Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression.
Parameters: | unconstrained : array or list
error_variance : array
transform_variance : boolean, optional
prefix : {‘s’,’d’,’c’,’z’}, optional
|
---|---|
Returns: | constrained : array or list
|
Notes
In the notation of [R100], 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 [R100].
Second, (\Sigma, P_1, \dots, P_p) are transformed into (\Sigma, \phi_1, \dots, \phi_p) via Lemmas 2.1 and 2.3 of [R100].
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.
References
[R100] | (1, 2, 3, 4) 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. |
[R101] | Ansley, Craig F, and Paul Newbold. 1979. “Multivariate Partial Autocorrelations.” In Proceedings of the Business and Economic Statistics Section, 349-53. American Statistical Association |