# True covariance matrix in Monte Carlo simulation?

In a linear model, such that Population model is

$$y_{it}=\ {\beta_{i1}f}_1+{\beta_{i2}f}_2+\cdots{\beta_{ik}f}_k\ +\ \varepsilon_{it} , i = 1,2,3...,p$$ and $$t=1,2,3,...T$$

and

$$\mathrm{\Sigma_y}\ =\ Bcov(f_t)B'\ +\ \Sigma_\varepsilon$$

Generate random sample using Multi-Variate Normal distribution from certain sample calibration, we have now true (kXT) of $$f$$ , (pXk) of $$beta$$ and pXT of $$\varepsilon$$, finally we have a sample $$y_i$$.

from $$y_i$$ we calculate sample Covariance Model $$S_y=\ \frac{1}{T}\sum_{i=1}^{T}{\left(y_i- \bar{y}\right)\left(y_j-\ \bar{y}\right)^\prime}=\left(s_{ij}\right)_{p\times p}$$

Now, I want to simulate this algorithm 1000 times and compare the frobenius distance from true covariance.

What is true covariance in this set-up? Does $$E\left[\left(y_i-Ey,y_j-Ey)\right)\right] = E\left[\left(\varepsilon_i,\varepsilon_j)\right)\right]=cov\left(\varepsilon_i,\varepsilon_j\right)$$? or $$\mathrm{\Sigma_y}\ =\ Bcov(f_t)B'\ +\ \Sigma_\varepsilon$$ as mentioned above, the reason why I aske silly question is becaues I can not distinguish sample covariance from population covariance in MC setup.

I will wait for good answers as soon as possible!

Full set up is available

High Dimensional Covariance Matrix Estimation Using a Factor Model(2008)pp.16