In this paper, we introduce the extended method of moments (XMM) estimator. This estimator accommodates a more general set of moment restrictions than the standard generalized method of moments (GMM) estimator. More specifically, the XMM differs from the GMM in that it can handle not only uniform conditional moment restrictions (i.e., valid for any value of the conditioning variable), but also local conditional moment restrictions valid for a given fixed value of the conditioning variable. The local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns. The estimated derivative prices are consistent for a large time series dimension, but a fixed number of cross sectionally observed derivative prices. The asymptotic properties of the XMM estimator are nonstandard, since the combination of uniform and local conditional moment restrictions induces different rates of convergence (parametric and nonparametric) for the parameters.

We develop an econometric methodology to infer the path of risk premia from a large unbalanced panel of individual stock returns. We estimate the time-varying risk premia implied by conditional linear asset pricing models where the conditioning includes both instruments common to all assets and asset-specific instruments. The estimator uses simple weighted two-pass cross-sectional regressions, and we show its consistency and asymptotic normality under increasing cross-sectional and time series dimensions. We address consistent estimation of the asymptotic variance by hard thresholding, and testing for asset pricing restrictions induced by the no-arbitrage assumption. We derive the restrictions given by a continuum of assets in a multi-period economy under an approximate factor structure robust to asset repackaging. The empirical analysis on returns for about ten thousand U.S. stocks from July 1964 to December 2009 shows that risk premia are large and volatile in crisis periods. They exhibit large positive and negative strays from time-invariant estimates, follow the macroeconomic cycles, and do not match risk premia estimates on standard sets of portfolios. The asset pricing restrictions are rejected for a conditional four-factor model capturing market, size, value, and momentum effects.