Updated Markov.ipynb notebook

Notebooks/Markov/Markov.ipynb

@@ -488,7 +488,7 @@
  "\n",
  "In exercise (1a) of [Problem Set 4](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Assignments/PS4/PS4.pdf), you will simulate this continuous AR(1) process.\n",
  "\n",
- "This AR(1) process can be approximated by an $N$-state Markov process. Exercises (1b) through (1f) of [Problem Set 4](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Assignments/PS4/PS4.pdf) have you estimate the values of a 5-state Markov process to closely approximate a calibration of the AR(1) process above. The following three methods describe the most common methods for approximating a continuous AR(1) process with a discrete Markov process. All three methods are coded up in the Python module [`ar1_approx.py`]()."
+ "This AR(1) process can be approximated by an $N$-state Markov process. Exercises (1b) through (1f) of [Problem Set 4](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Assignments/PS4/PS4.pdf) have you estimate the values of a 5-state Markov process to closely approximate a calibration of the AR(1) process above. The following three methods describe the most common methods for approximating a continuous AR(1) process with a discrete Markov process. All three methods are coded up in the Python module [`ar1_approx.py`](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Notebooks/Markov/ar1_approx.py)."
  ]
  },
  {
@@ -501,19 +501,23 @@
  " z_{t+1}\\sim N\\bigl(\\rho z_t + (1-\\rho)\\mu, \\sigma\\bigr)\n",
  "\\end{equation*}\n",
  "\n",
- "The code for this approach is in the Python module [`ar1_approx.addacooper()`]()."
+ "The code for this approach is in the Python module [`ar1_approx.addacooper()`](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Notebooks/Markov/ar1_approx.py)."
  ]
  },
  {
  "cell_type": "markdown",
  "metadata": {},
  "source": [
  "#### 6.1.2. Tauchen and Hussey method\n",
- "Tauchen (1986) showed that a discrete Markov process can approximate a continuous AR(1) process. Tauchen and Hussey (1991) uses Gaussian quadrature to choose an efficient number of nodes and weights to most closely approximate the integral required to estimate the appropriate transition probabilities and bin cutoffs. The code for this approach is in the Python module [`ar1_approx.tauchenhussey()`]().\n",
- "\n",
- "\n",
+ "Tauchen (1986) showed that a discrete Markov process can approximate a continuous AR(1) process. Tauchen and Hussey (1991) uses Gaussian quadrature to choose an efficient number of nodes and weights to most closely approximate the integral required to estimate the appropriate transition probabilities and bin cutoffs. The code for this approach is in the Python module [`ar1_approx.tauchenhussey()`](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Notebooks/Markov/ar1_approx.py)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
  "#### 6.1.3. Rouwenhurst method\n",
- "The method of Tauchen and Hussey (1991) introduces increased approximation error for AR(1) processes that have a persistence parameter that is close to 1 in absolute value $1 - \\abs{\\rho}<\\epsilon$, where $\\epsilon$ could be as large as 0.05. This is true for some stochastic processes used in macroeconomic models. Rouwenhorst (1995) introduces a method for calculating the Markov transition matrix that is robust to high-persistence AR(1) processes. The code for this approach is in the Python module [`ar1_approx.rouwen()`]()."
+ "The method of Tauchen and Hussey (1991) introduces increased approximation error for AR(1) processes that have a persistence parameter that is close to 1 in absolute value $1 - |\\rho|<\\epsilon$, where $\\epsilon$ could be as large as 0.05. This is true for some stochastic processes used in macroeconomic models. Rouwenhorst (1995) introduces a method for calculating the Markov transition matrix that is robust to high-persistence AR(1) processes. The code for this approach is in the Python module [`ar1_approx.rouwen()`](https://github.com/UC-MACSS/persp-research-econ_Spr19/blob/master/Notebooks/Markov/ar1_approx.py)."
  ]
  },
  {