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 $t=1,2,3,...T$


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

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


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