Linear Shrinkage

Linear Shrinkage

Linear Shrinkage can be used in both estimations of returns and covariance.

After knowing the potential problems that exist in the method- sample mean/ covariance, we must minimize the differences between our estimation and the ground truth, which gives a chance to introduce the Shrinkage Method. The target of this method is to shrink simple mean/ covariance to the structured estimation.

What is shrinkage?

In statistics, shrinkage is the reduction in the effects of sampling variation. In regression analysis, a fitted relationship appears to perform less well on a new data set than on the data set used for fitting.

Linear Shrinkage in the estimation of returns

Linear Shrinkage based on Sample Mean but shrink to Equal Mean. And its math forms can be viewed as follows:

\(r_i^* = (1-\delta) r_i + \delta \bar{r}\)

, where \(r_i\) is the estimation of Sample Mean and \(\bar{r}\) is the estimation of Equal Mean. \(\delta \) is the shrinkage rate with number between 0 and1.

Linear Shrinkage in the estimation of the covariance matrix

Similarly, the linear shrinkage method can be used in the estimation of the covariance matrix with forms below:

\(\Sigma^* =(1-\delta) \Sigma +\delta F\)

, where \(\Sigma\) represents the sample covariance and \(F\) represents Equal correlation.

Further Discussion

When talking about the selection of the optimal value for the shrinkage coefficient, Jorion points out shrinkage estimator can be thought as follows:

\(\delta =\frac{\lambda}{T+\lambda}\)

\(\lambda = \frac{(N+2)(T-1)}{(\bar{r}-r_M)\Sigma^\mathrm{-1} (\bar{r}-r_i)(T-N-2)}\)

, where \(T\) is the number of time periods of data, \(N\) is the number of asset returns time series, \(\Sigma\) is the covariance matrix of return time series, $r_i$ is the return of asset i and \(r_M\) is the sort of return reference.

References

[1] https://en.wikipedia.org/wiki/Shrinkage_(statistics)

[2] Ledoit, Olivier ; Wolf, Michael Honey, I shrunk the sample covariance matrix[J] Universitat Pompeu Fabra. Departament D’Economia I Empresa 2003.

[3] Tuan, Eric Journal of Real Estate Portfolio Management, Jan-Apr 2013, Vol.19(1), pp.89-101