class_docstr: Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors
  model
  
  Parameters
  ----------
  endog : array_like
      The observed time-series process :math:`y`
  exog : array_like, optional
      Array of exogenous regressors, shaped nobs x k.
  order : iterable or iterable of iterables, optional
      The (p,d,q) order of the model for the number of AR parameters,
      differences, and MA parameters. `d` must be an integer
      indicating the integration order of the process, while
      `p` and `q` may either be an integers indicating the AR and MA
      orders (so that all lags up to those orders are included) or else
      iterables giving specific AR and / or MA lags to include. Default is
      an AR(1) model: (1,0,0).
  seasonal_order : iterable, optional
      The (P,D,Q,s) order of the seasonal component of the model for the
      AR parameters, differences, MA parameters, and periodicity.
      `D` must be an integer indicating the integration order of the process,
      while `P` and `Q` may either be an integers indicating the AR and MA
      orders (so that all lags up to those orders are included) or else
      iterables giving specific AR and / or MA lags to include. `s` is an
      integer giving the periodicity (number of periods in season), often it
      is 4 for quarterly data or 12 for monthly data. Default is no seasonal
      effect.
  trend : str{'n','c','t','ct'} or iterable, optional
      Parameter controlling the deterministic trend polynomial :math:`A(t)`.
      Can be specified as a string where 'c' indicates a constant (i.e. a
      degree zero component of the trend polynomial), 't' indicates a
      linear trend with time, and 'ct' is both. Can also be specified as an
      iterable defining the non-zero polynomial exponents to include, in
      increasing order. For example, `[1,1,0,1]` denotes
      :math:`a + bt + ct^3`. Default is to not include a trend component.
  measurement_error : bool, optional
      Whether or not to assume the endogenous observations `endog` were
      measured with error. Default is False.
  time_varying_regression : bool, optional
      Used when an explanatory variables, `exog`, are provided
      to select whether or not coefficients on the exogenous regressors are
      allowed to vary over time. Default is False.
  mle_regression : bool, optional
      Whether or not to use estimate the regression coefficients for the
      exogenous variables as part of maximum likelihood estimation or through
      the Kalman filter (i.e. recursive least squares). If
      `time_varying_regression` is True, this must be set to False. Default
      is True.
  simple_differencing : bool, optional
      Whether or not to use partially conditional maximum likelihood
      estimation. If True, differencing is performed prior to estimation,
      which discards the first :math:`s D + d` initial rows but results in a
      smaller state-space formulation. See the Notes section for important
      details about interpreting results when this option is used. If False,
      the full SARIMAX model is put in state-space form so that all
      datapoints can be used in estimation. Default is False.
  enforce_stationarity : bool, optional
      Whether or not to transform the AR parameters to enforce stationarity
      in the autoregressive component of the model. Default is True.
  enforce_invertibility : bool, optional
      Whether or not to transform the MA parameters to enforce invertibility
      in the moving average component of the model. Default is True.
  hamilton_representation : bool, optional
      Whether or not to use the Hamilton representation of an ARMA process
      (if True) or the Harvey representation (if False). Default is False.
  concentrate_scale : bool, optional
      Whether or not to concentrate the scale (variance of the error term)
      out of the likelihood. This reduces the number of parameters estimated
      by maximum likelihood by one, but standard errors will then not
      be available for the scale parameter.
  trend_offset : int, optional
      The offset at which to start time trend values. Default is 1, so that
      if `trend='t'` the trend is equal to 1, 2, ..., nobs. Typically is only
      set when the model created by extending a previous dataset.
  use_exact_diffuse : bool, optional
      Whether or not to use exact diffuse initialization for non-stationary
      states. Default is False (in which case approximate diffuse
      initialization is used).
  **kwargs
      Keyword arguments may be used to provide default values for state space
      matrices or for Kalman filtering options. See `Representation`, and
      `KalmanFilter` for more details.
  
  Attributes
  ----------
  measurement_error : bool
      Whether or not to assume the endogenous
      observations `endog` were measured with error.
  state_error : bool
      Whether or not the transition equation has an error component.
  mle_regression : bool
      Whether or not the regression coefficients for
      the exogenous variables were estimated via maximum
      likelihood estimation.
  state_regression : bool
      Whether or not the regression coefficients for
      the exogenous variables are included as elements
      of the state space and estimated via the Kalman
      filter.
  time_varying_regression : bool
      Whether or not coefficients on the exogenous
      regressors are allowed to vary over time.
  simple_differencing : bool
      Whether or not to use partially conditional maximum likelihood
      estimation.
  enforce_stationarity : bool
      Whether or not to transform the AR parameters
      to enforce stationarity in the autoregressive
      component of the model.
  enforce_invertibility : bool
      Whether or not to transform the MA parameters
      to enforce invertibility in the moving average
      component of the model.
  hamilton_representation : bool
      Whether or not to use the Hamilton representation of an ARMA process.
  trend : str{'n','c','t','ct'} or iterable
      Parameter controlling the deterministic
      trend polynomial :math:`A(t)`. See the class
      parameter documentation for more information.
  polynomial_ar : ndarray
      Array containing autoregressive lag polynomial lags, ordered from
      lowest degree to highest. The polynomial begins with lag 0.
      Initialized with ones, unless a coefficient is constrained to be
      zero (in which case it is zero).
  polynomial_ma : ndarray
      Array containing moving average lag polynomial lags, ordered from
      lowest degree to highest. Initialized with ones, unless a coefficient
      is constrained to be zero (in which case it is zero).
  polynomial_seasonal_ar : ndarray
      Array containing seasonal moving average lag
      polynomial lags, ordered from lowest degree
      to highest. Initialized with ones, unless a
      coefficient is constrained to be zero (in which
      case it is zero).
  polynomial_seasonal_ma : ndarray
      Array containing seasonal moving average lag
      polynomial lags, ordered from lowest degree
      to highest. Initialized with ones, unless a
      coefficient is constrained to be zero (in which
      case it is zero).
  polynomial_trend : ndarray
      Array containing trend polynomial coefficients,
      ordered from lowest degree to highest. Initialized
      with ones, unless a coefficient is constrained to be
      zero (in which case it is zero).
  k_ar : int
      Highest autoregressive order in the model, zero-indexed.
  k_ar_params : int
      Number of autoregressive parameters to be estimated.
  k_diff : int
      Order of integration.
  k_ma : int
      Highest moving average order in the model, zero-indexed.
  k_ma_params : int
      Number of moving average parameters to be estimated.
  seasonal_periods : int
      Number of periods in a season.
  k_seasonal_ar : int
      Highest seasonal autoregressive order in the model, zero-indexed.
  k_seasonal_ar_params : int
      Number of seasonal autoregressive parameters to be estimated.
  k_seasonal_diff : int
      Order of seasonal integration.
  k_seasonal_ma : int
      Highest seasonal moving average order in the model, zero-indexed.
  k_seasonal_ma_params : int
      Number of seasonal moving average parameters to be estimated.
  k_trend : int
      Order of the trend polynomial plus one (i.e. the constant polynomial
      would have `k_trend=1`).
  k_exog : int
      Number of exogenous regressors.
  
  Notes
  -----
  The SARIMA model is specified :math:`(p, d, q) \times (P, D, Q)_s`.
  
  .. math::
  
      \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D y_t = A(t) +
          \theta_q (L) \tilde \theta_Q (L^s) \zeta_t
  
  In terms of a univariate structural model, this can be represented as
  
  .. math::
  
      y_t & = u_t + \eta_t \\
      \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) +
          \theta_q (L) \tilde \theta_Q (L^s) \zeta_t
  
  where :math:`\eta_t` is only applicable in the case of measurement error
  (although it is also used in the case of a pure regression model, i.e. if
  p=q=0).
  
  In terms of this model, regression with SARIMA errors can be represented
  easily as
  
  .. math::
  
      y_t & = \beta_t x_t + u_t \\
      \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) +
          \theta_q (L) \tilde \theta_Q (L^s) \zeta_t
  
  this model is the one used when exogenous regressors are provided.
  
  Note that the reduced form lag polynomials will be written as:
  
  .. math::
  
      \Phi (L) \equiv \phi_p (L) \tilde \phi_P (L^s) \\
      \Theta (L) \equiv \theta_q (L) \tilde \theta_Q (L^s)
  
  If `mle_regression` is True, regression coefficients are treated as
  additional parameters to be estimated via maximum likelihood. Otherwise
  they are included as part of the state with a diffuse initialization.
  In this case, however, with approximate diffuse initialization, results
  can be sensitive to the initial variance.
  
  This class allows two different underlying representations of ARMA models
  as state space models: that of Hamilton and that of Harvey. Both are
  equivalent in the sense that they are analytical representations of the
  ARMA model, but the state vectors of each have different meanings. For
  this reason, maximum likelihood does not result in identical parameter
  estimates and even the same set of parameters will result in different
  loglikelihoods.
  
  The Harvey representation is convenient because it allows integrating
  differencing into the state vector to allow using all observations for
  estimation.
  
  In this implementation of differenced models, the Hamilton representation
  is not able to accommodate differencing in the state vector, so
  `simple_differencing` (which performs differencing prior to estimation so
  that the first d + sD observations are lost) must be used.
  
  Many other packages use the Hamilton representation, so that tests against
  Stata and R require using it along with simple differencing (as Stata
  does).
  
  If `filter_concentrated = True` is used, then the scale of the model is
  concentrated out of the likelihood. A benefit of this is that there the
  dimension of the parameter vector is reduced so that numerical maximization
  of the log-likelihood function may be faster and more stable. If this
  option in a model with measurement error, it is important to note that the
  estimated measurement error parameter will be relative to the scale, and
  is named "snr.measurement_error" instead of "var.measurement_error". To
  compute the variance of the measurement error in this case one would
  multiply `snr.measurement_error` parameter by the scale.
  
  If `simple_differencing = True` is used, then the `endog` and `exog` data
  are differenced prior to putting the model in state-space form. This has
  the same effect as if the user differenced the data prior to constructing
  the model, which has implications for using the results:
  
  - Forecasts and predictions will be about the *differenced* data, not about
    the original data. (while if `simple_differencing = False` is used, then
    forecasts and predictions will be about the original data).
  - If the original data has an Int64Index, a new RangeIndex will be created
    for the differenced data that starts from one, and forecasts and
    predictions will use this new index.
  
  Detailed information about state space models can be found in [1]_. Some
  specific references are:
  
  - Chapter 3.4 describes ARMA and ARIMA models in state space form (using
    the Harvey representation), and gives references for basic seasonal
    models and models with a multiplicative form (for example the airline
    model). It also shows a state space model for a full ARIMA process (this
    is what is done here if `simple_differencing=False`).
  - Chapter 3.6 describes estimating regression effects via the Kalman filter
    (this is performed if `mle_regression` is False), regression with
    time-varying coefficients, and regression with ARMA errors (recall from
    above that if regression effects are present, the model estimated by this
    class is regression with SARIMA errors).
  - Chapter 8.4 describes the application of an ARMA model to an example
    dataset. A replication of this section is available in an example
    IPython notebook in the documentation.
  
  References
  ----------
  .. [1] Durbin, James, and Siem Jan Koopman. 2012.
     Time Series Analysis by State Space Methods: Second Edition.
     Oxford University Press.
class_docstr_tok: ["Seasonal","AutoRegressive","Integrated","Moving","Average","with","eXogenous","regressors","model","Parameters","--","--","--","--","--","endog",":","array_like","The","observed","time-series","process",":","math",":","`","y","`","exog",":","array_like",",","optional","Array","of","exogenous","regressors",",","shaped","nobs","x","k.","order",":","iterable","or","iterable","of","iterables",",","optional","The","(","p",",","d",",","q",")","order","of","the","model","for","the","number","of","AR","parameters",",","differences",",","and","MA","parameters",".","`","d","`","must","be","an","integer","indicating","the","integration","order","of","the","process",",","while","`","p","`","and","`","q","`","may","either","be","an","integers","indicating","the","AR","and","MA","orders","(","so","that","all","lags","up","to","those","orders","are","included",")","or","else","iterables","giving","specific","AR","and","\/","or","MA","lags","to","include",".","Default","is","an","AR","(","1",")","model",":","(","1,0,0",")",".","seasonal_order",":","iterable",",","optional","The","(","P",",","D",",","Q",",","s",")","order","of","the","seasonal","component","of","the","model","for","the","AR","parameters",",","differences",",","MA","parameters",",","and","periodicity",".","`","D","`","must","be","an","integer","indicating","the","integration","order","of","the","process",",","while","`","P","`","and","`","Q","`","may","either","be","an","integers","indicating","the","AR","and","MA","orders","(","so","that","all","lags","up","to","those","orders","are","included",")","or","else","iterables","giving","specific","AR","and","\/","or","MA","lags","to","include",".","`","s","`","is","an","integer","giving","the","periodicity","(","number","of","periods","in","season",")",",","often","it","is","4","for","quarterly","data","or","12","for","monthly","data",".","Default","is","no","seasonal","effect",".","trend",":","str","{","'n","'",",","'","c","'",",","'t","'",",","'ct","'","}","or","iterable",",","optional","Parameter","controlling","the","deterministic","trend","polynomial",":","math",":","`","A","(","t",")","`",".","Can","be","specified","as","a","string","where","'","c","'","indicates","a","constant","(","i.e",".","a","degree","zero","component","of","the","trend","polynomial",")",",","'t","'","indicates","a","linear","trend","with","time",",","and","'ct","'","is","both",".","Can","also","be","specified","as","an","iterable","defining","the","non-zero","polynomial","exponents","to","include",",","in","increasing","order",".","For","example",",","`","[","1,1,0,1","]","`","denotes",":","math",":","`","a","+","bt","+","ct^3","`",".","Default","is","to","not","include","a","trend","component",".","measurement_error",":","bool",",","optional","Whether","or","not","to","assume","the","endogenous","observations","`","endog","`","were","measured","with","error",".","Default","is","False",".","time_varying_regression",":","bool",",","optional","Used","when","an","explanatory","variables",",","`","exog","`",",","are","provided","to","select","whether","or","not","coefficients","on","the","exogenous","regressors","are","allowed","to","vary","over","time",".","Default","is","False",".","mle_regression",":","bool",",","optional","Whether","or","not","to","use","estimate","the","regression","coefficients","for","the","exogenous","variables","as","part","of","maximum","likelihood","estimation","or","through","the","Kalman","filter","(","i.e",".","recursive","least","squares",")",".","If","`","time_varying_regression","`","is","True",",","this","must","be","set","to","False",".","Default","is","True",".","simple_differencing",":","bool",",","optional","Whether","or","not","to","use","partially","conditional","maximum","likelihood","estimation",".","If","True",",","differencing","is","performed","prior","to","estimation",",","which","discards","the","first",":","math",":","`","s","D","+","d","`","initial","rows","but","results","in","a","smaller","state-space","formulation",".","See","the","Notes","section","for","important","details","about","interpreting","results","when","this","option","is","used",".","If","False",",","the","full","SARIMAX","model","is","put","in","state-space","form","so","that","all","datapoints","can","be","used","in","estimation",".","Default","is","False",".","enforce_stationarity",":","bool",",","optional","Whether","or","not","to","transform","the","AR","parameters","to","enforce","stationarity","in","the","autoregressive","component","of","the","model",".","Default","is","True",".","enforce_invertibility",":","bool",",","optional","Whether","or","not","to","transform","the","MA","parameters","to","enforce","invertibility","in","the","moving","average","component","of","the","model",".","Default","is","True",".","hamilton_representation",":","bool",",","optional","Whether","or","not","to","use","the","Hamilton","representation","of","an","ARMA","process","(","if","True",")","or","the","Harvey","representation","(","if","False",")",".","Default","is","False",".","concentrate_scale",":","bool",",","optional","Whether","or","not","to","concentrate","the","scale","(","variance","of","the","error","term",")","out","of","the","likelihood",".","This","reduces","the","number","of","parameters","estimated","by","maximum","likelihood","by","one",",","but","standard","errors","will","then","not","be","available","for","the","scale","parameter",".","trend_offset",":","int",",","optional","The","offset","at","which","to","start","time","trend","values",".","Default","is","1",",","so","that","if","`","trend='t","'","`","the","trend","is","equal","to","1",",","2",",","...",",","nobs",".","Typically","is","only","set","when","the","model","created","by","extending","a","previous","dataset",".","use_exact_diffuse",":","bool",",","optional","Whether","or","not","to","use","exact","diffuse","initialization","for","non-stationary","states",".","Default","is","False","(","in","which","case","approximate","diffuse","initialization","is","used",")",".","*","*","kwargs","Keyword","arguments","may","be","used","to","provide","default","values","for","state","space","matrices","or","for","Kalman","filtering","options",".","See","`","Representation","`",",","and","`","KalmanFilter","`","for","more","details",".","Attributes","--","--","--","--","--","measurement_error",":","bool","Whether","or","not","to","assume","the","endogenous","observations","`","endog","`","were","measured","with","error",".","state_error",":","bool","Whether","or","not","the","transition","equation","has","an","error","component",".","mle_regression",":","bool","Whether","or","not","the","regression","coefficients","for","the","exogenous","variables","were","estimated","via","maximum","likelihood","estimation",".","state_regression",":","bool","Whether","or","not","the","regression","coefficients","for","the","exogenous","variables","are","included","as","elements","of","the","state","space","and","estimated","via","the","Kalman","filter",".","time_varying_regression",":","bool","Whether","or","not","coefficients","on","the","exogenous","regressors","are","allowed","to","vary","over","time",".","simple_differencing",":","bool","Whether","or","not","to","use","partially","conditional","maximum","likelihood","estimation",".","enforce_stationarity",":","bool","Whether","or","not","to","transform","the","AR","parameters","to","enforce","stationarity","in","the","autoregressive","component","of","the","model",".","enforce_invertibility",":","bool","Whether","or","not","to","transform","the","MA","parameters","to","enforce","invertibility","in","the","moving","average","component","of","the","model",".","hamilton_representation",":","bool","Whether","or","not","to","use","the","Hamilton","representation","of","an","ARMA","process",".","trend",":","str","{","'n","'",",","'","c","'",",","'t","'",",","'ct","'","}","or","iterable","Parameter","controlling","the","deterministic","trend","polynomial",":","math",":","`","A","(","t",")","`",".","See","the","class","parameter","documentation","for","more","information",".","polynomial_ar",":","ndarray","Array","containing","autoregressive","lag","polynomial","lags",",","ordered","from","lowest","degree","to","highest",".","The","polynomial","begins","with","lag","0",".","Initialized","with","ones",",","unless","a","coefficient","is","constrained","to","be","zero","(","in","which","case","it","is","zero",")",".","polynomial_ma",":","ndarray","Array","containing","moving","average","lag","polynomial","lags",",","ordered","from","lowest","degree","to","highest",".","Initialized","with","ones",",","unless","a","coefficient","is","constrained","to","be","zero","(","in","which","case","it","is","zero",")",".","polynomial_seasonal_ar",":","ndarray","Array","containing","seasonal","moving","average","lag","polynomial","lags",",","ordered","from","lowest","degree","to","highest",".","Initialized","with","ones",",","unless","a","coefficient","is","constrained","to","be","zero","(","in","which","case","it","is","zero",")",".","polynomial_seasonal_ma",":","ndarray","Array","containing","seasonal","moving","average","lag","polynomial","lags",",","ordered","from","lowest","degree","to","highest",".","Initialized","with","ones",",","unless","a","coefficient","is","constrained","to","be","zero","(","in","which","case","it","is","zero",")",".","polynomial_trend",":","ndarray","Array","containing","trend","polynomial","coefficients",",","ordered","from","lowest","degree","to","highest",".","Initialized","with","ones",",","unless","a","coefficient","is","constrained","to","be","zero","(","in","which","case","it","is","zero",")",".","k_ar",":","int","Highest","autoregressive","order","in","the","model",",","zero-indexed",".","k_ar_params",":","int","Number","of","autoregressive","parameters","to","be","estimated",".","k_diff",":","int","Order","of","integration",".","k_ma",":","int","Highest","moving","average","order","in","the","model",",","zero-indexed",".","k_ma_params",":","int","Number","of","moving","average","parameters","to","be","estimated",".","seasonal_periods",":","int","Number","of","periods","in","a","season",".","k_seasonal_ar",":","int","Highest","seasonal","autoregressive","order","in","the","model",",","zero-indexed",".","k_seasonal_ar_params",":","int","Number","of","seasonal","autoregressive","parameters","to","be","estimated",".","k_seasonal_diff",":","int","Order","of","seasonal","integration",".","k_seasonal_ma",":","int","Highest","seasonal","moving","average","order","in","the","model",",","zero-indexed",".","k_seasonal_ma_params",":","int","Number","of","seasonal","moving","average","parameters","to","be","estimated",".","k_trend",":","int","Order","of","the","trend","polynomial","plus","one","(","i.e",".","the","constant","polynomial","would","have","`","k_trend=1","`",")",".","k_exog",":","int","Number","of","exogenous","regressors",".","Notes","--","--","-","The","SARIMA","model","is","specified",":","math",":","`","(","p",",","d",",","q",")","\\times","(","P",",","D",",","Q",")","_s","`",".","..","math",":",":","\\phi_p","(","L",")","\\tilde","\\phi_P","(","L^s",")","\\Delta^d","\\Delta_s^D","y_t","=","A","(","t",")","+","\\theta_q","(","L",")","\\tilde","\\theta_Q","(","L^s",")","\\zeta_t","In","terms","of","a","univariate","structural","model",",","this","can","be","represented","as","..","math",":",":","y_t","&","=","u_t","+","\\eta_t","\\\\","\\phi_p","(","L",")","\\tilde","\\phi_P","(","L^s",")","\\Delta^d","\\Delta_s^D","u_t","&","=","A","(","t",")","+","\\theta_q","(","L",")","\\tilde","\\theta_Q","(","L^s",")","\\zeta_t","where",":","math",":","`","\\eta_t","`","is","only","applicable","in","the","case","of","measurement","error","(","although","it","is","also","used","in","the","case","of","a","pure","regression","model",",","i.e",".","if","p=q=0",")",".","In","terms","of","this","model",",","regression","with","SARIMA","errors","can","be","represented","easily","as","..","math",":",":","y_t","&","=","\\beta_t","x_t","+","u_t","\\\\","\\phi_p","(","L",")","\\tilde","\\phi_P","(","L^s",")","\\Delta^d","\\Delta_s^D","u_t","&","=","A","(","t",")","+","\\theta_q","(","L",")","\\tilde","\\theta_Q","(","L^s",")","\\zeta_t","this","model","is","the","one","used","when","exogenous","regressors","are","provided",".","Note","that","the","reduced","form","lag","polynomials","will","be","written","as",":","..","math",":",":","\\Phi","(","L",")","\\equiv","\\phi_p","(","L",")","\\tilde","\\phi_P","(","L^s",")","\\\\","\\Theta","(","L",")","\\equiv","\\theta_q","(","L",")","\\tilde","\\theta_Q","(","L^s",")","If","`","mle_regression","`","is","True",",","regression","coefficients","are","treated","as","additional","parameters","to","be","estimated","via","maximum","likelihood",".","Otherwise","they","are","included","as","part","of","the","state","with","a","diffuse","initialization",".","In","this","case",",","however",",","with","approximate","diffuse","initialization",",","results","can","be","sensitive","to","the","initial","variance",".","This","class","allows","two","different","underlying","representations","of","ARMA","models","as","state","space","models",":","that","of","Hamilton","and","that","of","Harvey",".","Both","are","equivalent","in","the","sense","that","they","are","analytical","representations","of","the","ARMA","model",",","but","the","state","vectors","of","each","have","different","meanings",".","For","this","reason",",","maximum","likelihood","does","not","result","in","identical","parameter","estimates","and","even","the","same","set","of","parameters","will","result","in","different","loglikelihoods",".","The","Harvey","representation","is","convenient","because","it","allows","integrating","differencing","into","the","state","vector","to","allow","using","all","observations","for","estimation",".","In","this","implementation","of","differenced","models",",","the","Hamilton","representation","is","not","able","to","accommodate","differencing","in","the","state","vector",",","so","`","simple_differencing","`","(","which","performs","differencing","prior","to","estimation","so","that","the","first","d","+","sD","observations","are","lost",")","must","be","used",".","Many","other","packages","use","the","Hamilton","representation",",","so","that","tests","against","Stata","and","R","require","using","it","along","with","simple","differencing","(","as","Stata","does",")",".","If","`","filter_concentrated","=","True","`","is","used",",","then","the","scale","of","the","model","is","concentrated","out","of","the","likelihood",".","A","benefit","of","this","is","that","there","the","dimension","of","the","parameter","vector","is","reduced","so","that","numerical","maximization","of","the","log-likelihood","function","may","be","faster","and","more","stable",".","If","this","option","in","a","model","with","measurement","error",",","it","is","important","to","note","that","the","estimated","measurement","error","parameter","will","be","relative","to","the","scale",",","and","is","named","``","snr.measurement_error","''","instead","of","``","var.measurement_error","''",".","To","compute","the","variance","of","the","measurement","error","in","this","case","one","would","multiply","`","snr.measurement_error","`","parameter","by","the","scale",".","If","`","simple_differencing","=","True","`","is","used",",","then","the","`","endog","`","and","`","exog","`","data","are","differenced","prior","to","putting","the","model","in","state-space","form",".","This","has","the","same","effect","as","if","the","user","differenced","the","data","prior","to","constructing","the","model",",","which","has","implications","for","using","the","results",":","-","Forecasts","and","predictions","will","be","about","the","*","differenced","*","data",",","not","about","the","original","data",".","(","while","if","`","simple_differencing","=","False","`","is","used",",","then","forecasts","and","predictions","will","be","about","the","original","data",")",".","-","If","the","original","data","has","an","Int64Index",",","a","new","RangeIndex","will","be","created","for","the","differenced","data","that","starts","from","one",",","and","forecasts","and","predictions","will","use","this","new","index",".","Detailed","information","about","state","space","models","can","be","found","in","[","1","]","_",".","Some","specific","references","are",":","-","Chapter","3.4","describes","ARMA","and","ARIMA","models","in","state","space","form","(","using","the","Harvey","representation",")",",","and","gives","references","for","basic","seasonal","models","and","models","with","a","multiplicative","form","(","for","example","the","airline","model",")",".","It","also","shows","a","state","space","model","for","a","full","ARIMA","process","(","this","is","what","is","done","here","if","`","simple_differencing=False","`",")",".","-","Chapter","3.6","describes","estimating","regression","effects","via","the","Kalman","filter","(","this","is","performed","if","`","mle_regression","`","is","False",")",",","regression","with","time-varying","coefficients",",","and","regression","with","ARMA","errors","(","recall","from","above","that","if","regression","effects","are","present",",","the","model","estimated","by","this","class","is","regression","with","SARIMA","errors",")",".","-","Chapter","8.4","describes","the","application","of","an","ARMA","model","to","an","example","dataset",".","A","replication","of","this","section","is","available","in","an","example","IPython","notebook","in","the","documentation",".","References","--","--","--","--","--","..","[","1","]","Durbin",",","James",",","and","Siem","Jan","Koopman",".","2012",".","Time","Series","Analysis","by","State","Space","Methods",":","Second","Edition",".","Oxford","University","Press","."]
docstr: Initial selection matrix
docstr_tok: ["Initial","selection","matrix"]
code: def initial_selection(self):
      """Initial selection matrix"""
      if not (self.state_regression and self.time_varying_regression):
          if self.k_posdef > 0:
              selection = np.r_[
                  [0] * (self._k_states_diff),
                  [1] * (self._k_order > 0),
                  [0] * (self._k_order - 1),
                  [0] * ((1 - self.mle_regression) * self._k_exog),
              ][:, None]
  
              if len(selection) == 0:
                  selection = np.zeros((self.k_states, self.k_posdef))
          else:
              selection = np.zeros((self.k_states, 0))
      else:
          selection = np.zeros((self.k_states, self.k_posdef))
          # Typical state variance
          if self._k_order > 0:
              selection[0, 0] = 1
          # Time-varying regression coefficient variances
          for i in range(self._k_exog, 0, -1):
              selection[-i, -i] = 1
      return selection
  
imports: from warnings import warn
  import numpy
  import pandas
  from statsmodels.compat.pandas import Appender
  from statsmodels.tools.tools import Bunch
  from statsmodels.tools.data import _is_using_pandas
  from statsmodels.tools.decorators import cache_readonly
  import statsmodels.base.wrapper
  from statsmodels.tsa.arima.specification import SARIMAXSpecification
  from statsmodels.tsa.arima.params import SARIMAXParams
  from statsmodels.tsa.tsatools import lagmat
  from .initialization import Initialization
  from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper
  from .tools import companion_matrix, diff, is_invertible, constrain_stationary_univariate, unconstrain_stationary_univariate, prepare_exog, prepare_trend_spec, prepare_trend_data
label_desc: Use image node_id 8 for calling the SARIMAX obj's underlying member method code with example usage: obj.initial_selection() and returns: selection