# 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