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 the VECM-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.
Returns:

  • **Attributes**
  • 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 \(\beta\).
  • llf (float) – The model’s log-likelihood.
  • 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 [1] as \(\Sigma_{co}\) and its relationship to the ML-estimators can be seen in eq. (7.2.21) on p. 296 in [1].
  • 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 [1].
  • stderr_params (ndarray (d)) – Array containing the standard errors of \(\Pi\), \(\Gamma\), and estimated parameters related to deterministic terms.
  • stderr_coint (ndarray (neqs+num_det_coef_coint x coint_rank)) – Array containing the standard errors of \(\beta\) and estimated parameters related to deterministic terms inside the cointegration relation.
  • 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_alpha (ndarray (neqs x coint_rank))
  • 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_alpha (ndarray (neqs x coint_rank))
  • 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 via A[i] for \(i=0, \ldots, k_{ar}-1\).
  • cov_var_repr (ndarray (neqs**2 * k_ar x neqs**2 * k_ar)) – This matrix is called \(\Sigma^{co}_{\alpha}\) on p. 289 in [1]. It is needed e.g. for impulse-response-analysis.
  • fittedvalues (ndarray (nobs x neqs)) – The predicted in-sample values of the models’ endogenous variables.
  • resid (ndarray (nobs x neqs)) – The residuals.

References

[1](1, 2, 3, 4) Lütkepohl, H. 2005. New Introduction to Multiple Time Series Analysis. Springer.

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()