Achievable rate for MIMO channel with estimated channel matrix In a MIMO system, if the receiver has perfect CSI, i.e. knows that channel matrix $\boldsymbol{H}$, then the achievable rate is:
$$ R = \log \left|\boldsymbol{I}+\frac{P}{N_RN_T}\boldsymbol{H}\boldsymbol{H}^H\right|\tag1$$
where $P$ is the transmit power and $N_R,N_T$ the number of receive and transmit antennas respectively.
What happens if I don't know the channel, but estimate it? 
Will it simply be
$$ R = \log \left|\boldsymbol{I}+\frac{P}{N_RN_T}\boldsymbol{H}\hat{\boldsymbol{H}}^H\right|\tag2$$
where $\hat{\boldsymbol{H}}$ is the channel estimate?

I'm interested specifically in LMMSE estimation (and Gaussian channel).
I found the following relation which answers my question

$$R=\log\det \left(\boldsymbol{C}_{MMSE}^{-1}\right)$$

where $\boldsymbol{C}_{MMSE}$ is the covariance matrix of the error in the data $\boldsymbol{e}=\boldsymbol{x}-\hat{\boldsymbol{x}}$

I found the answer in the paper "An Iteratively Weighted MMSE Approach to Distributed Sum-Utility Maximization for a MIMO Interfering Broadcast Channel" but it is said to be well known.

Therefore, we need to assume some method for estimation the data using the estimated channel, for example, match-filter or MMSE estimator, and then calculate its error (as a function of the real and estimated channels).