This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

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The variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems. Classical methods of estimating the covariance matrices are based on the strict factor models, assuming independent idiosyncratic components. This assumption, however, is restrictive in practical applications. By assuming sparse error covariance matrix, we allow the presence of the cross-sectional correlation even after taking out common factors, and it enables us to combine the merits of both methods. We estimate the sparse covariance using the adaptive thresholding technique as in Cai and Liu (2011), taking into account the fact that direct observations of the idiosyncratic components are unavailable. The impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied.

The commercial asset value of sequestered forest carbon is based on protocols employed globally; however, their scientific basis has not been validated. We review and analyze commercial forest carbon protocols, claimed to have reduced net greenhouse gas emissions, issued by the California Air Resources Board and validated by the Climate Action Reserve (CARB-CAR). CARB-CAR forest carbon offsets, based on forest mensuration and model simulation, are compared to a global database of directly measured forest carbon sequestration, or net ecosystem exchange (NEE) of forest CO2. NEE is a meteorologically based method integrating CO2 fluxes between the atmosphere, forest and soils and is independent of the CARB-CAR methodology. Annual carbon accounting results for CAR681 are compared with NEE for the Ameriflux site, Howland Forest Maine, USA, (Ho-1), the only site where both methods were applied contemporaneously, invalidating CARB-CAR protocol offsets. We then test the null hypothesis that CARB-CAR project population data fall within global NEE population values for natural and managed forests measured in the field; net annual gC m−2yr−1 are compared for both protocols. Irrespective of geography, biome and project type, the CARB-CAR population mean is significantly different from the NEE population mean at the 95% confidence interval, rejecting the null hypothesis. The CARB-CAR population exhibits standard deviation ∼5× that of known interannual NEE ranges, is overcrediting biased, incapable of detecting forest transition to net positive CO2 emissions, and exceeds the 5% CARB compliance limit for invalidation. Exclusion of CO2 efflux via soil and ecosystem respiration precludes a valid net carbon accounting result for CARB-CAR and related protocols, consistent with our findings. Protocol invalidation risk extends to vendors and policy platforms such as the United Nations Program on Reducing Emissions from Deforestation and Forest Degradation (REDD+) and the Paris Agreement. We suggest that CARB-CAR and related protocols include NEE methodology for commercial forest carbon offsets to standardize methods, ensure in situ molecular specificity, verify claims of carbon emission reduction and harmonize carbon protocols for voluntary and compliance markets worldwide.

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