Dynamic factors in the presence of blocks

Abstract

Macroeconometric data often come under the form of large panels of time series, themselves decomposing into smaller but still quite large subpanels or blocks. We show how the dynamic factor analysis method proposed in Forni et al. (2000), combined with the identification method of Hallin and Liška (2007), allows for identifying and estimating joint and block-specific common factors. This leads to a more sophisticated analysis of the structures of dynamic interrelations within and between the blocks in such datasets, along with an informative decomposition of explained variances. The method is illustrated with an analysis of a dataset of Industrial Production Indices for France, Germany, and Italy.

Introduction

In many fields–macroeconometrics, finance, environmental sciences, chemometrics, …–information comes under the form of a large number of observed time series or panel data. Panel data consist of series of observations (length $T$) made on $n$ individuals or “cross-sectional items” that have been put together on purpose, because, mainly, they carry some information about some common feature or unobservable process of interest, or are expected to do so. This “commonness” is a distinctive feature of panel data: mutually independent cross-sectional items, in that respect, do not constitute a panel (or then, a degenerate one). Cross-sectional heterogeneity is another distinctive feature of panels: $n$ (possibly non-independent) replications of the same time series would be another form of degeneracy. Moreover, the impact of item-specific or idiosyncratic effects, which have the role of a nuisance, very often dominate, quantitatively, that of the common features one is interested in.

Finally, all individuals in a panel are exposed to the influence of unobservable or unrecorded covariates, which create complex interdependencies, both in the cross-sectional as in the time dimension, which cannot be modelled, as this would require questionable modelling assumptions and a prohibitive number of nuisance parameters. These interdependencies may affect all (or almost all) items in the panel, in which case they are “common”; they also may be specific to a small number of items, hence “idiosyncratic”.

The idea of separating “common” and “idiosyncratic” effects is thus at the core of panel data analysis. The same idea is the cornerstone of factor analysis. There is little surprise, thus, to see a time series version of factor analysis emerging as a powerful tool in the context of panel data. This dynamic version of factor models, however, requires adequate definitions of “common” and “idiosyncratic”. This definition should not simply allow for identifying the decomposition of the observation into a “common” component and an “idiosyncratic” one, but also should provide an adequate translation of the intuitive meanings of “common” and “idiosyncratic”.

Denote by $X_{it}$ the observation of item $i(i=1,\dots ,n)$ at time $t(t=1,\dots ,T)$; the factor model decomposition of this observation takes the form

$$X_{it}={\chi }_{it}+{\xi }_{it}\text{,}i=1,\dots ,n,t=1,\dots ,T\text{,}$$

where the common component ${\chi }_{it}$ and an the idiosyncratic one ${\xi }_{it}$ are mutually orthogonal (at all leads and lags) but unobservable. Some authors identify this decomposition by requiring the idiosyncratic components to be “small” or “negligible”, as in dimension reduction techniques. Some others require that the $n$ idiosyncratic processes be mutually orthogonal white noises. Such characterizations do not reflect the fundamental nature of factor models: idiosyncratic components indeed can be “large” and strongly autocorrelated, while white noise can be common. For instance, in a model of the form $X_{it}={\chi }_t+{\xi }_{it}$, where ${\chi }_t$ is white noise and orthogonal to ${\xi }_{it}={\epsilon }_{it}+a_i{\epsilon }_{i,t-1}$, with i.i.d. ${\epsilon }_{it}$’s, the white noise component ${\chi }_t$, which is present in all cross-sectional items, very much qualifies as being “common”, while the cross-sectionally independent autocorrelated ${\xi }_{it}$’s, being item-specific, exhibit all the attributes one would like to see in an “idiosyncratic” component.

A possible characterization of commonness/idiosyncrasy is obtained by requiring the common component to account for all cross-sectional correlations, leading to possibly autocorrelated but cross-sectionally orthogonal idiosyncratic components. This yields the so-called “exact factor models” considered, for instance, by Sargent and Sims (1977) and Geweke (1977). These exact models, however, are too restrictive in most real life applications, where it often happens that two (or a small number of) cross-sectional items, being neighbours in some broad sense, exhibit cross-sectional correlation also in variables that are orthogonal, at all leads and lags, to all other observations throughout the panel. A “weak” or “approximate factor model”, allowing for mildly cross-sectionally correlated idiosyncratic components, therefore also has been proposed (Chamberlain, 1983, Chamberlain and Rothschild, 1983), in which, however, the common and idiosyncratic components are only asymptotically (as $n\to \infty$) identified. Under its most general form, the characterization of idiosyncrasy, in this weak factor model, can be based on the behavior, as $n\to \infty$, of the eigenvalues of the spectral density matrices of the unobservable idiosyncratic components, but also (Forni and Lippi, 2001) on the asymptotic behavior of the eigenvalues of the spectral density matrices of the observations themselves: see Section 2 for details. This general characterization is the one we are adopting here.

Finally, once the common and idiosyncratic components are identified, two types of factor models can be found in the literature, depending on the way factors are driving the common components. In static factor models, it is assumed that common components are of the form

$${\chi }_{it}=\sum _{l=1}^q{b}_{il}{f}_{lt}\text{,}i=1,\dots ,n,t=1,\dots ,T\text{,}$$

that is, the ${\chi }_{it}$’s are driven by $q$ factors $f_{1t},\dots ,f_{qt}$ which are loaded instantaneously. This static approach is the one adopted by Chamberlain (1983), Chamberlain and Rothschild (1983), Stock and Watson, 1989, Stock and Watson, 2002a, Stock and Watson, 2002b, Stock and Watson, 2005, Bai and Ng, 2002, Bai and Ng, 2007, and a large number of applied studies. The so-called general dynamic model decomposes common components into

$${\chi }_{it}=\sum _{l=1}^q{b}_{il}\left(L\right)u_{lt}\text{,}i=1,\dots ,n,t=1,\dots ,T\text{,}$$

where $u_{1t},\dots ,u_{qt}$, the unobservable common shocks, are loaded via one-sided linear filters $b_{il}\left(L\right)$. That “fully dynamic” approach (the terminology is not unified and the adjective “dynamic” is often used in an ambiguous way) goes back, under exact factor form, to Chamberlain (1983) and Chamberlain and Rothschild (1983), but was developed, mainly, by Forni et al., 2000, Forni et al., 2003, Forni et al., 2004, Forni et al., 2005, Forni et al., 2009, Forni and Lippi, 2001 and Hallin and Liška (2007).

The static model (1) clearly is a particular case of the general dynamic one (2). Its main advantage is simplicity. On the other hand, both models share the same assumption on the asympotic behavior of spectral eigenvalues—the plausibility of which is confirmed by empirical evidence. But the static model (1) places an additional and rather severe restriction on the data-generating process, while the dynamic one (2), as shown by Forni and Lippi (2001), does not—we refer to Section 2 for details. Moreover, the synchronization of clocks and calendars across the panel is often quite approximative, so that the concept of “instantaneous loading” itself may be questionable.

Both the static and the general dynamic models are receiving increasing attention in finance and macroeconometric applications where information usually is scattered through a (very) large number $n$ of interrelated time series ($n$ values of the order of several hundreds, or even one thousand, are not uncommon). Classical multivariate time series techniques are totally helpless in the presence of such values of $n$, and factor model methods, to the best of our knowledge, are the only ones that can handle such datasets. In macroeconomics, factor models are used in business cycle analysis (Forni and Reichlin, 1998, Giannone et al., 2006), in the identification of economy-wide and global shocks, in the construction of indices and forecasts exploiting the information scattered in a huge number of interrelated series (Altissimo et al., 2001), in the monitoring of economic policy (Giannone et al., 2005), and in monetary policy applications (Bernanke and Boivin, 2003, Favero et al., 2005). In finance, factor models are at the heart of the extensions proposed by Chamberlain and Rothschild (1983) and Ingersol (1984) of the classical arbitrage pricing theory; they also have been considered in performance evaluation and risk measurement (Chapters 5 and 6 of Campbell et al., 1997), and in the statistical analysis of the structure of stock returns (Yao, 2008).

Factor models in the recent years also generated a huge amount of applied work: see d’Agostino and Giannone (2005), Artis et al. (2005), Bruneau et al. (2007), Den Reijer (2005), Dreger and Schumacher (2004), Schumacher (2007), Nieuwenhuyzen (2004), Schneider and Spitzner (2004), Giannone and Matheson (2007), and Stock and Watson (2002b) for applications to data from UK, France, the Netherlands, Germany, Belgium, Austria, New Zealand, and the US, respectively; Altissimo et al. (2001), Angelini et al. (2001), Forni et al. (2003), and Marcellino et al. (2003) for the Euro area and Aiolfi et al. (2006) for South American data—to quote only a few. Dynamic factor models also have entered the practice of a number of economic and financial institutions, including several central banks and national statistical offices, who are using them in their current analysis and prediction of economic activity. A real time coincident indicator of the EURO area business cycle (EuroCOIN), based on Forni et al. (2000), is published monthly by the London-based Center for Economic Policy Research and the Banca d’Italia: see (http://www.cepr.org/data/EuroCOIN/). A similar index, based on the same methods, is established for the US economy by the Federal Reserve Bank of Chicago.

Although heterogeneous, panel data very often are obtained by pooling together several “blocks” which themselves constitute “large” subpanels. In macroeconometrics, for instance, data typically are organized either by country or sectoral origin: the database which is used in the construction of EuroCOIN, the monthly indicator of the euro area business cycle published by CEPR, includes almost 1000 time series that cover six European countries and are organized into eleven subpanels including industrial production, producer prices, monetary aggregates, etc. Depending on the objectives of the analysis, such a panel could be divided into six blocks (one for each country), or into eleven blocks (one for each subpanel). When these blocks are large enough, several dynamic factor models can be considered and analyzed, allowing for a refined analysis of interblock relations. In the simple two-block case, “marginal common factors” can be defined for each block, and need not coincide with the “joint common factors” resulting from pooling the two blocks.

The objective of this paper is to provide a theoretical basis for that type of analysis. For simplicity, we start with the simple case of two blocks. We show (Section 2) how a factorization of the Hilbert space spanned by the $n$ observed series leads to a decomposition of each of them into four mutually orthogonal (at all leads and lags) components: a strongly idiosyncratic one, a strongly common one, a weakly common, and a weakly idiosyncratic one. In Sections 3 Identifying the factor structure; population results, 4 Recovering the factor structure; estimation results, we show how projections onto appropriate subspaces provide consistent data-driven reconstructions of those various components. Section 5 is devoted to the general case of $K\ge 2$ blocks, allowing for various decompositions of each observation into mutually orthogonal (at all leads and lags) components. The tools we are using throughout are Brillinger’s theory of dynamic principal components, a key result (Proposition 2) by Forni et al. (2000), and the identification method developed by Hallin and Liška (2007). Proofs are concentrated in Appendix.

The potential of the method is briefly illustrated, in Section 6, with a panel of Industrial Production Index data for France and Germany ($K=2$ blocks, $q=3$ factors), then France, Germany and Italy ($K=3$ blocks, $q=4$ factors). Simple as it is, the analysis of that dataset reveals some striking facts. For instance, both Germany and Italy exhibit a “national common factor” which is idiosyncratic to the other two countries, while France’s common factors are included in the space spanned by Germany’s. The (estimated) percentages of explained variation associated with the various components also are quite illuminating: Germany, with 25% of common variation, is the “most common” out of the three countries. But it also is, with only 6.4% of its total variation, the “least strongly common” one. France has the highest proportion (82.4%) of marginal idiosyncratic variation but also the highest proportions of strongly and weakly idiosyncratic variations (72.6% and 9.8%, respectively). We do not attempt here to provide an economic interpretation for such facts. Nor do we apply the method to a more sophisticated dataset.² But we feel that the simple application we are proposing provides sufficient evidence of the potential power of the method, both from a structural as from a quantitative point of view.³