, variance, transform_variance=False, prefix=None)[source]

Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression.

  • unconstrained (array or list) – Arbitrary matrices to be transformed to stationary coefficient matrices of the VAR. If a list, should be a list of length order, where each element is an array sized k_endog x k_endog. If an array, should be the matrices horizontally concatenated and sized k_endog x k_endog * order.

  • error_variance (array) – The variance / covariance matrix of the error term. Should be sized k_endog x k_endog. This is used as input in the algorithm even if is not transformed by it (when transform_variance is False). The error term variance is required input when transformation is used either to force an autoregressive component to be stationary or to force a moving average component to be invertible.

  • transform_variance (boolean, optional) – Whether or not to transform the error variance term. This option is not typically used, and the default is False.

  • prefix ({'s','d','c','z'}, optional) – The appropriate BLAS prefix to use for the passed datatypes. Only use if absolutely sure that the prefix is correct or an error will result.


constrained – Transformed coefficient matrices leading to a stationary VAR representation. Will match the type of the passed unconstrained variable (so if a list was passed, a list will be returned).

Return type

array or list


In the notation of 1, the arguments (variance, unconstrained) are written as \((\Sigma, A_1, \dots, A_p)\), where \(p\) is the order of the vector autoregression, and is here determined by the length of the unconstrained argument.

There are two steps in the constraining algorithm.

First, \((A_1, \dots, A_p)\) are transformed into \((P_1, \dots, P_p)\) via Lemma 2.2 of 1.

Second, \((\Sigma, P_1, \dots, P_p)\) are transformed into \((\Sigma, \phi_1, \dots, \phi_p)\) via Lemmas 2.1 and 2.3 of 1.

If transform_variance=True, then only Lemma 2.1 is applied in the second step.

While this function can be used even in the univariate case, it is much slower, so in that case constrain_stationary_univariate is preferred.



Ansley, Craig F., and Robert Kohn. 1986. “A Note on Reparameterizing a Vector Autoregressive Moving Average Model to Enforce Stationarity.” Journal of Statistical Computation and Simulation 24 (2): 99-106.


Ansley, Craig F, and Paul Newbold. 1979. “Multivariate Partial Autocorrelations.” In Proceedings of the Business and Economic Statistics Section, 349-53. American Statistical Association