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123 changes: 2 additions & 121 deletions README.rst
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Expand Up @@ -21,126 +21,7 @@ It's as simple as:
Documentation
-------------

There is no formal documentation (yet).

An small-scale nonlinear New Keynesian model with ZLB is provided `as an example <https://github.com/gboehl/econpizza/blob/master/econpizza/examples/nk.yaml>`_. Here is how to simulate it and plot some nonlinear impulse responses:

.. code-block:: python
import numpy as np
import matplotlib.pyplot as plt
import econpizza as ep
from econpizza import example_nk
# use the NK model again
mod = ep.load(example_nk)
# increase the discount factor by .02 (this is NOT percentage deviation!)
shk = ('e_beta', .02)
# use the stacking method. As above, you could also feed in the initial state instead
x, x_lin, flag = ep.find_path_stacked(mod, shock=shk)
# plotting. x_lin is the linearized first-order solution
for i,v in enumerate(mod['variables']):
plt.figure()
plt.plot(x[:,i])
plt.plot(x_lin[:,i])
plt.title(v)
The impulse responses are the usual dynamics of a nonlinear DSGE.

The `yaml files <https://github.com/gboehl/econpizza/tree/master/econpizza/examples>`_ follow a simple structure:

1. define all variables and shocks
2. provide the nonlinear equations. Note that the dash at the beginning of each line is *not* a minus!
3. provide the parameters and values.
4. optionally provide some steady state values and/or values for initial guesses
5. optionally provide some auxilliary equations that are not directly part of the nonlinear system (see the `yaml for the BH model <https://github.com/gboehl/econpizza/blob/master/econpizza/examples/bh.yaml>`_)


Alternative Boehl-Hommes method
-------------------------------

An alternative method is implemented, similar to the one introduced in Boehl & Hommes (2021), where we use it to solve for chaotic asset price dynamics. It can be understood as a policy function iteration where the initial state is the only fixed grid point and all other grid points are chosen endogenously (as in a "reverse" EGM) to map the expected trajectory.

The main advantage (in terms of robustness) over Fair-Taylor comes from exploiting the property that most determined perfect forsight models be a contraction mapping both, forward and backwards. The model is given by

.. code-block::
f(x_{t-1}, x_t, x_{t+1}) = 0.
We iterate on the expected trajectory itself instead of the policy function. We hence require

.. code-block::
d f(x_{t-1}, x_t, x_{t+1} ) < d x_{t-1},
d f(x_{t-1}, x_t, x_{t+1} ) < d x_{t+1}.
This is also the weakness of the method: not every DSGE model (that is Blanchard-Kahn determined) sense is such backward-and-forward contraction. In most cases the algorithm converges anyways, but convergence is not guaranteed.

.. code-block:: python
import numpy as np
import matplotlib.pyplot as plt
import econpizza as ep
from econpizza import example_nk
# load the example. The steady state is automatically solved for
# example_nk is nothing else but the path to the yaml, hence you could also use `filename = 'path_to/model.yaml'`
mod = ep.load(example_nk)
# get the steady state as an initial state
state = mod['stst'].copy()
# increase the discount factor by one percent
state['beta'] *= 1.02
# simulate the model
x, _, flag = ep.find_path(mod, state.values())
# plotting
for i,v in enumerate(mod['variables']):
plt.figure()
plt.plot(x[:,i])
plt.title(v)
Lets go for a second, numerically more challenging example: the chaotic rational expectations model of Boehl & Hommes (2021)

.. code-block:: python
import numpy as np
import matplotlib.pyplot as plt
import econpizza as ep
from econpizza import example_bh
# parse the yaml
mod = ep.load(example_bh, raise_errors=False)
# B-K conditions will complain because the model is not determined around the steady state. This is not a problem
# choose an interesting initial state
state = np.zeros(len(mod['variables']))
state[:-1] = [.1, .2, 0.]
# solve and simulate. The lower eps is not actually necessary
x, _, flag = ep.find_path(mod, state, T=1000, max_horizon=1000, tol=1e-8)
# plotting
for i,v in enumerate(mod['variables']):
plt.figure()
plt.plot(x[:,i])
plt.title(v)
This will give you:

.. image:: https://github.com/gboehl/econpizza/blob/master/docs/p_and_n.png?raw=true
:width: 800
:alt: Dynamics of prices and fractions
There is some `documentation <https://econpizza.readthedocs.io/en/latest/index.html>`_ out there.


Citation
Expand All @@ -158,7 +39,7 @@ Citation
}
We appreciate citations for **econpizza** because it helps us to find out how people have been using the package and it motivates further work.
I appreciate citations for **econpizza** because it helps me to find out how people have been using the package and it motivates further work.


References
Expand Down
79 changes: 79 additions & 0 deletions docs/boehl_hommes.rst
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Boehl-Hommes method
===================

An alternative method is implemented, similar to the one introduced in Boehl & Hommes (2021), where we use it to solve for chaotic asset price dynamics. It can be understood as a policy function iteration where the initial state is the only fixed grid point and all other grid points are chosen endogenously (as in a "reverse" EGM) to map the expected trajectory.

The main advantage (in terms of robustness) over Fair-Taylor comes from exploiting the property that most determined perfect forsight models be a contraction mapping both, forward and backwards. The model is given by

.. code-block::
f(x_{t-1}, x_t, x_{t+1}) = 0.
We iterate on the expected trajectory itself instead of the policy function. We hence require

.. code-block::
d f(x_{t-1}, x_t, x_{t+1} ) < d x_{t-1},
d f(x_{t-1}, x_t, x_{t+1} ) < d x_{t+1}.
This is also the weakness of the method: not every DSGE model (that is Blanchard-Kahn determined) sense is such backward-and-forward contraction. In most cases the algorithm converges anyways, but convergence is not guaranteed.

.. code-block:: python
import numpy as np
import matplotlib.pyplot as plt
import econpizza as ep
from econpizza import example_nk
# load the example. The steady state is automatically solved for
# example_nk is nothing else but the path to the yaml, hence you could also use `filename = 'path_to/model.yaml'`
mod = ep.load(example_nk)
# get the steady state as an initial state
state = mod['stst'].copy()
# increase the discount factor by one percent
state['beta'] *= 1.02
# simulate the model
x, _, flag = ep.find_path(mod, state.values())
# plotting
for i,v in enumerate(mod['variables']):
plt.figure()
plt.plot(x[:,i])
plt.title(v)
Lets go for a second, numerically more challenging example: the chaotic rational expectations model of Boehl & Hommes (2021)

.. code-block:: python
import numpy as np
import matplotlib.pyplot as plt
import econpizza as ep
from econpizza import example_bh
# parse the yaml
mod = ep.load(example_bh, raise_errors=False)
# B-K conditions will complain because the model is not determined around the steady state. This is not a problem
# choose an interesting initial state
state = np.zeros(len(mod['variables']))
state[:-1] = [.1, .2, 0.]
# solve and simulate. The lower eps is not actually necessary
x, _, flag = ep.find_path(mod, state, T=1000, max_horizon=1000, tol=1e-8)
# plotting
for i,v in enumerate(mod['variables']):
plt.figure()
plt.plot(x[:,i])
plt.title(v)
This will give you:

.. image:: https://github.com/gboehl/econpizza/blob/master/docs/p_and_n.png?raw=true
:width: 800
:alt: Dynamics of prices and fractions
2 changes: 2 additions & 0 deletions docs/index.rst
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Expand Up @@ -9,5 +9,7 @@
:maxdepth: 2

readme
tutorial
boehl_hommes
modules
indices
41 changes: 41 additions & 0 deletions docs/tutorial.rst
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Quickstart
==========

An small-scale nonlinear New Keynesian model with ZLB is provided `as an example <https://github.com/gboehl/econpizza/blob/master/econpizza/examples/nk.yaml>`_. Here is how to simulate it and plot some nonlinear impulse responses:

.. code-block:: python
import numpy as np
import matplotlib.pyplot as plt
import econpizza as ep
from econpizza import example_nk
# use the NK model again
mod = ep.load(example_nk)
# increase the discount factor by .02 (this is NOT percentage deviation!)
shk = ('e_beta', .02)
# use the stacking method. As above, you could also feed in the initial state instead
x, x_lin, flag = ep.find_path_stacked(mod, shock=shk)
# plotting. x_lin is the linearized first-order solution
for i,v in enumerate(mod['variables']):
plt.figure()
plt.plot(x[:,i])
plt.plot(x_lin[:,i])
plt.title(v)
The impulse responses are the usual dynamics of a nonlinear DSGE.

The `yaml files <https://github.com/gboehl/econpizza/tree/master/econpizza/examples>`_ follow a simple structure:

1. define all variables and shocks
2. provide the nonlinear equations. Note that the dash at the beginning of each line is *not* a minus!
3. provide the parameters and values.
4. optionally provide some steady state values and/or values for initial guesses
5. optionally provide some auxilliary equations that are not directly part of the nonlinear system (see the `yaml for the BH model <https://github.com/gboehl/econpizza/blob/master/econpizza/examples/bh.yaml>`_)

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