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In this paper, the Cramer-Rao lower bound (CRLB) for spatial correlation matrices is derived based on a rigorous model of the doubly selective fading channel for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. Adopting an orthogonal pilot pattern for multiple transmitting antennas and assuming independent samples along the time, the sample auto-correlation matrix of the channel response is complex Wishart distributed. Then, the maximum likelihood estimator (MLE) and the analytic expression of CRLB are derived by assuming that temporal and frequency correlations are known. Furthermore, lower bounds of total mean squared error (TMSE) and average mean squared error (AvgMSE) are deduced from CRLB for asymptotically infinite and finite signal-to-noise ratios (SNR's), respectively. According to the lower bound of AvgMSE, the amount of samples and the order of frequency selectivity show dominant impact on the accuracy of estimation. Besides, the number of pilot tones, SNR and normalized maximum Doppler spread together influence the effective order frequency selectivity. Numerical simulations demonstrate the analytic results.