Minimum distance factor analysis of time series: Theory and application

The standard approach to factor analysis (FA) of time series assumes a parametric probability model for the dynamics of the factors and obtains the maximum likelihood (ML) estimator of the parameters, setting up a state-space model and applying the Kalman filter (KF) to evaluate the likelihood function. An estimate of the factor scores is obtained as a by-product. This parametric approach has two problems. First, if the model is misspecified, then the misspecified ML estimator, or the quasi-ML (QML) estimator, may not be consistent. Second, ML estimation of the state-space model may be difficult when the number of variables is large. An alternative approach is to apply a minimum distance (MD) estimator to the autocovariance structure of the observable variables, i.e., minimum distance factor analysis (MD-FA). For stationary sequences, an MD estimator is consistent and uniformly asymptotically normal (CUAN) under mild conditions on dependence and moments. A consistent estimator of the asymptotic variance-covariance matrix of the MD estimator is available. Given an estimate of the model parameters, the generalized least squares (GLS) estimator of the factor scores is applicable. With stronger conditions, the results extend to nonstationary sequences. When the dynamics of the factors is misspecified, the QML estimator may not be consistent. Such misspecification does not matter to MD estimators. For a particular type of misspecification that may occur in time series analysis of business cycles, however, Monte Carlo experiments show that the QML estimator is consistent and more efficient than MD estimators. MD estimators are still useful when the QML estimator is difficult to implement. An important application of FA is to construct an index from several variables. The traditional composite index (CI) of business cycles is essentially the simple average of the standardized growth rate series of the business cycle indicators (BCI). MD-FA of the BCIs gives a more efficient CI, with larger weights on more informative BCIs. The weights on the BCIs for the new CI are similar to those for the Stock-Watson index (SWI). Hence the new CI is a simpler alternative to the SWI.