If you are a VAR (vector autoregression) junkie like I am, this paper by Stock and Watson might interest you, but otherwise you will want to skip this. I just want the link easily accessible. Seriously, unless you are a VAR junkie and need a fix, scroll on by…
This paper considers VAR models incorporating many time series that interact through a few dynamic factors. Several econometric issues are addressed including estimation of the number of dynamic factors and tests for the factor restrictions imposed on the VAR. Structural VAR identification based on timing restrictions, long run restrictions, and restrictions on factor loadings are discussed and practical computational methods suggested. Empirical analysis using U.S. data suggest several (7) dynamic factors, rejection of the exact dynamic factor model but support for an approximate factor model, and sensible results for a SVAR that identifies money policy shocks using timing restrictions.
…In this paper, we examine VAR methods that can be used to identify the space of structural shocks when there are hundreds of economic time series variables that potentially contain information about these underlying shocks. … The premise of the dynamic factor model (DFM) is that there are a small number of unobserved common dynamic factors that produce the observed comovements of economic time series. These common dynamic factors are driven by the common structural economic shocks, which are the relevant shocks that one must identify for the purposes of conducting policy analysis. Even if the number of common shocks is small, because the dynamic factors are unobserved this model implies that the innovations from conventional VAR analysis with a small or moderate number of variables will fail to span the space of the structural shocks to the dynamic factors. Instead, these shocks are only revealed when one looks at a very large number of variables and distills from them the small number of common sources of comovement.
There is a body of empirical evidence that the dynamic factor model, with a small number of factors, captures the main comovements of postwar U.S. macroeconomic time series data. Sims and Sargent (1977) examine a small system and conclude that two dynamic factors can explain 80% or more of the variance of major economic variables, including the unemployment rate, industrial production growth, employment growth, and wholesale price inflation; moreover, one of these dynamic factors is primarily associated with the real variables, while the other is primarily associated with prices. Empirical work using methods developed for many-variable systems has supported the view that only a few – perhaps two – dynamic factors explain much of the predictable variation in major macroeconomic aggregates (e.g. Stock and Watson (1999, 2002a), Giannone, Reichlin, and Sala (2004)). These new methods for estimating and analyzing dynamic factor models, combined with the empirical evidence that perhaps only a few dynamic factors are needed to explain the comovement of macroeconomic variables, has motivated recent research on how best to integrate factor methods into VAR and SVAR analysis (Bernanke and Boivin (2003), Bernanke, Boivin, and Eliasz (2005; BBE hereafter), Favero and Marcellino (2001), Favero, Marcellino, and Neglia (2004), Giannone, Reichlin, and Sala (2002, 2004), and Forni, Giannone, Lippi, and Reichlin (2004)); we return to this recent literature in Sections 2 and 5.
This paper has three objectives. The first is to provide a unifying framework that explicates the implications of DFMs for VAR analysis, both reduced-form (including forecasting applications) and structural. In particular we list a number of testable overidentifying restrictions that are central to the simplifications provided by introducing factors into VARs.
Our second objective is to examine empirically these implications of the DFM for VAR analysis. Is there support for the exact factor model restrictions or, if not, for an approximate factor model such as that of Chamberlain and Rothschild (1983)? If so, how many factors are needed: two, as suggested by Sargent and Sims (1977) and more recent literature, or more? Another implication of the DFM is that, once factors are included in the VAR, impulse responses with respect to structural shocks should not change upon the inclusion of additional observable variables; but is this borne out empirically?
Our third objective is to provide a unified framework and some new econometric methods for structural VAR analysis using dynamic factors. These methods build on the important initial work by Giannoni, Reichlin, and Sala (2002) and BBE (2005) on the formulation and estimation of structural VARs using factors obtained from large data sets, and we adopt BBE’s term and refer to these system as FAVARs (Factor-Augmented VARs). We consider a variety of identifying schemes, including schemes based on the timing of shocks (as considered by BBE), on long run restrictions (as considered by Giannoni, Reichlin, and Sala (2002)), and on restrictions on the factor loading matrices (as considered by Kose, Otrok and Whiteman (2003), among others). We present feasible estimation strategies for imposing the potentially numerous overidentifying restrictions.
We have three main empirical findings, which are based on an updated version of the Stock-Watson (2002a) data set (the version used here has 132 monthly U.S. variables, 1959 – 2003). First, it appears that the number of dynamic factors present in our data set exceeds two; we estimate the number to be seven. This estimate is robust to details of the model specification and estimation method, and it substantially exceeds the estimates appearing in the earlier literature; we suggest that this estimate is not spurious but rather reflects the narrow scope of the data sets, combined with methodological limitations, in the early studies that suggested only one or two factors.
Second, we find that many of the implications of the DFM for the full 132- variable VAR are rejected, however these rejections are almost entirely associated with coefficients that are statistically significantly different from zero but are very small in an economic or practical sense.
Third, we illustrate the structural FAVAR methods by an empirical reexamination of the BBE identification scheme, using different estimation procedures. We find generally similar results to BBE, which in many cases accord with standard macroeconomic theory; but we also find many rejections of the overidentifying restrictions. These rejections suggest specific ways in which the BBE identifying assumptions fail, something not possible in exactly identified SVAR analysis…