Ledoit & Wolf (2003) shrinkage approach

http://www.ledoit.net/honey.pdf In the case where you have one sample covariance matrix S and an optimal shrinkage parameter and you want to estimate the covariance matrix resulting from the Ledoit and Wolf approach how do you estimate this matrix given that you want to use the constant (average) correlation model as a prior?

for example:

delta = 0.5;
S = [0.0309 0.0056 0.0011;
0.0056 0.0739 0.0148;
0.0011 0.0148 0.0489]


The calculations are explained in Appendix A on page 12.

The three estimates of correlation are:

$$r_{12}=\frac{s_{12}}{\sqrt{s_{11} s_{22}}}=\frac{0.0056}{\sqrt{0.0309 \times 0.0739}}=0.117189$$

$$r_{23}=\frac{0.0148}{\sqrt{0.0739 \times 0.0489}}=0.246198$$

$$r_{13}=\frac{0.0011}{\sqrt{0.0309 \times 0.0489}}=0.028298$$

Next we will impose that the common correlation $$\bar{r}$$ is the average of these 3 values.

$$\bar{r}=\frac{1}{3}(0.117189+0.246198+0.028298)=0.130562$$

Now I can show you what the matrix F (the constant correlation covariance matrix) looks like. The diagonal entries $$f_{ii}$$ are taken unchanged from your sample matrix, the off diagonal are found from $$f_{ij}=\bar{r}\sqrt{\sigma_{ii} \sigma_{jj}}$$

F=[0.0309,0.006239,0.005075;
0.006239,0.0739,0.007849;
0.005075,0.007849,0.0489]


This the shrinkage target.

To find the final result of the Ledoit Wolf approach we have to take a 50/50 compromise (element by element) between your matrix $$S$$ and the matrix $$F$$.

$$\delta S + (1-\delta) F = 0.5 S + 0.5 F$$

(Apologies, I am doing these calculations by hand so there is the possibility of an error).