Using nonparametric techniques, we develop a methodology for estimating and testing conditional alphas and betas and long-run alphas and betas, which are the averages of conditional alphas and betas, respectively, across time. The estimators and tests can be implemented for a single asset or jointly across portfolios. The traditional Gibbons, Ross, and Shanken (1989) test arises as a special case of no time variation in the alphas and factor loadings and homoskedasticity. As applications of the methodology, we estimate conditional CAPM and multifactor models on book-to-market and momentum decile portfolios. We reject the null that long-run alphas are equal to zero even though there is substantial variation in the conditional factor loadings of these portfolios.

We develop a new approach to approximating asset prices in the context of continuous-time models. For any pricing model that lacks a closed-form solution, we provide a closed-form approximate solution, which relies on the expansion of the intractable model around an “auxiliary” one. We derive an expression for the difference between the true (but unknown) price and the auxiliary one, which we approximate in closed-form, and use to create increasingly improved refinements to the initial mispricing induced by the auxiliary model. The approach is intuitive, simple to implement, and leads to fast and extremely accurate approximations. We illustrate this method in a variety of contexts including option pricing with stochastic volatility, computation of Greeks, and the term structure of interest rates.

The control function approach is a convenient method of estimation in simultaneous equation systems. This requires that the system can be expressed in triangular form with variables satisfying a conditional mean independence restriction. Linear simultaneous models with additive errors can always be expressed in this form. However, in nonlinear nonadditive simultaneous systems, conditional independence requires a strong additional restriction known as control function separability. We argue that nonadditive models are a key characteristic of simultaneous models of economic behavior with unobserved heterogeneity. We review alternative “system” approaches and document the biases that occur when the control function approach is used inappropriately.

This paper studies a shape-invariant Engel curve system with endogenous total expenditure, in which the shape-invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric Engel curves. We focus on the identification and estimation of both the nonparametric shapes of the Engel curves and the parametric specification of the demographic scaling parameters. The identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions, allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support. Root-n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of “low-level” sufficient conditions. Our empirical application using the U.K. Family Expenditure Survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of Engel curves.