# Explaining degrees of freedom when testing the number of common factors

To test the adequacy of the $m$ common factor model, the null hypothesis is $H_0: \Sigma = LL'+\Psi$, where $\Sigma_{p\times p}$, $L_{p\times m}$ is the loading factors matrix, and $\Psi_{p\times p}$ is a diagonal matrix with variances of the specific factors in the main diagonal.

When using the MLR statistic, under $H_0$, we get an approximate chi-squared distribution, with degrees of freedom $$dim(\Theta)-dim(\Theta_0)=\frac{1}{2}p(p+1)-[p(m+1)-\frac{1}{2}m(m-1)]$$

I can explain all terms, except the last one, $\frac{1}{2}m(m-1)$. Where does it come from?

For example, the $1/2 p(p+1)$ is the number of parameters in $\Sigma$ (symmetric), $pm$ is for $L$, $p$ is for $\Psi$

For the matrices $L, \Psi$ to be uniquely determined there is an extra condition which is imposed when finding the ML estimators: $$L'\Psi^{-1}L=\Delta$$ where $\Delta$ is a diagonal matrix. Well, the zeros in the off the main diagonal impose $\frac{1}{2}m(m-1)$ conditions. And these need to be subtracted to $dim(\Theta_0)$.