We reconsider the microeconomic foundations of financial economics. Motivated by the importance of Knightian uncertainty in markets, we present a model that does not carry any probabilistic structure ex ante, yet is based on a common order. We derive the fundamental equivalence of economic viability of asset prices and absence of arbitrage. We also obtain a modified version of the fundamental theorem of asset pricing using the notion of sublinear pricing measures. Different versions of the efficient market hypothesis are related to the assumptions one is willing to impose on the common order.

We study deep neural networks and their use in semiparametric inference. We establish novel nonasymptotic high probability bounds for deep feedforward neural nets. These deliver rates of convergence that are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second‐step inference after first‐step estimation with deep learning, a result also new to the literature. Our nonasymptotic high probability bounds, and the subsequent semiparametric inference, treat the current standard architecture: fully connected feedforward neural networks (multilayer perceptrons), with the now‐common rectified linear unit activation function, unbounded weights, and a depth explicitly diverging with the sample size. We discuss other architectures as well, including fixed‐width, very deep networks. We establish the nonasymptotic bounds for these deep nets for a general class of nonparametric regression‐type loss functions, which includes as special cases least squares, logistic regression, and other generalized linear models. We then apply our theory to develop semiparametric inference, focusing on causal parameters for concreteness, and demonstrate the effectiveness of deep learning with an empirical application to direct mail marketing.

Heteroskedasticity‐ and autocorrelation‐robust (HAR) inference in time series regression typically involves kernel estimation of the long‐run variance. Conventional wisdom holds that, for a given kernel, the choice of truncation parameter trades off a test's null rejection rate and power, and that this tradeoff differs across kernels. We formalize this intuition: using higher‐order expansions, we provide a unified size‐power frontier for both kernel and weighted orthonormal series tests using nonstandard “fixed‐ b ” critical values. We also provide a frontier for the subset of these tests for which the fixed‐ b distribution is t or F . These frontiers are respectively achieved by the QS kernel and equal‐weighted periodogram. The frontiers have simple closed‐form expressions, which show that the price paid for restricting attention to tests with t and F critical values is small. The frontiers are derived for the Gaussian multivariate location model, but simulations suggest the qualitative findings extend to stochastic regressors.