This study presents a systematic framework for applying random matrix theory (RMT) to high-dimensional covariance estimation, addressing the limitations of classical asymptotic methods when sample sizes are modest relative to dimensionality. Novel spectral corrections for sample covariance matrices are derived, enabling more reliable hypothesis testing and parameter estimation in such settings. The theoretical results are validated through simulations and applied to a balanced panel of monthly agricultural yield observations across Burundi’s 18 provinces over 120 months, yielding a dimensionality ratio of 0.15. A spiked population model with three dominant factors is employed, justified by the expectation that yields are driven by a small number of latent variables, including regional climate patterns and soil quality gradients. The empirical spectral density of the sample covariance matrix closely conforms to the Marčenko–Pastur law for the bulk eigenvalues, with a Kolmogorov–Smirnov test statistic of 0.042 confirming no significant deviation at the 5% level. However, the largest eigenvalues deviate markedly from the theoretical support, and application of the Tracy–Widom test confirms the presence of spiked factors. The core methodological contribution lies in characterising the bias of sample eigenvectors under the spiked model, demonstrating that sample eigenvectors are not consistent estimators of their population counterparts in this high-dimensional regime. The derived corrections mitigate this eigenvector misalignment, improving the accuracy of subsequent inference. These findings advance the theoretical foundations of high-dimensional multivariate statistics and provide practical tools for researchers working with limited sample sizes, with direct applicability to agricultural data analysis in Burundi and analogous contexts elsewhere.