U md@s dZddlZddlmZddlZddlZddlm Z ddl m Z ddl m Z ddlmZddlmmZddlmZd d lmZmZd d lmZmZmZd d lmZd d lm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&GdddeZ'GdddeZ(GdddeZ)e*e)e(dS)zs Vector Autoregressive Moving Average with eXogenous regressors model Author: Chad Fulton License: Simplified-BSD N)warn)Appender)Bunch)_is_using_pandas) var_model)EstimationWarning)INVERT_UNIVARIATESOLVE_LU)MLEModel MLEResultsMLEResultsWrapper)Initialization) is_invertibleconcat prepare_exog!constrain_stationary_multivariate#unconstrain_stationary_multivariateprepare_trend_specprepare_trend_datac seZdZdZdfd d Zd d d Zed dZeddZeddZ ddZ ddZ fddZ d!ddZ ejddZeejjd"fdd ZZS)#VARMAXuM Vector Autoregressive Moving Average with eXogenous regressors model Parameters ---------- endog : array_like The observed time-series process :math:`y`, , shaped nobs x k_endog. exog : array_like, optional Array of exogenous regressors, shaped nobs x k. order : iterable The (p,q) order of the model for the number of AR and MA parameters to use. 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 a constant trend component. error_cov_type : {'diagonal', 'unstructured'}, optional The structure of the covariance matrix of the error term, where "unstructured" puts no restrictions on the matrix and "diagonal" requires it to be a diagonal matrix (uncorrelated errors). Default is "unstructured". measurement_error : bool, optional Whether or not to assume the endogenous observations `endog` were measured with error. 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. 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. **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 ---------- order : iterable The (p,q) order of the model for the number of AR and MA parameters to use. trend : str{'n','c','t','ct'} or iterable 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`. error_cov_type : {'diagonal', 'unstructured'}, optional The structure of the covariance matrix of the error term, where "unstructured" puts no restrictions on the matrix and "diagonal" requires it to be a diagonal matrix (uncorrelated errors). Default is "unstructured". measurement_error : bool, optional Whether or not to assume the endogenous observations `endog` were measured with error. 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. Notes ----- Generically, the VARMAX model is specified (see for example chapter 18 of [1]_): .. math:: y_t = A(t) + A_1 y_{t-1} + \dots + A_p y_{t-p} + B x_t + \epsilon_t + M_1 \epsilon_{t-1} + \dots M_q \epsilon_{t-q} where :math:`\epsilon_t \sim N(0, \Omega)`, and where :math:`y_t` is a `k_endog x 1` vector. Additionally, this model allows considering the case where the variables are measured with error. Note that in the full VARMA(p,q) case there is a fundamental identification problem in that the coefficient matrices :math:`\{A_i, M_j\}` are not generally unique, meaning that for a given time series process there may be multiple sets of matrices that equivalently represent it. See Chapter 12 of [1]_ for more information. Although this class can be used to estimate VARMA(p,q) models, a warning is issued to remind users that no steps have been taken to ensure identification in this case. References ---------- .. [1] Lütkepohl, Helmut. 2007. New Introduction to Multiple Time Series Analysis. Berlin: Springer. 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Notes ----- We need special handling for simulating or forecasting with `exog` or trend, because if we had these then the last predicted_state has been set to NaN since we did not have the appropriate `exog` to create it. Since we handle trend in the same way as `exog`, we still have this issue when only trend is used without `exog`. rNr!rzPProvided exogenous values are not of the appropriate shape. 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