Dynamic factor models

Journal of Econometrics 119 (2004) 223 – 230

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Guest editorial

Dynamic factor models Factor models in social science originate from the need to condense large-scale statistical data on many variables into a much smaller number of indices or “common factors”, with as little loss of information as possible. These indices are more often than not considered latent, even though one may also test for an hypothetically prescribed simple structure involving a number of observed variables as candidate common factors. In both cases, being latent or not, factors are conceived to reduce the dimension of the statistical model. As is well explained by Bartholomew (1987), a second approach to factor models is more theoretical and arises naturally in social science contexts where one wants to think about some quantities for which no measuring instrument exists. Business sentiment, quality of life, and general intelligence are such hypothetical variables that may be extracted as common factors from a set of answers to a questionnaire used in a sample survey. Actually, Factor Analysis was invented by psychologists for measurement of general intelligence. In the beginning of the twentieth century, in the study of human abilities, psychologists, including Burt and Spearman, were concerned with the fact that people performing well in one test of mental ability also tended to do well in others. This motivated the hypothesis of a common latent factor, called general intelligence. Of course, the scores on di5erent items were not perfectly correlated but it could be hypothesized that the variation in performance from one item to another was only due to additional random elements, called speci7c because they were mutually independent. In other words, one common factor was able to summarize the dependence between a number of scores. In the 1930s, Thurstone and his associates in Chicago proposed to replace Spearman’s single general factor by a limited number of common factors representing di5erent abilities. In this initial approach to factor analysis, the dimension reduction is cross sectional in nature, through a postulated conditional independence, given the common factors, of a large number of scores measured on a population of children. This idea was soon revisited in 7nance where factor or multi-index models were originally introduced to simplify the computation of the covariance matrix of returns in a mean-variance portfolio allocation framework. By postulating conditional independence of contemporaneous returns of a large number n of assets given a small number K of factors, one dramatically reduces the number of parameters one needs to estimate to capture the cross-sectional dependence between returns and to exploit it for risk diversi7cation. Ross (1976) pushes this idea to its very limit (literally) by showing that perfect diversi7cation of speci7c (or idiosyncratic) risk could be reached thanks to a number of assets going to in7nity. Then, by virtue of the no arbitrage principle, only the risk c 2003 Elsevier B.V. All rights reserved. 0304-4076/$ - see front matter
doi:10.1016/S0304-4076(03)00195-7

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related to the common factors should be compensated in equilibrium. This gives rise to the so-called Arbitrage Pricing Theory (APT hereafter). More generally, people in Finance term “linear factor model” any pricing model where expectations of returns are constrained to be linear functions of their regression coeBcients (the “betas”) with respect to some factors as in the APT. Equivalently, (see, e.g., Cochrane, 2001), the pricing kernel in those models is linear in the factors. Then, the factor structure facilitates the estimation of both the expectations and the covariance matrix of a large set of returns. This is particularly important in dynamic contexts where expectations and covariances of returns are conditional on the past and, therefore, time varying. Conditionally heteroskedastic factor models (Engle et al., 1990; King et al., 1994) provide a parsimonious parameterization of time varying risks and risk premiums. Independently, the factor model was also proposed in the same period for (macro) economic time series analysis (Geweke, 1977; Sargent and Sims, 1977). As is well explained in the 7rst paper of this special issue (Forni, Hallin, Lippi, and Reichlin), a vector of n time series is decomposed into two mutually orthogonal components, a common component driven by a small number of common shocks or factors and an idiosyncratic component, driven by n variable-speci7c shocks which are at least “locally” mutually orthogonal as in the original factor analysis model. In some respect, the factor model should be called “dynamic” only when the di5erent series are “hit” by di5erent lags of the common shocks, while it is rather “static” when all series under study are hit by the common shocks at the same time. But the important point to notice is that, irrespective of the quali7cation static or dynamic, the factor analysis principle is still applied here to get a drastic reduction of the cross-sectional dimension. Even though the principle is now applied to a n-dimensional time series of macroeconomic variables rather than to a n-dimensional vector of scores observed on a population, the correlation relationships that are summarized by the reduced number of factors are the transversal ones. Such problems are extremely common in macroeconomics like construction of coincident and leading indicators, forecasting, identi7cation of shocks and propagation mechanisms, co-cyclical variables in business cycles, etc. By contrast, in time series state-space models, dimension is reduced longitudinally by assuming conditional serial independence between consecutive observations given a small number of state variables also called factors. In macroeconometrics, the state of the system will represent the various unobserved components such as trends, seasonalities, or cycles (see, e.g., Harvey and Shephard (1993) for a survey of the linear state-space literature in econometrics). Given the state path, the time series is “shocked” by an irregular component without any speci7c dynamic structure of interest. In 7nancial econometrics, the stochastic volatility literature (see, e.g., Ghysels et al. (1997) for a survey) typically starts from the idea that the remaining dynamics in returns innovations can be summarized by a dynamic and latent scaling factor called the volatility process. Moreover, the recent 7nance literature has stressed the importance to describe the volatility dynamics itself through several latent components (or factors) including some transitory ones and possible also more persistent ones. Then, the state-space model is nonlinear even though a log-transformation may allow to use Kalman 7ltering (Harvey et al., 1994) for a Gaussian quasi-likelihood inference.

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Nonlinear objects of interest in 7nance like volatility factors, conditional betas, durations between transactions, etc., have recently given rise to a burgeoning set of nonlinear state-space approaches. Models of time series subject to changes in regime (Hamilton, 1993) are also popular examples of nonlinear state-space models. Following a common terminology, we de7nitely consider nonlinear state-space models as examples of dynamic factor models since, as in the factor analysis model, a reduced number of variables (called states or factors) are used to reduce the dimension of a large set of multivariate correlations of interest, either cross-sectionally (factor analysis approach) or longitudinally (state-space approach). It is all the more important to consider these two dimensions with a uni7ed viewpoint that one should realize that they are more often than not simultaneously present in time series econometrics. For instance, when following Geweke (1977), Forni, Hallin, Lippi, and Reichlin adopt a (generalized) factor analysis approach not only for the stationary covariance matrix but also for the whole set of values of the spectral density matrix, they also summarize, by the same token, the set of serial correlations. Similarly, to describe the volatility dynamics of a large set of n returns (Doz and Renault, 2003), parsimony can be reached only by using a conditionally heteroskedastic factor model (reduction of the cross-sectional dimension) jointly with a parsimonious stochastic volatility model for the factors (reduction of the longitudinal dimension). More generally, all of the modern asset pricing literature focuses on the double dimension reduction issue by considering pricing kernels that are spanned by a reduced number of factors and the dynamics of which is captured by a reduced number of state variables. Most of the papers gathered in this special Annals issue were presented at a conference held at Universite Libre de Bruxelles, Belgium on November 25 –26th 1999 (20eme Rencontre Franco—Belges de Statisticiens) called to bring together researchers with common interests in dynamic factor models. As planned and anticipated by the conference organizers, the papers submitted and accepted, after revision, for publication in this issue are aimed at not only at extending the standard factor analysis to economic time series or macroeconomic panels that require handling a large number n of long time series, but also to propose new nonlinear state-space models or more generally new tools for dimension reduction in time series. The 7rst two papers of this special issue are both mainly interested in a cross sectional reduction of the dimension in a dynamic context. They both involve authors who have been at the forefront of developing extensions of the factor analysis model well suited for economic time series (see, in particular, Forni et al. (2000), hereafter FHLR, and King et al. (1994), hereafter KSW). FHLR’s generalized factor model was one of the 7rst attempts (see also Stock and Watson, 1999) to adapt in the study of macroeconomic time series a modeling principle that can be traced back to Ross’s (1976) APT and more precisely to its statistical formalization by Chamberlain and Rothschild (1983). In a theoretical model with a countably in7nite collection of variables (n going to in7nity), it is possible to mimic perfectly the behavior of the common factors by well-chosen linear combinations of this in7nite number of variables. This powerful identi7cation tool allows to go even further by relaxing the factor analysis orthogonality conditions between idiosyncratic components. In the generalized K-factor model, one only assumes that K eigenvalues of the covariance matrix of the n variables diverge as

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n tends to in7nity, while the remaining ones are bounded. Chamberlain and Rothschild (1983) and Chamberlain (1983) had shown that this kind of assumption allows for making a tight relationship between traditional factor analysis and the much more user friendly principal components analysis. Still in the APT context, KSW extended the factor analysis model to time varying conditional covariance matrices. The 7rst paper is by Mario Forni, Marc Hallin, Marco Lippi, and Lucrezia Reichlin and pushes the analysis of FHLR one step further by deriving rates of convergence for their “estimators” (or factor scores) of unobservable common factors when both the dimension n of the time series and the length T of the observation period tend to in7nity along speci7c paths (n; T (n)). The consistency result in FHLR merely asserted the existence of paths (n; T (n)) such their estimators converge. In order to obtain operational results as consistency rates as well as a characterization of the paths along which they can be achieved, one has to specify “how common” the common shocks are. More precisely, the speed at which information about the common shocks enters the system as the cross sectional dimension increases, crucially depends on the rate of divergence of the diverging dynamic eigenvalues. It is important to notice that the K diverging eigenvalues are termed “dynamic” because the assumption of K eigenvalues going to in7nity with n, while the other n − K ones remain bounded, is maintained not only for the unconditional covariance matrix but also for all the values of the spectral density matrix on [ − ; ]. Therefore, the idiosyncratic shocks are orthogonal to the common factors at all leads and lags. The paper by Enrique Sentana also addresses the issue of identi7cation of common factors when the cross-sectional dimension tends to in7nity. In his case, this dimension is given by the number n of primitive assets for which a time series of T returns is observed. In contrast to the 7rst paper, the factor analysis principle is not applied to the spectral density matrix but, as in KSW, to the conditional covariance matrix (conditional on the past). Roughly speaking, one could say that the dynamic factor model of the 7rst paper focuses on dynamic principal components useful for forecasting the mean value of the processes of interest (through K coincident indicators) while the conditional factor model of the second paper is more interested in risk forecasting through K main directions of conditional heteroskedasticity. In other words, as explained above, while focusing on a cross-sectional model of the covariance matrix in the spirit of Ross’s APT, this conditional APT provides some longitudinal dimension reduction as well as a tool for the assessment of expected returns through risk premiums. In this context, looking for n-consistent estimators of the common factors amounts to trying to mimic the behavior of the common factors perfectly using the returns of K well diversi7ed portfolios. Enrique Sentana’s analysis is innovating in two respects. First, he proves that, while many portfolios converge to the factors as the number of assets increases, the Kalman 7lter portfolios are the ones with both minimum tracking error variability and maximum correlation with the common factors. Second, the methods proposed so far mainly considered passive (i.e., static) portfolios, as opposed to active (i.e., dynamic) investment strategies, which would use the information available at the time agents’ decisions are taken to form portfolios. The purpose of this paper is to 7ll in this gap in the context of the dynamic version of the APT developed in KSW.

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The possibly nonstationary dynamic factor models as considered in the third paper, co-authored by Daniel Pe˜na and Pilar Poncela, are tightly related to the dynamic factor models of the 7rst paper. The dimension reduction is now more longitudinal in nature, since common trends are supposed to summarize the dominant properties of the dynamics of the n time series (for a 7xed dimension n which need not be large). Yet, it still follows that the advantage of the factor model in terms of mean square forecast errors increases with the dimension n of the time series vector and with the strength of the dynamic relationships between the components. Actually (see, e.g., Escribano and Pe˜na, 1994), when the common factors are nonstationary, the forecasting problem is very related to forecasting problems in the presence of cointegration relationships. The paper shows both through a Monte Carlo study and using a real example that one can obtain a substantial reduction in mean square forecast error from the factor model with respect to alternative forecasting approaches. The theoretical explanation that is provided is that the factor model forecasts incorporate a pooling term similar to the one derived from hierarchical Bayesian models. This pooling terms can be, in some particular cases, identical to the shrinkage term proposed by Garcia-Ferrer et al. (1987). The focus of interest of the fourth paper by Serge Darolles, Jean-Pierre Florens and Christian GouriOeroux is de7nitely a reduction of the longitudinal dimension of correlations. This paper estimates the part of the longitudinal information in a stationary process Xt that can be summarized by the nonlinear canonical directions, that is to say pairs of functions f and g such that f(Xt ) and g(Xt+h ) are the most correlated. All this is done in a fully nonparametric setting. In particular, the authors focus on the implications of time-reversibility of a process. For scalar di5usions, time-reversibility is essentially tantamount to stationarity. This remark has already been used by Hansen et al. (1998) for identifying scalar di5usions and by Florens et al. (1998) for testing for embeddability by stationary scalar di5usions. The present paper pushes the analysis one step further by deducing a test procedure of the time reversibility property for any multivariate stationary process from the comparison of the unconstrained and constrained (by the time reversibility hypothesis) estimators of the canonical directions. Since these directions are unknown nonlinear functions, their estimators, both constrained and unconstrained, must be nonparametric. Another contribution of this paper is to derive the asymptotic properties of kernel-based estimators of the canonical decomposition. As recently put forward by Chen et al. (2000), it is particularly relevant for the identi7cation of common factors (in terms of long term principal components) among n continuous time jointly Markov processes to know if the time reversibility hypothesis can be accepted. The contribution of the last three papers in this issue is to propose new nonlinear state space models which are particularly well suited for 7nancial time series. The paper by Nour Meddahi and Eric Renault focuses on a linear state-space representation of the conditional variance process of a 7nancial asset return. In the same way that GARCH (Engle, 1982; Bollerslev, 1986) are nonlinear time series models built from a linear ARMA representation of squared innovations, this paper proposes to work with a linear state space model for them. In this way, it introduces a semiparametric volatility model, termed Square-Root Stochastic AutoRegressive Volatility (SR-SARV), which is

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characterized by autoregressive dynamics of the stochastic variance. This models nests the standard GARCH model as well as its various asymmetric extensions (Engle and Ng, 1993; Glosten et al., 1989; Heston and Nandi, 2000). It is actually a competitor of the weak GARCH extension proposed by Drost and Nijman (1993). The two classes of processes share the advantage to be closed with respect to temporal aggregation and to be consistent with discrete time sampling in continuous time stochastic volatility models with linear mean reversion in the volatility process. Following the terminology proposed by Barndor5-Nielsen and Shephard (2001), any positive Non-Gaussian Ornstein-Uhlenbeck based continuous time model of volatility gives rise in discrete time to a SR-SARV process and, in the particular case of no leverage e5ect, to a weak GARCH process (Drost and Werker, 1996). The main advantage of SR-SARV with respect to weak GARCH is to capture a genuine conditional variance and not only a linear projection on the past of squared returns thus, to provide conditional moments restrictions for the purpose of inference. Since no linear projections on squared returns are at play, the approach is able to accommodate in7nite fourth moments, non zero conditional skewness or leverage, and even in7nite unconditional variance in the IGARCH spirit. The Stochastic Conditional Duration (SCD) model introduced in the paper by Luc Bauwens and David Veredas extends the logarithmic ACD model of Bauwens and Giot (2000) (following the ACD model of Engle and Russell (1998)) in a similar way as the SR-SARV model extends the GARCH class. While the GARCH speci7cation is a particular case of autoregressive dynamics of the stochastic variance, the log-ACD speci7cation is a particular instance of autoregressive dynamics of the log-expected duration. However, Bauwens and Veredas focus on some parametric speci7cations of this nonlinear state-space model of durations between trades. Basically, the latent state process (interpreted as the expected duration) is perturbed multiplicatively by a Weibull or gamma distributed innovation. The model yields a wide range of shapes of hazard functions. By adapting the idea put forward by Harvey et al. (1994) for stochastic volatility models, the estimation of the parameters is performed by quasi-maximum likelihood, after transforming the original nonlinear state-space model in a linear although non-Gaussian form, and using the Kalman 7lter. It is important to realize that a byproduct of the SCD speci7cation with respect to standard ACD is to provide some Rexibility to entail unconstrained variation of the conditional mean and conditional variance of durations. The 7nal paper, written by Eric Ghysels, Christian Gourieroux, and Joann Jasiak pushes this remark even further by arguing that at least two time varying factors are required to accommodate the complex duration dynamics of 7nancial markets. Intuitively, one factor may drive the conditional mean, while a second one drives the conditional (under)-overdispersion. In other words, the duration process itself is endowed with a stochastic volatility factor, which motivates the title of the paper. In this nonlinear state-space model of durations, the observable process is simply a well-suited nonlinear transformation of a latent bivariate Gaussian autoregressive process. The paper presents the distributional properties of this Stochastic Volatility Duration (SVD) model and compares its performance to the one of ACD models in an empirical study of intertrade durations. The versatility of this new approach suggests

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several new diagnostic tools for risk analysis such as the conditional overdispersion and Time at Risk. To conclude, we would like to take this opportunity to thank all the referees who helped us with the long although interesting task of reviewing the papers and providing us with timely and insightful reports. This Annals issue started when two of us (Christophe Croux and Bas Werker) were at the same place, Universite Libre de Bruxelles, where the conference on factor models was organized, and ended when a second pair of co-editors (Eric Renault and Bas Werker) were both aBliated or associated to CIREQ, research center in Montreal. We would like to acknowledge the continuous support from these two institutions for this project. Last, but absolutely not least, we would also like to thank the Editors of the Journal of Econometrics for their support. References Barndor5-Nielsen, O.E., Shephard, N., 2001. Non-Gaussian OU based models and some of their uses in 7nancial economics. Journal of the Royal Statistical Society B 63, 167–241. Bartholomew, D.J., 1987. Latent Variable Models and Factor Analysis. In: GriBn’s Statistical Monograph, Vol. 40. Oxford University Press, Oxford. Bauwens, L., Giot, P., 2000. The logarithmic ACD model: an application to the bid-ask quote process of three NYSE stocks. Annales d’Economie et de Statistique 60, 117–149. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Chamberlain, G., 1983. Funds, factors, and diversi7cation in arbitrage pricing models. Econometrica 51, 1305–1324. Chamberlain, G., Rothschild, M., 1983. Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica 51, 1281–1304. Chen, X., Hansen, L.P., Scheinkman, J., 2000. Principal components and the long run, unpublished manuscript. Cochrane, J.H., 2001. Asset Pricing. Princeton University Press, Princeton, NJ. Doz, C., Renault, E., 2003. Conditionally heteroskedastic factor models: identi7cation and instrumental variables estimation, unpublished manuscript. Drost, F.C., Nijman, Th.E., 1993. Temporal aggregation of GARCH processes. Econometrica 61, 909–927. Drost, F.C., Werker, B.J.M., 1996. Closing the GARCH gap: continuous time GARCH modeling. Journal of Econometrics 74, 31–58. Engle, R.F., 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inRation. Econometrica 50, 987–1007. Engle, R.F., Ng, V.K., 1993. Measuring and testing the impact of news on volatility. Journal of Finance 48, 1749–1778. Engle, R.F., Ng, V.K., Rothschild, M., 1990. Asset pricing with a factor ARCH structure: empirical estimates for Treasury bills. Journal of Econometrics 45, 213–237. Engle, R.F., Russell, J.R., 1998. Autoregressive conditional duration: a new approach for irregularly spaced transaction data. Econometrica 66, 1127–1162. Escribano, A., Pe˜na, D., 1994. Cointegration and common factors. Journal of Time Series Analysis 15, 577–586. Florens, J.P., Renault, E., Touzi, N., 1998. Testing for embeddability by stationary reversible continuous-time Markov processes. Econometric Theory 14, 744–769. Garcia-Ferrer, A., High7eld, R.A., Palm, F., Zellner, A., 1987. Macroeconomic forecasting using pooled international data. Journal of Business and Economic Statistics 5, 53–67. Geweke, J., 1977. The dynamic factor analysis of economic time series. In: Aigner, D.J., Goldberger, A.S. (Eds.), Latent Variables in Socio-Economic Models. North-Holland, Amsterdam.

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Christophe Croux K.U. Leuven and ECARES Belgium Eric Renault Department des Sciences Economiques; CIRANO and CIREQ Universite de Montreal; 2020 Rue Universite; 25e Etage Montreal; Quebec; Canada H3A 2A5 E-mail address: [email protected] Bas Werker CentER at Tilburg University, The Netherlands