用python進(jìn)行時(shí)間序列分析的方法

# -*- coding:utf-8 -*-
import numpy as np
import pandas as pdfrom datetime import datetimeimport matplotlib.pylab as plt
# 讀取數(shù)據(jù)，pd.read_csv默認(rèn)生成DataFrame對(duì)象，需將其轉(zhuǎn)換成Series對(duì)象df = pd.read_csv('AirPassengers.csv', encoding='utf-8', index_col='date')df.index = pd.to_datetime(df.index) # 將字符串索引轉(zhuǎn)換成時(shí)間索引ts = df['x'] # 生成pd.Series對(duì)象# 查看數(shù)據(jù)格式ts.head()ts.head().index

ts['1949']

ts['1949-1' : '1949-6']

# -*- coding:utf-8 -*-
from statsmodels.tsa.stattools import adfuller
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
# 移動(dòng)平均圖
def draw_trend(timeSeries, size):
  f = plt.figure(facecolor='white')
  # 對(duì)size個(gè)數(shù)據(jù)進(jìn)行移動(dòng)平均
  rol_mean = timeSeries.rolling(window=size).mean()
  # 對(duì)size個(gè)數(shù)據(jù)進(jìn)行加權(quán)移動(dòng)平均
  rol_weighted_mean = pd.ewma(timeSeries, span=size)

  timeSeries.plot(color='blue', label='Original')
  rolmean.plot(color='red', label='Rolling Mean')
  rol_weighted_mean.plot(color='black', label='Weighted Rolling Mean')
  plt.legend(loc='best')
  plt.title('Rolling Mean')
  plt.show()

def draw_ts(timeSeries):  f = plt.figure(facecolor='white')
  timeSeries.plot(color='blue')
  plt.show()

'''　　Unit Root Test
  The null hypothesis of the Augmented Dickey-Fuller is that there is a unit
  root, with the alternative that there is no unit root. That is to say the
  bigger the p-value the more reason we assert that there is a unit root
'''
def testStationarity(ts):
  dftest = adfuller(ts)
  # 對(duì)上述函數(shù)求得的值進(jìn)行語義描述
  dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','#Lags Used','Number of Observations Used'])
  for key,value in dftest[4].items():
    dfoutput['Critical Value (%s)'%key] = value
  return dfoutput

# 自相關(guān)和偏相關(guān)圖，默認(rèn)階數(shù)為31階
def draw_acf_pacf(ts, lags=31):
  f = plt.figure(facecolor='white')
  ax1 = f.add_subplot(211)
  plot_acf(ts, lags=31, ax=ax1)
  ax2 = f.add_subplot(212)
  plot_pacf(ts, lags=31, ax=ax2)
  plt.show()

ts_log = np.log(ts)
test_stationarity.draw_ts(ts_log)

test_stationarity.draw_trend(ts_log, 12)

diff_12 = ts_log.diff(12)
diff_12.dropna(inplace=True)
diff_12_1 = diff_12.diff(1)
diff_12_1.dropna(inplace=True)
test_stationarity.testStationarity(diff_12_1)

from statsmodels.tsa.seasonal import seasonal_decompose
decomposition = seasonal_decompose(ts_log, model="additive")

trend = decomposition.trend
seasonal = decomposition.seasonal
residual = decomposition.resid

rol_mean = ts_log.rolling(window=12).mean()
rol_mean.dropna(inplace=True)
ts_diff_1 = rol_mean.diff(1)
ts_diff_1.dropna(inplace=True)
test_stationarity.testStationarity(ts_diff_1)

ts_diff_2 = ts_diff_1.diff(1)
ts_diff_2.dropna(inplace=True)

from statsmodels.tsa.arima_model import ARMA
model = ARMA(ts_diff_2, order=(1, 1)) 
result_arma = model.fit( disp=-1, method='css')

predict_ts = result_arma.predict()
# 一階差分還原diff_shift_ts = ts_diff_1.shift(1)diff_recover_1 = predict_ts.add(diff_shift_ts)# 再次一階差分還原
rol_shift_ts = rol_mean.shift(1)
diff_recover = diff_recover_1.add(rol_shift_ts)
# 移動(dòng)平均還原
rol_sum = ts_log.rolling(window=11).sum()
rol_recover = diff_recover*12 - rol_sum.shift(1)
# 對(duì)數(shù)還原
log_recover = np.exp(rol_recover)
log_recover.dropna(inplace=True)

ts = ts[log_recover.index] # 過濾沒有預(yù)測的記錄plt.figure(facecolor='white')
log_recover.plot(color='blue', label='Predict')
ts.plot(color='red', label='Original')
plt.legend(loc='best')
plt.title('RMSE: %.4f'% np.sqrt(sum((log_recover-ts)**2)/ts.size))
plt.show()

# 差分操作
def diff_ts(ts, d):
  global shift_ts_list
  # 動(dòng)態(tài)預(yù)測第二日的值時(shí)所需要的差分序列
  global last_data_shift_list
  shift_ts_list = []
  last_data_shift_list = []
  tmp_ts = ts
  for i in d:
    last_data_shift_list.append(tmp_ts[-i])
    print last_data_shift_list
    shift_ts = tmp_ts.shift(i)
    shift_ts_list.append(shift_ts)
    tmp_ts = tmp_ts - shift_ts
  tmp_ts.dropna(inplace=True)
  return tmp_ts

# 還原操作
def predict_diff_recover(predict_value, d):
  if isinstance(predict_value, float):
    tmp_data = predict_value
    for i in range(len(d)):
      tmp_data = tmp_data + last_data_shift_list[-i-1]
  elif isinstance(predict_value, np.ndarray):
    tmp_data = predict_value[0]
    for i in range(len(d)):
      tmp_data = tmp_data + last_data_shift_list[-i-1]
  else:
    tmp_data = predict_value
    for i in range(len(d)):
      try:
        tmp_data = tmp_data.add(shift_ts_list[-i-1])
      except:
        raise ValueError('What you input is not pd.Series type!')
    tmp_data.dropna(inplace=True)
  return tmp_data

diffed_ts = diff_ts(ts_log, d=[12, 1])
model = arima_model(diffed_ts)
model.certain_model(1, 1)
predict_ts = model.properModel.predict()
diff_recover_ts = predict_diff_recover(predict_ts, d=[12, 1])
log_recover = np.exp(diff_recover_ts)

def proper_model(data_ts, maxLag):
  init_bic = sys.maxint
  init_p = 0
  init_q = 0
  init_properModel = None
  for p in np.arange(maxLag):
    for q in np.arange(maxLag):
      model = ARMA(data_ts, order=(p, q))
      try:
        results_ARMA = model.fit(disp=-1, method='css')
      except:
        continue
      bic = results_ARMA.bic
      if bic < init_bic:
        init_p = p
        init_q = q
        init_properModel = results_ARMA
        init_bic = bic
  return init_bic, init_p, init_q, init_properModel

from dateutil.relativedelta import relativedeltadef _add_new_data(ts, dat, type='day'):
if type == 'day':
    new_index = ts.index[-1] + relativedelta(days=1)
  elif type == 'month':
    new_index = ts.index[-1] + relativedelta(months=1)
  ts[new_index] = dat

def add_today_data(model, ts, data, d, type='day'):
  _add_new_data(ts, data, type) # 為原始序列添加數(shù)據(jù)
  # 為滯后序列添加新值
  d_ts = diff_ts(ts, d)
  model.add_today_data(d_ts[-1], type)

def forecast_next_day_data(model, type='day'):
  if model == None:
    raise ValueError('No model fit before')
  fc = model.forecast_next_day_value(type)
  return predict_diff_recover(fc, [12, 1])

ts_train = ts_log[:'1956-12']
ts_test = ts_log['1957-1':]

diffed_ts = diff_ts(ts_train, [12, 1])
forecast_list = []
for i, dta in enumerate(ts_test):
  if i%7 == 0:
    model = arima_model(diffed_ts)
    model.certain_model(1, 1)
  forecast_data = forecast_next_day_data(model, type='month')
  forecast_list.append(forecast_data)
  add_today_data(model, ts_train, dta, [12, 1], type='month')

predict_ts = pd.Series(data=forecast_list, index=ts['1957-1':].index)log_recover = np.exp(predict_ts)original_ts = ts['1957-1':]

# -*-coding:utf-8-*-
import pandas as pd
import numpy as np
from statsmodels.tsa.arima_model import ARMA
import sys
from dateutil.relativedelta import relativedelta
from copy import deepcopy
import matplotlib.pyplot as plt

class arima_model:

  def __init__(self, ts, maxLag=9):
    self.data_ts = ts
    self.resid_ts = None
    self.predict_ts = None
    self.maxLag = maxLag
    self.p = maxLag
    self.q = maxLag
    self.properModel = None
    self.bic = sys.maxint

  # 計(jì)算最優(yōu)ARIMA模型，將相關(guān)結(jié)果賦給相應(yīng)屬性
  def get_proper_model(self):
    self._proper_model()
    self.predict_ts = deepcopy(self.properModel.predict())
    self.resid_ts = deepcopy(self.properModel.resid)

  # 對(duì)于給定范圍內(nèi)的p,q計(jì)算擬合得最好的arima模型，這里是對(duì)差分好的數(shù)據(jù)進(jìn)行擬合，故差分恒為0
  def _proper_model(self):
    for p in np.arange(self.maxLag):
      for q in np.arange(self.maxLag):
        # print p,q,self.bic
        model = ARMA(self.data_ts, order=(p, q))
        try:
          results_ARMA = model.fit(disp=-1, method='css')
        except:
          continue
        bic = results_ARMA.bic
        # print 'bic:',bic,'self.bic:',self.bic
        if bic < self.bic:
          self.p = p
          self.q = q
          self.properModel = results_ARMA
          self.bic = bic
          self.resid_ts = deepcopy(self.properModel.resid)
          self.predict_ts = self.properModel.predict()

  # 參數(shù)確定模型
  def certain_model(self, p, q):
      model = ARMA(self.data_ts, order=(p, q))
      try:
        self.properModel = model.fit( disp=-1, method='css')
        self.p = p
        self.q = q
        self.bic = self.properModel.bic
        self.predict_ts = self.properModel.predict()
        self.resid_ts = deepcopy(self.properModel.resid)
      except:
        print 'You can not fit the model with this parameter p,q, ' \
           'please use the get_proper_model method to get the best model'

  # 預(yù)測第二日的值
  def forecast_next_day_value(self, type='day'):
    # 我修改了statsmodels包中arima_model的源代碼，添加了constant屬性，需要先運(yùn)行forecast方法，為constant賦值
    self.properModel.forecast()
    if self.data_ts.index[-1] != self.resid_ts.index[-1]:
      raise ValueError('''The index is different in data_ts and resid_ts, please add new data to data_ts.
      If you just want to forecast the next day data without add the real next day data to data_ts,
      please run the predict method which arima_model included itself''')
    if not self.properModel:
      raise ValueError('The arima model have not computed, please run the proper_model method before')
    para = self.properModel.params

    # print self.properModel.params
    if self.p == 0:  # It will get all the value series with setting self.data_ts[-self.p:] when p is zero
      ma_value = self.resid_ts[-self.q:]
      values = ma_value.reindex(index=ma_value.index[::-1])
    elif self.q == 0:
      ar_value = self.data_ts[-self.p:]
      values = ar_value.reindex(index=ar_value.index[::-1])
    else:
      ar_value = self.data_ts[-self.p:]
      ar_value = ar_value.reindex(index=ar_value.index[::-1])
      ma_value = self.resid_ts[-self.q:]
      ma_value = ma_value.reindex(index=ma_value.index[::-1])
      values = ar_value.append(ma_value)

    predict_value = np.dot(para[1:], values) + self.properModel.constant[0]
    self._add_new_data(self.predict_ts, predict_value, type)
    return predict_value

  # 動(dòng)態(tài)添加數(shù)據(jù)函數(shù)，針對(duì)索引是月份和日分別進(jìn)行處理
  def _add_new_data(self, ts, dat, type='day'):
    if type == 'day':
      new_index = ts.index[-1] + relativedelta(days=1)
    elif type == 'month':
      new_index = ts.index[-1] + relativedelta(months=1)
    ts[new_index] = dat

  def add_today_data(self, dat, type='day'):
    self._add_new_data(self.data_ts, dat, type)
    if self.data_ts.index[-1] != self.predict_ts.index[-1]:
      raise ValueError('You must use the forecast_next_day_value method forecast the value of today before')
    self._add_new_data(self.resid_ts, self.data_ts[-1] - self.predict_ts[-1], type)

if __name__ == '__main__':
  df = pd.read_csv('AirPassengers.csv', encoding='utf-8', index_col='date')
  df.index = pd.to_datetime(df.index)
  ts = df['x']

  # 數(shù)據(jù)預(yù)處理
  ts_log = np.log(ts)
  rol_mean = ts_log.rolling(window=12).mean()
  rol_mean.dropna(inplace=True)
  ts_diff_1 = rol_mean.diff(1)
  ts_diff_1.dropna(inplace=True)
  ts_diff_2 = ts_diff_1.diff(1)
  ts_diff_2.dropna(inplace=True)

  # 模型擬合
  model = arima_model(ts_diff_2)
  # 這里使用模型參數(shù)自動(dòng)識(shí)別
  model.get_proper_model()
  print 'bic:', model.bic, 'p:', model.p, 'q:', model.q
  print model.properModel.forecast()[0]
  print model.forecast_next_day_value(type='month')

  # 預(yù)測結(jié)果還原
  predict_ts = model.properModel.predict()
  diff_shift_ts = ts_diff_1.shift(1)
  diff_recover_1 = predict_ts.add(diff_shift_ts)
  rol_shift_ts = rol_mean.shift(1)
  diff_recover = diff_recover_1.add(rol_shift_ts)
  rol_sum = ts_log.rolling(window=11).sum()
  rol_recover = diff_recover*12 - rol_sum.shift(1)
  log_recover = np.exp(rol_recover)
  log_recover.dropna(inplace=True)

  # 預(yù)測結(jié)果作圖
  ts = ts[log_recover.index]
  plt.figure(facecolor='white')
  log_recover.plot(color='blue', label='Predict')
  ts.plot(color='red', label='Original')
  plt.legend(loc='best')
  plt.title('RMSE: %.4f'% np.sqrt(sum((log_recover-ts)**2)/ts.size))
  plt.show()

# Note: The information criteria add 1 to the number of parameters
#    whenever the model has an AR or MA term since, in principle,
#    the variance could be treated as a free parameter and restricted
#    This code does not allow this, but it adds consistency with other
#    packages such as gretl and X12-ARIMA
 
from __future__ import absolute_import
from statsmodels.compat.python import string_types, range
# for 2to3 with extensions
 
from datetime import datetime
 
import numpy as np
from scipy import optimize
from scipy.stats import t, norm
from scipy.signal import lfilter
from numpy import dot, log, zeros, pi
from numpy.linalg import inv
 
from statsmodels.tools.decorators import (cache_readonly,
                     resettable_cache)
import statsmodels.tsa.base.tsa_model as tsbase
import statsmodels.base.wrapper as wrap
from statsmodels.regression.linear_model import yule_walker, GLS
from statsmodels.tsa.tsatools import (lagmat, add_trend,
                   _ar_transparams, _ar_invtransparams,
                   _ma_transparams, _ma_invtransparams,
                   unintegrate, unintegrate_levels)
from statsmodels.tsa.vector_ar import util
from statsmodels.tsa.ar_model import AR
from statsmodels.tsa.arima_process import arma2ma
from statsmodels.tools.numdiff import approx_hess_cs, approx_fprime_cs
from statsmodels.tsa.base.datetools import _index_date
from statsmodels.tsa.kalmanf import KalmanFilter
 
_armax_notes = """
 
    Notes
    -----
    If exogenous variables are given, then the model that is fit is
 
    .. math::
 
      \\phi(L)(y_t - X_t\\beta) = \\theta(L)\epsilon_t
 
    where :math:`\\phi` and :math:`\\theta` are polynomials in the lag
    operator, :math:`L`. This is the regression model with ARMA errors,
    or ARMAX model. This specification is used, whether or not the model
    is fit using conditional sum of square or maximum-likelihood, using
    the `method` argument in
    :meth:`statsmodels.tsa.arima_model.%(Model)s.fit`. Therefore, for
    now, `css` and `mle` refer to estimation methods only. This may
    change for the case of the `css` model in future versions.
"""
 
_arma_params = """\
  endog : array-like
    The endogenous variable.
  order : iterable
    The (p,q) order of the model for the number of AR parameters,
    differences, and MA parameters to use.
  exog : array-like, optional
    An optional arry of exogenous variables. This should *not* include a
    constant or trend. You can specify this in the `fit` method."""
 
_arma_model = "Autoregressive Moving Average ARMA(p,q) Model"
 
_arima_model = "Autoregressive Integrated Moving Average ARIMA(p,d,q) Model"
 
_arima_params = """\
  endog : array-like
    The endogenous variable.
  order : iterable
    The (p,d,q) order of the model for the number of AR parameters,
    differences, and MA parameters to use.
  exog : array-like, optional
    An optional arry of exogenous variables. This should *not* include a
    constant or trend. You can specify this in the `fit` method."""
 
_predict_notes = """
    Notes
    -----
    Use the results predict method instead.
"""
 
_results_notes = """
    Notes
    -----
    It is recommended to use dates with the time-series models, as the
    below will probably make clear. However, if ARIMA is used without
    dates and/or `start` and `end` are given as indices, then these
    indices are in terms of the *original*, undifferenced series. Ie.,
    given some undifferenced observations::
 
     1970Q1, 1
     1970Q2, 1.5
     1970Q3, 1.25
     1970Q4, 2.25
     1971Q1, 1.2
     1971Q2, 4.1
 
    1970Q1 is observation 0 in the original series. However, if we fit an
    ARIMA(p,1,q) model then we lose this first observation through
    differencing. Therefore, the first observation we can forecast (if
    using exact MLE) is index 1. In the differenced series this is index
    0, but we refer to it as 1 from the original series.
"""
 
_predict = """
    %(Model)s model in-sample and out-of-sample prediction
 
    Parameters
    ----------
    %(params)s
    start : int, str, or datetime
      Zero-indexed observation number at which to start forecasting, ie.,
      the first forecast is start. Can also be a date string to
      parse or a datetime type.
    end : int, str, or datetime
      Zero-indexed observation number at which to end forecasting, ie.,
      the first forecast is start. Can also be a date string to
      parse or a datetime type. However, if the dates index does not
      have a fixed frequency, end must be an integer index if you
      want out of sample prediction.
    exog : array-like, optional
      If the model is an ARMAX and out-of-sample forecasting is
      requested, exog must be given. Note that you'll need to pass
      `k_ar` additional lags for any exogenous variables. E.g., if you
      fit an ARMAX(2, q) model and want to predict 5 steps, you need 7
      observations to do this.
    dynamic : bool, optional
      The `dynamic` keyword affects in-sample prediction. If dynamic
      is False, then the in-sample lagged values are used for
      prediction. If `dynamic` is True, then in-sample forecasts are
      used in place of lagged dependent variables. The first forecasted
      value is `start`.
    %(extra_params)s
 
    Returns
    -------
    %(returns)s
    %(extra_section)s
"""
 
_predict_returns = """predict : array
      The predicted values.
 
"""
 
_arma_predict = _predict % {"Model" : "ARMA",
              "params" : """
      params : array-like
      The fitted parameters of the model.""",
              "extra_params" : "",
              "returns" : _predict_returns,
              "extra_section" : _predict_notes}
 
_arma_results_predict = _predict % {"Model" : "ARMA", "params" : "",
                  "extra_params" : "",
                  "returns" : _predict_returns,
                  "extra_section" : _results_notes}
 
_arima_predict = _predict % {"Model" : "ARIMA",
               "params" : """params : array-like
      The fitted parameters of the model.""",
               "extra_params" : """typ : str {'linear', 'levels'}
 
      - 'linear' : Linear prediction in terms of the differenced
       endogenous variables.
      - 'levels' : Predict the levels of the original endogenous
       variables.\n""", "returns" : _predict_returns,
               "extra_section" : _predict_notes}
 
_arima_results_predict = _predict % {"Model" : "ARIMA",
                   "params" : "",
                   "extra_params" :
                   """typ : str {'linear', 'levels'}
 
      - 'linear' : Linear prediction in terms of the differenced
       endogenous variables.
      - 'levels' : Predict the levels of the original endogenous
       variables.\n""",
                   "returns" : _predict_returns,
                   "extra_section" : _results_notes}
 
_arima_plot_predict_example = """    Examples
    --------
    >>> import statsmodels.api as sm
    >>> import matplotlib.pyplot as plt
    >>> import pandas as pd
    >>>
    >>> dta = sm.datasets.sunspots.load_pandas().data[['SUNACTIVITY']]
    >>> dta.index = pd.DatetimeIndex(start='1700', end='2009', freq='A')
    >>> res = sm.tsa.ARMA(dta, (3, 0)).fit()
    >>> fig, ax = plt.subplots()
    >>> ax = dta.ix['1950':].plot(ax=ax)
    >>> fig = res.plot_predict('1990', '2012', dynamic=True, ax=ax,
    ...            plot_insample=False)
    >>> plt.show()
 
    .. plot:: plots/arma_predict_plot.py
"""
 
_plot_predict = ("""
    Plot forecasts
           """ + '\n'.join(_predict.split('\n')[2:])) % {
           "params" : "",
             "extra_params" : """alpha : float, optional
      The confidence intervals for the forecasts are (1 - alpha)%
    plot_insample : bool, optional
      Whether to plot the in-sample series. Default is True.
    ax : matplotlib.Axes, optional
      Existing axes to plot with.""",
           "returns" : """fig : matplotlib.Figure
      The plotted Figure instance""",
           "extra_section" : ('\n' + _arima_plot_predict_example +
                     '\n' + _results_notes)
           }
 
_arima_plot_predict = ("""
    Plot forecasts
           """ + '\n'.join(_predict.split('\n')[2:])) % {
           "params" : "",
             "extra_params" : """alpha : float, optional
      The confidence intervals for the forecasts are (1 - alpha)%
    plot_insample : bool, optional
      Whether to plot the in-sample series. Default is True.
    ax : matplotlib.Axes, optional
      Existing axes to plot with.""",
           "returns" : """fig : matplotlib.Figure
      The plotted Figure instance""",
        "extra_section" : ('\n' + _arima_plot_predict_example +
                  '\n' +
                  '\n'.join(_results_notes.split('\n')[:3]) +
               ("""
    This is hard-coded to only allow plotting of the forecasts in levels.
""") +
               '\n'.join(_results_notes.split('\n')[3:]))
           }
 
 
def cumsum_n(x, n):
  if n:
    n -= 1
    x = np.cumsum(x)
    return cumsum_n(x, n)
  else:
    return x
 
 
def _check_arima_start(start, k_ar, k_diff, method, dynamic):
  if start < 0:
    raise ValueError("The start index %d of the original series "
             "has been differenced away" % start)
  elif (dynamic or 'mle' not in method) and start < k_ar:
    raise ValueError("Start must be >= k_ar for conditional MLE "
             "or dynamic forecast. Got %d" % start)
 
 
def _get_predict_out_of_sample(endog, p, q, k_trend, k_exog, start, errors,
                trendparam, exparams, arparams, maparams, steps,
                method, exog=None):
  """
  Returns endog, resid, mu of appropriate length for out of sample
  prediction.
  """
  if q:
    resid = np.zeros(q)
    if start and 'mle' in method or (start == p and not start == 0):
      resid[:q] = errors[start-q:start]
    elif start:
      resid[:q] = errors[start-q-p:start-p]
    else:
      resid[:q] = errors[-q:]
  else:
    resid = None
 
  y = endog
  if k_trend == 1:
    # use expectation not constant
    if k_exog > 0:
      #TODO: technically should only hold for MLE not
      # conditional model. See #274.
      # ensure 2-d for conformability
      if np.ndim(exog) == 1 and k_exog == 1:
        # have a 1d series of observations -> 2d
        exog = exog[:, None]
      elif np.ndim(exog) == 1:
        # should have a 1d row of exog -> 2d
        if len(exog) != k_exog:
          raise ValueError("1d exog given and len(exog) != k_exog")
        exog = exog[None, :]
      X = lagmat(np.dot(exog, exparams), p, original='in', trim='both')
      mu = trendparam * (1 - arparams.sum())
      # arparams were reversed in unpack for ease later
      mu = mu + (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
    else:
      mu = trendparam * (1 - arparams.sum())
      mu = np.array([mu]*steps)
  elif k_exog > 0:
    X = np.dot(exog, exparams)
    #NOTE: you shouldn't have to give in-sample exog!
    X = lagmat(X, p, original='in', trim='both')
    mu = (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
  else:
    mu = np.zeros(steps)
 
  endog = np.zeros(p + steps - 1)
 
  if p and start:
    endog[:p] = y[start-p:start]
  elif p:
    endog[:p] = y[-p:]
 
  return endog, resid, mu
 
 
def _arma_predict_out_of_sample(params, steps, errors, p, q, k_trend, k_exog,
                endog, exog=None, start=0, method='mle'):
  (trendparam, exparams,
   arparams, maparams) = _unpack_params(params, (p, q), k_trend,
                     k_exog, reverse=True)
 #  print 'params:',params
 #  print 'arparams:',arparams,'maparams:',maparams
  endog, resid, mu = _get_predict_out_of_sample(endog, p, q, k_trend, k_exog,
                         start, errors, trendparam,
                         exparams, arparams,
                         maparams, steps, method,
                         exog)
#  print 'mu[-1]:',mu[-1], 'mu[0]:',mu[0]
  forecast = np.zeros(steps)
  if steps == 1:
    if q:
      return mu[0] + np.dot(arparams, endog[:p]) + np.dot(maparams,
                                resid[:q]), mu[0]
    else:
      return mu[0] + np.dot(arparams, endog[:p]), mu[0]
 
  if q:
    i = 0 # if q == 1
  else:
    i = -1
 
  for i in range(min(q, steps - 1)):
    fcast = (mu[i] + np.dot(arparams, endog[i:i + p]) +
         np.dot(maparams[:q - i], resid[i:i + q]))
    forecast[i] = fcast
    endog[i+p] = fcast
 
  for i in range(i + 1, steps - 1):
    fcast = mu[i] + np.dot(arparams, endog[i:i+p])
    forecast[i] = fcast
    endog[i+p] = fcast
 
  #need to do one more without updating endog
  forecast[-1] = mu[-1] + np.dot(arparams, endog[steps - 1:])
  return forecast, mu[-1] #Modified by me, the former is return forecast
 
 
def _arma_predict_in_sample(start, end, endog, resid, k_ar, method):
  """
  Pre- and in-sample fitting for ARMA.
  """
  if 'mle' in method:
    fittedvalues = endog - resid # get them all then trim
  else:
    fittedvalues = endog[k_ar:] - resid
 
  fv_start = start
  if 'mle' not in method:
    fv_start -= k_ar # start is in terms of endog index
  fv_end = min(len(fittedvalues), end + 1)
  return fittedvalues[fv_start:fv_end]
 
 
def _validate(start, k_ar, k_diff, dates, method):
  if isinstance(start, (string_types, datetime)):
    start = _index_date(start, dates)
    start -= k_diff
  if 'mle' not in method and start < k_ar - k_diff:
    raise ValueError("Start must be >= k_ar for conditional "
             "MLE or dynamic forecast. Got %s" % start)
 
  return start
 
 
def _unpack_params(params, order, k_trend, k_exog, reverse=False):
  p, q = order
  k = k_trend + k_exog
  maparams = params[k+p:]
  arparams = params[k:k+p]
  trend = params[:k_trend]
  exparams = params[k_trend:k]
  if reverse:
    return trend, exparams, arparams[::-1], maparams[::-1]
  return trend, exparams, arparams, maparams
 
 
def _unpack_order(order):
  k_ar, k_ma, k = order
  k_lags = max(k_ar, k_ma+1)
  return k_ar, k_ma, order, k_lags
 
 
def _make_arma_names(data, k_trend, order, exog_names):
  k_ar, k_ma = order
  exog_names = exog_names or []
  ar_lag_names = util.make_lag_names([data.ynames], k_ar, 0)
  ar_lag_names = [''.join(('ar.', i)) for i in ar_lag_names]
  ma_lag_names = util.make_lag_names([data.ynames], k_ma, 0)
  ma_lag_names = [''.join(('ma.', i)) for i in ma_lag_names]
  trend_name = util.make_lag_names('', 0, k_trend)
  exog_names = trend_name + exog_names + ar_lag_names + ma_lag_names
  return exog_names
 
 
def _make_arma_exog(endog, exog, trend):
  k_trend = 1 # overwritten if no constant
  if exog is None and trend == 'c':  # constant only
    exog = np.ones((len(endog), 1))
  elif exog is not None and trend == 'c': # constant plus exogenous
    exog = add_trend(exog, trend='c', prepend=True)
  elif exog is not None and trend == 'nc':
    # make sure it's not holding constant from last run
    if exog.var() == 0:
      exog = None
    k_trend = 0
  if trend == 'nc':
    k_trend = 0
  return k_trend, exog
 
 
def _check_estimable(nobs, n_params):
  if nobs <= n_params:
    raise ValueError("Insufficient degrees of freedom to estimate")
 
 
class ARMA(tsbase.TimeSeriesModel):
 
  __doc__ = tsbase._tsa_doc % {"model" : _arma_model,
                 "params" : _arma_params, "extra_params" : "",
                 "extra_sections" : _armax_notes %
                 {"Model" : "ARMA"}}
 
  def __init__(self, endog, order, exog=None, dates=None, freq=None,
         missing='none'):
    super(ARMA, self).__init__(endog, exog, dates, freq, missing=missing)
    exog = self.data.exog # get it after it's gone through processing
    _check_estimable(len(self.endog), sum(order))
    self.k_ar = k_ar = order[0]
    self.k_ma = k_ma = order[1]
    self.k_lags = max(k_ar, k_ma+1)
    self.constant = 0 #Added by me
    if exog is not None:
      if exog.ndim == 1:
        exog = exog[:, None]
      k_exog = exog.shape[1] # number of exog. variables excl. const
    else:
      k_exog = 0
    self.k_exog = k_exog
 
  def _fit_start_params_hr(self, order):
    """
    Get starting parameters for fit.
 
    Parameters
    ----------
    order : iterable
      (p,q,k) - AR lags, MA lags, and number of exogenous variables
      including the constant.
 
    Returns
    -------
    start_params : array
      A first guess at the starting parameters.
 
    Notes
    -----
    If necessary, fits an AR process with the laglength selected according
    to best BIC. Obtain the residuals. Then fit an ARMA(p,q) model via
    OLS using these residuals for a first approximation. Uses a separate
    OLS regression to find the coefficients of exogenous variables.
 
    References
    ----------
    Hannan, E.J. and Rissanen, J. 1982. "Recursive estimation of mixed
      autoregressive-moving average order." `Biometrika`. 69.1.
    """
    p, q, k = order
    start_params = zeros((p+q+k))
    endog = self.endog.copy() # copy because overwritten
    exog = self.exog
    if k != 0:
      ols_params = GLS(endog, exog).fit().params
      start_params[:k] = ols_params
      endog -= np.dot(exog, ols_params).squeeze()
    if q != 0:
      if p != 0:
        # make sure we don't run into small data problems in AR fit
        nobs = len(endog)
        maxlag = int(round(12*(nobs/100.)**(1/4.)))
        if maxlag >= nobs:
          maxlag = nobs - 1
        armod = AR(endog).fit(ic='bic', trend='nc', maxlag=maxlag)
        arcoefs_tmp = armod.params
        p_tmp = armod.k_ar
        # it's possible in small samples that optimal lag-order
        # doesn't leave enough obs. No consistent way to fix.
        if p_tmp + q >= len(endog):
          raise ValueError("Proper starting parameters cannot"
                   " be found for this order with this "
                   "number of observations. Use the "
                   "start_params argument.")
        resid = endog[p_tmp:] - np.dot(lagmat(endog, p_tmp,
                           trim='both'),
                        arcoefs_tmp)
        if p < p_tmp + q:
          endog_start = p_tmp + q - p
          resid_start = 0
        else:
          endog_start = 0
          resid_start = p - p_tmp - q
        lag_endog = lagmat(endog, p, 'both')[endog_start:]
        lag_resid = lagmat(resid, q, 'both')[resid_start:]
        # stack ar lags and resids
        X = np.column_stack((lag_endog, lag_resid))
        coefs = GLS(endog[max(p_tmp + q, p):], X).fit().params
        start_params[k:k+p+q] = coefs
      else:
        start_params[k+p:k+p+q] = yule_walker(endog, order=q)[0]
    if q == 0 and p != 0:
      arcoefs = yule_walker(endog, order=p)[0]
      start_params[k:k+p] = arcoefs
 
    # check AR coefficients
    if p and not np.all(np.abs(np.roots(np.r_[1, -start_params[k:k + p]]
                      )) < 1):
      raise ValueError("The computed initial AR coefficients are not "
               "stationary\nYou should induce stationarity, "
               "choose a different model order, or you can\n"
               "pass your own start_params.")
    # check MA coefficients
    elif q and not np.all(np.abs(np.roots(np.r_[1, start_params[k + p:]]
                       )) < 1):
      return np.zeros(len(start_params))  #modified by me
      raise ValueError("The computed initial MA coefficients are not "
               "invertible\nYou should induce invertibility, "
               "choose a different model order, or you can\n"
               "pass your own start_params.")
 
    # check MA coefficients
    # print start_params
    return start_params
 
  def _fit_start_params(self, order, method):
    if method != 'css-mle': # use Hannan-Rissanen to get start params
      start_params = self._fit_start_params_hr(order)
    else: # use CSS to get start params
      func = lambda params: -self.loglike_css(params)
      #start_params = [.1]*(k_ar+k_ma+k_exog) # different one for k&#63;
      start_params = self._fit_start_params_hr(order)
      if self.transparams:
        start_params = self._invtransparams(start_params)
      bounds = [(None,)*2]*sum(order)
      mlefit = optimize.fmin_l_bfgs_b(func, start_params,
                      approx_grad=True, m=12,
                      pgtol=1e-7, factr=1e3,
                      bounds=bounds, iprint=-1)
      start_params = self._transparams(mlefit[0])
    return start_params
 
  def score(self, params):
    """
    Compute the score function at params.
 
    Notes
    -----
    This is a numerical approximation.
    """
    return approx_fprime_cs(params, self.loglike, args=(False,))
 
  def hessian(self, params):
    """
    Compute the Hessian at params,
 
    Notes
    -----
    This is a numerical approximation.
    """
    return approx_hess_cs(params, self.loglike, args=(False,))
 
  def _transparams(self, params):
    """
    Transforms params to induce stationarity/invertability.
 
    Reference
    ---------
    Jones(1980)
    """
    k_ar, k_ma = self.k_ar, self.k_ma
    k = self.k_exog + self.k_trend
    newparams = np.zeros_like(params)
 
    # just copy exogenous parameters
    if k != 0:
      newparams[:k] = params[:k]
 
    # AR Coeffs
    if k_ar != 0:
      newparams[k:k+k_ar] = _ar_transparams(params[k:k+k_ar].copy())
 
    # MA Coeffs
    if k_ma != 0:
      newparams[k+k_ar:] = _ma_transparams(params[k+k_ar:].copy())
    return newparams
 
  def _invtransparams(self, start_params):
    """
    Inverse of the Jones reparameterization
    """
    k_ar, k_ma = self.k_ar, self.k_ma
    k = self.k_exog + self.k_trend
    newparams = start_params.copy()
    arcoefs = newparams[k:k+k_ar]
    macoefs = newparams[k+k_ar:]
    # AR coeffs
    if k_ar != 0:
      newparams[k:k+k_ar] = _ar_invtransparams(arcoefs)
 
    # MA coeffs
    if k_ma != 0:
      newparams[k+k_ar:k+k_ar+k_ma] = _ma_invtransparams(macoefs)
    return newparams
 
  def _get_predict_start(self, start, dynamic):
    # do some defaults
    method = getattr(self, 'method', 'mle')
    k_ar = getattr(self, 'k_ar', 0)
    k_diff = getattr(self, 'k_diff', 0)
    if start is None:
      if 'mle' in method and not dynamic:
        start = 0
      else:
        start = k_ar
      self._set_predict_start_date(start) # else it's done in super
    elif isinstance(start, int):
      start = super(ARMA, self)._get_predict_start(start)
    else: # should be on a date
      #elif 'mle' not in method or dynamic: # should be on a date
      start = _validate(start, k_ar, k_diff, self.data.dates,
               method)
      start = super(ARMA, self)._get_predict_start(start)
    _check_arima_start(start, k_ar, k_diff, method, dynamic)
    return start
 
  def _get_predict_end(self, end, dynamic=False):
    # pass through so predict works for ARIMA and ARMA
    return super(ARMA, self)._get_predict_end(end)
 
  def geterrors(self, params):
    """
    Get the errors of the ARMA process.
 
    Parameters
    ----------
    params : array-like
      The fitted ARMA parameters
    order : array-like
      3 item iterable, with the number of AR, MA, and exogenous
      parameters, including the trend
    """
 
    #start = self._get_predict_start(start) # will be an index of a date
    #end, out_of_sample = self._get_predict_end(end)
    params = np.asarray(params)
    k_ar, k_ma = self.k_ar, self.k_ma
    k = self.k_exog + self.k_trend
 
    method = getattr(self, 'method', 'mle')
    if 'mle' in method: # use KalmanFilter to get errors
      (y, k, nobs, k_ar, k_ma, k_lags, newparams, Z_mat, m, R_mat,
       T_mat, paramsdtype) = KalmanFilter._init_kalman_state(params,
                                  self)
 
      errors = KalmanFilter.geterrors(y, k, k_ar, k_ma, k_lags, nobs,
                      Z_mat, m, R_mat, T_mat,
                      paramsdtype)
      if isinstance(errors, tuple):
        errors = errors[0] # non-cython version returns a tuple
    else: # use scipy.signal.lfilter
      y = self.endog.copy()
      k = self.k_exog + self.k_trend
      if k > 0:
        y -= dot(self.exog, params[:k])
 
      k_ar = self.k_ar
      k_ma = self.k_ma
 
      (trendparams, exparams,
       arparams, maparams) = _unpack_params(params, (k_ar, k_ma),
                         self.k_trend, self.k_exog,
                         reverse=False)
      b, a = np.r_[1, -arparams], np.r_[1, maparams]
      zi = zeros((max(k_ar, k_ma)))
      for i in range(k_ar):
        zi[i] = sum(-b[:i+1][::-1]*y[:i+1])
      e = lfilter(b, a, y, zi=zi)
      errors = e[0][k_ar:]
    return errors.squeeze()
 
  def predict(self, params, start=None, end=None, exog=None, dynamic=False):
    method = getattr(self, 'method', 'mle') # don't assume fit
    #params = np.asarray(params)
 
    # will return an index of a date
    start = self._get_predict_start(start, dynamic)
    end, out_of_sample = self._get_predict_end(end, dynamic)
    if out_of_sample and (exog is None and self.k_exog > 0):
      raise ValueError("You must provide exog for ARMAX")
 
    endog = self.endog
    resid = self.geterrors(params)
    k_ar = self.k_ar
 
    if out_of_sample != 0 and self.k_exog > 0:
      if self.k_exog == 1 and exog.ndim == 1:
        exog = exog[:, None]
        # we need the last k_ar exog for the lag-polynomial
      if self.k_exog > 0 and k_ar > 0:
        # need the last k_ar exog for the lag-polynomial
        exog = np.vstack((self.exog[-k_ar:, self.k_trend:], exog))
 
    if dynamic:
      #TODO: now that predict does dynamic in-sample it should
      # also return error estimates and confidence intervals
      # but how&#63; len(endog) is not tot_obs
      out_of_sample += end - start + 1
      pr, ct = _arma_predict_out_of_sample(params, out_of_sample, resid,
                        k_ar, self.k_ma, self.k_trend,
                        self.k_exog, endog, exog,
                        start, method)
      self.constant = ct
      return pr
 
    predictedvalues = _arma_predict_in_sample(start, end, endog, resid,
                         k_ar, method)
    if out_of_sample:
      forecastvalues, ct = _arma_predict_out_of_sample(params, out_of_sample,
                             resid, k_ar,
                             self.k_ma,
                             self.k_trend,
                             self.k_exog, endog,
                             exog, method=method)
      self.constant = ct
      predictedvalues = np.r_[predictedvalues, forecastvalues]
    return predictedvalues
  predict.__doc__ = _arma_predict
 
  def loglike(self, params, set_sigma2=True):
    """
    Compute the log-likelihood for ARMA(p,q) model
 
    Notes
    -----
    Likelihood used depends on the method set in fit
    """
    method = self.method
    if method in ['mle', 'css-mle']:
      return self.loglike_kalman(params, set_sigma2)
    elif method == 'css':
      return self.loglike_css(params, set_sigma2)
    else:
      raise ValueError("Method %s not understood" % method)
 
  def loglike_kalman(self, params, set_sigma2=True):
    """
    Compute exact loglikelihood for ARMA(p,q) model by the Kalman Filter.
    """
    return KalmanFilter.loglike(params, self, set_sigma2)
 
  def loglike_css(self, params, set_sigma2=True):
    """
    Conditional Sum of Squares likelihood function.
    """
    k_ar = self.k_ar
    k_ma = self.k_ma
    k = self.k_exog + self.k_trend
    y = self.endog.copy().astype(params.dtype)
    nobs = self.nobs
    # how to handle if empty&#63;
    if self.transparams:
      newparams = self._transparams(params)
    else:
      newparams = params
    if k > 0:
      y -= dot(self.exog, newparams[:k])
    # the order of p determines how many zeros errors to set for lfilter
    b, a = np.r_[1, -newparams[k:k + k_ar]], np.r_[1, newparams[k + k_ar:]]
    zi = np.zeros((max(k_ar, k_ma)), dtype=params.dtype)
    for i in range(k_ar):
      zi[i] = sum(-b[:i + 1][::-1] * y[:i + 1])
    errors = lfilter(b, a, y, zi=zi)[0][k_ar:]
 
    ssr = np.dot(errors, errors)
    sigma2 = ssr/nobs
    if set_sigma2:
      self.sigma2 = sigma2
    llf = -nobs/2.*(log(2*pi) + log(sigma2)) - ssr/(2*sigma2)
    return llf
 
  def fit(self, start_params=None, trend='c', method="css-mle",
      transparams=True, solver='lbfgs', maxiter=50, full_output=1,
      disp=5, callback=None, **kwargs):
    """
    Fits ARMA(p,q) model using exact maximum likelihood via Kalman filter.
 
    Parameters
    ----------
    start_params : array-like, optional
      Starting parameters for ARMA(p,q). If None, the default is given
      by ARMA._fit_start_params. See there for more information.
    transparams : bool, optional
      Whehter or not to transform the parameters to ensure stationarity.
      Uses the transformation suggested in Jones (1980). If False,
      no checking for stationarity or invertibility is done.
    method : str {'css-mle','mle','css'}
      This is the loglikelihood to maximize. If "css-mle", the
      conditional sum of squares likelihood is maximized and its values
      are used as starting values for the computation of the exact
      likelihood via the Kalman filter. If "mle", the exact likelihood
      is maximized via the Kalman Filter. If "css" the conditional sum
      of squares likelihood is maximized. All three methods use
      `start_params` as starting parameters. See above for more
      information.
    trend : str {'c','nc'}
      Whether to include a constant or not. 'c' includes constant,
      'nc' no constant.
    solver : str or None, optional
      Solver to be used. The default is 'lbfgs' (limited memory
      Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
      'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
      (conjugate gradient), 'ncg' (non-conjugate gradient), and
      'powell'. By default, the limited memory BFGS uses m=12 to
      approximate the Hessian, projected gradient tolerance of 1e-8 and
      factr = 1e2. You can change these by using kwargs.
    maxiter : int, optional
      The maximum number of function evaluations. Default is 50.
    tol : float
      The convergence tolerance. Default is 1e-08.
    full_output : bool, optional
      If True, all output from solver will be available in
      the Results object's mle_retvals attribute. Output is dependent
      on the solver. See Notes for more information.
    disp : bool, optional
      If True, convergence information is printed. For the default
      l_bfgs_b solver, disp controls the frequency of the output during
      the iterations. disp < 0 means no output in this case.
    callback : function, optional
      Called after each iteration as callback(xk) where xk is the current
      parameter vector.
    kwargs
      See Notes for keyword arguments that can be passed to fit.
 
    Returns
    -------
    statsmodels.tsa.arima_model.ARMAResults class
 
    See also
    --------
    statsmodels.base.model.LikelihoodModel.fit : for more information
      on using the solvers.
    ARMAResults : results class returned by fit
 
    Notes
    ------
    If fit by 'mle', it is assumed for the Kalman Filter that the initial
    unkown state is zero, and that the inital variance is
    P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
    r, order = 'F')
 
    """
    k_ar = self.k_ar
    k_ma = self.k_ma
 
    # enforce invertibility
    self.transparams = transparams
 
    endog, exog = self.endog, self.exog
    k_exog = self.k_exog
    self.nobs = len(endog) # this is overwritten if method is 'css'
 
    # (re)set trend and handle exogenous variables
    # always pass original exog
    k_trend, exog = _make_arma_exog(endog, self.exog, trend)
 
    # Check has something to estimate
    if k_ar == 0 and k_ma == 0 and k_trend == 0 and k_exog == 0:
      raise ValueError("Estimation requires the inclusion of least one "
             "AR term, MA term, a constant or an exogenous "
             "variable.")
 
    # check again now that we know the trend
    _check_estimable(len(endog), k_ar + k_ma + k_exog + k_trend)
 
    self.k_trend = k_trend
    self.exog = exog  # overwrites original exog from __init__
 
    # (re)set names for this model
    self.exog_names = _make_arma_names(self.data, k_trend, (k_ar, k_ma),
                      self.exog_names)
    k = k_trend + k_exog
 
    # choose objective function
    if k_ma == 0 and k_ar == 0:
      method = "css" # Always CSS when no AR or MA terms
 
    self.method = method = method.lower()
 
    # adjust nobs for css
    if method == 'css':
      self.nobs = len(self.endog) - k_ar
 
    if start_params is not None:
      start_params = np.asarray(start_params)
 
    else: # estimate starting parameters
      start_params = self._fit_start_params((k_ar, k_ma, k), method)
 
    if transparams: # transform initial parameters to ensure invertibility
      start_params = self._invtransparams(start_params)
 
    if solver == 'lbfgs':
      kwargs.setdefault('pgtol', 1e-8)
      kwargs.setdefault('factr', 1e2)
      kwargs.setdefault('m', 12)
      kwargs.setdefault('approx_grad', True)
    mlefit = super(ARMA, self).fit(start_params, method=solver,
                    maxiter=maxiter,
                    full_output=full_output, disp=disp,
                    callback=callback, **kwargs)
    params = mlefit.params
 
    if transparams: # transform parameters back
      params = self._transparams(params)
 
    self.transparams = False # so methods don't expect transf.
 
    normalized_cov_params = None # TODO: fix this
    armafit = ARMAResults(self, params, normalized_cov_params)
    armafit.mle_retvals = mlefit.mle_retvals
    armafit.mle_settings = mlefit.mle_settings
    armafit.mlefit = mlefit
    return ARMAResultsWrapper(armafit)
 
 
#NOTE: the length of endog changes when we give a difference to fit
#so model methods are not the same on unfit models as fit ones
#starting to think that order of model should be put in instantiation...
class ARIMA(ARMA):
 
  __doc__ = tsbase._tsa_doc % {"model" : _arima_model,
                 "params" : _arima_params, "extra_params" : "",
                 "extra_sections" : _armax_notes %
                 {"Model" : "ARIMA"}}
 
  def __new__(cls, endog, order, exog=None, dates=None, freq=None,
        missing='none'):
    p, d, q = order
    if d == 0: # then we just use an ARMA model
      return ARMA(endog, (p, q), exog, dates, freq, missing)
    else:
      mod = super(ARIMA, cls).__new__(cls)
      mod.__init__(endog, order, exog, dates, freq, missing)
      return mod
 
  def __init__(self, endog, order, exog=None, dates=None, freq=None,
         missing='none'):
    p, d, q = order
    if d > 2:
      #NOTE: to make more general, need to address the d == 2 stuff
      # in the predict method
      raise ValueError("d > 2 is not supported")
    super(ARIMA, self).__init__(endog, (p, q), exog, dates, freq, missing)
    self.k_diff = d
    self._first_unintegrate = unintegrate_levels(self.endog[:d], d)
    self.endog = np.diff(self.endog, n=d)
    #NOTE: will check in ARMA but check again since differenced now
    _check_estimable(len(self.endog), p+q)
    if exog is not None:
      self.exog = self.exog[d:]
    if d == 1:
      self.data.ynames = 'D.' + self.endog_names
    else:
      self.data.ynames = 'D{0:d}.'.format(d) + self.endog_names
    # what about exog, should we difference it automatically before
    # super call&#63;
 
  def _get_predict_start(self, start, dynamic):
    """
    """
    #TODO: remove all these getattr and move order specification to
    # class constructor
    k_diff = getattr(self, 'k_diff', 0)
    method = getattr(self, 'method', 'mle')
    k_ar = getattr(self, 'k_ar', 0)
    if start is None:
      if 'mle' in method and not dynamic:
        start = 0
      else:
        start = k_ar
    elif isinstance(start, int):
        start -= k_diff
        try: # catch when given an integer outside of dates index
          start = super(ARIMA, self)._get_predict_start(start,
                                 dynamic)
        except IndexError:
          raise ValueError("start must be in series. "
                   "got %d" % (start + k_diff))
    else: # received a date
      start = _validate(start, k_ar, k_diff, self.data.dates,
               method)
      start = super(ARIMA, self)._get_predict_start(start, dynamic)
    # reset date for k_diff adjustment
    self._set_predict_start_date(start + k_diff)
    return start
 
  def _get_predict_end(self, end, dynamic=False):
    """
    Returns last index to be forecast of the differenced array.
    Handling of inclusiveness should be done in the predict function.
    """
    end, out_of_sample = super(ARIMA, self)._get_predict_end(end, dynamic)
    if 'mle' not in self.method and not dynamic:
      end -= self.k_ar
 
    return end - self.k_diff, out_of_sample
 
  def fit(self, start_params=None, trend='c', method="css-mle",
      transparams=True, solver='lbfgs', maxiter=50, full_output=1,
      disp=5, callback=None, **kwargs):
    """
    Fits ARIMA(p,d,q) model by exact maximum likelihood via Kalman filter.
 
    Parameters
    ----------
    start_params : array-like, optional
      Starting parameters for ARMA(p,q). If None, the default is given
      by ARMA._fit_start_params. See there for more information.
    transparams : bool, optional
      Whehter or not to transform the parameters to ensure stationarity.
      Uses the transformation suggested in Jones (1980). If False,
      no checking for stationarity or invertibility is done.
    method : str {'css-mle','mle','css'}
      This is the loglikelihood to maximize. If "css-mle", the
      conditional sum of squares likelihood is maximized and its values
      are used as starting values for the computation of the exact
      likelihood via the Kalman filter. If "mle", the exact likelihood
      is maximized via the Kalman Filter. If "css" the conditional sum
      of squares likelihood is maximized. All three methods use
      `start_params` as starting parameters. See above for more
      information.
    trend : str {'c','nc'}
      Whether to include a constant or not. 'c' includes constant,
      'nc' no constant.
    solver : str or None, optional
      Solver to be used. The default is 'lbfgs' (limited memory
      Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
      'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
      (conjugate gradient), 'ncg' (non-conjugate gradient), and
      'powell'. By default, the limited memory BFGS uses m=12 to
      approximate the Hessian, projected gradient tolerance of 1e-8 and
      factr = 1e2. You can change these by using kwargs.
    maxiter : int, optional
      The maximum number of function evaluations. Default is 50.
    tol : float
      The convergence tolerance. Default is 1e-08.
    full_output : bool, optional
      If True, all output from solver will be available in
      the Results object's mle_retvals attribute. Output is dependent
      on the solver. See Notes for more information.
    disp : bool, optional
      If True, convergence information is printed. For the default
      l_bfgs_b solver, disp controls the frequency of the output during
      the iterations. disp < 0 means no output in this case.
    callback : function, optional
      Called after each iteration as callback(xk) where xk is the current
      parameter vector.
    kwargs
      See Notes for keyword arguments that can be passed to fit.
 
    Returns
    -------
    `statsmodels.tsa.arima.ARIMAResults` class
 
    See also
    --------
    statsmodels.base.model.LikelihoodModel.fit : for more information
      on using the solvers.
    ARIMAResults : results class returned by fit
 
    Notes
    ------
    If fit by 'mle', it is assumed for the Kalman Filter that the initial
    unkown state is zero, and that the inital variance is
    P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
    r, order = 'F')
 
    """
    arima_fit = super(ARIMA, self).fit(start_params, trend,
                      method, transparams, solver,
                      maxiter, full_output, disp,
                      callback, **kwargs)
    normalized_cov_params = None # TODO: fix this&#63;
    arima_fit = ARIMAResults(self, arima_fit._results.params,
                 normalized_cov_params)
    arima_fit.k_diff = self.k_diff
    return ARIMAResultsWrapper(arima_fit)
 
  def predict(self, params, start=None, end=None, exog=None, typ='linear',
        dynamic=False):
    # go ahead and convert to an index for easier checking
    if isinstance(start, (string_types, datetime)):
      start = _index_date(start, self.data.dates)
    if typ == 'linear':
      if not dynamic or (start != self.k_ar + self.k_diff and
                start is not None):
        return super(ARIMA, self).predict(params, start, end, exog,
                         dynamic)
      else:
        # need to assume pre-sample residuals are zero
        # do this by a hack
        q = self.k_ma
        self.k_ma = 0
        predictedvalues = super(ARIMA, self).predict(params, start,
                               end, exog,
                               dynamic)
        self.k_ma = q
        return predictedvalues
    elif typ == 'levels':
      endog = self.data.endog
      if not dynamic:
        predict = super(ARIMA, self).predict(params, start, end,
                           dynamic)
 
        start = self._get_predict_start(start, dynamic)
        end, out_of_sample = self._get_predict_end(end)
        d = self.k_diff
        if 'mle' in self.method:
          start += d - 1 # for case where d == 2
          end += d - 1
          # add each predicted diff to lagged endog
          if out_of_sample:
            fv = predict[:-out_of_sample] + endog[start:end+1]
            if d == 2: #TODO: make a general solution to this
              fv += np.diff(endog[start - 1:end + 1])
            levels = unintegrate_levels(endog[-d:], d)
            fv = np.r_[fv,
                  unintegrate(predict[-out_of_sample:],
                        levels)[d:]]
          else:
            fv = predict + endog[start:end + 1]
            if d == 2:
              fv += np.diff(endog[start - 1:end + 1])
        else:
          k_ar = self.k_ar
          if out_of_sample:
            fv = (predict[:-out_of_sample] +
               endog[max(start, self.k_ar-1):end+k_ar+1])
            if d == 2:
              fv += np.diff(endog[start - 1:end + 1])
            levels = unintegrate_levels(endog[-d:], d)
            fv = np.r_[fv,
                  unintegrate(predict[-out_of_sample:],
                        levels)[d:]]
          else:
            fv = predict + endog[max(start, k_ar):end+k_ar+1]
            if d == 2:
              fv += np.diff(endog[start - 1:end + 1])
      else:
        #IFF we need to use pre-sample values assume pre-sample
        # residuals are zero, do this by a hack
        if start == self.k_ar + self.k_diff or start is None:
          # do the first k_diff+1 separately
          p = self.k_ar
          q = self.k_ma
          k_exog = self.k_exog
          k_trend = self.k_trend
          k_diff = self.k_diff
          (trendparam, exparams,
           arparams, maparams) = _unpack_params(params, (p, q),
                             k_trend,
                             k_exog,
                             reverse=True)
          # this is the hack
          self.k_ma = 0
 
          predict = super(ARIMA, self).predict(params, start, end,
                             exog, dynamic)
          if not start:
            start = self._get_predict_start(start, dynamic)
            start += k_diff
          self.k_ma = q
          return endog[start-1] + np.cumsum(predict)
        else:
          predict = super(ARIMA, self).predict(params, start, end,
                             exog, dynamic)
          return endog[start-1] + np.cumsum(predict)
      return fv
 
    else: # pragma : no cover
      raise ValueError("typ %s not understood" % typ)
 
  predict.__doc__ = _arima_predict
 
 
class ARMAResults(tsbase.TimeSeriesModelResults):
  """
  Class to hold results from fitting an ARMA model.
 
  Parameters
  ----------
  model : ARMA instance
    The fitted model instance
  params : array
    Fitted parameters
  normalized_cov_params : array, optional
    The normalized variance covariance matrix
  scale : float, optional
    Optional argument to scale the variance covariance matrix.
 
  Returns
  --------
  **Attributes**
 
  aic : float
    Akaike Information Criterion
    :math:`-2*llf+2* df_model`
    where `df_model` includes all AR parameters, MA parameters, constant
    terms parameters on constant terms and the variance.
  arparams : array
    The parameters associated with the AR coefficients in the model.
  arroots : array
    The roots of the AR coefficients are the solution to
    (1 - arparams[0]*z - arparams[1]*z**2 -...- arparams[p-1]*z**k_ar) = 0
    Stability requires that the roots in modulus lie outside the unit
    circle.
  bic : float
    Bayes Information Criterion
    -2*llf + log(nobs)*df_model
    Where if the model is fit using conditional sum of squares, the
    number of observations `nobs` does not include the `p` pre-sample
    observations.
  bse : array
    The standard errors of the parameters. These are computed using the
    numerical Hessian.
  df_model : array
    The model degrees of freedom = `k_exog` + `k_trend` + `k_ar` + `k_ma`
  df_resid : array
    The residual degrees of freedom = `nobs` - `df_model`
  fittedvalues : array
    The predicted values of the model.
  hqic : float
    Hannan-Quinn Information Criterion
    -2*llf + 2*(`df_model`)*log(log(nobs))
    Like `bic` if the model is fit using conditional sum of squares then
    the `k_ar` pre-sample observations are not counted in `nobs`.
  k_ar : int
    The number of AR coefficients in the model.
  k_exog : int
    The number of exogenous variables included in the model. Does not
    include the constant.
  k_ma : int
    The number of MA coefficients.
  k_trend : int
    This is 0 for no constant or 1 if a constant is included.
  llf : float
    The value of the log-likelihood function evaluated at `params`.
  maparams : array
    The value of the moving average coefficients.
  maroots : array
    The roots of the MA coefficients are the solution to
    (1 + maparams[0]*z + maparams[1]*z**2 + ... + maparams[q-1]*z**q) = 0
    Stability requires that the roots in modules lie outside the unit
    circle.
  model : ARMA instance
    A reference to the model that was fit.
  nobs : float
    The number of observations used to fit the model. If the model is fit
    using exact maximum likelihood this is equal to the total number of
    observations, `n_totobs`. If the model is fit using conditional
    maximum likelihood this is equal to `n_totobs` - `k_ar`.
  n_totobs : float
    The total number of observations for `endog`. This includes all
    observations, even pre-sample values if the model is fit using `css`.
  params : array
    The parameters of the model. The order of variables is the trend
    coefficients and the `k_exog` exognous coefficients, then the
    `k_ar` AR coefficients, and finally the `k_ma` MA coefficients.
  pvalues : array
    The p-values associated with the t-values of the coefficients. Note
    that the coefficients are assumed to have a Student's T distribution.
  resid : array
    The model residuals. If the model is fit using 'mle' then the
    residuals are created via the Kalman Filter. If the model is fit
    using 'css' then the residuals are obtained via `scipy.signal.lfilter`
    adjusted such that the first `k_ma` residuals are zero. These zero
    residuals are not returned.
  scale : float
    This is currently set to 1.0 and not used by the model or its results.
  sigma2 : float
    The variance of the residuals. If the model is fit by 'css',
    sigma2 = ssr/nobs, where ssr is the sum of squared residuals. If
    the model is fit by 'mle', then sigma2 = 1/nobs * sum(v**2 / F)
    where v is the one-step forecast error and F is the forecast error
    variance. See `nobs` for the difference in definitions depending on the
    fit.
  """
  _cache = {}
 
  #TODO: use this for docstring when we fix nobs issue
 
  def __init__(self, model, params, normalized_cov_params=None, scale=1.):
    super(ARMAResults, self).__init__(model, params, normalized_cov_params,
                     scale)
    self.sigma2 = model.sigma2
    nobs = model.nobs
    self.nobs = nobs
    k_exog = model.k_exog
    self.k_exog = k_exog
    k_trend = model.k_trend
    self.k_trend = k_trend
    k_ar = model.k_ar
    self.k_ar = k_ar
    self.n_totobs = len(model.endog)
    k_ma = model.k_ma
    self.k_ma = k_ma
    df_model = k_exog + k_trend + k_ar + k_ma
    self._ic_df_model = df_model + 1
    self.df_model = df_model
    self.df_resid = self.nobs - df_model
    self._cache = resettable_cache()
    self.constant = 0 #Added by me
 
  @cache_readonly
  def arroots(self):
    return np.roots(np.r_[1, -self.arparams])**-1
 
  @cache_readonly
  def maroots(self):
    return np.roots(np.r_[1, self.maparams])**-1
 
  @cache_readonly
  def arfreq(self):
    r"""
    Returns the frequency of the AR roots.
 
    This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
    roots.
    """
    z = self.arroots
    if not z.size:
      return
    return np.arctan2(z.imag, z.real) / (2*pi)
 
  @cache_readonly
  def mafreq(self):
    r"""
    Returns the frequency of the MA roots.
 
    This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
    roots.
    """
    z = self.maroots
    if not z.size:
      return
    return np.arctan2(z.imag, z.real) / (2*pi)
 
  @cache_readonly
  def arparams(self):
    k = self.k_exog + self.k_trend
    return self.params[k:k+self.k_ar]
 
  @cache_readonly
  def maparams(self):
    k = self.k_exog + self.k_trend
    k_ar = self.k_ar
    return self.params[k+k_ar:]
 
  @cache_readonly
  def llf(self):
    return self.model.loglike(self.params)
 
  @cache_readonly
  def bse(self):
    params = self.params
    hess = self.model.hessian(params)
    if len(params) == 1: # can't take an inverse, ensure 1d
      return np.sqrt(-1./hess[0])
    return np.sqrt(np.diag(-inv(hess)))
 
  def cov_params(self): # add scale argument&#63;
    params = self.params
    hess = self.model.hessian(params)
    return -inv(hess)
 
  @cache_readonly
  def aic(self):
    return -2 * self.llf + 2 * self._ic_df_model
 
  @cache_readonly
  def bic(self):
    nobs = self.nobs
    return -2 * self.llf + np.log(nobs) * self._ic_df_model
 
  @cache_readonly
  def hqic(self):
    nobs = self.nobs
    return -2 * self.llf + 2 * np.log(np.log(nobs)) * self._ic_df_model
 
  @cache_readonly
  def fittedvalues(self):
    model = self.model
    endog = model.endog.copy()
    k_ar = self.k_ar
    exog = model.exog # this is a copy
    if exog is not None:
      if model.method == "css" and k_ar > 0:
        exog = exog[k_ar:]
    if model.method == "css" and k_ar > 0:
      endog = endog[k_ar:]
    fv = endog - self.resid
    # add deterministic part back in
    #k = self.k_exog + self.k_trend
    #TODO: this needs to be commented out for MLE with constant
    #if k != 0:
    #  fv += dot(exog, self.params[:k])
    return fv
 
  @cache_readonly
  def resid(self):
    return self.model.geterrors(self.params)
 
  @cache_readonly
  def pvalues(self):
  #TODO: same for conditional and unconditional&#63;
    df_resid = self.df_resid
    return t.sf(np.abs(self.tvalues), df_resid) * 2
 
  def predict(self, start=None, end=None, exog=None, dynamic=False):
    return self.model.predict(self.params, start, end, exog, dynamic)
  predict.__doc__ = _arma_results_predict
 
  def _forecast_error(self, steps):
    sigma2 = self.sigma2
    ma_rep = arma2ma(np.r_[1, -self.arparams],
             np.r_[1, self.maparams], nobs=steps)
 
    fcasterr = np.sqrt(sigma2 * np.cumsum(ma_rep**2))
    return fcasterr
 
  def _forecast_conf_int(self, forecast, fcasterr, alpha):
    const = norm.ppf(1 - alpha / 2.)
    conf_int = np.c_[forecast - const * fcasterr,
             forecast + const * fcasterr]
 
    return conf_int
 
  def forecast(self, steps=1, exog=None, alpha=.05):
    """
    Out-of-sample forecasts
 
    Parameters
    ----------
    steps : int
      The number of out of sample forecasts from the end of the
      sample.
    exog : array
      If the model is an ARMAX, you must provide out of sample
      values for the exogenous variables. This should not include
      the constant.
    alpha : float
      The confidence intervals for the forecasts are (1 - alpha) %
 
    Returns
    -------
    forecast : array
      Array of out of sample forecasts
    stderr : array
      Array of the standard error of the forecasts.
    conf_int : array
      2d array of the confidence interval for the forecast
    """
    if exog is not None:
      #TODO: make a convenience function for this. we're using the
      # pattern elsewhere in the codebase
      exog = np.asarray(exog)
      if self.k_exog == 1 and exog.ndim == 1:
        exog = exog[:, None]
      elif exog.ndim == 1:
        if len(exog) != self.k_exog:
          raise ValueError("1d exog given and len(exog) != k_exog")
        exog = exog[None, :]
      if exog.shape[0] != steps:
        raise ValueError("new exog needed for each step")
      # prepend in-sample exog observations
      exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
               exog))
 
    forecast, ct = _arma_predict_out_of_sample(self.params,
                        steps, self.resid, self.k_ar,
                        self.k_ma, self.k_trend,
                        self.k_exog, self.model.endog,
                        exog, method=self.model.method)
    self.constant = ct
 
    # compute the standard errors
    fcasterr = self._forecast_error(steps)
    conf_int = self._forecast_conf_int(forecast, fcasterr, alpha)
 
    return forecast, fcasterr, conf_int
 
  def summary(self, alpha=.05):
    """Summarize the Model
 
    Parameters
    ----------
    alpha : float, optional
      Significance level for the confidence intervals.
 
    Returns
    -------
    smry : Summary instance
      This holds the summary table and text, which can be printed or
      converted to various output formats.
 
    See Also
    --------
    statsmodels.iolib.summary.Summary
    """
    from statsmodels.iolib.summary import Summary
    model = self.model
    title = model.__class__.__name__ + ' Model Results'
    method = model.method
    # get sample TODO: make better sample machinery for estimation
    k_diff = getattr(self, 'k_diff', 0)
    if 'mle' in method:
      start = k_diff
    else:
      start = k_diff + self.k_ar
    if self.data.dates is not None:
      dates = self.data.dates
      sample = [dates[start].strftime('%m-%d-%Y')]
      sample += ['- ' + dates[-1].strftime('%m-%d-%Y')]
    else:
      sample = str(start) + ' - ' + str(len(self.data.orig_endog))
 
    k_ar, k_ma = self.k_ar, self.k_ma
    if not k_diff:
      order = str((k_ar, k_ma))
    else:
      order = str((k_ar, k_diff, k_ma))
    top_left = [('Dep. Variable:', None),
          ('Model:', [model.__class__.__name__ + order]),
          ('Method:', [method]),
          ('Date:', None),
          ('Time:', None),
          ('Sample:', [sample[0]]),
          ('', [sample[1]])
          ]
 
    top_right = [
           ('No. Observations:', [str(len(self.model.endog))]),
           ('Log Likelihood', ["%#5.3f" % self.llf]),
           ('S.D. of innovations', ["%#5.3f" % self.sigma2**.5]),
           ('AIC', ["%#5.3f" % self.aic]),
           ('BIC', ["%#5.3f" % self.bic]),
           ('HQIC', ["%#5.3f" % self.hqic])]
 
    smry = Summary()
    smry.add_table_2cols(self, gleft=top_left, gright=top_right,
               title=title)
    smry.add_table_params(self, alpha=alpha, use_t=False)
 
    # Make the roots table
    from statsmodels.iolib.table import SimpleTable
 
    if k_ma and k_ar:
      arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
      mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
      stubs = arstubs + mastubs
      roots = np.r_[self.arroots, self.maroots]
      freq = np.r_[self.arfreq, self.mafreq]
    elif k_ma:
      mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
      stubs = mastubs
      roots = self.maroots
      freq = self.mafreq
    elif k_ar:
      arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
      stubs = arstubs
      roots = self.arroots
      freq = self.arfreq
    else: # 0,0 model
      stubs = []
    if len(stubs): # not 0, 0
      modulus = np.abs(roots)
      data = np.column_stack((roots.real, roots.imag, modulus, freq))
      roots_table = SimpleTable(data,
                   headers=['      Real',
                        '     Imaginary',
                        '     Modulus',
                        '    Frequency'],
                   title="Roots",
                   stubs=stubs,
                   data_fmts=["%17.4f", "%+17.4fj",
                         "%17.4f", "%17.4f"])
 
      smry.tables.append(roots_table)
    return smry
 
  def summary2(self, title=None, alpha=.05, float_format="%.4f"):
    """Experimental summary function for ARIMA Results
 
    Parameters
    -----------
    title : string, optional
      Title for the top table. If not None, then this replaces the
      default title
    alpha : float
      significance level for the confidence intervals
    float_format: string
      print format for floats in parameters summary
 
    Returns
    -------
    smry : Summary instance
      This holds the summary table and text, which can be printed or
      converted to various output formats.
 
    See Also
    --------
    statsmodels.iolib.summary2.Summary : class to hold summary
      results
 
    """
    from pandas import DataFrame
    # get sample TODO: make better sample machinery for estimation
    k_diff = getattr(self, 'k_diff', 0)
    if 'mle' in self.model.method:
      start = k_diff
    else:
      start = k_diff + self.k_ar
    if self.data.dates is not None:
      dates = self.data.dates
      sample = [dates[start].strftime('%m-%d-%Y')]
      sample += [dates[-1].strftime('%m-%d-%Y')]
    else:
      sample = str(start) + ' - ' + str(len(self.data.orig_endog))
 
    k_ar, k_ma = self.k_ar, self.k_ma
 
    # Roots table
    if k_ma and k_ar:
      arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
      mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
      stubs = arstubs + mastubs
      roots = np.r_[self.arroots, self.maroots]
      freq = np.r_[self.arfreq, self.mafreq]
    elif k_ma:
      mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
      stubs = mastubs
      roots = self.maroots
      freq = self.mafreq
    elif k_ar:
      arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
      stubs = arstubs
      roots = self.arroots
      freq = self.arfreq
    else: # 0, 0 order
      stubs = []
 
    if len(stubs):
      modulus = np.abs(roots)
      data = np.column_stack((roots.real, roots.imag, modulus, freq))
      data = DataFrame(data)
      data.columns = ['Real', 'Imaginary', 'Modulus', 'Frequency']
      data.index = stubs
 
    # Summary
    from statsmodels.iolib import summary2
    smry = summary2.Summary()
 
    # Model info
    model_info = summary2.summary_model(self)
    model_info['Method:'] = self.model.method
    model_info['Sample:'] = sample[0]
    model_info['  '] = sample[-1]
    model_info['S.D. of innovations:'] = "%#5.3f" % self.sigma2**.5
    model_info['HQIC:'] = "%#5.3f" % self.hqic
    model_info['No. Observations:'] = str(len(self.model.endog))
 
    # Parameters
    params = summary2.summary_params(self)
    smry.add_dict(model_info)
    smry.add_df(params, float_format=float_format)
    if len(stubs):
      smry.add_df(data, float_format="%17.4f")
    smry.add_title(results=self, title=title)
 
    return smry
 
  def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
           alpha=.05, plot_insample=True, ax=None):
    from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
    _ = _import_mpl()
    fig, ax = create_mpl_ax(ax)
 
 
    # use predict so you set dates
    forecast = self.predict(start, end, exog, dynamic)
    # doing this twice. just add a plot keyword to predict&#63;
    start = self.model._get_predict_start(start, dynamic=False)
    end, out_of_sample = self.model._get_predict_end(end, dynamic=False)
 
    if out_of_sample:
      steps = out_of_sample
      fc_error = self._forecast_error(steps)
      conf_int = self._forecast_conf_int(forecast[-steps:], fc_error,
                        alpha)
 
 
    if hasattr(self.data, "predict_dates"):
      from pandas import TimeSeries
      forecast = TimeSeries(forecast, index=self.data.predict_dates)
      ax = forecast.plot(ax=ax, label='forecast')
    else:
      ax.plot(forecast)
 
    x = ax.get_lines()[-1].get_xdata()
    if out_of_sample:
      label = "{0:.0%} confidence interval".format(1 - alpha)
      ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
              color='gray', alpha=.5, label=label)
 
    if plot_insample:
      ax.plot(x[:end + 1 - start], self.model.endog[start:end+1],
          label=self.model.endog_names)
 
    ax.legend(loc='best')
 
    return fig
  plot_predict.__doc__ = _plot_predict
 
 
class ARMAResultsWrapper(wrap.ResultsWrapper):
  _attrs = {}
  _wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
                  _attrs)
  _methods = {}
  _wrap_methods = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_methods,
                   _methods)
wrap.populate_wrapper(ARMAResultsWrapper, ARMAResults)
 
 
class ARIMAResults(ARMAResults):
  def predict(self, start=None, end=None, exog=None, typ='linear',
        dynamic=False):
    return self.model.predict(self.params, start, end, exog, typ, dynamic)
  predict.__doc__ = _arima_results_predict
 
  def _forecast_error(self, steps):
    sigma2 = self.sigma2
    ma_rep = arma2ma(np.r_[1, -self.arparams],
             np.r_[1, self.maparams], nobs=steps)
 
    fcerr = np.sqrt(np.cumsum(cumsum_n(ma_rep, self.k_diff)**2)*sigma2)
    return fcerr
 
  def _forecast_conf_int(self, forecast, fcerr, alpha):
    const = norm.ppf(1 - alpha/2.)
    conf_int = np.c_[forecast - const*fcerr, forecast + const*fcerr]
    return conf_int
 
  def forecast(self, steps=1, exog=None, alpha=.05):
    """
    Out-of-sample forecasts
 
    Parameters
    ----------
    steps : int
      The number of out of sample forecasts from the end of the
      sample.
    exog : array
      If the model is an ARIMAX, you must provide out of sample
      values for the exogenous variables. This should not include
      the constant.
    alpha : float
      The confidence intervals for the forecasts are (1 - alpha) %
 
    Returns
    -------
    forecast : array
      Array of out of sample forecasts
    stderr : array
      Array of the standard error of the forecasts.
    conf_int : array
      2d array of the confidence interval for the forecast
 
    Notes
    -----
    Prediction is done in the levels of the original endogenous variable.
    If you would like prediction of differences in levels use `predict`.
    """
    if exog is not None:
      if self.k_exog == 1 and exog.ndim == 1:
        exog = exog[:, None]
      if exog.shape[0] != steps:
        raise ValueError("new exog needed for each step")
      # prepend in-sample exog observations
      exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
               exog))
    forecast, ct = _arma_predict_out_of_sample(self.params, steps, self.resid,
                        self.k_ar, self.k_ma,
                        self.k_trend, self.k_exog,
                        self.model.endog,
                        exog, method=self.model.method)
 
    #self.constant = ct
    d = self.k_diff
    endog = self.model.data.endog[-d:]
    forecast = unintegrate(forecast, unintegrate_levels(endog, d))[d:]
 
    # get forecast errors
    fcerr = self._forecast_error(steps)
    conf_int = self._forecast_conf_int(forecast, fcerr, alpha)
    return forecast, fcerr, conf_int
 
  def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
           alpha=.05, plot_insample=True, ax=None):
    from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
    _ = _import_mpl()
    fig, ax = create_mpl_ax(ax)
 
    # use predict so you set dates
    forecast = self.predict(start, end, exog, 'levels', dynamic)
    # doing this twice. just add a plot keyword to predict&#63;
    start = self.model._get_predict_start(start, dynamic=dynamic)
    end, out_of_sample = self.model._get_predict_end(end, dynamic=dynamic)
 
    if out_of_sample:
      steps = out_of_sample
      fc_error = self._forecast_error(steps)
      conf_int = self._forecast_conf_int(forecast[-steps:], fc_error,
                        alpha)
 
    if hasattr(self.data, "predict_dates"):
      from pandas import TimeSeries
      forecast = TimeSeries(forecast, index=self.data.predict_dates)
      ax = forecast.plot(ax=ax, label='forecast')
    else:
      ax.plot(forecast)
 
    x = ax.get_lines()[-1].get_xdata()
    if out_of_sample:
      label = "{0:.0%} confidence interval".format(1 - alpha)
      ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
              color='gray', alpha=.5, label=label)
 
    if plot_insample:
      import re
      k_diff = self.k_diff
      label = re.sub("D\d*\.", "", self.model.endog_names)
      levels = unintegrate(self.model.endog,
                 self.model._first_unintegrate)
      ax.plot(x[:end + 1 - start],
          levels[start + k_diff:end + k_diff + 1], label=label)
 
    ax.legend(loc='best')
 
    return fig
 
  plot_predict.__doc__ = _arima_plot_predict
 
 
class ARIMAResultsWrapper(ARMAResultsWrapper):
  pass
wrap.populate_wrapper(ARIMAResultsWrapper, ARIMAResults)
 
 
if __name__ == "__main__":
  import statsmodels.api as sm
 
  # simulate arma process
  from statsmodels.tsa.arima_process import arma_generate_sample
  y = arma_generate_sample([1., -.75], [1., .25], nsample=1000)
  arma = ARMA(y)
  res = arma.fit(trend='nc', order=(1, 1))
 
  np.random.seed(12345)
  y_arma22 = arma_generate_sample([1., -.85, .35], [1, .25, -.9],
                  nsample=1000)
  arma22 = ARMA(y_arma22)
  res22 = arma22.fit(trend='nc', order=(2, 2))
 
  # test CSS
  arma22_css = ARMA(y_arma22)
  res22css = arma22_css.fit(trend='nc', order=(2, 2), method='css')
 
  data = sm.datasets.sunspots.load()
  ar = ARMA(data.endog)
  resar = ar.fit(trend='nc', order=(9, 0))
 
  y_arma31 = arma_generate_sample([1, -.75, -.35, .25], [.1],
                  nsample=1000)
 
  arma31css = ARMA(y_arma31)
  res31css = arma31css.fit(order=(3, 1), method="css", trend="nc",
               transparams=True)
 
  y_arma13 = arma_generate_sample([1., -.75], [1, .25, -.5, .8],
                  nsample=1000)
  arma13css = ARMA(y_arma13)
  res13css = arma13css.fit(order=(1, 3), method='css', trend='nc')
 
# check css for p < q and q < p
  y_arma41 = arma_generate_sample([1., -.75, .35, .25, -.3], [1, -.35],
                  nsample=1000)
  arma41css = ARMA(y_arma41)
  res41css = arma41css.fit(order=(4, 1), trend='nc', method='css')
 
  y_arma14 = arma_generate_sample([1, -.25], [1., -.75, .35, .25, -.3],
                  nsample=1000)
  arma14css = ARMA(y_arma14)
  res14css = arma14css.fit(order=(4, 1), trend='nc', method='css')
 
  # ARIMA Model
  from statsmodels.datasets import webuse
  dta = webuse('wpi1')
  wpi = dta['wpi']
 
  mod = ARIMA(wpi, (1, 1, 1)).fit()