This paper introduces a nonlinear shrinkage estimator of the covariance matrix that does not require recovering the population eigenvalues first. We estimate the sample spectral density and its Hilbert transform directly by smoothing the sample eigenvalues with a variable-bandwidth kernel. Relative to numerically inverting the so-called QuEST function, the main advantages of direct kernel estimation are: (1) it is much easier to comprehend because it is analogous to kernel density estimation; (2) it is only twenty lines of code in Matlab — as opposed to thousands — which makes it more verifiable and customizable; (3) it is 200 times faster without significant loss of accuracy; and (4) it can handle matrices of a dimension larger by a factor of ten. Even for dimension 10, 000, the code runs in less than two minutes on a desktop computer; this makes the power of nonlinear shrinkage as accessible to applied statisticians as the one of linear shrinkage.
Generous behaviour is known to increase happiness, which could thereby motivate generosity. In this study, we use functional magnetic resonance imaging and a public pledge for future generosity to investigate the brain mechanisms that link generous behaviour with increases in happiness. Participants promised to spend money over the next 4 weeks either on others (experimental group) or on themselves (control group). Here, we report that, compared to controls, participants in the experimental group make more generous choices in an independent decision-making task and show stronger increases in self-reported happiness. Generous decisions engage the temporo-parietal junction (TPJ) in the experimental more than in the control group and differentially modulate the connectivity between TPJ and ventral striatum. Importantly, striatal activity during generous decisions is directly related to changes in happiness. These results demonstrate that top–down control of striatal activity plays a fundamental role in linking commitment-induced generosity with happiness.
During competitive interactions, humans have to estimate the impact of their own actions on their opponent's strategy. Here we provide evidence that neural computations in the right temporoparietal junction (rTPJ) and interconnected structures are causally involved in this process. By combining inhibitory continuous theta-burst transcranial magnetic stimulation with model-based functional MRI, we show that disrupting neural excitability in the rTPJ reduces behavioral and neural indices of mentalizing-related computations, as well as functional connectivity of the rTPJ with ventral and dorsal parts of the medial prefrontal cortex. These results provide a causal demonstration that neural computations instantiated in the rTPJ are neurobiological prerequisites for the ability to integrate opponent beliefs into strategic choice, through system-level interaction within the valuation and mentalizing networks.
We consider optimal redistribution in a model where individuals can self-select into one of several possible sectors based on heterogeneity in a multidimensional skill vector. We first show that when the government does not observe the sectoral choice or underlying skills of its citizens, the constrained Pareto frontier can be implemented with a single nonlinear income tax. We then characterize this optimal tax schedule. If sectoral inputs are complements, a many-sector model with self-selection leads to optimal income taxes that are less progressive than the corresponding taxes in a standard single-sector model under natural conditions. However, they are more progressive than in canonical multisector economies with discrete types and without occupational choice or overlapping sectoral wage distributions.
