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.
[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))
---------------------------------------------------------------------------
ModuleNotFoundError Traceback (most recent call last)
<ipython-input-1-3fd38b6bfa68> 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:
\[\begin{split} P(S_t = s_t | S_{t-1} = s_{t-1}) =
\begin{bmatrix}
p_{00} & p_{10} \\
p_{01} & p_{11}
\end{bmatrix}\end{split}\]
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.
[2]:
# Get the RGNP data to replicate Hamilton
dta = pd.read_stata('https://www.stata-press.com/data/r14/rgnp.dta').iloc[1:]
dta.index = pd.DatetimeIndex(dta.date, freq='QS')
dta_hamilton = dta.rgnp
# 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()
---------------------------------------------------------------------------
ConnectionRefusedError Traceback (most recent call last)
/usr/lib/python3.7/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1318 h.request(req.get_method(), req.selector, req.data, headers,
-> 1319 encode_chunked=req.has_header('Transfer-encoding'))
1320 except OSError as err: # timeout error
/usr/lib/python3.7/http/client.py in request(self, method, url, body, headers, encode_chunked)
1251 """Send a complete request to the server."""
-> 1252 self._send_request(method, url, body, headers, encode_chunked)
1253
/usr/lib/python3.7/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
1297 body = _encode(body, 'body')
-> 1298 self.endheaders(body, encode_chunked=encode_chunked)
1299
/usr/lib/python3.7/http/client.py in endheaders(self, message_body, encode_chunked)
1246 raise CannotSendHeader()
-> 1247 self._send_output(message_body, encode_chunked=encode_chunked)
1248
/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/python3.7/http/client.py in connect(self)
1413
-> 1414 super().connect()
1415
/usr/lib/python3.7/http/client.py in connect(self)
937 self.sock = self._create_connection(
--> 938 (self.host,self.port), self.timeout, self.source_address)
939 self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)
/usr/lib/python3.7/socket.py in create_connection(address, timeout, source_address)
727 try:
--> 728 raise err
729 finally:
/usr/lib/python3.7/socket.py in create_connection(address, timeout, source_address)
715 sock.bind(source_address)
--> 716 sock.connect(sa)
717 # Break explicitly a reference cycle
ConnectionRefusedError: [Errno 111] Connection refused
During handling of the above exception, another exception occurred:
URLError Traceback (most recent call last)
<ipython-input-2-7ccef5c6c7dd> in <module>
1 # Get the RGNP data to replicate Hamilton
----> 2 dta = pd.read_stata('https://www.stata-press.com/data/r14/rgnp.dta').iloc[1:]
3 dta.index = pd.DatetimeIndex(dta.date, freq='QS')
4 dta_hamilton = dta.rgnp
5
/usr/lib/python3/dist-packages/pandas/util/_decorators.py in wrapper(*args, **kwargs)
206 else:
207 kwargs[new_arg_name] = new_arg_value
--> 208 return func(*args, **kwargs)
209
210 return wrapper
/usr/lib/python3/dist-packages/pandas/util/_decorators.py in wrapper(*args, **kwargs)
206 else:
207 kwargs[new_arg_name] = new_arg_value
--> 208 return func(*args, **kwargs)
209
210 return wrapper
/usr/lib/python3/dist-packages/pandas/io/stata.py in read_stata(filepath_or_buffer, convert_dates, convert_categoricals, encoding, index_col, convert_missing, preserve_dtypes, columns, order_categoricals, chunksize, iterator)
225 columns=columns,
226 order_categoricals=order_categoricals,
--> 227 chunksize=chunksize,
228 )
229
/usr/lib/python3/dist-packages/pandas/util/_decorators.py in wrapper(*args, **kwargs)
206 else:
207 kwargs[new_arg_name] = new_arg_value
--> 208 return func(*args, **kwargs)
209
210 return wrapper
/usr/lib/python3/dist-packages/pandas/util/_decorators.py in wrapper(*args, **kwargs)
206 else:
207 kwargs[new_arg_name] = new_arg_value
--> 208 return func(*args, **kwargs)
209
210 return wrapper
/usr/lib/python3/dist-packages/pandas/io/stata.py in __init__(self, path_or_buf, convert_dates, convert_categoricals, index_col, convert_missing, preserve_dtypes, columns, order_categoricals, encoding, chunksize)
1088 path_or_buf = _stringify_path(path_or_buf)
1089 if isinstance(path_or_buf, str):
-> 1090 path_or_buf, encoding, _, should_close = get_filepath_or_buffer(path_or_buf)
1091
1092 if isinstance(path_or_buf, (str, bytes)):
/usr/lib/python3/dist-packages/pandas/io/common.py in get_filepath_or_buffer(filepath_or_buffer, encoding, compression, mode)
194
195 if _is_url(filepath_or_buffer):
--> 196 req = urlopen(filepath_or_buffer)
197 content_encoding = req.headers.get("Content-Encoding", None)
198 if content_encoding == "gzip":
/usr/lib/python3.7/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
220 else:
221 opener = _opener
--> 222 return opener.open(url, data, timeout)
223
224 def install_opener(opener):
/usr/lib/python3.7/urllib/request.py in open(self, fullurl, data, timeout)
523 req = meth(req)
524
--> 525 response = self._open(req, data)
526
527 # post-process response
/usr/lib/python3.7/urllib/request.py in _open(self, req, data)
541 protocol = req.type
542 result = self._call_chain(self.handle_open, protocol, protocol +
--> 543 '_open', req)
544 if result:
545 return result
/usr/lib/python3.7/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
501 for handler in handlers:
502 func = getattr(handler, meth_name)
--> 503 result = func(*args)
504 if result is not None:
505 return result
/usr/lib/python3.7/urllib/request.py in https_open(self, req)
1360 def https_open(self, req):
1361 return self.do_open(http.client.HTTPSConnection, req,
-> 1362 context=self._context, check_hostname=self._check_hostname)
1363
1364 https_request = AbstractHTTPHandler.do_request_
/usr/lib/python3.7/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1319 encode_chunked=req.has_header('Transfer-encoding'))
1320 except OSError as err: # timeout error
-> 1321 raise URLError(err)
1322 r = h.getresponse()
1323 except:
URLError: <urlopen error [Errno 111] Connection refused>
[3]:
res_hamilton.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-3-37be29eb8e24> in <module>
----> 1 res_hamilton.summary()
NameError: name 'res_hamilton' is not defined
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.
[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>
1 fig, axes = plt.subplots(2, figsize=(7,7))
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])
NameError: name 'res_hamilton' is not defined
From the estimated transition matrix we can calculate the expected duration of a recession versus an expansion.
[5]:
print(res_hamilton.expected_durations)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-5-517c0129988b> in <module>
----> 1 print(res_hamilton.expected_durations)
NameError: name 'res_hamilton' is not defined
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:
\[\begin{split}\begin{align}
y_t & = \varepsilon_t \\
\varepsilon_t & \sim N(0, \sigma_{S_t}^2)
\end{align}\end{split}\]
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 hypothesized 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).
[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.loc[:'1986'] - raw.loc[:'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()
---------------------------------------------------------------------------
ProxySchemeUnknown Traceback (most recent call last)
<ipython-input-6-aec9dc729724> 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)
410
411 try:
--> 412 conn = self.get_connection(request.url, proxies)
413 except LocationValueError as e:
414 raise InvalidURL(e, request=request)
/usr/lib/python3/dist-packages/requests/adapters.py in get_connection(self, url, proxies)
307 raise InvalidProxyURL("Please check proxy URL. It is malformed"
308 " and could be missing the host.")
--> 309 proxy_manager = self.proxy_manager_for(proxy)
310 conn = proxy_manager.connection_from_url(url)
311 else:
/usr/lib/python3/dist-packages/requests/adapters.py in proxy_manager_for(self, proxy, **proxy_kwargs)
197 maxsize=self._pool_maxsize,
198 block=self._pool_block,
--> 199 **proxy_kwargs)
200
201 return manager
/usr/lib/python3/dist-packages/urllib3/poolmanager.py in proxy_from_url(url, **kw)
468
469 def proxy_from_url(url, **kw):
--> 470 return ProxyManager(proxy_url=url, **kw)
/usr/lib/python3/dist-packages/urllib3/poolmanager.py in __init__(self, proxy_url, num_pools, headers, proxy_headers, **connection_pool_kw)
418
419 if proxy.scheme not in ("http", "https"):
--> 420 raise ProxySchemeUnknown(proxy.scheme)
421
422 self.proxy = proxy
ProxySchemeUnknown: Not supported proxy scheme None
[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.
[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:
\[\begin{split} P(S_t = s_t | S_{t-1} = s_{t-1}) =
\begin{bmatrix}
p_{00,t} & p_{10,t} \\
p_{01,t} & p_{11,t}
\end{bmatrix}\end{split}\]
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 \(x_{t-1}\).
[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));
---------------------------------------------------------------------------
ProxySchemeUnknown 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)
410
411 try:
--> 412 conn = self.get_connection(request.url, proxies)
413 except LocationValueError as e:
414 raise InvalidURL(e, request=request)
/usr/lib/python3/dist-packages/requests/adapters.py in get_connection(self, url, proxies)
307 raise InvalidProxyURL("Please check proxy URL. It is malformed"
308 " and could be missing the host.")
--> 309 proxy_manager = self.proxy_manager_for(proxy)
310 conn = proxy_manager.connection_from_url(url)
311 else:
/usr/lib/python3/dist-packages/requests/adapters.py in proxy_manager_for(self, proxy, **proxy_kwargs)
197 maxsize=self._pool_maxsize,
198 block=self._pool_block,
--> 199 **proxy_kwargs)
200
201 return manager
/usr/lib/python3/dist-packages/urllib3/poolmanager.py in proxy_from_url(url, **kw)
468
469 def proxy_from_url(url, **kw):
--> 470 return ProxyManager(proxy_url=url, **kw)
/usr/lib/python3/dist-packages/urllib3/poolmanager.py in __init__(self, proxy_url, num_pools, headers, proxy_headers, **connection_pool_kw)
418
419 if proxy.scheme not in ("http", "https"):
--> 420 raise ProxySchemeUnknown(proxy.scheme)
421
422 self.proxy = proxy
ProxySchemeUnknown: Not supported proxy scheme None
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.
[10]:
mod_filardo = sm.tsa.MarkovAutoregression(
dta_filardo.iloc[2:]['dlip'], k_regimes=2, order=4, switching_ar=False,
exog_tvtp=sm.add_constant(dta_filardo.iloc[1:-1]['dmdlleading']))
np.random.seed(12345)
res_filardo = mod_filardo.fit(search_reps=20)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-10-b0ee7e59fc01> in <module>
1 mod_filardo = sm.tsa.MarkovAutoregression(
----> 2 dta_filardo.iloc[2:]['dlip'], k_regimes=2, order=4, switching_ar=False,
3 exog_tvtp=sm.add_constant(dta_filardo.iloc[1:-1]['dmdlleading']))
4
5 np.random.seed(12345)
NameError: name 'dta_filardo' is not defined
[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.
[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:
[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.
