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='n', 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¶

Deterministic terms outside the cointegration relation.

exog_coint¶

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¶

Cointegration rank, equals the rank of the matrix \(\Pi\) and the number of columns of \(\alpha\) and \(\beta\).

alpha¶

Estimate for the parameter \(\alpha\) of a VECM.

beta¶

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='n'¶

"n" - 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=None¶

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

y_lag1=None¶

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

delta_x=None¶

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

model=None¶

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.

nobs¶

Number of observations (excluding the presample).

Type:¶: int

model¶

Type:¶: see Parameters

y_all¶

Type:¶: see endog in Parameters

exog¶

Type:¶: see Parameters

exog_coint¶

Type:¶: see Parameters

names¶

Type:¶: see Parameters

dates¶

Type:¶: see Parameters

neqs¶

Number of variables in the time series.

Type:¶: int

k_ar¶

Type:¶: see Parameters

deterministic¶

Type:¶: see Parameters

seasons¶

Type:¶: see Parameters

first_season¶

Type:¶: see Parameters

alpha¶

Type:¶: see Parameters

beta¶

Type:¶: see Parameters

gamma¶

Type:¶: see Parameters

sigma_u¶

Type:¶: see Parameters

det_coef_coint¶

Estimated coefficients for the all deterministic terms inside the cointegration relation.

Type:¶: ndarray (#(determinist. terms inside the coint. rel.) x coint_rank)

const_coint¶

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.

Type:¶: ndarray (1 x coint_rank)

lin_trend_coint¶

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.

Type:¶: ndarray (1 x coint_rank)

exog_coint_coefs¶

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.

Type:¶: ndarray (exog_coint.shape[1] x coint_rank) or None

det_coef¶

Estimated coefficients for the all deterministic terms outside the cointegration relation.

Type:¶: ndarray (neqs x #(deterministic terms outside the coint. rel.))

const¶

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.

Type:¶: ndarray (neqs x 1) or (neqs x 0)

seasonal¶

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.

Type:¶: ndarray (neqs x seasons)

lin_trend¶

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.

Type:¶: ndarray (neqs x 1) or (neqs x 0)

exog_coefs¶

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.

Type:¶: ndarray (neqs x exog_coefs.shape[1])

_delta_y_1_T¶

Type:¶: see delta_y_1_T in Parameters

_y_lag1¶

Type:¶: see y_lag1 in Parameters

_delta_x¶

Type:¶: see delta_x in Parameters

coint_rank¶

Cointegration rank, equals the rank of the matrix \(\Pi\) and the number of columns of \(\alpha\) and \(\beta\).

Type:¶: int

llf¶

The model’s log-likelihood.

Type:¶: float

cov_params¶

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].

Type:¶: ndarray (d x d)

cov_params_wo_det¶

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].

Type:¶: ndarray

stderr_params¶

Array containing the standard errors of \(\Pi\), \(\Gamma\), and estimated parameters related to deterministic terms.

Type:¶: ndarray (d)

stderr_coint¶

Array containing the standard errors of \(\beta\) and estimated parameters related to deterministic terms inside the cointegration relation.

Type:¶: ndarray (neqs+num_det_coef_coint x coint_rank)

stderr_alpha¶

The standard errors of \(\alpha\).

Type:¶: ndarray (neqs x coint_rank)

stderr_beta¶

The standard errors of \(\beta\).

Type:¶: ndarray (neqs x coint_rank)

stderr_det_coef_coint¶

The standard errors of estimated the parameters related to deterministic terms inside the cointegration relation.

Type:¶: ndarray (num_det_coef_coint x coint_rank)

stderr_gamma¶

The standard errors of \(\Gamma_1, \ldots, \Gamma_{k_{ar}-1}\).

Type:¶: ndarray (neqs x neqs*(k_ar-1))

stderr_det_coef¶

The standard errors of estimated the parameters related to deterministic terms outside the cointegration relation.

Type:¶: ndarray (neqs x det. terms outside the coint. relation)

tvalues_alpha¶

Type:¶: ndarray (neqs x coint_rank)

tvalues_beta¶

Type:¶: ndarray (neqs x coint_rank)

tvalues_det_coef_coint¶

Type:¶: ndarray (num_det_coef_coint x coint_rank)

tvalues_gamma¶

Type:¶: ndarray (neqs x neqs*(k_ar-1))

tvalues_det_coef¶

Type:¶: ndarray (neqs x det. terms outside the coint. relation)

pvalues_alpha¶

Type:¶: ndarray (neqs x coint_rank)

pvalues_beta¶

Type:¶: ndarray (neqs x coint_rank)

pvalues_det_coef_coint¶

Type:¶: ndarray (num_det_coef_coint x coint_rank)

pvalues_gamma¶

Type:¶: ndarray (neqs x neqs*(k_ar-1))

pvalues_det_coef¶

Type:¶: ndarray (neqs x det. terms outside the coint. relation)

var_rep¶

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 \(i=0, \ldots, k_{ar}-1\).

Type:¶: (k_ar x neqs x neqs)

cov_var_repr¶

This matrix is called \(\Sigma^{co}_{\alpha}\) on p. 289 in [1]. It is needed e.g. for impulse-response-analysis.

Type:¶: ndarray (neqs**2 * k_ar x neqs**2 * k_ar)

fittedvalues¶

The predicted in-sample values of the models’ endogenous variables.

Type:¶: ndarray (nobs x neqs)

resid¶

The residuals.

Type:¶: ndarray (nobs x neqs)

References

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`