initial condition $X_0$ in
-
-```{code-block} python3
-initial_conditions = np.linspace(8, 0, J)
+```{solution-start} sd_ex1
+:class: dropdown
 ```
-
-For each $X_0$ in this set,
-
-1. Generate $k$ time-series of length $n$, each starting at $X_0$ and obeying {eq}`statd_tar`.
-1. Create a boxplot representing $n$ distributions, where the $t$-th distribution shows the $k$ observations of $X_t$.
-
-Use $\theta = 0.9, n = 20, k = 5000, J = 8$
-
-## Solutions
-
-### Exercise 1
-
 Look-ahead estimation of a TAR stationary density, where the TAR model
 is
 
@@ -925,7 +851,35 @@ ax.legend(loc='upper left')
 plt.show()
 ```
 
-### Exercise 2
+
+```{solution-end}
+```
+
+
+(statd_ex2)=
+```{exercise-start}
+:label: sd_ex2
+```
+
+Replicate the figure on global convergence {ref}`shown above <statd_egs>`.
+
+The densities come from the stochastic growth model treated {ref}`at the start of the lecture <solow_swan>`.
+
+Begin with the code found {ref}`above <stoch_growth>`.
+
+Use the same parameters.
+
+For the four initial distributions, use the shifted beta distributions
+
+```{code-block} python3
+ψ_0 = beta(5, 5, scale=0.5, loc=i*2)
+```
+```{exercise-end}
+```
+
+```{solution-start} sd_ex2
+:class: dropdown
+```
 
 Here's one program that does the job
 
@@ -974,7 +928,79 @@ for i in range(4):
 plt.show()
 ```
 
-### Exercise 3
+```{solution-end}
+```
+
+(statd_ex3)=
+```{exercise-start}
+:label: sd_ex3
+```
+
+A common way to compare distributions visually is with [boxplots](https://en.wikipedia.org/wiki/Box_plot).
+
+To illustrate, let's generate three artificial data sets and compare them with a boxplot.
+
+The three data sets we will use are:
+
+$$
+\{ X_1, \ldots, X_n \} \sim LN(0, 1), \;\;
+\{ Y_1, \ldots, Y_n \} \sim N(2, 1), \;\;
+\text{ and } \;
+\{ Z_1, \ldots, Z_n \} \sim N(4, 1), \;
+$$
+
+Here is the code and figure:
+
+```{code-cell} python3
+n = 500
+x = np.random.randn(n)        # N(0, 1)
+x = np.exp(x)                 # Map x to lognormal
+y = np.random.randn(n) + 2.0  # N(2, 1)
+z = np.random.randn(n) + 4.0  # N(4, 1)
+
+fig, ax = plt.subplots(figsize=(10, 6.6))
+ax.boxplot([x, y, z])
+ax.set_xticks((1, 2, 3))
+ax.set_ylim(-2, 14)
+ax.set_xticklabels(('$X$', '$Y$', '$Z$'), fontsize=16)
+plt.show()
+```
+
+Each data set is represented by a box, where the top and bottom of the box are the third and first quartiles of the data, and the red line in the center is the median.
+
+The boxes give some indication as to
+
+* the location of probability mass for each sample
+* whether the distribution is right-skewed (as is the lognormal distribution), etc
+
+Now let's put these ideas to use in a simulation.
+
+Consider the threshold autoregressive model in {eq}`statd_tar`.
+
+We know that the distribution of $X_t$ will converge to {eq}`statd_tar_ts` whenever $|\theta| < 1$.
+
+Let's observe this convergence from different initial conditions using
+boxplots.
+
+In particular, the exercise is to generate J boxplot figures, one for each initial condition $X_0$ in
+
+```{code-block} python3
+initial_conditions = np.linspace(8, 0, J)
+```
+
+For each $X_0$ in this set,
+
+1. Generate $k$ time-series of length $n$, each starting at $X_0$ and obeying {eq}`statd_tar`.
+1. Create a boxplot representing $n$ distributions, where the $t$-th distribution shows the $k$ observations of $X_t$.
+
+Use $\theta = 0.9, n = 20, k = 5000, J = 8$
+
+```{exercise-end}
+```
+
+```{solution-start} sd_ex3
+:class: dropdown
+```
 
 Here's a possible solution.
 
@@ -984,7 +1010,7 @@ series for one boxplot all at once
 ```{code-cell} python3
 n = 20
 k = 5000
-J = 6
+J = 8
 
 θ = 0.9
 d = np.sqrt(1 - θ**2)
@@ -1008,6 +1034,9 @@ for j in range(J):
 plt.show()
 ```
 
+```{solution-end}
+```
+
 ## Appendix
 
 (statd_appendix)=