Determining the Number of Factors in Approximate Factor Models

In this paper we develop some econometric theory for factor models of large dimensions. The focus is the determination of the number of factors, which is an unresolved issue in the rapidly growing literature on multifactor models. We propose some panel C(p) criteria and show that the number of factors can be consistently estimated using the criteria. The theory is developed under the framework of large cross-sections (N) and large time dimensions (T). No restriction is imposed on the relation between N and T. Simulations show that the proposed criteria yield almost precise estimates of the number of factors for configurations of the panel data encountered in practice.

[1]  C. L. Mallows.  Some comments on C_p , 1973 .

[2]  S. Ross.  The arbitrage theory of capital asset pricing , 1976 .

[3]  Thomas J. Sargent,et al.  Business cycle modeling without pretending to have too much a priori economic theory , 1976 .

[4]  John Geweke,et al.  Estimating regression models of finite but unknown order , 1981 .

[5]  M. Rothschild,et al.  Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets , 1982 .

[6]  Dorothy T. Thayer,et al.  EM algorithms for ML factor analysis , 1982 .

[7]  Gary Chamberlain,et al.  FUNDS, FACTORS, AND DIVERSIFICATION IN ARBITRAGE PRICING MODELS , 1983 .

[8]  M. Rothschild,et al.  Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets , 1983 .

[9]  Phoebus J. Dhrymes,et al.  A Critical Reexamination of the Empirical Evidence on the Arbitrage Pricing Theory , 1984 .

[10]  Gregory Connor,et al.  Performance Measurement with the Arbitrage Pricing Theory: A New Framework for Analysis , 1985 .

[11]  J. Schmee.  An Introduction to Multivariate Statistical Analysis , 1986 .

[12]  Clive W. J. Granger.  Implications of Aggregation with Common Factors , 1987 .

[13]  Bruce N. Lehmann,et al.  The empirical foundations of the arbitrage pricing theory , 1988 .

[14]  Z. Bai,et al.  On the limit of the largest eigenvalue of the large dimensional sample covariance matrix , 1988 .

[15]  Gregory Connor,et al.  Risk and Return in an Equilibrium Apt: Application of a New Test Methodology , 1988 .

[16]  G. Schwert,et al.  Tests for Unit Roots: a Monte Carlo Investigation , 1988 .

[17]  Arthur Lewbel,et al.  The Rank of Demand Systems: Theory and Nonparametric Estimation , 1991 .

[18]  Gregory Connor,et al.  A Test for the Number of Factors in an Approximate Factor Model , 1993 .

[19]  C. Mallows.  More comments on C p , 1995 .

[20]  Serena Ng,et al.  A Semi-Parametric Factor Model for Interest Rates , 1996 .

[21]  A. Lo,et al.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1996, Macroeconomic Dynamics.

[22]  Allan W. Gregory,et al.  Measuring World Business Cycles , 1997 .

[23]  Stephen G. Donald.  INFERENCE CONCERNING THE NUMBER OF FACTORS IN A MULTIVARIATE NONPARAMETRIC RELATIONSHIP , 1997 .

[24]  Stephen G. Donald,et al.  Inferring the rank of a matrix , 1997 .

[25]  E. Ghysels,et al.  A Semiparametric Factor Model of Interest Rates and Tests of the Affine Term Structure , 1998, Review of Economics and Statistics.

[26]  Marco Lippi,et al.  Aggregation and the Microfoundations of Dynamic Macroeconomics , 1998 .

[27]  Liuren Wu,et al.  Predictable Changes in Yields and Forward Rates , 1998 .

[28]  Lucrezia Reichlin,et al.  Let's Get Real: A Factor Analytical Approach to Disaggregated Business Cycle Dynamics , 1998 .

[29]  Allan W. Gregory,et al.  Common and country-specific fluctuations in productivity, investment, and the current account , 1999 .

[30]  J. W. Silverstein,et al.  EXACT SEPARATION OF EIGENVALUES OF LARGE DIMENSIONAL SAMPLE COVARIANCE MATRICES , 1999 .

[31]  J. Cochrane,et al.  Portfolio Advice for a Multifactor World , 1999 .

[32]  Marco Lippi,et al.  Reference cycles: the NBER methodology revisited , 2000 .

[33]  M. Hallin,et al.  The Generalized Dynamic-Factor Model: Identification and Estimation , 2000, Review of Economics and Statistics.

[34]  Colin L. Mallows,et al.  Some Comments on Cp , 2000, Technometrics.

[35]  Marco Lippi,et al.  THE GENERALIZED DYNAMIC FACTOR MODEL: REPRESENTATION THEORY , 2001, Econometric Theory.