# 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 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)) – 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]) Gives the covariance matrix of the corresponding VAR-representation. Return the in-sample values of endog calculated by the model. irf([periods]) 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. Return the difference between observed and fitted values. Standard errors of beta and deterministic terms inside the cointegration relation. 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.