Sentana-Fiorentini (2001) Identification, Estimation and Testing of Conditionally Heteroskedastic Factor Models

Summary

This paper studies conditionally heteroskedastic factor models — models in which each observed series is a linear combination of a few dynamic (GARCH-type) common factors plus an idiosyncratic term — and asks how the time-varying volatility of the factors changes the standard inference of static factor analysis. The headline result is that dynamic heteroskedasticity alleviates the identification problem: the rotational indeterminacy that plagues static factor models is (partly) resolved once one accounts for the fact that the factors have distinct time-varying variances. The authors show that traditional unconditional ML estimation of the static (unconditional) variance parameters remains consistent when the loadings are identified from the unconditional distribution — but the standard errors must be robustified for the neglected conditional heteroskedasticity — develop a simple LM test for ARCH effects in the common factors, discuss two-step estimation of the conditional-variance parameters, and run a Monte Carlo study. The results bear on dynamic APT models, simultaneous equations, and structural VARs. (Journal of Econometrics 102(2): 143–164; a heavily revised version of Sentana 1992.)

Key Claims

• The model. Multivariate volatility is captured by a small number of dynamic heteroskedastic common factors (each with its own ARCH/GARCH conditional variance) plus idiosyncratic noise: $y_t = C f_t + u_t$, where the $f_t$ (and possibly $u_t$) are conditionally heteroskedastic. Such multi-index models have a long tradition in finance (APT).
• Conditional heteroskedasticity aids identification. In a static factor model the loadings $C$ are identified only up to an orthogonal rotation. When the common factors have different, time-varying conditional variances, that dynamic variation supplies additional identifying restrictions, so the loadings become (more) identified — the rotational indeterminacy is alleviated. This has direct implications for identifying dynamic APT factors, simultaneous-equation systems, structural VARs, and oblique factor models with constant conditional covariances.
• Unconditional ML stays consistent, but needs robust SEs. The traditional ML estimator of the static (unconditional) variance parameters — derived as if observations were serially independent — remains consistent provided the loadings are identified from the unconditional distribution; however, because it ignores the conditional heteroskedasticity, its standard errors must be robustified (a sandwich/quasi-ML correction).
• A diagnostic LM test for factor ARCH. They derive a simple preliminary LM test for the presence of ARCH effects in the common factors, computable from the residuals of the static factor fit — a cheap check for whether the conditional-heteroskedasticity machinery is needed.
• Two-step estimation. Given the (consistently estimated, robustified) static parameters, the remaining conditional-variance parameters can be estimated in a second step, avoiding full joint ML while retaining consistency.
• Finite-sample evidence. A detailed Monte Carlo exercise assesses the identification gains, the consistency/robust-SE claims, the LM test, and the two-step estimator in samples of realistic size.

Concepts Introduced or Extended

• Factor Model
• GARCH
• Structural Identification

Entities Mentioned

• Enrique Sentana
• Gabriele Fiorentini

Quotes

"We show that identification problems are alleviated when variation in factor variances is accounted for. Our results apply to dynamic APT models and other structural models."

"Traditional ML estimation of unconditional variance parameters remains consistent if the factor loadings are identified from the unconditional distribution, but their standard errors must be robustified."

My Take

The elegant and durable idea here is that time-varying second moments are an identifying resource, not just a nuisance: the very GARCH dynamics that complicate estimation are what pin down the factor rotation that static factor analysis leaves free. That insight — heteroskedasticity as identification — later became central well beyond factor models (it is the logic behind identification-through-heteroskedasticity in structural VARs, à la Rigobon). For this wiki the paper is the rigorous bridge between the factor-model and GARCH literatures and a caution for applied work: you can often get away with static ML for the loadings, but only with robustified standard errors, and you should test for factor ARCH first. Its practical cost is the usual one for latent-factor GARCH — the full likelihood is heavy, which is exactly why the consistency-of-static-ML and two-step results matter.