SARIMAX: Model selection, missing data

The example mirrors Durbin and Koopman (2012), Chapter 8.4 in application of Box-Jenkins methodology to fit ARMA models. The novel feature is the ability of the model to work on datasets with missing values.

[1]:
%matplotlib inline
[2]:
import numpy as np
import pandas as pd
from scipy.stats import norm
import statsmodels.api as sm
import matplotlib.pyplot as plt
[3]:
import requests
from io import BytesIO
from zipfile import ZipFile

# Download the dataset
dk = requests.get('http://www.ssfpack.com/files/DK-data.zip').content
f = BytesIO(dk)
zipped = ZipFile(f)
df = pd.read_table(
    BytesIO(zipped.read('internet.dat')),
    skiprows=1, header=None, sep='\s+', engine='python',
    names=['internet','dinternet']
)
---------------------------------------------------------------------------
ConnectionRefusedError                    Traceback (most recent call last)
/usr/lib/python3/dist-packages/urllib3/connection.py in _new_conn(self)
    157         try:
--> 158             conn = connection.create_connection(
    159                 (self._dns_host, self.port), self.timeout, **extra_kw)

/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 111] 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)
    596             # Make the request on the httplib connection object.
--> 597             httplib_response = self._make_request(conn, method, url,
    598                                                   timeout=timeout_obj,

/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/python3.8/http/client.py in request(self, method, url, body, headers, encode_chunked)
   1229         """Send a complete request to the server."""
-> 1230         self._send_request(method, url, body, headers, encode_chunked)
   1231

/usr/lib/python3.8/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
   1275             body = _encode(body, 'body')
-> 1276         self.endheaders(body, encode_chunked=encode_chunked)
   1277

/usr/lib/python3.8/http/client.py in endheaders(self, message_body, encode_chunked)
   1224             raise CannotSendHeader()
-> 1225         self._send_output(message_body, encode_chunked=encode_chunked)
   1226

/usr/lib/python3.8/http/client.py in _send_output(self, message_body, encode_chunked)
   1003         del self._buffer[:]
-> 1004         self.send(msg)
   1005

/usr/lib/python3.8/http/client.py in send(self, data)
    943             if self.auto_open:
--> 944                 self.connect()
    945             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)
    166         except SocketError as e:
--> 167             raise NewConnectionError(
    168                 self, "Failed to establish a new connection: %s" % e)

NewConnectionError: <urllib3.connection.HTTPConnection object at 0x7efca32c90a0>: Failed to establish a new connection: [Errno 111] 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)
    438             if not chunked:
--> 439                 resp = conn.urlopen(
    440                     method=request.method,

/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)
    636
--> 637             retries = retries.increment(method, url, error=e, _pool=self,
    638                                         _stacktrace=sys.exc_info()[2])

/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://www.ssfpack.com/files/DK-data.zip (Caused by ProxyError('Cannot connect to proxy.', NewConnectionError('<urllib3.connection.HTTPConnection object at 0x7efca32c90a0>: Failed to establish a new connection: [Errno 111] Connection refused')))

During handling of the above exception, another exception occurred:

ProxyError                                Traceback (most recent call last)
<ipython-input-3-074aec8a1161> in <module>
      4
      5 # Download the dataset
----> 6 dk = requests.get('http://www.ssfpack.com/files/DK-data.zip').content
      7 f = BytesIO(dk)
      8 zipped = ZipFile(f)

/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://www.ssfpack.com/files/DK-data.zip (Caused by ProxyError('Cannot connect to proxy.', NewConnectionError('<urllib3.connection.HTTPConnection object at 0x7efca32c90a0>: Failed to establish a new connection: [Errno 111] Connection refused')))

Model Selection

As in Durbin and Koopman, we force a number of the values to be missing.

[4]:
# Get the basic series
dta_full = df.dinternet[1:].values
dta_miss = dta_full.copy()

# Remove datapoints
missing = np.r_[6,16,26,36,46,56,66,72,73,74,75,76,86,96]-1
dta_miss[missing] = np.nan
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-4-70c0b0b5593e> in <module>
      1 # Get the basic series
----> 2 dta_full = df.dinternet[1:].values
      3 dta_miss = dta_full.copy()
      4
      5 # Remove datapoints

NameError: name 'df' is not defined

Then we can consider model selection using the Akaike information criteria (AIC), but running the model for each variant and selecting the model with the lowest AIC value.

There are a couple of things to note here:

  • When running such a large batch of models, particularly when the autoregressive and moving average orders become large, there is the possibility of poor maximum likelihood convergence. Below we ignore the warnings since this example is illustrative.
  • We use the option enforce_invertibility=False, which allows the moving average polynomial to be non-invertible, so that more of the models are estimable.
  • Several of the models do not produce good results, and their AIC value is set to NaN. This is not surprising, as Durbin and Koopman note numerical problems with the high order models.
[5]:
import warnings

aic_full = pd.DataFrame(np.zeros((6,6), dtype=float))
aic_miss = pd.DataFrame(np.zeros((6,6), dtype=float))

warnings.simplefilter('ignore')

# Iterate over all ARMA(p,q) models with p,q in [0,6]
for p in range(6):
    for q in range(6):
        if p == 0 and q == 0:
            continue

        # Estimate the model with no missing datapoints
        mod = sm.tsa.statespace.SARIMAX(dta_full, order=(p,0,q), enforce_invertibility=False)
        try:
            res = mod.fit(disp=False)
            aic_full.iloc[p,q] = res.aic
        except:
            aic_full.iloc[p,q] = np.nan

        # Estimate the model with missing datapoints
        mod = sm.tsa.statespace.SARIMAX(dta_miss, order=(p,0,q), enforce_invertibility=False)
        try:
            res = mod.fit(disp=False)
            aic_miss.iloc[p,q] = res.aic
        except:
            aic_miss.iloc[p,q] = np.nan
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-5-bcbba42f81f2> in <module>
     13
     14         # Estimate the model with no missing datapoints
---> 15         mod = sm.tsa.statespace.SARIMAX(dta_full, order=(p,0,q), enforce_invertibility=False)
     16         try:
     17             res = mod.fit(disp=False)

NameError: name 'dta_full' is not defined

For the models estimated over the full (non-missing) dataset, the AIC chooses ARMA(1,1) or ARMA(3,0). Durbin and Koopman suggest the ARMA(1,1) specification is better due to parsimony.

\[\begin{split}\text{Replication of:}\\ \textbf{Table 8.1} ~~ \text{AIC for different ARMA models.}\\ \newcommand{\r}[1]{{\color{red}{#1}}} \begin{array}{lrrrrrr} \hline q & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline p & {} & {} & {} & {} & {} & {} \\ 0 & 0.00 & 549.81 & 519.87 & 520.27 & 519.38 & 518.86 \\ 1 & 529.24 & \r{514.30} & 516.25 & 514.58 & 515.10 & 516.28 \\ 2 & 522.18 & 516.29 & 517.16 & 515.77 & 513.24 & 514.73 \\ 3 & \r{511.99} & 513.94 & 515.92 & 512.06 & 513.72 & 514.50 \\ 4 & 513.93 & 512.89 & nan & nan & 514.81 & 516.08 \\ 5 & 515.86 & 517.64 & nan & nan & nan & nan \\ \hline \end{array}\end{split}\]

For the models estimated over missing dataset, the AIC chooses ARMA(1,1)

\[\begin{split}\text{Replication of:}\\ \textbf{Table 8.2} ~~ \text{AIC for different ARMA models with missing observations.}\\ \begin{array}{lrrrrrr} \hline q & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline p & {} & {} & {} & {} & {} & {} \\ 0 & 0.00 & 488.93 & 464.01 & 463.86 & 462.63 & 463.62 \\ 1 & 468.01 & \r{457.54} & 459.35 & 458.66 & 459.15 & 461.01 \\ 2 & 469.68 & nan & 460.48 & 459.43 & 459.23 & 460.47 \\ 3 & 467.10 & 458.44 & 459.64 & 456.66 & 459.54 & 460.05 \\ 4 & 469.00 & 459.52 & nan & 463.04 & 459.35 & 460.96 \\ 5 & 471.32 & 461.26 & nan & nan & 461.00 & 462.97 \\ \hline \end{array}\end{split}\]

Note: the AIC values are calculated differently than in Durbin and Koopman, but show overall similar trends.

Postestimation

Using the ARMA(1,1) specification selected above, we perform in-sample prediction and out-of-sample forecasting.

[6]:
# Statespace
mod = sm.tsa.statespace.SARIMAX(dta_miss, order=(1,0,1))
res = mod.fit(disp=False)
print(res.summary())
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-6-8dd911607628> in <module>
      1 # Statespace
----> 2 mod = sm.tsa.statespace.SARIMAX(dta_miss, order=(1,0,1))
      3 res = mod.fit(disp=False)
      4 print(res.summary())

NameError: name 'dta_miss' is not defined
[7]:
# In-sample one-step-ahead predictions, and out-of-sample forecasts
nforecast = 20
predict = res.get_prediction(end=mod.nobs + nforecast)
idx = np.arange(len(predict.predicted_mean))
predict_ci = predict.conf_int(alpha=0.5)

# Graph
fig, ax = plt.subplots(figsize=(12,6))
ax.xaxis.grid()
ax.plot(dta_miss, 'k.')

# Plot
ax.plot(idx[:-nforecast], predict.predicted_mean[:-nforecast], 'gray')
ax.plot(idx[-nforecast:], predict.predicted_mean[-nforecast:], 'k--', linestyle='--', linewidth=2)
ax.fill_between(idx, predict_ci[:, 0], predict_ci[:, 1], alpha=0.15)

ax.set(title='Figure 8.9 - Internet series');
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-7-394a7033843a> in <module>
      1 # In-sample one-step-ahead predictions, and out-of-sample forecasts
      2 nforecast = 20
----> 3 predict = res.get_prediction(end=mod.nobs + nforecast)
      4 idx = np.arange(len(predict.predicted_mean))
      5 predict_ci = predict.conf_int(alpha=0.5)

NameError: name 'res' is not defined