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:

(*)$${x}_{it}={\alpha}_{i}{u}_{t}+{\beta}_{i}{u}_{t-1}+{\xi}_{it},\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}i=1,\dots ,n,\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}t=1,\dots ,\phantom{\rule{0.2em}{0ex}}T,$$

where the observable variables ${x}_{it}$ are driven by the white noise ${u}_{t}$, which is common to all the variables, the common shock, and by the idiosyncratic component ${\xi}_{it}$. The common shock ${u}_{t}$ is orthogonal to the idiosyncratic components ${\xi}_{it}$, the idiosyncratic components are mutually orthogonal (or weakly correlated). Last, the variations of the common shock ${u}_{t}$ affect the variable ${x}_{it}$
*dynamically*, that is, through the lag polynomial ${\alpha}_{i}+{\beta}_{i}L$. 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 ($\ast $) becomes

$$\begin{array}{lll}{x}_{it}\hfill & =\hfill & {\alpha}_{i}{F}_{1t}+{\beta}_{i}{F}_{2t}+{\xi}_{it},\hfill \\ {F}_{1t}\hfill & =\hfill & {u}_{t},\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}{F}_{2t}={u}_{t-1}.\hfill \end{array}$$

Instead of the dynamic equation ($\ast $) there is now a static equation, while instead of the white noise ${u}_{t}$ there are now two factors, also called *static factors*, which are dynamically linked:

$${F}_{1t}={u}_{t},\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}{F}_{2t}={F}_{\mathrm{1,}t-1}\mathrm{.}$$

This transformation into a static representation, whose general form is

$${x}_{it}={\lambda}_{i1}{F}_{1t}+\cdots +{\lambda}_{ir}{F}_{rt}+{\xi}_{it},$$

is extremely convenient for estimation and forecasting of high-dimensional dynamic factor models. In particular, the factors ${F}_{jt}$ and the loadings ${\lambda}_{ij}$ can be consistently estimated from the principal components of the observable variables ${x}_{it}$.

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.