Factor Models

Abstract

This paper introduces a novel approach for dealing with the ‘curse of dimensionality’ in the case of large linear dynamic systems. Restrictions on the coefficients of an unrestricted VAR are proposed that are binding only in a limit as the number of endogenous variables tends to infinity. It is shown that under such restrictions, an infinite-dimensional VAR (or IVAR) can be arbitrarily well characterized by a large number of finite-dimensional models in the spirit of the global VAR model proposed in Pesaran et al. (JBES, 2004). The paper also considers IVAR models with dominant individual units and shows that this will lead to a dynamic factor model with the dominant unit acting as the factor. The problems of estimation and inference in a stationary IVAR with unknown number of unobserved common factors are also investigated. A cross section augmented least squares estimator is proposed and its asymptotic distribution is derived. Satisfactory small sample properties are documented by Monte Carlo experiments. An empirical application to modelling of real GDP growth and investmentoutput ratios provides an illustration of the proposed approach. Considerable heterogeneities across countries and significant presence of dominant effects are found. The results also suggest that increase in investment as a share of GDP predict higher growth rate of GDP per capita for non-negligible fraction of countries and vice versa. JEL Code: C10, C33, C51, O40. Keywords: large N and T panels, weak and strong cross section dependence, VAR, global VAR, factor models, capital accumulation and growth. Alexander Chudik Cambridge University Faculty of Economics Austin Robinson Building Sidgwick Avenue Cambridge, CB3 9DD United Kingdom ac474@cam.ac.uk M. Hashem Pesaran Cambridge University Faculty of Economics Austin Robinson Building Sidgwick Avenue Cambridge, CB3 9DD United Kingdom mhp1@econ.cam.ac.uk November 19, 2007 We are grateful to Elisa Tosetti for helpful comments and suggestions. 1 Introduction Following the seminal work of Sims (1980), vector autoregressive models (VARs) are widely used in macroeconometrics and …nance. VARs provide a ‡exible framework for the analysis of complex dynamics and interactions that exist between variables in the national and global economy. However, the application of the approach in practice is often limited to a handful of variables which could lead to misleading inference if important variables are omitted merely to accommodate the VAR modelling strategy. Number of parameters to be estimated grows at the quadratic rate with the number of variables, which is limited by the size of typical data sets to no more than 5 to 7. In many empirical applications, this is not satisfactory. Some restrictions must be imposed for the analysis of large systems. To deal with this so-called ‘curse of dimensionality’, two di¤erent approaches have been suggested in the literature: (i) shrinkage of the parameter space and (ii) shrinkage of the data. Parameter space can be shrunk by imposing a set of restrictions, which could be for instance obtained from a theoretical structural model, directly on the parameters. Alternatively, one could use techniques, where prior distributions are imposed on the parameters to be estimated. Bayesian VAR (BVAR) proposed by Doan, Litterman and Sims (1984), for example, use what has become known as ‘Minnesota’priors to shrink the parameters space.1 In most applications, BVARs have been applied to relatively small systems2 (e.g. Leeper, Sims, and Zha, 1996, considered 13 and 18 variable BVAR) and the focus has been mainly on forecasting.3 The second approach to mitigating the curse of dimensionality is to shrink the data, along the lines of index models. Geweke (1977) and Sargent and Sims (1977) introduced dynamic factor models, which have been more recently generalized to allow for weak cross sectional dependence by Forni and Lippi (2001) and Forni et al. (2000, 2004). Empirical evidence suggest that few dynamic factors are needed to explain the co-movement of macroeconomic variables: Stock and Watson (1999, 2002), Giannoni, Reichlin and Sala (2005) conclude that only few, perhaps two, factors explain much of the predictable variations, while Stock and Watson (2005) estimate as much as seven factors. This has led to the development of factor augmented VAR (FAVAR) models by Bernanke, Boivin, and Eliasz (2005) and Stock and Watson (2005), among others. This paper proposes to deal with the curse of dimensionality by shrinking the parameter space in the limit as the number of endogenous variables (N) tends to in…nity. Under this set up, the in…nite-dimensional VAR (or IVAR) could be arbitrarily well approximated by a set of …nite-dimensional small-scale models that can be consistently estimated separately, which is in the spirit of global VAR (GVAR) models proposed in Pesaran, Schuermann and Weiner (2004, PSW). By imposing restrictions on the parameters of IVAR model that are binding only in the limit, we e¤ectively end up with shrinking of the data. The paper thus provides a link between the two existing approaches to mitigating the curse of dimensionality in the literature and discusses the conditions under which it is appropriate to shrink the data using static or dynamic factor approaches. This also provides theoretical justi…cation for factor models in a large systems with all variables being determined endogenously. We link our analysis to dynamic factor models by showing that dominant unit becomes (in the limit) a dynamic common factor for the remaining units in the system. Static factor models are also obtained as a special case of IVAR. In addition to the limiting restrictions proposed in this paper, other exact or Bayesian type restrictions can also be easily imposed. One of the main motivations behind the proposed approach is to develop a theoretical econometric underpinning for global macroeconomic modelling that allows for variety of channels through which the 1Other types of priors have also been considered in the literature. See, for example, Del Negro and Schorfheide (2004) for a recent reference. 2Few exceptions include Giacomini and White (2006) and De Mol, Giannone and Reichlin (2006). 3Bayesian VARs are known to produce better forecasts than unrestricted VARs and, in many situations, ARIMA or structural models. See Litterman (1986), and Canova (1995) for further references.