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

Auxiliary array for internal computations. It will be calculated if not given as parameter.

y_lag1 : ndarray or None, default: None

Auxiliary array for internal computations. It will be calculated if not given as parameter.

delta_x : ndarray or None, default: None

Auxiliary 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

R216(1,2,3,4,5)

Lütkepohl, H. 2005. New Introduction to Multiple Time Series Analysis. Springer.

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 [R216] as \(\Sigma_{co}\) and its relationship to the ML-estimators can be seen in eq. (7.2.21) on p. 296 in [R216].

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 [R216].

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 [R216]. 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.

Methods

conf_int_alpha([alpha])

conf_int_beta([alpha])

conf_int_det_coef([alpha])

conf_int_det_coef_coint([alpha])

conf_int_gamma([alpha])

irf([periods])

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.

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.

Methods

conf_int_alpha([alpha])

conf_int_beta([alpha])

conf_int_det_coef([alpha])

conf_int_det_coef_coint([alpha])

conf_int_gamma([alpha])

irf([periods])

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.

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.

Properties

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.

llf

Compute the VECM’s loglikelihood.

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

tvalues_alpha

tvalues_beta

tvalues_det_coef

tvalues_det_coef_coint

tvalues_gamma

var_rep