The second related literature is the large body of work that uses approximate factor structures to study asset prices. Contributions include Chamberlain and Rothschild (1983), Connor and Korajczyk (1986, 1988, 1993), Mei (1993), Schneewwiss and Mathes (1995), Bekker et. al. (1996), Geweke and Zhou (1996), and Zhou (1997); also see the survey in Campbell, Lo and McKinley (1996, chapter 6)).

The work in these literatures most closely related to the present paper is by Connor and Korajczyk (1986, 1988, 1993) and Forni and Reichlin (1996, 1997, 1998); both consider the determination of the number of factors and their estimation in large systems. Working within a static approximate factor model that allows some cross-sectional dependence among the idiosyncratic errors, Connor and Korajczyk (1986, 1988, 1993) show that factors estimated by principal components are consistent (at a given date) as N-*oo with T fixed. They apply their methods to evaluating the arbitrage pricing theory of asset prices. Forni and Reichlin (1998), working with a dynamic factor model with mutually uncorrelated idiosynchratic errors, show that cross sectional averages consistently estimate a certain scalar linear combination of the factors. They use this insight in Forni and Reichlin (1996, 1998) to motivate heuristically a dynamic principal components procedure for estimating the vector common factors and for studying the dynamic properties of the factors. Forni and Reichlin (1997) suggest an alternative estimator, which they motivate by dynamic principal components, although no proofs of consistency of the estimated factors are provided. * In the first applications of this large cross section approach to macroeconomic data, they apply their methods to large regional and sectoral data sets, for example Forni and Reichlin (1998) analyze productivity and output for 450 U.S. industries.

Relative to this literature, the current paper makes four main methodological contributions, which are motivated by our focus on real-time economic forecasting. First, consistency of the estimated factors is shown when the factor loadings are time varying and the number of factors tends to infinity. Second, the estimated factors are shown to be uniformly (in the time subscript) consistent, not just for a given date, and rate of consistency is given. Third, estimation methods that are computationally feasible for large N are presented for both balanced and unbalanced panels. Fourth, results are given on the use of information criteria to select the number of factors for forecasting. The empirical contribution of this paper is to demonstrate the potential for substantial improvements in macroeconomic time series forecasts using these methods.