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Dimensionality Reduction for Stationary Time Series via Stochastic Nonconvex Optimization
Stochastic optimization naturally arises in machine learning. Efficient
algorithms with provable guarantees, however, are still largely missing, when
the objective function is nonconvex and the data points are dependent. This
paper studies this fundamental challenge through a streaming PCA problem for
stationary time series data. Specifically, our goal is to estimate the
principle component of time series data with respect to the covariance matrix
of the stationary distribution. Computationally, we propose a variant of Oja's
algorithm combined with downsampling to control the bias of the stochastic
gradient caused by the data dependency. Theoretically, we quantify the
uncertainty of our proposed stochastic algorithm based on diffusion
approximations. This allows us to prove the asymptotic rate of convergence and
further implies near optimal asymptotic sample complexity. Numerical
experiments are provided to support our analysis
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Micro-data in Macroeconomics
This dissertation contains three essays on Macroeconomics. Detailed micro-level data is used in all three essays. The first chapter studies wealth inequality problems. More specif- ically, it focuses on capital return inequality among university endowments. It combines university-level data on endowment size, capital returns, and portfolio allocations into a unified dataset. Using panel data regression, I show a strong impact of size on investment return. Everything else the same, the biggest endowment has a capital return 8 percent higher than the smallest endowment. However, after adjusting for risk using Sharpe ratios, the strong positive correlation turns negligible or even negative. This result suggests that the higher return of bigger endowments can be attributed to risk compensation rather than to an informational premium.
The second and the third chapters employ firm-level data to study macroeconomic pro- ductivity. The second chapter documents the sectoral growth paths of measured total factor productivity (TFP) in southern Europe during the boom that proceeded the great contraction (1996 to 2007). Using both aggregate and firm-level panel data, I show that TFP in sectors that displayed fast expansion, such as construction, dropped significantly, while in non- expanding sectors, such as manufacturing, it stayed stable. I evaluate the relevance of two alternative explanations of this phenomenon: capital misallocation (the increase in capital was directed to less productive firms) and labor quality mismeasurement (lower quality of incoming labor was not fully captured in the TFP calculation). I find that the misalloca- tion channel is almost negligible. Moreover, worker-firm matched data shows that labor quality did deteriorate in the expanding sectors but not in the others, giving credence to the labor-quality mismeasurement hypothesis. A model featuring both the misallocation and the mismeasurement channels and calibrated to match the micro-level productivity distri- bution and labor quality distribution predicts that the drop in true TFP was small if labor quality is measured properly.
The third chapter documents the total factor productivity growth path in China from 1998 to 2015 using both the aggregate and the firm-level data. We find that measured TFP growth is positive from 1998 to 2011, before turning flat and even negative. A care- ful comparison between state-owned enterprises (SOEs) and private firms reveals that the slowing down of TFP growth of SOEs is the major contributor to the TFP growth reversal of the whole manufacturing sector. The reversal is not due to changes in the composition of production in different sub-sectors, but mostly due to changes within existing firms
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