statsmodels.tsa.vector_ar.vecm.VECMResults¶
-
class
statsmodels.tsa.vector_ar.vecm.
VECMResults
(endog, exog, exog_coint, k_ar, coint_rank, alpha, beta, gamma, sigma_u, deterministic='nc', seasons=0, first_season=0, delta_y_1_T=None, y_lag1=None, delta_x=None, model=None, names=None, dates=None)[source]¶ Class for holding estimation related results of a vector error correction model (VECM).
Parameters: endog : ndarray (neqs x nobs_tot)
Array of observations.
exog : ndarray (nobs_tot x neqs) or None
Deterministic terms outside the cointegration relation.
exog_coint : ndarray (nobs_tot x neqs) or None
Deterministic terms inside the cointegration relation.
k_ar : int, >= 1
Lags in the VAR representation. This implies that the number of lags in the VEC representation (=lagged differences) equals \(k_{ar} - 1\).
coint_rank : int, 0 <= coint_rank <= neqs
Cointegration rank, equals the rank of the matrix \(\Pi\) and the number of columns of \(\alpha\) and \(\beta\).
alpha : ndarray (neqs x coint_rank)
Estimate for the parameter \(\alpha\) of a VECM.
beta : ndarray (neqs x coint_rank)
Estimate for the parameter \(\beta\) of a VECM.
gamma : ndarray (neqs x neqs*(k_ar-1))
Array containing the estimates of the \(k_{ar}-1\) parameter matrices \(\Gamma_1, \dots, \Gamma_{k_{ar}-1}\) of a VECM(\(k_{ar}-1\)). The submatrices are stacked horizontally from left to right.
sigma_u : ndarray (neqs x neqs)
Estimate of white noise process covariance matrix \(\Sigma_u\).
deterministic : str {
"nc"
,"co"
,"ci"
,"lo"
,"li"
}"nc"
- no deterministic terms"co"
- constant outside the cointegration relation"ci"
- constant within the cointegration relation"lo"
- linear trend outside the cointegration relation"li"
- linear trend within the cointegration relation
Combinations of these are possible (e.g.
"cili"
or"colo"
for linear trend with intercept). See the docstring of theVECM
-class for more information.seasons : int, default: 0
Number of periods in a seasonal cycle. 0 means no seasons.
first_season : int, default: 0
Season of the first observation.
delta_y_1_T : ndarray or None, default: None
Auxilliary array for internal computations. It will be calculated if not given as parameter.
y_lag1 : ndarray or None, default: None
Auxilliary array for internal computations. It will be calculated if not given as parameter.
delta_x : ndarray or None, default: None
Auxilliary array for internal computations. It will be calculated if not given as parameter.
model :
VECM
An instance of the
VECM
-class.names : list of str
Each str in the list represents the name of a variable of the time series.
dates : array-like
For example a DatetimeIndex of length nobs_tot.
References
[R82] (1, 2, 3, 4) Lütkepohl, H. 2005. New Introduction to Multiple Time Series Analysis. Springer. Attributes
llf
()Compute the VECM’s loglikelihood. stderr_params
()stderr_coint
()Standard errors of beta and deterministic terms inside the cointegration relation. tvalues_alpha
()pvalues_alpha
()cov_var_repr
()Gives the covariance matrix of the corresponding VAR-representation. fittedvalues
()Return the in-sample values of endog calculated by the model. resid
()Return the difference between observed and fitted values. nobs (int) Number of observations (excluding the presample). model (see Parameters) y_all (see endog in Parameters) exog (see Parameters) exog_coint (see Parameters) names (see Parameters) dates (see Parameters) neqs (int) Number of variables in the time series. k_ar (see Parameters) deterministic (see Parameters) seasons (see Parameters) first_season (see Parameters) alpha (see Parameters) beta (see Parameters) gamma (see Parameters) sigma_u (see Parameters) det_coef_coint (ndarray (#(determinist. terms inside the coint. rel.) x coint_rank)) Estimated coefficients for the all deterministic terms inside the cointegration relation. const_coint (ndarray (1 x coint_rank)) If there is a constant deterministic term inside the cointegration relation, then const_coint is the first row of det_coef_coint. Otherwise it’s an ndarray of zeros. lin_trend_coint (ndarray (1 x coint_rank)) If there is a linear deterministic term inside the cointegration relation, then lin_trend_coint contains the corresponding estimated coefficients. As such it represents the corresponding row of det_coef_coint. If there is no linear deterministic term inside the cointegration relation, then lin_trend_coint is an ndarray of zeros. exog_coint_coefs (ndarray (exog_coint.shape[1] x coint_rank) or None) If deterministic terms inside the cointegration relation are passed via the exog_coint parameter, then exog_coint_coefs contains the corresponding estimated coefficients. As such exog_coint_coefs represents the last rows of det_coef_coint. If no deterministic terms were passed via the exog_coint parameter, this attribute is None. det_coef (ndarray (neqs x #(deterministic terms outside the coint. rel.))) Estimated coefficients for the all deterministic terms outside the cointegration relation. const (ndarray (neqs x 1) or (neqs x 0)) If a constant deterministic term outside the cointegration is specified within the deterministic parameter, then const is the first column of det_coef_coint. Otherwise it’s an ndarray of size zero. seasonal (ndarray (neqs x seasons)) If the seasons parameter is > 0, then seasonal contains the estimated coefficients corresponding to the seasonal terms. Otherwise it’s an ndarray of size zero. lin_trend (ndarray (neqs x 1) or (neqs x 0)) If a linear deterministic term outside the cointegration is specified within the deterministic parameter, then lin_trend contains the corresponding estimated coefficients. As such it represents the corresponding column of det_coef_coint. If there is no linear deterministic term outside the cointegration relation, then lin_trend is an ndarray of size zero. exog_coefs (ndarray (neqs x exog_coefs.shape[1])) If deterministic terms outside the cointegration relation are passed via the exog parameter, then exog_coefs contains the corresponding estimated coefficients. As such exog_coefs represents the last columns of det_coef. If no deterministic terms were passed via the exog parameter, this attribute is an ndarray of size zero. _delta_y_1_T (see delta_y_1_T in Parameters) _y_lag1 (see y_lag1 in Parameters) _delta_x (see delta_x in Parameters) coint_rank (int) Cointegration rank, equals the rank of the matrix \(\Pi\) and the number of columns of \(\alpha\) and \(\beta\). cov_params (ndarray (d x d)) Covariance matrix of the parameters. The number of rows and columns, d (used in the dimension specification of this argument), is equal to neqs * (neqs+num_det_coef_coint + neqs*(k_ar-1)+number of deterministic dummy variables outside the cointegration relation). For the case with no deterministic terms this matrix is defined on p. 287 in [R82] as \(\Sigma_{co}\) and its relationship to the ML-estimators can be seen in eq. (7.2.21) on p. 296 in [R82]. cov_params_wo_det (ndarray) Covariance matrix of the parameters \(\tilde{\Pi}, \tilde{\Gamma}\) where \(\tilde{\Pi} = \tilde{\alpha} \tilde{\beta'}\). Equals cov_params without the rows and columns related to deterministic terms. This matrix is defined as \(\Sigma_{co}\) on p. 287 in [R82]. stderr_alpha ( ndarray (neqs x coint_rank)) The standard errors of \(\alpha\). stderr_beta (ndarray (neqs x coint_rank)) The standard errors of \(\beta\). stderr_det_coef_coint (ndarray (num_det_coef_coint x coint_rank)) The standard errors of estimated the parameters related to deterministic terms inside the cointegration relation. stderr_gamma (ndarray (neqs x neqs*(k_ar-1))) The standard errors of \(\Gamma_1, \ldots, \Gamma_{k_{ar}-1}\). stderr_det_coef (ndarray (neqs x det. terms outside the coint. relation)) The standard errors of estimated the parameters related to deterministic terms outside the cointegration relation. tvalues_beta (ndarray (neqs x coint_rank)) tvalues_det_coef_coint (ndarray (num_det_coef_coint x coint_rank)) tvalues_gamma (ndarray (neqs x neqs*(k_ar-1))) tvalues_det_coef (ndarray (neqs x det. terms outside the coint. relation)) pvalues_beta (ndarray (neqs x coint_rank)) pvalues_det_coef_coint (ndarray (num_det_coef_coint x coint_rank)) pvalues_gamma (ndarray (neqs x neqs*(k_ar-1))) pvalues_det_coef (ndarray (neqs x det. terms outside the coint. relation)) var_rep ((k_ar x neqs x neqs)) KxK parameter matrices \(A_i\) of the corresponding VAR representation. If the return value is assigned to a variable A
, these matrices can be accessed viaA[i]
for \(i=0, \ldots, k_{ar}-1\).Methods
conf_int_alpha
([alpha])conf_int_beta
([alpha])conf_int_det_coef
([alpha])conf_int_det_coef_coint
([alpha])conf_int_gamma
([alpha])cov_params_default
()cov_params_wo_det
()cov_var_repr
()Gives the covariance matrix of the corresponding VAR-representation. fittedvalues
()Return the in-sample values of endog calculated by the model. irf
([periods])llf
()Compute the VECM’s loglikelihood. ma_rep
([maxn])orth_ma_rep
([maxn, P])Compute orthogonalized MA coefficient matrices. plot_data
([with_presample])Plot the input time series. plot_forecast
(steps[, alpha, plot_conf_int, …])Plot the forecast. predict
([steps, alpha, exog_fc, exog_coint_fc])Calculate future values of the time series. pvalues_alpha
()pvalues_beta
()pvalues_det_coef
()pvalues_det_coef_coint
()pvalues_gamma
()resid
()Return the difference between observed and fitted values. stderr_alpha
()stderr_beta
()stderr_coint
()Standard errors of beta and deterministic terms inside the cointegration relation. stderr_det_coef
()stderr_det_coef_coint
()stderr_gamma
()stderr_params
()summary
([alpha])Return a summary of the estimation results. test_granger_causality
(caused[, causing, signif])Test for Granger-causality. test_inst_causality
(causing[, signif])Test for instantaneous causality. test_normality
([signif])Test assumption of normal-distributed errors using Jarque-Bera-style omnibus \(\\chi^2\) test. test_whiteness
([nlags, signif, adjusted])Test the whiteness of the residuals using the Portmanteau test. tvalues_alpha
()tvalues_beta
()tvalues_det_coef
()tvalues_det_coef_coint
()tvalues_gamma
()var_rep
()