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()