covarianceShrinkage

README

covarianceShrinkage

A Go package implementing Ledoit-Wolf linear and nonlinear shrinkage for improving the estimation of large covariance matrices. Particularly useful in high-dimensional statistics and portfolio optimization, where the number of assets exceeds the number of observations.

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Installation

go get github.com/dgelody/covarianceShrinkage

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Functions

CovCor

Signature:

func CovCor(X *mat.Dense, S *mat.SymDense) (FMatrix *mat.SymDense, norm float64, err error) 

Description:

Linear shrinkage towards constant-correlation matrix; the target preserves the diagonal of the sample covariance matrix and all correlation coefficients are the same. See Ledoit and Wolf (2004a).

Parameters:

• X - matrix of n random variables by t observations. The data used to create the covariance matrix. Should be differenced if the covariance matrix is the covariance of the differences.
• S - n by n covariance matrix. If nil, S will be the sample covariance matrix

Returns:

• F - n by n shrinkage estimator of the covariance matrix
• norm - maximum absolute difference between the covariance matrix S and the shrinkage estimator of the covariance matrix F
• err - error

QuadraticInverseShrinkage

Signature:

func QuadraticInverseShrinkage(X *mat.Dense, S *mat.SymDense, demeanData bool) (F *mat.Dense, norm float64, err error)

Description:

Nonlinear shrinkage derived under Frobenius loss and its two cousins, Inverse Stein’s loss and Minimum Variance loss, called quadratic-inverse shrinkage (QIS). Preserves the variance of the original covariance matrix. See Ledoit and Wolf (2022, Section 4.5).

Parameters:

• X - matrix of t observations by n random variables. The data used to create the covariance matrix. Should be differenced if the covariance matrix is the covariance of the differences.
• S - n by n covariance matrix. If nil, S will be the sample covariance matrix
• demeanData - if true then demean the data, otherwise no demeaning takes place

Returns:

• F - n by n shrinkage estimator of the covariance matrix
• norm - maximum absolute difference between the covariance matrix S and the shrinkage estimator of the covariance matrix F
• err - error

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Testing

Run all unit tests with coverage:

go test -v -cover ./...

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References

(a) Ledoit, O. and Wolf, M. (2004a). Honey, I shrunk the sample covariance matrix. Journal of Portfolio Management, 30(4):110–119.

(b) Ledoit, O. and Wolf, M. (2004b). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2):365–411.

(c) Ledoit, O. and Wolf, M. (2022). Quadratic shrinkage for large covariance matrices. Bernoulli, 28(3): 1519-1547.

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License

MIT License. See the LICENSE file for details.

Documentation

Index

• func CovCor(X *mat.Dense, S *mat.SymDense) (F *mat.SymDense, norm float64, err error)
• func FrobeniusNorm(A, B []float64) float64
• func MaxAbsDiff(A, B []float64) float64
• func QuadraticInverseShrinkage(X *mat.Dense, S *mat.SymDense, demeanData bool) (F *mat.Dense, norm float64, err error)
• func SampleCovarianceMatrix(n int, t int, X []float64) []float64
• type Target

Functions

func CovCor

func CovCor(X *mat.Dense, S *mat.SymDense) (F *mat.SymDense, norm float64, err error)

Returns a covariance estimate using linear shrinkage. The target preserves the diagonal of the sample covariance matrix and all correlation coefficients are the same.

Arguments: X *mat.Dense : matrix of n random variables by t observations. The data used to create the covariance matrix. Should be differenced if the covariance matrix is the covariance of the differences.

S *mat.SymDense: nxn covariance matrix. If nil, S will be the sample covariance matrix

Returns: F *mat.SymDense: nxn shrinkage estimator of the covariance matrix norm float64: maximum absolute difference between covariance and shrinkage matrix err error

func FrobeniusNorm

func FrobeniusNorm(A, B []float64) float64

Computes the Frobenius norm between matrices A and B

Arguments: A []float64: matrix A B []float64: matrix B

Returns: float64: Frobenius norm

func MaxAbsDiff

func MaxAbsDiff(A, B []float64) float64

Computes the maximum absolute difference between elements of matrices A and B

Arguments: A []float64: matrix A B []float64: matrix B

Returns: float64: maximum absolute difference between elements

func QuadraticInverseShrinkage

func QuadraticInverseShrinkage(X *mat.Dense, S *mat.SymDense, demeanData bool) (F *mat.Dense, norm float64, err error)

deameanData bool: if true then demean the data, otherwise no demeaning takes place

Returns: F *mat.Dense: the Quadratic-Inverse-Shrinkage covariance estimator norm float64: maximum absolute difference between covariance and shrinkage matrix err error

func SampleCovarianceMatrix

func SampleCovarianceMatrix(n int, t int, X []float64) []float64

Computes the sample covariance matrix

Arguments: n int: number of assets t int: number of periods X []float64: matrix of n assets by t periods

Returns: []float64: nxn sample covariance matrix

Types

type Target

type Target int