3.2 Models and Methodology
3.2.3 Notation and Preliminaries
De…ne Ft0; 0i and r denote the true common factors, the factor loadings, and the true number of factors, respectively, with the assumption that
r does not depend on N and T . So, at a given date t;
Chamberlain and Rothschild (1983) show that as the r largest eigen-values of T 1PT
t=1XtXt0 will go to in…nity as N and T ! 1 , while the (r + 1)th eigenvalue remains bounded. Also, the information in the data about the common factors will be of order N , while the informa-tion about idiosyncratic factors will remain …nite.
The main objective of Bai and Ng (2002) is to determine the true number of factors, r. In classical factor analysis (e.g., Anderson (1984)), N is assumed …xed, the factors are independent of the errors et, and the covariance of et is diagonal. Normalizing the covariance matrix of Ft to be an identity matrix, then = 0 00 + , where and are the covariance matrices of Xt and et, respectively. Under these assumptions, a root-T consistent and asymptotically normal estimator of , as the sample covariance matrix b= (1=T ) PT
t=1(X X)(Xt
X)0can be obtained.
The essentials of classical factor analysis carry over to the case of large N but …xed T since the N N problem can be turned into a T T problem, as noted by Connor and Korajczyk (1993) and others.
Bai and Ng (2002) develop asymptotic results for consistent estimation of the number of factors when N and T ! 1:
Let tr(A) denote the trace of A. The norm of the matrix A is then
kAk = tr(A0A) 1=2. The following assumptions are made in Bai and Ng (2002):
Assumptions 1:
1. Factors: E kFt0k < 1; and T 1PT
t=1Ft0Ft00 p! F; as T ! 1, for some r r positive de…nite matrix F:
2. Factor Loadings: k ik < 1; and o0 o=N D ! 0; as N ! 1; for some r r positive de…nite matrix D:
3. Weak Dependence between Factors and Idiosyncratic Errors:
E N1
4. Time and Cross-Section Dependence and Heteroskedasticity: There exists a positive constant M < 1, such that for all N and T . i ) E(eit) = 0; Ejeitj8 M ;
Assumption 1 is standard for factor models. Assumption 2 ensures that each factor has a nontrivial contribution to the variance of Xt. Bai and Ng (2002) only consider nonrandom factor loadings for simplicity.
Their results still hold when the i are random, provided they are in-dependent of the factors and idiosyncratic errors, and E k ik4 M:
When the factors and idiosyncratic errors are independent, which is a standard assumption for conventional factor models, Assumption 3
is implied by Assumptions 1 and 4. Assumption 4 allows for limited time-series and cross-section dependence in the idiosyncratic compo-nent. Heteroskedasticity in both the time and cross-section dimensions is also allowed.
At this point, it is crucial to point out the three main classi…cations of factor models in the literature.
Strict Factor Model
In traditional factor analysis ((Sargent and Sims (1977) and (Geweke (1977)), it is assumed that there is no cross-correlation among the idio-syncratic components at any lead and lag. This assumption allows for identi…cation of common and idiosyncratic components but represents a strong restriction. Chamberlain and Rothschild (1983) and Cham-berlain (1983) propose an approximate static factor model in which the factor term is of the form ft = Axt, where xtis an r 1factor process.
Since no lagged values of xtis involved explicitly, xtis coined as a static factor.
Hence, the allowance for some correlation in the idiosyncratic com-ponents sets up the model to have an approximate factor structure. It is more general than a strict factor model, which assumes eitis uncorre-lated across i, the framework in which the APT theory of Ross (1976) is based.
Approximate Factor Model
When idiosyncratic noise is allowed to be mildly cross-correlated, then 3:2 is regarded as an approximate factor model. In other words, an ap-proximate factor model exists where serial dependence and heteroskedas-ticity of et, and for weak dependence between factors and idiosyncratic
series are allowed. Bai and Ng (2002) consider an approximate factor model while allowing weak-form serial (and cross-sectional) dependence in the idiosyncratic component as long as cross section and time series dimension; N and T are large. This is because dependence due to the factor structure asymptotically dominates any weak dependence in the idiosyncratic component, and hence well designed criteria (Bai and Ng (2002)) can eventually detect strong dependence due to the factor structure as both N and T grow.
Dynamic Factor Model
The dynamic-factor model was proposed by Sargent and Sims (1977) and Geweke (1977). It assumes that in the decomposition 3:1 each component of ft is a sum of r uncorrelated moving average processes driven, respectively, by r common factors. Furthermore it requires that ft and "t are uncorrelated with each other, and all the idiosyncratic components (i.e. the components of "t) are also uncorrelated.
Forni, Hallin, Lippi and Reichlin (2005) call their model as general-ized dynamic factor model. Their model encompasses as a special case the ‘approximate factor model’of Chamberlain (1983) and Chamber-lain and Rothschild (1983), which allows for correlated idiosyncratic components, but is static. Also, it generalizes the factor model of Sar-gent and Sims (1977) and Geweke (1977), which is dynamic, but has orthogonal idiosyncratic components. This approach deals with large panels of time series, i.e. when the number of variables becomes large compared to the number of observations. Each time series is repre-sented as the sum of two components: the common component and the idiosyncratic component.
Since the idiosyncratic components are correlated, the model cannot be estimated on the basis of traditional methods. The authors propose
a method, yielding consistent estimates of the components as both the cross-section and the time dimensions go to in…nity at some rate. The common components are computed as the projections of the observa-tions onto the leads and lags of the dynamic principal components of the observations and the idiosyncratic components are derived as the orthogonal residuals. The method is applied to a panel including sev-eral macroeconomic indicators for each of the EURO countries, in order to obtain an index describing the state of the economy in the EURO area. The European coincident indicator is de…ned as the common component of the European GDP.
The generalized dynamic factor model exploits the dynamic covari-ance structure of the data, i.e. the relation between di¤erent variables at di¤erent points in time. This makes an important di¤erence to the forecasting model proposed by Stock and Watson (2002). Their forecast is based on a projection onto the space spanned by the static princi-pal components of the data. Thus, being based on contemporaneous covariances only, their approach fails to exploit the dynamic relations between the variables of the panel. Forni, Hallin, Lippi and Reichlin (2005) work out the theoretical advantage of the dynamic approach compared to the static one.