High-dimensional dynamic factor models have their origin in macroeconomics, more specifically in empirical research on business cycles. The central idea, going back to the work of Burns and Mitchell in the 1940s, is that the fluctuations of all the macro and sectoral variables in the economy are driven by a “reference cycle,” that is, a one-dimensional latent cause of variation. After a fairly long process of generalization and formalization, the literature settled at the beginning of the 2000s on a model in which (a) both n, the number of variables in the data set, and T, the number of observations for each variable, may be large; (b) all the variables in the data set depend dynamically on a fixed, independent of n, number of common shocks, plus variable-specific, usually called idiosyncratic, components. The structure of the model can be exemplified as follows:
(*)xit=αiut+βiut−1+ξit,i=1,…,n,t=1,…,T,
where the observable variables xit are driven by the white noise ut, which is common to all the variables, the common shock, and by the idiosyncratic component ξit. The common shock ut is orthogonal to the idiosyncratic components ξit, the idiosyncratic components are mutually orthogonal (or weakly correlated). Last, the variations of the common shock ut affect the variable xitdynamically, that is, through the lag polynomial αi+βiL. Asymptotic results for high-dimensional factor models, consistency of estimators of the common shocks in particular, are obtained for both n and T tending to infinity.
The time-domain approach to these factor models is based on the transformation of dynamic equations into static representations. For example, equation (∗) becomes
xit=αiF1t+βiF2t+ξit,F1t=ut,F2t=ut−1.
Instead of the dynamic equation (∗) there is now a static equation, while instead of the white noise ut there are now two factors, also called static factors, which are dynamically linked:
F1t=ut,F2t=F1,t−1.
This transformation into a static representation, whose general form is
xit=λi1F1t+⋯+λirFrt+ξit,
is extremely convenient for estimation and forecasting of high-dimensional dynamic factor models. In particular, the factors Fjt and the loadings λij can be consistently estimated from the principal components of the observable variables xit.
Assumption allowing consistent estimation of the factors and loadings are discussed in detail. Moreover, it is argued that in general the vector of the factors is singular; that is, it is driven by a number of shocks smaller than its dimension. This fact has very important consequences. In particular, singularity implies that the fundamentalness problem, which is hard to solve in structural vector autoregressive (VAR) analysis of macroeconomic aggregates, disappears when the latter are studied as part of a high-dimensional dynamic factor model.
Constrained Factor Models for High-Dimensional Matrix-Variate Time Series
