  We analyze the spectral properties of large, time-lagged correlation matrices
using the tools of random matrix theory. We compare predictions of the
one-dimensional spectra, based on approaches already proposed in the
literature. Employing the methods of free random variables and diagrammatic
techniques, we solve a general random matrix problem, namely the spectrum of a
matrix $\frac{1}{T}XAX^{\dagger}$, where $X$ is an $N\times T$ Gaussian random
matrix and $A$ is \textit{any} $T\times T$, not necessarily symmetric
(Hermitian) matrix. As a particular application, we present the spectral
features of the large lagged correlation matrices as a function of the depth of
the time-lag. We also analyze the properties of left and right eigenvector
correlations for the time-lagged matrices. We positively verify our results by
the numerical simulations.
