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)
    158         try:
--> 159             conn = connection.create_connection(
    160                 (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)
     83     if err is not None:
---> 84         raise err
     85

/usr/lib/python3/dist-packages/urllib3/util/connection.py in create_connection(address, timeout, source_address, socket_options)
     73                 sock.bind(source_address)
---> 74             sock.connect(sa)
     75             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)
    669             # Make the request on the httplib connection object.
--> 670             httplib_response = self._make_request(
    671                 conn,

/usr/lib/python3/dist-packages/urllib3/connectionpool.py in _make_request(self, conn, method, url, timeout, chunked, **httplib_request_kw)
    391         else:
--> 392             conn.request(method, url, **httplib_request_kw)
    393

/usr/lib/python3.8/http/client.py in request(self, method, url, body, headers, encode_chunked)
   1254         """Send a complete request to the server."""
-> 1255         self._send_request(method, url, body, headers, encode_chunked)
   1256

/usr/lib/python3.8/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
   1300             body = _encode(body, 'body')
-> 1301         self.endheaders(body, encode_chunked=encode_chunked)
   1302

/usr/lib/python3.8/http/client.py in endheaders(self, message_body, encode_chunked)
   1249             raise CannotSendHeader()
-> 1250         self._send_output(message_body, encode_chunked=encode_chunked)
   1251

/usr/lib/python3.8/http/client.py in _send_output(self, message_body, encode_chunked)
   1009         del self._buffer[:]
-> 1010         self.send(msg)
   1011

/usr/lib/python3.8/http/client.py in send(self, data)
    949             if self.auto_open:
--> 950                 self.connect()
    951             else:

/usr/lib/python3/dist-packages/urllib3/connection.py in connect(self)
    186     def connect(self):
--> 187         conn = self._new_conn()
    188         self._prepare_conn(conn)

/usr/lib/python3/dist-packages/urllib3/connection.py in _new_conn(self)
    170         except SocketError as e:
--> 171             raise NewConnectionError(
    172                 self, "Failed to establish a new connection: %s" % e

NewConnectionError: <urllib3.connection.HTTPConnection object at 0x7f046fba13a0>: 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)
    723
--> 724             retries = retries.increment(
    725                 method, url, error=e, _pool=self, _stacktrace=sys.exc_info()[2]

/usr/lib/python3/dist-packages/urllib3/util/retry.py in increment(self, method, url, response, error, _pool, _stacktrace)
    438         if new_retry.is_exhausted():
--> 439             raise MaxRetryError(_pool, url, error or ResponseError(cause))
    440

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 0x7f046fba13a0>: 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)
     74
     75     kwargs.setdefault('allow_redirects', True)
---> 76     return request('get', url, params=params, **kwargs)
     77
     78

/usr/lib/python3/dist-packages/requests/api.py in request(method, url, **kwargs)
     59     # cases, and look like a memory leak in others.
     60     with sessions.Session() as session:
---> 61         return session.request(method=method, url=url, **kwargs)
     62
     63

/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)
    528         }
    529         send_kwargs.update(settings)
--> 530         resp = self.send(prep, **send_kwargs)
    531
    532         return resp

/usr/lib/python3/dist-packages/requests/sessions.py in send(self, request, **kwargs)
    641
    642         # Send the request
--> 643         r = adapter.send(request, **kwargs)
    644
    645         # 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 0x7f046fba13a0>: 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