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date: 14 August 2020

Time-Domain Approach in High-Dimensional Dynamic Factor Models

Summary and Keywords

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+βiut1+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=ut1.

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,t1.

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.

Keywords: large data sets, curse of dimensionality, factor models, dynamic principal components, singular stochastic vectors, macroeconometrics

Introduction

A prerequisite for this article is a basic knowledge of the time-domain analysis of weakly stationary stochastic processes, including fundamental and nonfundamental representations, and of principal component analysis. All the definitions and results are illustrated by elementary examples in which the number of factors is one or two and the dynamics are limited to moving averages of order one.

The present article is closely linked to Lippi (2018), “Frequency-Domain Approach to High-Dimensional Dynamic Factor Models.” The latter, in the section “Early Literature on Dynamic Factor Models,” contains a short review of the literature on dynamic factor models from the 1940s, with the seminal contribution in Burns and Mitchell (1946), to the 1970s, with Sargent and Sims (1977) and related papers, to the introduction of infinite cross-sections, first in Chamberlain and Rothschild (1983) and Chamberlain (1983) and then in Forni and Reichlin (1996), Forni and Lippi (1997), Stock and Watson (1998), and Forni and Reichlin (1998).

Another important observation regarding the relationship between this article and Lippi (2018) is that “frequency domain” and “time domain” have become customary in the literature on dynamic factor models in reference to the methods outlined in Lippi (2018) and in this article, respectively. However, there is no conceptual difference between the models and the role of the spectral methods is purely technical, as shown in Hallin and Lippi (2013). Much more important is the fact that some simple models cannot be put in the static representation (see later discussion) and therefore cannot be studied with this article’s techniques.

General Assumptions and Representation Theorem” gives the basic definition of the high-dimensional dynamic factor model and state the main representation theorem. A static equation links the n observable variables xit to the r factors Fjt and the idiosyncratic components ξit. The representation theorem proves that the number of factors is equal to the number of eigenvalues of the variance-covariance matrix of the x’s, which diverge as n tends to infinity. Thus the unobserved structure of the model is “revealed” by properties of the observable variables. The model and the theorem are illustrated by means of an elementary example.

“Dynamics” shows that a rich variety of dynamic models, driven by a q-dimensional white-noise vector of common shocks ut, can be accommodated in the static representation of the previous section. This is done by replacing lags of ut in the primitive form with factors Fjt in the static representation. A consequence is that in general the vector Ft=(F1tFrt) is singular; that is, its dimension is larger than the dimension of its driving white noise ut. A theorem on singular autoregressive–moving-average vector processes ensures that Ft hasafinite-degreevectorautoregressive(VAR)representation. The same theorem implies that ut is fundamental for Ft.

“Estimation” provides motivation for the use of principal components to construct estimators of the factor space. The estimation theory for the number of static factors r, for the number of dynamic factors q, and for the factors themselves is reviewed. “Forecasting” presents the forecasting equation implicit in the structure of the dynamic factor model.

The Factor Model and Structural Macroeconomic Analysis” shows that the high-dimensional dynamic factor models can be used for structural macroeconomic analysis. The common and idiosyncratic components of variables like the gross domestic product (GDP), industrial production, inflation, and so on are interpreted as the relevant macroeconomic indicators and measurement errors, respectively. The usual identification techniques of the structural shocks can be applied with no modification. Thus, unlike in structural VARs (SVAR), no fundamentalness problem arises.

Usually, the results on estimation and forecasting of factor models are applied to the data set obtained by transformations by which stationarity is achieved, differencing in particular. However, “Non-Stationary Processes” briefly reviews papers in which the non-stationarity of the data set is directly taken into account. “Conclusions and Open Problems for Future Research” concludes and outlines some ideas for future developments.

As a rule, expressions like xit or μnjx are written without a comma between n and t or n and n. However, to avoid possible confusion, this article uses xi,t+1, xi+1,t, μn,j+1x, and so on. The end of “Examples” and “Observations” is marked with the symbol □.

General Assumptions and Representation Theorem

The data sets considered in the literature on high-dimensional dynamic factor models are of the form

{Xit,1<t<T,1<i<n},
(1)

where n, the size of the cross-section, is comparable to, or even larger than, T. The corresponding theoretical assumption is that X is a finite realization of a double-indexed stochastic process

{xit,t,i}.
(2)

It is assumed that the vector process {(xi1tximt),t} is weakly stationary for all m-tuples i1,...,im, which implies that all the processes xit are weakly stationary. Without loss of generality, it is assumed that all the processes xit are zero-mean.

In empirical applications to macroeconomic data sets, most of the variables are nonstationary, I(1) in particular. Usually, the results on estimation and forecasting of factor models are applied to the data set obtained by transformations by which stationarity is achieved, differencing in particular. However, the section “Non-Stationary Processes” briefly reviews papers in which the non-stationarity of the data set is directly taken into account.

A particular form of the factor model illustrated in this article is the following. Suppose that the observed processes xit can be represented as follows:

xit=χit+ξit,χit=λi1F1t++λirFrt,
(3)

where:

  1. (a) The vector process Ft=(F1tFrt) is weakly stationary and zero-mean;

  2. (b) ξi1tξi2t for all t;

  3. (c) ξitFt.

The unobserved processes χit and ξit are called the common components and the idiosyncratic components of the processes xit, respectively. The unobserved processes Fjt,j=1,,r, are called the common factors. The unobserved coefficients λij are called the loadings of the factors. Note that under (b) and (c) the covariance among the x’s is completely accounted for by the covariance among the common components.

Note also that in factor models in which n is a given, finite number, assumption (b) is necessary for identification. As seen later in Theorem 1, a weaker form of (b) is sufficient when n is allowed to tend to infinity.

The most elementary specification of model (3) is the following.

Example 1

Let r=1, only one common factor, and

xit=λFt+ξit,
(4)

where λ0. For the covariance among the x’s:

cov(xi1t,xi2t)=cov(χi1t,χi2t)=λ2σF2.

By taking the arithmetic mean of the first n processes:(1/n)i=1nxit=λFt+(1/n)i=1nξit, that is

1ni=1nxitλFt=1ni=1nξit.

Assuming that the variances of the idiosyncratic components are bounded, σξi2σξ2, i,

var(1ni=1nξit)1nσξ2.

Thus, in mean square,

limn1ni=1nxit=λFt.
(5)

This example is as simple as it is important to grasp the idea leading to estimates for the unobserved components. Here n, the size of the cross-section, and T, the number of observations for each time series, both tend to infinity. A consequence of n tending to infinity is that the idiosyncratic components are averaged away and the common components are estimated consistently, which is not the case if the model only contains a finite number of processes xit.□

Observation 1

The discussion in Example 1, the limit in equation (5) in particular, involves population covariances and mean-square convergence. Examples based on population covariances are used in this article only for heuristic purposes; see also the introduction of principal components in the “Estimation” section, for example.□

The result presented here as Theorem 1 states that the processes xit have a representation like (3), with a slightly different definition of the idiosyncratic components, if and only if their covariances satisfy the condition specified in Assumption 1. Thus the existence of a structure like (3) can be established by a criterion implying only properties of the observable processes xit.

Define xnt=(x1tx2txnt). Γnkx is the covariance between xnt and xn,tk:

Γnkx=E(xtxtk).
(6)

The eigenvalues of the variance-covariance matrix Γn0x are real and non-negative. They are denoted by μnjx,j=1,,n, with the assumption that μnjxμn,j+1x,j=1,,n1. In the same way χnt,ξnt,Γnkχ,Γnkξ is defined and the eigenvalues μnjχ and μnjξ. Again, this is assuming that μnjχμn,j+1χ and for j=1,,n1.

Assumption 1

There exists an integer s0 such that μn,s+1xR for some positive real R and all n.

If Assumption 1 holds, let r be the minimum of all integers s such that μn,s+1x is bounded for n. Of course if r>0 then limnμnjx= for 0<jr.

In Example 1, assuming that σξi2=σξ2 for all i, it is easily seen that

Γn0x=λ2σF2(11)(11)+σξ2In,

with eigenvalues

μn1x=nλ2σF2+σξ2,μn2x=σξ2,μn3x=σξ2,,μnn=σξ2,

so that Assumption 1 is fulfilled and r=1.

Observation 2

Assumption 1 is a fairly serious restriction. A simple example in which no finite s can be determined can be found in Lippi (2018) and the section “The Finite-Dimension Assumption and the Static Method.”

Theorem 1

Suppose that Assumption 1 holds and r>0. Then there exists an r-dimensional weakly stationary vector process Ft=(F1tFrt) and loadings λij, i, j=1,...,r, such that

xit=χit+ξit,χit=λi1F1t++λirFrt,
(7)

where:

  1. (i) limnμnjχ=, for jr.

  2. (ii) There exists a positive real W such that μn1ξW for all n.

  3. (iii) ξitandFt are orthogonal for all t. As a consequence ξit and the common components χjt are orthogonal for all i and j.

Conversely, if the processes xit have representation (7) with a r˜-dimensional vector F˜t, and (i), (ii), (iii) hold, then Assumption 1 is fulfilled and r=r˜.

This representation theorem was proved in Chamberlain and Rothschild (1983). It is a basic result in the literature on high-dimensional dynamic factor models taking the time-domain approach. Both the criteria to determine the number of factors and the construction of estimators for the common components and factors rely directly or indirectly on Theorem 1 (see “Estimation”).

Condition (ii) is the definition of an idiosyncratic component in this literature. Note that it is an asymptotic definition. It includes the case in which different idiosyncratic component are orthogonal (see assumption (b) earlier) with a bound for the idiosyncratic variances σξi2. However, it also includes nondiagonal variance-covariance matrices for the idiosyncratic components. For example, local or sectoral covariances for the idiosyncratic components produce a block-diagonal shape for Γn0ξ. Imposing a bound for the maximum eigenvalues of the blocks, condition (ii) is fulfilled.

Theorem 1 can be obviously adapted to r=0, with (7) holding with χit=0, so that the x’s themselves are idiosyncratic.

Observation 3

Condition (i) in Theorem 1 is crucial. It ensures that no representation with a smaller number of factors is possible. For example, suppose that χit=λi1F1t+λi2F2t, and that λi2=αλi1 for all i. Then, of course, setting F˜t=F1t+αF2t produces χit=λi1(F1t+αF2t)=λi1F˜t, which is a representation with one factor. It is easy to see that in this case only μn1χ, but also that only μn1χ, that is, that r is equal to one, not two.

Observation 4

It is important to point out that if the infinite-dimensional vector (2) has a factor structure (i.e., if Assumption 1 holds), then r, the number of factors, is identified as the number of eigenvalues of Γn0x tending to infinity. Moreover, the components χit and ξit are identified as well; see Chamberlain (1983), Chamberlain and Rothschild (1983), and Forni and Lippi (2001). In particular, the space spanned by the common components χit and that spanned by the factors Ft coincide. However, the factors Ft and the corresponding loadings are identified only up to a linear transformation. For, if Ft is a vector of factors and xit=λiFt, where λi=(λi·λir), then

xit=(λiH1)(HFt)=λ*Ft*,

where H is any non-singular r×r matrix. As a consequence, if it is convenient, it can be assumed that the factors are orthonormal, that is, mutually orthogonal and unit-variance.

Dynamics

The factor model (7) can accommodate several interesting dynamic structures. For example, suppose that all the x’s are driven by a common cyclical macroeconomic variable Ct in the following way:

xit=αiCt+βiCt1+ξit,
(8)

and assume that

(1aLbL2)Ct=ut,

where L is the lag operator and ut is a scalar white noise. Here each variable xit loads Ct and Ct1 with individual weights, so that, for example, some of the variables have βi=0 and are therefore coincident with the cycle Ct and others have αi=0 and are therefore lagging. Setting F1t=Ct and F2t=Ct1,

xit=aiF1t+βiF2t+ξit,

which is a representation of the form (7).

More in general, it can be assumed that the common macroeconomic variable is a vector

ft=M(L)ut,

where ut is a q-dimensional white-noise vector and M(L) is a square-summable q×q matrix (infinite) polynomial, and that

xit=αift+βift1+ξit,

where αi and βi are q×1. Here it is the macroeconomic vector ft that is loaded dynamically by the variables xit. Setting Ft=(ftft1), representation (7) is

xit=(αiβi)Ft+ξit,

with r=2q.

In both cases, a scalar or a vector cyclical macroeconomic variable, the vector Ft is singular in that it has dimension 2q but is driven by a q-dimensional white-noise vector. In the first case:

Ft=(CtCt1)=((1aLbL2)1(1aLbL2)1L)ut.

In the second:

A general representation of the factor model, in which the dynamic structure is explicit, is the following:

xit=χit+ξitχit=λiFtFt=D(L)ut,
(9)

where Ft is r dimensional, ut is q-dimensional white noise, and D(L) is an r×q square summable matrix polynomial. Note that representation (9) completes representation (3) with an equation linking Ft to ut. In view of (9) the terms static factors for Ft and dynamic or primitive factors for ut are often used.

The argument employed in Observation 4 to show that Ft is not identified applies to ut as well. Thus, with no loss of generality, it is assumed throughout that ut is an orthonormal white-noise vector.

As shown in these examples, the static factors Ft and the static representation xit=λiFt+ξit usually result from a transformation of the primitive dynamic form of the model. As seen in the section “Estimation,” the static representation is a very convenient tool to construct estimators and forecasts for the variables xit (see sections on “Estimation” and “Forecasting”). However, when the model is employed for structural analysis the dynamic factors ut and the dynamic relationship between ut and the variables xit play a prominent role (see “The Factor Model and Structural Macroeconomic Analysis”).

The examples also show that typically r>q; that is, the vector Ft is singular. This property has very important consequences, as shown in the example and in Theorem 2.

Example 2

Consider again model (8), simplified by assuming Ct=ut, so that F1t=ut, F2t=ut1, and

(F1tF2t)=(1L)ut.

This is a singular MA(1) representation. The unusual fact is that the MA matrix (1 L) has a square, stable left inverse of degree 1, namely

(10L1)(1L)=(10),

so that

(10L1)(F1tF2t)=(10)ut,

which is a AR(1) representation.

This unexpected result, that a singular moving average of order one has a finite-degree autoregressive representation, is just an example of the following general result (see Anderson & Deistler, 2008).

Theorem 2

Assume that the r-dimensional vector Ft has the representation

Ft=D(L)ut,

where ut is a q-dimensional white noise and q<r, so that the vector Ft is singular. Moreover, assume that the entries of D(L) are rational functions of L:dij(L)=hij(L)/kij(L), where hij and kij,i=1,,r,j=1,,q, are polynomials Then, for generic values of the coefficients of hij and kij, that is, for all values of the coefficients with the exception of a lower-dimensional subset, Ft has an autoregressive representation

G(L)Ft=D(0)ut,

where (i) G(L) is an r×r, stable, finite-degree polynomial matrix, (ii) the rank of D(0) is q. (For a more general definition of genericity, see Anderson & Deistler, 2008.)

Using Theorem 2, under the assumption that the entries of D(L) are rational functions, representation (9) can generically be rewritten as

xit=χit+ξitχit=λiFtG(L)Ft=Rut
(10)

where R is r×q and G(L) is r×r, stable and finite-degree. Therefore, if an estimate of Ft is available, the shocks ut can be estimated by means of a singular VAR for Ft.

The matrix R has generically full rank q. As a consequence, generically the vector ut belongs to the space spanned by Fts,s0; that is, by definition, ut is generically fundamental for Ft.

Example 3

The following elementary example, in which r=2 and q=1, can illustrate this result:

F1t=ut+2ut1=(1+2L)utF2t=ut+3ut1=(1+3L)ut.
(11)

As the polynomials 1+2L and 1+3L are not invertible, ut is not fundamental for F1t or for F2t. However, multiplying the first equation by 3, the second by 2, and subtracting: ut=3F1t2F2t, so that ut belongs to space spanned by current and past values of the vector Ft, that is, ut is fundamental for the vector Ft.

This section on the dynamics of the factor model concludes by observing that no special assumption has been imposed on the autocovariance of the idiosyncratic components ξit.

Estimation

The section “General Assumptions and Representation Theorem” discussed estimation of the common components by means of the elementary model (4). In that case the arithmetic average of the observable variables xit converges in mean square to λFt as n. Slightly complicating the model by assuming that xit=λiFt+ξit, thus allowing the loadings to be variable specific, the average becomes

1nxit=λ¯nFt+1ni=1nξit,

where λ¯n=(1/n)i=1nλi. The idiosyncratic average converges to zero in mean square, but an additional assumption is needed to ensure that λ¯n is bounded away from zero as n.

Further complications arise if r=2:

xit=λi1F1t+λi2F2t+ξit.

To recover the two factors two distinct averages are needed. For example,

x¯nt1=1ni=1nxitandx¯nt2=1ni=1n(1)nxit,

but an assumption is still needed ensuring that the variance-covariance matrix of x¯nt1 and x¯nt2 remains bounded away from singularity (its determinant is bounded away from zero) as n.

Principal Components

In general, in a model with r factors, r1, r averages are needed:

xntj=i=1naij,nxit,

j=1,...,r, such that:

  1. (i) The variance-covariance matrix of the vector ynt=(xnt1xntr) is bounded away from singularity as n.

  2. (ii) limni=1naij,nξit=0 in mean square.

A very convenient solution of this problem consists in setting xntj=(μnjx)1Pnj,tx,j=1,,r, where Pnj,tx is the j-th principal component of the vector xnt. Precisely,

Pnj,tx=pnjxxnt=pnj,1xx1t++pnj,nXntx,

where pnjx is a row eigenvector of Γn0x corresponding to the j-th eigenvalue, such that i=1n(pnj,ix)2=1

Observation 5

In Lippi (2018) both eigenvectors of the spectral density of the vector xnt, are considered, that is, dynamic eigenvectors and eigenvectors of the variance-covariance matrix of xnt (i.e., static eigenvectors). In that article, to avoid confusion the static eigenvectors are denoted as pnjS,x. As no confusion can arise here, pnjx. is used.□

Regarding (i), as is well known, the variance-covariance matrix of the principal-component vector (Pn1,txPnr,tx) is the r×r diagonal matrix whose entry (k,k) is μnkx. As a consequence, the variance-covariance matrix of

ynt=((μn1x)1Pn1,tx(μnrx)1Pnr,tx)=((μn1x)1pn1xxnt(μnrx)1pnrxxnt)

is the identity matrix Ir and is therefore bounded away from singularity. Using μnjx as n for j=1,...,r, and μn1ξW for all n, it is easy to show (ii), that is, that

limn(μnjx)1pnjxξnj=0,

for j=1,,r.

In conclusion, the first r normalized principal components are an orthonormal basis “converging” in mean square to the space spanned by the factors Fjt,j=1,...,r. (the average of the idiosyncratic components by means of each of the normalized eigenvectors tends to zero in mean square). This is the rationale for expressions like “the normalized principal components converge to the factors.” Though suggestive, such statements should be taken with extreme care. Indeed, as seen in Observation 4, any basis in the factor space provides a vector of static factors.

Estimating the Factors and the Loadings

Principal components, as presented earlier, are unfeasible as estimators of the factors for two reasons. First, an estimate of Γn0x, call it Γ^n0x is observed, which depends on the observed (n,T) sample of the processes xit. Second, the number r of factors is not known.

The number of factors is preliminary to estimation of the principal components. The literature on the determination of r starts with Bai and Ng (2002), who use an information criterion with the variance explained by the factors on one hand, a penalty for adding factors on the other. They show consistency of the criterion as n and T tend to infinity. Improvement and alternatives to Bai and Ng’s criterion are Onatski (2010), Alessi, Barigozzi, and Capasso (2010), and Ahn and Horenstein (2013). A test for the number of factors is proposed in Onatski (2009).

Regarding the number of dynamic factors q, consistent criteria are given in Hallin and Liška (2007), Amengual and Watson (2007), and Bai and Ng (2007).

Once r has been determined, using the estimated principal components of the x’s (by means of the eigenvectors of Γ^n0x), estimates of the factors and the loadings are obtained. Standard consistency, that is, convergence in probability, as n and T tend to infinity, is proven in Stock and Watson (2002a, 2002b) and Bai and Ng (2002). In particular, Bai and Ng obtain the rate of convergence max(T1,n1) for the squared distance between the estimated factors and the factor space.

Once the factors have been estimated and q has been determined, estimation of the last equation in (10) provides an estimate of the dynamic shocks ut and G(L), thus completing the estimation of the high-dimensional dynamic factor model.

Doz, Giannone, and Reichlin (2011) construct a consistent Gaussian quasi-maximum likelihood estimator as an alternative to principal components. Given n, they estimate a factor model by maximum likelihood under the assumption that the idiosyncratic components are mutually orthogonal. They show that the bias resulting by “forcing” this orthogonality into the finite-n model becomes negligible as n tends to infinity.

Forecasting

Suppose that one wants to forecast the variable x1t. If the factors and the loadings were known then separately forecasting the factors Ft and the idiosyncratic component ξ1t, using past values of Ft and ξ1t respectively, produces a forecast of x1t which outperforms forecasting x1t by its past values. Equivalently, the orthogonal projection of x1,t+h,h>0, on Fts,ξ1,ts,s0, outperforms the projection on x1,ts, or, lastly, the projection

Proj(x1t|Fts,x1,ts,s0)
(12)

outperforms the projection on x1,ts,s0. Again, projection (12) is unfeasible and is replaced in empirical situations by

x^1,t+h=α^(L)F^t+β^(L)x1t,

where α^(L) is a finite-degree 1×r matrix polynomial and β^(L) is a finite-degree polynomial. For this specification, see Stock and Watson (2002a).

The literature on forecasting with factor models is vast and cannot be reviewed here. Many papers have produced forecast comparisons including univariate models and different specifications of the dynamic factor model. Different datasets have been used for the United States and Europe in particular. A recent paper reviewing this literature is Stock and Watson (2016).

Regarding the method described in this article versus its “competitor,” presented in Lippi (2018), see Forni, Hallin, Lippi, and Zaffaroni (2017), containing results based on simulated data, and Forni, Giovannelli, Lippi, and Soccorsi (2018), based on the U.S. monthly macroeconomic data set known as the Stock and Watson data set. Both comparisons are illustrated in detail in Lippi (2018); see the section “The Static and Dynamic Methods Compared: Results Based on Simulated and Empirical Data.”

The Factor Model and Structural Macroeconomic Analysis

The high-dimensional dynamic factor models have been mainly developed with the purpose of analyzing and forecasting macroeconomic data sets. In this context, the idiosyncratic components are interpreted as causes of variation of the x’s that are specific to one or just a few variables, like regional or sectoral shocks, plus measurement errors. In particular, for the big aggregates like income, consumption, and investment, in which all local or sectoral shocks have been averaged out, the variable ξit can be interpreted as only containing measurement error. On the other hand, the common factors Ft and ut, being pervasive, that is, affecting all the variables xit, can be interpreted as macroeconomic causes of variation.

This interpretation of common and idiosyncratic components of macroeconomic variables has been used to argue that dynamic factor models offer an alternative with important advantages to structural VAR analysis. For heuristic purposes a very simple model is considered here, assuming that the variables χ1t and χ2t are the rate of growth of the GDP and aggregate consumption, respectively, and are driven by a demand shock v1t and a supply shock v2t:

(χ1tχ2t)=B2(L)(v1tv2t),
(13)

where B2(L) is a 2×2 matrix of rational functions of L. Moreover, we assume that the supply shock has no contemporaneous effect on Consumption, so that b2,22(0)=0, that is, the (2, 2) entry of B2(L) vanishes for L=0.

The variables χit, i=1,2, are observed with a measurement error: xit=χit+ξit,i=1,2. Moreover, it is supposed that x1t and x2t are the first two variables of a large data set with n variables, generated by a dynamic factor model with q=2, r>q and that all the common components are driven by vt. This means xnt=χnt+ξnt,

χnt=Bn(L)vt,
(14)

where Bn(L) is an n×2 matrix of rational functions in L whose upper 2×2 matrix is B2(L). The variables xit can also be represented as in (10). Using the second and third equation,

χnt=ΛFt=Λ[G(L)]1Rut=Cn(L)ut,
(15)

where Λ=(λ1λn).

The shocks v1t and v2t in representation (14) have an economic interpretation as demand and supply shocks, while the entries of Bn(L) are the corresponding impulse-response functions. On the other hand, representation (15) is “estimated” starting with the determination of r and q. However, Bn(L) and Cn(L) are singular, so that by Theorem 2, one can safely assume that both are fundamental. This implies that there exists a 2×2 orthogonal matrix K such that ut=Kvt (see, e.g., Forni, Giannone, Lippi, & Reichlin, 2009, Proposition 2), so that

χnt=Bn(L)vt=Cn(L)ut=[CnK]vt.

Because vt is orthonormal white noise, Bn(L)=Cn(L)K, so that the (2, 2) entry of Cn(L)K vanishes. Equivalently, determining K such that the (2, 2) entry of Cn(L)K vanishes, the structural impulse response functions Bn(L), B2(L) in particular are obtained.

In conclusion, estimating the common components of the x’s and imposing the identifying restriction on the (2, 2) entry results in an estimate of the structural model (13).

On the other hand, SVAR analysis starts with the estimation of an autoregression

A2(L)(x1tx2t)=(ϵ1tϵ2t).

Then the residual vector ϵt=(ϵ1tϵ2t) is transformed into a vector wt such that ϵt=Uwt, where U is a non-singular matrix, such that

  1. (i) wt is orthonormal,

  2. (ii) the impulse response functions A2(L)1U=E2(L), such that

(x1tx2t)=E2(L)(w1tw2t),
(16)

fulfill restrictions suggested by economic logic. In this case, assuming knowledge of the underlying model (13), the restriction e22(0)=0 will be imposed. As is well known, this restriction just-identifies the matrix U and therefore the matrix E2(L).

Next is a comparison of the results obtained with the SVAR, that is (16), with those obtained with the dynamic factor model, that is (13):

  1. 1. First, both the estimated matrix E2(L) and the shocks wt are “contaminated” by the measurement errors ξit,i=1,2.

  2. 2. Assuming that the the errors ξit,i=1,2, are small, one might be led to believe that D(L) and wt are a good and easily estimated approximation of B2(L) and vt, respectively. To see that this is not necessarily the case, observe that the matrix E2(L) is by definition invertible, so that representation (16) is fundamental. On the other hand, Theorem 2 ensures that representation (14) is fundamental but says nothing about representation (13).

  3. 3. This is easily understood using model (11), where the scalar shock ut is fundamental for the two-dimensional vector Ft but nonfundamental for F1t or F2t. Even assuming that F1t is observed without error, if one estimates an AR for F1t, a(L)F1t=u˜t, the resulting MA representation F1t=a(L)1u˜tis not an approximation of the “structural” representation F1t=(1+2L)ut.

In general, the difficulty with SVAR analysis is that the “structural” moving average xt=E(L)wt results by inverting the estimated autoregressive matrix A(L) and multiplying A(L)1 by a non-singular matrix: E(L)=A(L)1U. Thus E(L) is obviously invertible, E(L)1=U1A(L); that is, wt is by definition fundamental for xt.

However, fundamentalness, which has a motivation in Theorem 2 for singular representations like (13), has no motivation in SVAR analysis, which is based on non-singular vectors. This point was made several years ago (see Hansen & Sargent, 1980; Lippi & Reichlin, 1993, 1994) and is now a topic for a growing literature (see, e.g., Alessi, Barigozzi, & Capasso, 2011; Fernández-Villaverde, Rubio-Ramírez, Sargent, & Watson, 2007). The following simple model may help the reader to grasp the point of fundamentalness in SVAR analysis.

Example 4

Suppose that xt is the rate of change of aggregate productivity and that xt evolves according to the following equation:

xt=p1vt+p2vt1,
(17)

where ut is the shock to technology, pi>0,i=1,2 and p1+p2=1. Thus ut determines the increase of productivity p1ut at time t and the increase p2ut at time t+1. Equation (17) has the interpretation of a learning-by-doing process taking two time periods to complete. Rewriting (17) as

xt=Vt+PVt1,

where Vt=p1vt and P=p2/p1, one has a standard MA(1). If P<1 the MA(1) is invertible and estimating an autoregression for xt would produce an approximation of the structural shock Vt. On the other hand, if P>1 an autoregression would not produce an approximation of Vt. In the first case Vt is fundamental for xt, and in the second case Vt is nonfundamental. On the other hand, the econometrician observes the autocovariances of xt, so that he or she only knows that xt is an MA(1). Therefore, either additional information is available on p1 and p2, that is on P, or the model is not identified. Moreover, the usual practice of estimating autoregressions produces an approximation of the structural shock Vt only if the learning-by-doing process has p1>p2.□

In conclusion, under the assumption that the idiosyncratic components of the large macroeconomic variables are measurement errors, the dynamic factor model produces estimates of the “true” underlying variables χit, of the structural shocks and impulse-response functions. In particular, singularity of the vector χt and Theorem 2 imply that the shocks estimated from representation (10) differ from the structural shocks by a non-singular matrix.

For the comparison of SVAR analysis and structural analysis based on high-dimensional dynamic factor models, see Stock and Watson (2005) and Forni et al. (2009).

Non-Stationary Processes

As observed in the section “General Assumptions and Representation Theorem,” almost all the observed variables xit in macroeconomic datasets are non-stationary. Assume for simplicity that all the variables xit are I(1), that the factors are I(1) and the idiosyncratic components are I(0). In this case the factors Ft can be directly estimated as the principal components of the I(1) variables xit (Bai, 2004; Peña & Poncela, 2006). If the idiosyncratic components are I(1) or I(0) the principal components of the stationary series (1L)xit provide an estimate of the differenced factors (1L)Ft and of the loadings The factors Ft can then be recovered by integration (see, e.g., Bai & Ng, 2004). In any case, the common practice consisting in taking principal components of the variables (1L)xit provides consistent estimates of the factors and the loadings.

Thus, as far as estimating the factors and the loadings is concerned, the practice consisting in estimating the dynamic factor model using the differences of the variables xit is correct. A difficulty with respect to the stationary case arises with the estimation of ut and of the impulse-response functions of the variables xit with respect to ut (or structural shocks obtained by a linear transformation of ut). This requires the estimation of an autoregressive model for the I(1) factors Ft. The latter, under the assumption of singularity, are obviously cointegrated with cointegration rank at least rq. To see this, consider representation (9). As Ft is I(1) The moving average for Ft becomes

(1L)Ft=D(L)ut=D(1)ut+(1L)D˜(L)ut.

The r-dimensional vector Ft is cointegrated if the equation

cD(1)=0,

where c is an r-dimensional row vector and has nontrivial solutions. But D(1) is r×q, so that its rank is less or equal to q; that is, the cointegration rank of Ft is at least rq.

Cointegration of Ft implies that a VAR in the differences (1L)Ft is misspecified. Rather, the VAR should be estimated either in the levels or specified as an error correction mechanism. On cointegration of singular vectors and related error correction mechanisms, see Barigozzi, Lippi, and Luciani (2016a) and Banerjee, Marcellino, and Masten (2014, 2017). On estimation of the high-dimensional dynamic factor models with I(1) factors and I(1) idiosyncratic components, see Barigozzi, Lippi, and Luciani (2016b).

Conclusions and Open Problems for Future Research

Forecasting with high-dimensional dynamic factor models has been developed in a vast literature, with several variants and refinements of the standard method based on principal components. Recent papers comparing different proposals seem to show that great improvements are unlikely within the present approach, that is, by and large, principal-components methods applied to data sets containing between 100 and 300 series; see the papers cited in “Estimation.” Rather, it is possible that data sets of the big-data size may help to produce indicators leading to substantial improvements in the predictive power of dynamic factor models. For an exploration of possible use of big data in macroeconomic analysis, see Ng (2017).

Regarding the application of dynamic factor models to macroeconomic analysis, important papers are Bernanke and Boivin (2003), Bernanke, Boivin, and Eliasz (2005), and Boivin, Giannoni, and Mihov (2009). In Forni and Gambetti (2010) a data set containing 112 U.S. monthly macroeconomic series is used to study the effects of monetary policy on the common components of some key variables, that is, on the variables obtained by removing measurement errors; see “The Factor Model and Structural Macroeconomic Analysis.” Monetary policy shocks are identified using a standard recursive scheme, in which the first impact on both industrial production and prices is zero. They find that some puzzling results obtained using SVAR methods disappear when the observed variables are replaced by their common components. First, the maximal effect on the common component of bilateral real exchange rates is observed on impact, so that the ”delayed overshooting” puzzle disappears. Second, after a contractionary shock, prices fall at all horizons, so that the price puzzle does not occur. Finally, monetary policy has a sizable effect on both real and nominal variables.

Systematic application of factor-model techniques to macroeconomic analysis, as described in the sections “The Factor Model and Structural Macroeconomic Analysis” and “Non-Stationary Processes,” which has been insufficiently explored so far, is an extremely promising research field.

Further Reading

Some papers cited in this article and Lippi (2018) are selected here as basic contributions on dynamic factor models.

For the idea of latent dynamic factors in macroeconomic time series, see Sargent and Sims (1977). Chamberlain and Rothschild (1983) and Chamberlain (1983) introduce the definitions of common and idiosyncratic components for a large cross-section of stochastic variables. They also establish the relationship between principal components and factor analysis.

Forni, Hallin, Lippi, and Reichlin (2000), Forni and Lippi (2001), Stock and Watson (2002a, 2002b), and Bai and Ng (2002) extend Chamberlain and Rothschild’s (1983) contributions to a time-series context, formalizing the large-dimensional dynamic factor model.

The determination of the number of static and dynamic factors was first studied in Bai and Ng (2002) and Hallin and Liška (2007), respectively.

Regarding forecasting with dynamic factor models, see Stock and Watson (2016), Forni et al. (2017), Forni et al. (2018), and Lippi (2018).

Stock and Watson (2005) and Forni et al. (2009) argue that dynamic factor models can be used for structural analysis as an alternative to SVAR models.

Lastly, regarding the fundamentalness problem in empirical macroeconomics, see Alessi et al. (2011). For fundamentalness and singular vectors, see Anderson and Deistler (2008).

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