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## Frequency-Domain Approach in High-Dimensional Dynamic Factor Models

High-Dimensional Dynamic Factor Models have their origin in macroeconomics, precisely in empirical research on Business Cycles. The central idea, going back to the work of Burns and Mitchell in the years 1940, 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 year 2000 on a model in which (1) both n the number of variables in the dataset and T , the number of observations for each variable, may be large, and (2) all the variables in the dataset depend dynamically on a fixed independent of n , a number of “common factors,” plus variable-specific, usually called “idiosyncratic,” components. The structure of the model can be exemplified as follows: x i t = α i u t + β i u t − 1 + ξ i t , i = 1, … , n , t = 1, … , T , (*) where the observable variables x i t are driven by the white noise u t , which is common to all the variables, the common factor, and by the idiosyncratic component ξ i t . The common factor u t is orthogonal to the idiosyncratic components ξ i t , the idiosyncratic components are mutually orthogonal (or weakly correlated). Lastly, the variations of the common factor u t affect the variable x i t dynamically, that is through the lag polynomial α i + β i L . Asymptotic results for High-Dimensional Factor Models, particularly consistency of estimators of the common factors, are obtained for both n and T tending to infinity. Model ( ∗ ) , generalized to allow for more than one common factor and a rich dynamic loading of the factors, has been studied in a fairly vast literature, with many applications based on macroeconomic datasets: (a) forecasting of inflation, industrial production, and unemployment; (b) structural macroeconomic analysis; and (c) construction of indicators of the Business Cycle. This literature can be broadly classified as belonging to the time- or the frequency-domain approach. The works based on the second are the subject of the present chapter. We start with a brief description of early work on Dynamic Factor Models. Formal definitions and the main Representation Theorem follow. The latter determines the number of common factors in the model by means of the spectral density matrix of the vector ( x 1 t x 2 t ⋯ x n t ) . Dynamic principal components, based on the spectral density of the x ’s, are then used to construct estimators of the common factors. These results, obtained in early 2000, are compared to the literature based on the time-domain approach, in which the covariance matrix of the x ’s and its (static) principal components are used instead of the spectral density and dynamic principal components. Dynamic principal components produce two-sided estimators, which are good within the sample but unfit for forecasting. The estimators based on the time-domain approach are simple and one-sided. However, they require the restriction of finite dimension for the space spanned by the factors. Recent papers have constructed one-sided estimators based on the frequency-domain method for the unrestricted model. These results exploit results on stochastic processes of dimension n that are driven by a q -dimensional white noise, with q < n , that is, singular vector stochastic processes. The main features of this literature are described with some detail. Lastly, we report and comment the results of an empirical paper, the last in a long list, comparing predictions obtained with time- and frequency-domain methods. The paper uses a large monthly U.S. dataset including the Great Moderation and the Great Recession.

## Time-Domain Approach in High-Dimensional Dynamic Factor Models

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 i t = α i u t + β i u t − 1 + ξ i t , i = 1 , … , n , t = 1 , … , T , where the observable variables x i t are driven by the white noise u t , which is common to all the variables, the common shock, and by the idiosyncratic component ξ i t . The common shock u t is orthogonal to the idiosyncratic components ξ i t , the idiosyncratic components are mutually orthogonal (or weakly correlated). Last, the variations of the common shock u t affect the variable x i t dynamically, that is, through the lag polynomial α i + β 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 ( ∗ ) becomes x i t = α i F 1 t + β i F 2 t + ξ i t , F 1 t = u t , F 2 t = u t − 1 . Instead of the dynamic equation ( ∗ ) 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 1 t = u t , F 2 t = F 1, t − 1 . This transformation into a static representation, whose general form is x i t = λ i 1 F 1 t + ⋯ + λ i r F r t + ξ i t , is extremely convenient for estimation and forecasting of high-dimensional dynamic factor models. In particular, the factors F j t and the loadings λ i j can be consistently estimated from the principal components of the observable variables x i t . 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.