On the concept of matrix derivative

Magnus, Jan R.

doi:10.1016/j.jmva.2010.05.005

Cited by 47 publications

(19 citation statements)

References 4 publications

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“…This circumstance has been noted recently by Magnus [14], who has demonstrated various deleterious consequences of adopting the alternative non-vectorial definitions.…”

Section: The Concept Of a Matrix Derivativementioning

confidence: 92%

“…Notwithstanding this ample provision of sources, there continues to be confusion and difficulty in this area.A recent paper of Magnus [14], which testifies to these problems, cites several instances in which inappropriate definitions of matrix derivatives have been adopted, and it gives examples that highlight the perverse effects of adopting such definitions.…”

Section: Introductionmentioning

confidence: 99%

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On Kronecker products, tensor products and matrix differential calculus

Pollock

2013

International Journal of Computer Mathematics

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The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. The vectorization operations and the commutation transformations that are common in multivariate statistical analysis alter the positional relationship of the matrix elements. These elements correspond to numbers that are liable to be stored in contiguous memory cells of a computer, which should remain undisturbed. It is suggested that, in the absence of an adequate index notation that enables the manipulations to be performed without disturbing the data, even the most clear-headed of computer programmers is liable to perform wholly unnecessary and time-wasting operations that shift data between memory cells.

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“…This circumstance has been noted recently by Magnus [14], who has demonstrated various deleterious consequences of adopting the alternative non-vectorial definitions.…”

Section: The Concept Of a Matrix Derivativementioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

On Kronecker products, tensor products and matrix differential calculus

Pollock

2013

International Journal of Computer Mathematics

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“…The first order conditions in (3) are nonlinear in g in general, and a numerical optimization method, such as the Newton-Raphson algorithm, is required to obtain the desiredĝ ML . Observe that ∂l(g) ∂g results in a K-dimensional column vector of partial derivatives, each with respect to one component of g. For matrix derivations, see [19]. With a calculatedĝ ML , the estimator (1) takes the form…”

Section: A Direct Nonresponse Weighting Adjustmentmentioning

confidence: 99%

Comparisons of Some Weighting Methods for Non-response Adjustment

Rota

Laitila

2015

LJS

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Sample and population auxiliary information have been demonstrated to be useful and yield approximately equal resultsin large samples. Several functional forms of weights are suggested in the literature. This paper studies the properties of calibrationestimators when the functional form of response probability is assumed to be known. The focus is on the difference between populationand sample level auxiliary information, the latter being demonstrated to be more appropriate for estimating the coefficients in theresponse probability model. Results also suggest a two-step procedure, using sample information for model coefficient estimation inthe first step and calibration estimation of the study variable total in the second step.

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“…A selection matrix can be premultiplied to consider only a subset of controls as observables, see the An and Schorfheide (2007) model in Appendix A.2 for an example. 5 For recent literature in favor of this definition, see Magnus (2010) and Pollock (2013). The online documentation of the code also contains additional material which clarifies the concept.…”

Section: Pruned State-space Systemmentioning

confidence: 99%

Identification of DSGE models—The effect of higher-order approximation and pruning

Mutschler

2015

Journal of Economic Dynamics and Control

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a b s t r a c tThis paper shows how to check rank criteria for a local identification of nonlinear DSGE models, given higher-order approximations and pruning. This approach imposes additional restrictions on (higher-order) moments and polyspectra, which can be used to identify parameters that are unidentified in a first-order approximation. The identification procedures are demonstrated by means of the Kim (2003) and the An and Schorfheide (2007) models. Both models are identifiable with a second-order approximation. Furthermore, analytical derivatives of unconditional moments, cumulants and corresponding polyspectra up to fourth order are derived for the pruned state-space.

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On the concept of matrix derivative

Abstract: a b s t r a c tWe discuss how to generalize the concept of vector derivative to matrix derivative, propose two definitions, a 'broad' and a 'narrow' one, compare the two definitions, and argue in favor of the narrow definition.

Cited by 47 publications

References 4 publications

On Kronecker products, tensor products and matrix differential calculus

On Kronecker products, tensor products and matrix differential calculus

Comparisons of Some Weighting Methods for Non-response Adjustment

Identification of DSGE models—The effect of higher-order approximation and pruning

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