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']
)

---------------------------------------------------------------------------
ProxySchemeUnknown                        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)
    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

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

SARIMAX: Model selection, missing data¶

Model Selection¶

Postestimation¶

Table of Contents

Table of Contents