Markov switching autoregression models

This notebook provides an example of the use of Markov switching models in Statsmodels to replicate a number of results presented in Kim and Nelson (1999). It applies the Hamilton (1989) filter the Kim (1994) smoother.

This is tested against the Markov-switching models from E-views 8, which can be found at http://www.eviews.com/EViews8/ev8ecswitch_n.html#MarkovAR or the Markov-switching models of Stata 14 which can be found at http://www.stata.com/manuals14/tsmswitch.pdf.

In [1]:
%matplotlib inline

import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
import requests
from io import BytesIO

# NBER recessions
from pandas_datareader.data import DataReader
from datetime import datetime
usrec = DataReader('USREC', 'fred', start=datetime(1947, 1, 1), end=datetime(2013, 4, 1))
/build/statsmodels-0.8.0/.pybuild/cpython3_3.7_statsmodels/build/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.
  from pandas.core import datetools
/usr/lib/python3/dist-packages/scipy/_lib/_util.py:18: RuntimeWarning: invalid value encountered in multiply
  out = np.ones(shape, dtype=bool) * value
---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
<ipython-input-1-67b3d8188f6e> in <module>()
      9 
     10 # NBER recessions
---> 11 from pandas_datareader.data import DataReader
     12 from datetime import datetime
     13 usrec = DataReader('USREC', 'fred', start=datetime(1947, 1, 1), end=datetime(2013, 4, 1))

ModuleNotFoundError: No module named 'pandas_datareader'

Hamilton (1989) switching model of GNP

This replicates Hamilton's (1989) seminal paper introducing Markov-switching models. The model is an autoregressive model of order 4 in which the mean of the process switches between two regimes. It can be written:

$$ y_t = \mu_{S_t} + \phi_1 (y_{t-1} - \mu_{S_{t-1}}) + \phi_2 (y_{t-2} - \mu_{S_{t-2}}) + \phi_3 (y_{t-3} - \mu_{S_{t-3}}) + \phi_4 (y_{t-4} - \mu_{S_{t-4}}) + \varepsilon_t $$

Each period, the regime transitions according to the following matrix of transition probabilities:

$$ P(S_t = s_t | S_{t-1} = s_{t-1}) = \begin{bmatrix} p_{00} & p_{10} \\ p_{01} & p_{11} \end{bmatrix} $$

where $p_{ij}$ is the probability of transitioning from regime $i$, to regime $j$.

The model class is MarkovAutoregression in the time-series part of Statsmodels. In order to create the model, we must specify the number of regimes with k_regimes=2, and the order of the autoregression with order=4. The default model also includes switching autoregressive coefficients, so here we also need to specify switching_ar=False to avoid that.

After creation, the model is fit via maximum likelihood estimation. Under the hood, good starting parameters are found using a number of steps of the expectation maximization (EM) algorithm, and a quasi-Newton (BFGS) algorithm is applied to quickly find the maximum.

In [2]:
# Get the RGNP data to replicate Hamilton
from statsmodels.tsa.regime_switching.tests.test_markov_autoregression import rgnp
dta_hamilton = pd.Series(rgnp, index=pd.date_range('1951-04-01', '1984-10-01', freq='QS'))

# Plot the data
dta_hamilton.plot(title='Growth rate of Real GNP', figsize=(12,3))

# Fit the model
mod_hamilton = sm.tsa.MarkovAutoregression(dta_hamilton, k_regimes=2, order=4, switching_ar=False)
res_hamilton = mod_hamilton.fit()
In [3]:
res_hamilton.summary()
/usr/lib/python3/dist-packages/scipy/_lib/_util.py:18: RuntimeWarning: invalid value encountered in multiply
  out = np.ones(shape, dtype=bool) * value
Out[3]:
Markov Switching Model Results
Dep. Variable: y No. Observations: 131
Model: MarkovAutoregression Log Likelihood -181.263
Date: Sat, 21 Nov 2020 AIC 380.527
Time: 06:55:53 BIC 406.404
Sample: 04-01-1952 HQIC 391.042
- 10-01-1984
Covariance Type: approx
Regime 0 parameters
coef std err z P>|z| [0.025 0.975]
const -0.3588 0.265 -1.356 0.175 -0.877 0.160
Regime 1 parameters
coef std err z P>|z| [0.025 0.975]
const 1.1635 0.075 15.614 0.000 1.017 1.310
Non-switching parameters
coef std err z P>|z| [0.025 0.975]
sigma2 0.5914 0.103 5.761 0.000 0.390 0.793
ar.L1 0.0135 0.120 0.112 0.911 -0.222 0.249
ar.L2 -0.0575 0.138 -0.418 0.676 -0.327 0.212
ar.L3 -0.2470 0.107 -2.310 0.021 -0.457 -0.037
ar.L4 -0.2129 0.111 -1.926 0.054 -0.430 0.004
Regime transition parameters
coef std err z P>|z| [0.025 0.975]
p[0->0] 0.7547 0.097 7.819 0.000 0.565 0.944
p[1->0] 0.0959 0.038 2.542 0.011 0.022 0.170

We plot the filtered and smoothed probabilities of a recession. Filtered refers to an estimate of the probability at time $t$ based on data up to and including time $t$ (but excluding time $t+1, ..., T$). Smoothed refers to an estimate of the probability at time $t$ using all the data in the sample.

For reference, the shaded periods represent the NBER recessions.

In [4]:
fig, axes = plt.subplots(2, figsize=(7,7))
ax = axes[0]
ax.plot(res_hamilton.filtered_marginal_probabilities[0])
ax.fill_between(usrec.index, 0, 1, where=usrec['USREC'].values, color='k', alpha=0.1)
ax.set_xlim(dta_hamilton.index[4], dta_hamilton.index[-1])
ax.set(title='Filtered probability of recession')

ax = axes[1]
ax.plot(res_hamilton.smoothed_marginal_probabilities[0])
ax.fill_between(usrec.index, 0, 1, where=usrec['USREC'].values, color='k', alpha=0.1)
ax.set_xlim(dta_hamilton.index[4], dta_hamilton.index[-1])
ax.set(title='Smoothed probability of recession')

fig.tight_layout()
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-4-9b61339d54f4> in <module>()
      2 ax = axes[0]
      3 ax.plot(res_hamilton.filtered_marginal_probabilities[0])
----> 4 ax.fill_between(usrec.index, 0, 1, where=usrec['USREC'].values, color='k', alpha=0.1)
      5 ax.set_xlim(dta_hamilton.index[4], dta_hamilton.index[-1])
      6 ax.set(title='Filtered probability of recession')

NameError: name 'usrec' is not defined

From the estimated transition matrix we can calculate the expected duration of a recession versus an expansion.

In [5]:
print(res_hamilton.expected_durations)
[ 4.07604792 10.42589263]

In this case, it is expected that a recession will last about one year (4 quarters) and an expansion about two and a half years.

Kim, Nelson, and Startz (1998) Three-state Variance Switching

This model demonstrates estimation with regime heteroskedasticity (switching of variances) and no mean effect. The dataset can be reached at http://econ.korea.ac.kr/~cjkim/MARKOV/data/ew_excs.prn.

The model in question is:

$$ \begin{align} y_t & = \varepsilon_t \\ \varepsilon_t & \sim N(0, \sigma_{S_t}^2) \end{align} $$

Since there is no autoregressive component, this model can be fit using the MarkovRegression class. Since there is no mean effect, we specify trend='nc'. There are hypotheized to be three regimes for the switching variances, so we specify k_regimes=3 and switching_variance=True (by default, the variance is assumed to be the same across regimes).

In [6]:
# Get the dataset
ew_excs = requests.get('http://econ.korea.ac.kr/~cjkim/MARKOV/data/ew_excs.prn').content
raw = pd.read_table(BytesIO(ew_excs), header=None, skipfooter=1, engine='python')
raw.index = pd.date_range('1926-01-01', '1995-12-01', freq='MS')

dta_kns = raw.ix[:'1986'] - raw.ix[:'1986'].mean()

# Plot the dataset
dta_kns[0].plot(title='Excess returns', figsize=(12, 3))

# Fit the model
mod_kns = sm.tsa.MarkovRegression(dta_kns, k_regimes=3, trend='nc', switching_variance=True)
res_kns = mod_kns.fit()
---------------------------------------------------------------------------
ConnectionRefusedError                    Traceback (most recent call last)
/usr/lib/python3/dist-packages/urllib3/connection.py in _new_conn(self)
    158             conn = connection.create_connection(
--> 159                 (self._dns_host, self.port), self.timeout, **extra_kw)
    160 

/usr/lib/python3/dist-packages/urllib3/util/connection.py in create_connection(address, timeout, source_address, socket_options)
     79     if err is not None:
---> 80         raise err
     81 

/usr/lib/python3/dist-packages/urllib3/util/connection.py in create_connection(address, timeout, source_address, socket_options)
     69                 sock.bind(source_address)
---> 70             sock.connect(sa)
     71             return sock

ConnectionRefusedError: [Errno 146] Connection refused

During handling of the above exception, another exception occurred:

NewConnectionError                        Traceback (most recent call last)
/usr/lib/python3/dist-packages/urllib3/connectionpool.py in urlopen(self, method, url, body, headers, retries, redirect, assert_same_host, timeout, pool_timeout, release_conn, chunked, body_pos, **response_kw)
    599                                                   body=body, headers=headers,
--> 600                                                   chunked=chunked)
    601 

/usr/lib/python3/dist-packages/urllib3/connectionpool.py in _make_request(self, conn, method, url, timeout, chunked, **httplib_request_kw)
    353         else:
--> 354             conn.request(method, url, **httplib_request_kw)
    355 

/usr/lib/python3.7/http/client.py in request(self, method, url, body, headers, encode_chunked)
   1243         """Send a complete request to the server."""
-> 1244         self._send_request(method, url, body, headers, encode_chunked)
   1245 

/usr/lib/python3.7/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
   1289             body = _encode(body, 'body')
-> 1290         self.endheaders(body, encode_chunked=encode_chunked)
   1291 

/usr/lib/python3.7/http/client.py in endheaders(self, message_body, encode_chunked)
   1238             raise CannotSendHeader()
-> 1239         self._send_output(message_body, encode_chunked=encode_chunked)
   1240 

/usr/lib/python3.7/http/client.py in _send_output(self, message_body, encode_chunked)
   1025         del self._buffer[:]
-> 1026         self.send(msg)
   1027 

/usr/lib/python3.7/http/client.py in send(self, data)
    965             if self.auto_open:
--> 966                 self.connect()
    967             else:

/usr/lib/python3/dist-packages/urllib3/connection.py in connect(self)
    180     def connect(self):
--> 181         conn = self._new_conn()
    182         self._prepare_conn(conn)

/usr/lib/python3/dist-packages/urllib3/connection.py in _new_conn(self)
    167             raise NewConnectionError(
--> 168                 self, "Failed to establish a new connection: %s" % e)
    169 

NewConnectionError: <urllib3.connection.HTTPConnection object at 0xffd29d1668>: Failed to establish a new connection: [Errno 146] Connection refused

During handling of the above exception, another exception occurred:

MaxRetryError                             Traceback (most recent call last)
/usr/lib/python3/dist-packages/requests/adapters.py in send(self, request, stream, timeout, verify, cert, proxies)
    448                     retries=self.max_retries,
--> 449                     timeout=timeout
    450                 )

/usr/lib/python3/dist-packages/urllib3/connectionpool.py in urlopen(self, method, url, body, headers, retries, redirect, assert_same_host, timeout, pool_timeout, release_conn, chunked, body_pos, **response_kw)
    637             retries = retries.increment(method, url, error=e, _pool=self,
--> 638                                         _stacktrace=sys.exc_info()[2])
    639             retries.sleep()

/usr/lib/python3/dist-packages/urllib3/util/retry.py in increment(self, method, url, response, error, _pool, _stacktrace)
    397         if new_retry.is_exhausted():
--> 398             raise MaxRetryError(_pool, url, error or ResponseError(cause))
    399 

MaxRetryError: HTTPConnectionPool(host='127.0.0.1', port=9): Max retries exceeded with url: http://econ.korea.ac.kr/~cjkim/MARKOV/data/ew_excs.prn (Caused by ProxyError('Cannot connect to proxy.', NewConnectionError('<urllib3.connection.HTTPConnection object at 0xffd29d1668>: Failed to establish a new connection: [Errno 146] Connection refused')))

During handling of the above exception, another exception occurred:

ProxyError                                Traceback (most recent call last)
<ipython-input-6-9e237cd253ae> in <module>()
      1 # Get the dataset
----> 2 ew_excs = requests.get('http://econ.korea.ac.kr/~cjkim/MARKOV/data/ew_excs.prn').content
      3 raw = pd.read_table(BytesIO(ew_excs), header=None, skipfooter=1, engine='python')
      4 raw.index = pd.date_range('1926-01-01', '1995-12-01', freq='MS')
      5 

/usr/lib/python3/dist-packages/requests/api.py in get(url, params, **kwargs)
     73 
     74     kwargs.setdefault('allow_redirects', True)
---> 75     return request('get', url, params=params, **kwargs)
     76 
     77 

/usr/lib/python3/dist-packages/requests/api.py in request(method, url, **kwargs)
     58     # cases, and look like a memory leak in others.
     59     with sessions.Session() as session:
---> 60         return session.request(method=method, url=url, **kwargs)
     61 
     62 

/usr/lib/python3/dist-packages/requests/sessions.py in request(self, method, url, params, data, headers, cookies, files, auth, timeout, allow_redirects, proxies, hooks, stream, verify, cert, json)
    531         }
    532         send_kwargs.update(settings)
--> 533         resp = self.send(prep, **send_kwargs)
    534 
    535         return resp

/usr/lib/python3/dist-packages/requests/sessions.py in send(self, request, **kwargs)
    644 
    645         # Send the request
--> 646         r = adapter.send(request, **kwargs)
    647 
    648         # Total elapsed time of the request (approximately)

/usr/lib/python3/dist-packages/requests/adapters.py in send(self, request, stream, timeout, verify, cert, proxies)
    508 
    509             if isinstance(e.reason, _ProxyError):
--> 510                 raise ProxyError(e, request=request)
    511 
    512             if isinstance(e.reason, _SSLError):

ProxyError: HTTPConnectionPool(host='127.0.0.1', port=9): Max retries exceeded with url: http://econ.korea.ac.kr/~cjkim/MARKOV/data/ew_excs.prn (Caused by ProxyError('Cannot connect to proxy.', NewConnectionError('<urllib3.connection.HTTPConnection object at 0xffd29d1668>: Failed to establish a new connection: [Errno 146] Connection refused')))
In [7]:
res_kns.summary()
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-7-e40786696a94> in <module>()
----> 1 res_kns.summary()

NameError: name 'res_kns' is not defined

Below we plot the probabilities of being in each of the regimes; only in a few periods is a high-variance regime probable.

In [8]:
fig, axes = plt.subplots(3, figsize=(10,7))

ax = axes[0]
ax.plot(res_kns.smoothed_marginal_probabilities[0])
ax.set(title='Smoothed probability of a low-variance regime for stock returns')

ax = axes[1]
ax.plot(res_kns.smoothed_marginal_probabilities[1])
ax.set(title='Smoothed probability of a medium-variance regime for stock returns')

ax = axes[2]
ax.plot(res_kns.smoothed_marginal_probabilities[2])
ax.set(title='Smoothed probability of a high-variance regime for stock returns')

fig.tight_layout()
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-8-0cf0c0fb70ab> in <module>()
      2 
      3 ax = axes[0]
----> 4 ax.plot(res_kns.smoothed_marginal_probabilities[0])
      5 ax.set(title='Smoothed probability of a low-variance regime for stock returns')
      6 

NameError: name 'res_kns' is not defined

Filardo (1994) Time-Varying Transition Probabilities

This model demonstrates estimation with time-varying transition probabilities. The dataset can be reached at http://econ.korea.ac.kr/~cjkim/MARKOV/data/filardo.prn.

In the above models we have assumed that the transition probabilities are constant across time. Here we allow the probabilities to change with the state of the economy. Otherwise, the model is the same Markov autoregression of Hamilton (1989).

Each period, the regime now transitions according to the following matrix of time-varying transition probabilities:

$$ P(S_t = s_t | S_{t-1} = s_{t-1}) = \begin{bmatrix} p_{00,t} & p_{10,t} \\ p_{01,t} & p_{11,t} \end{bmatrix} $$

where $p_{ij,t}$ is the probability of transitioning from regime $i$, to regime $j$ in period $t$, and is defined to be:

$$ p_{ij,t} = \frac{\exp\{ x_{t-1}' \beta_{ij} \}}{1 + \exp\{ x_{t-1}' \beta_{ij} \}} $$

Instead of estimating the transition probabilities as part of maximum likelihood, the regression coefficients $\beta_{ij}$ are estimated. These coefficients relate the transition probabilities to a vector of pre-determined or exogenous regressors $x_{t-1}$.

In [9]:
# Get the dataset
filardo = requests.get('http://econ.korea.ac.kr/~cjkim/MARKOV/data/filardo.prn').content
dta_filardo = pd.read_table(BytesIO(filardo), sep=' +', header=None, skipfooter=1, engine='python')
dta_filardo.columns = ['month', 'ip', 'leading']
dta_filardo.index = pd.date_range('1948-01-01', '1991-04-01', freq='MS')

dta_filardo['dlip'] = np.log(dta_filardo['ip']).diff()*100
# Deflated pre-1960 observations by ratio of std. devs.
# See hmt_tvp.opt or Filardo (1994) p. 302
std_ratio = dta_filardo['dlip']['1960-01-01':].std() / dta_filardo['dlip'][:'1959-12-01'].std()
dta_filardo['dlip'][:'1959-12-01'] = dta_filardo['dlip'][:'1959-12-01'] * std_ratio

dta_filardo['dlleading'] = np.log(dta_filardo['leading']).diff()*100
dta_filardo['dmdlleading'] = dta_filardo['dlleading'] - dta_filardo['dlleading'].mean()

# Plot the data
dta_filardo['dlip'].plot(title='Standardized growth rate of industrial production', figsize=(13,3))
plt.figure()
dta_filardo['dmdlleading'].plot(title='Leading indicator', figsize=(13,3));
---------------------------------------------------------------------------
ConnectionRefusedError                    Traceback (most recent call last)
/usr/lib/python3/dist-packages/urllib3/connection.py in _new_conn(self)
    158             conn = connection.create_connection(
--> 159                 (self._dns_host, self.port), self.timeout, **extra_kw)
    160 

/usr/lib/python3/dist-packages/urllib3/util/connection.py in create_connection(address, timeout, source_address, socket_options)
     79     if err is not None:
---> 80         raise err
     81 

/usr/lib/python3/dist-packages/urllib3/util/connection.py in create_connection(address, timeout, source_address, socket_options)
     69                 sock.bind(source_address)
---> 70             sock.connect(sa)
     71             return sock

ConnectionRefusedError: [Errno 146] Connection refused

During handling of the above exception, another exception occurred:

NewConnectionError                        Traceback (most recent call last)
/usr/lib/python3/dist-packages/urllib3/connectionpool.py in urlopen(self, method, url, body, headers, retries, redirect, assert_same_host, timeout, pool_timeout, release_conn, chunked, body_pos, **response_kw)
    599                                                   body=body, headers=headers,
--> 600                                                   chunked=chunked)
    601 

/usr/lib/python3/dist-packages/urllib3/connectionpool.py in _make_request(self, conn, method, url, timeout, chunked, **httplib_request_kw)
    353         else:
--> 354             conn.request(method, url, **httplib_request_kw)
    355 

/usr/lib/python3.7/http/client.py in request(self, method, url, body, headers, encode_chunked)
   1243         """Send a complete request to the server."""
-> 1244         self._send_request(method, url, body, headers, encode_chunked)
   1245 

/usr/lib/python3.7/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
   1289             body = _encode(body, 'body')
-> 1290         self.endheaders(body, encode_chunked=encode_chunked)
   1291 

/usr/lib/python3.7/http/client.py in endheaders(self, message_body, encode_chunked)
   1238             raise CannotSendHeader()
-> 1239         self._send_output(message_body, encode_chunked=encode_chunked)
   1240 

/usr/lib/python3.7/http/client.py in _send_output(self, message_body, encode_chunked)
   1025         del self._buffer[:]
-> 1026         self.send(msg)
   1027 

/usr/lib/python3.7/http/client.py in send(self, data)
    965             if self.auto_open:
--> 966                 self.connect()
    967             else:

/usr/lib/python3/dist-packages/urllib3/connection.py in connect(self)
    180     def connect(self):
--> 181         conn = self._new_conn()
    182         self._prepare_conn(conn)

/usr/lib/python3/dist-packages/urllib3/connection.py in _new_conn(self)
    167             raise NewConnectionError(
--> 168                 self, "Failed to establish a new connection: %s" % e)
    169 

NewConnectionError: <urllib3.connection.HTTPConnection object at 0xffd2a25c88>: Failed to establish a new connection: [Errno 146] Connection refused

During handling of the above exception, another exception occurred:

MaxRetryError                             Traceback (most recent call last)
/usr/lib/python3/dist-packages/requests/adapters.py in send(self, request, stream, timeout, verify, cert, proxies)
    448                     retries=self.max_retries,
--> 449                     timeout=timeout
    450                 )

/usr/lib/python3/dist-packages/urllib3/connectionpool.py in urlopen(self, method, url, body, headers, retries, redirect, assert_same_host, timeout, pool_timeout, release_conn, chunked, body_pos, **response_kw)
    637             retries = retries.increment(method, url, error=e, _pool=self,
--> 638                                         _stacktrace=sys.exc_info()[2])
    639             retries.sleep()

/usr/lib/python3/dist-packages/urllib3/util/retry.py in increment(self, method, url, response, error, _pool, _stacktrace)
    397         if new_retry.is_exhausted():
--> 398             raise MaxRetryError(_pool, url, error or ResponseError(cause))
    399 

MaxRetryError: HTTPConnectionPool(host='127.0.0.1', port=9): Max retries exceeded with url: http://econ.korea.ac.kr/~cjkim/MARKOV/data/filardo.prn (Caused by ProxyError('Cannot connect to proxy.', NewConnectionError('<urllib3.connection.HTTPConnection object at 0xffd2a25c88>: Failed to establish a new connection: [Errno 146] Connection refused')))

During handling of the above exception, another exception occurred:

ProxyError                                Traceback (most recent call last)
<ipython-input-9-e3772af85a7a> in <module>()
      1 # Get the dataset
----> 2 filardo = requests.get('http://econ.korea.ac.kr/~cjkim/MARKOV/data/filardo.prn').content
      3 dta_filardo = pd.read_table(BytesIO(filardo), sep=' +', header=None, skipfooter=1, engine='python')
      4 dta_filardo.columns = ['month', 'ip', 'leading']
      5 dta_filardo.index = pd.date_range('1948-01-01', '1991-04-01', freq='MS')

/usr/lib/python3/dist-packages/requests/api.py in get(url, params, **kwargs)
     73 
     74     kwargs.setdefault('allow_redirects', True)
---> 75     return request('get', url, params=params, **kwargs)
     76 
     77 

/usr/lib/python3/dist-packages/requests/api.py in request(method, url, **kwargs)
     58     # cases, and look like a memory leak in others.
     59     with sessions.Session() as session:
---> 60         return session.request(method=method, url=url, **kwargs)
     61 
     62 

/usr/lib/python3/dist-packages/requests/sessions.py in request(self, method, url, params, data, headers, cookies, files, auth, timeout, allow_redirects, proxies, hooks, stream, verify, cert, json)
    531         }
    532         send_kwargs.update(settings)
--> 533         resp = self.send(prep, **send_kwargs)
    534 
    535         return resp

/usr/lib/python3/dist-packages/requests/sessions.py in send(self, request, **kwargs)
    644 
    645         # Send the request
--> 646         r = adapter.send(request, **kwargs)
    647 
    648         # Total elapsed time of the request (approximately)

/usr/lib/python3/dist-packages/requests/adapters.py in send(self, request, stream, timeout, verify, cert, proxies)
    508 
    509             if isinstance(e.reason, _ProxyError):
--> 510                 raise ProxyError(e, request=request)
    511 
    512             if isinstance(e.reason, _SSLError):

ProxyError: HTTPConnectionPool(host='127.0.0.1', port=9): Max retries exceeded with url: http://econ.korea.ac.kr/~cjkim/MARKOV/data/filardo.prn (Caused by ProxyError('Cannot connect to proxy.', NewConnectionError('<urllib3.connection.HTTPConnection object at 0xffd2a25c88>: Failed to establish a new connection: [Errno 146] Connection refused')))

The time-varying transition probabilities are specified by the exog_tvtp parameter.

Here we demonstrate another feature of model fitting - the use of a random search for MLE starting parameters. Because Markov switching models are often characterized by many local maxima of the likelihood function, performing an initial optimization step can be helpful to find the best parameters.

Below, we specify that 20 random perturbations from the starting parameter vector are examined and the best one used as the actual starting parameters. Because of the random nature of the search, we seed the random number generator beforehand to allow replication of the result.

In [10]:
mod_filardo = sm.tsa.MarkovAutoregression(
    dta_filardo.ix[2:, 'dlip'], k_regimes=2, order=4, switching_ar=False,
    exog_tvtp=sm.add_constant(dta_filardo.ix[1:-1, 'dmdlleading']))

np.random.seed(12345)
res_filardo = mod_filardo.fit(search_reps=20)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-10-7d8be23b1f6f> in <module>()
      1 mod_filardo = sm.tsa.MarkovAutoregression(
----> 2     dta_filardo.ix[2:, 'dlip'], k_regimes=2, order=4, switching_ar=False,
      3     exog_tvtp=sm.add_constant(dta_filardo.ix[1:-1, 'dmdlleading']))
      4 
      5 np.random.seed(12345)

NameError: name 'dta_filardo' is not defined
In [11]:
res_filardo.summary()
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-11-254b3810b2f9> in <module>()
----> 1 res_filardo.summary()

NameError: name 'res_filardo' is not defined

Below we plot the smoothed probability of the economy operating in a low-production state, and again include the NBER recessions for comparison.

In [12]:
fig, ax = plt.subplots(figsize=(12,3))

ax.plot(res_filardo.smoothed_marginal_probabilities[0])
ax.fill_between(usrec.index, 0, 1, where=usrec['USREC'].values, color='gray', alpha=0.2)
ax.set_xlim(dta_filardo.index[6], dta_filardo.index[-1])
ax.set(title='Smoothed probability of a low-production state');
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-12-1a1095e831fe> in <module>()
      1 fig, ax = plt.subplots(figsize=(12,3))
      2 
----> 3 ax.plot(res_filardo.smoothed_marginal_probabilities[0])
      4 ax.fill_between(usrec.index, 0, 1, where=usrec['USREC'].values, color='gray', alpha=0.2)
      5 ax.set_xlim(dta_filardo.index[6], dta_filardo.index[-1])

NameError: name 'res_filardo' is not defined

Using the time-varying transition probabilities, we can see how the expected duration of a low-production state changes over time:

In [13]:
res_filardo.expected_durations[0].plot(
    title='Expected duration of a low-production state', figsize=(12,3));
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-13-a30b5c1ed40c> in <module>()
----> 1 res_filardo.expected_durations[0].plot(
      2     title='Expected duration of a low-production state', figsize=(12,3));

NameError: name 'res_filardo' is not defined

During recessions, the expected duration of a low-production state is much higher than in an expansion.