Exact and Approximate Factor Models: Residual Correlation

Summary

The document distinguishes exact and approximate factor models by the structure allowed for residual covariance. In an exact factor model, the residual covariance matrix is diagonal, so residual movements are uncorrelated across assets. In an approximate factor model, that matrix need not be diagonal, although its norm is assumed to remain bounded.

This distinction affects how much shared movement the model assigns to its observed factors. Exact models treat those factors as accounting for common influences, leaving independent asset-specific variation. Approximate models allow residual correlations, which can capture shared movement due to omitted or unknown influences. The explanation is conceptual and does not compare estimation procedures or empirical performance; the appropriate model depends on the assumptions and purpose of a given analysis.

Key ideas

• An exact factor model assumes a diagonal residual covariance matrix.
• An approximate factor model permits residual correlations subject to a boundedness condition.
• Exact models attribute common asset movements to the included factors.
• Approximate models can retain shared residual movements associated with omitted or unknown influences.

Tags

# Answer by nbbo2 (score 0)


# Can someone briefly explain me what's the difference between Exact Factor Model (EFM) and Approximate Factor Model (AFM)?












I'm reading De Nard et al. paper "Portfolio Selection in Large Dimensions" and the authors talk about these two models, but I don't have any background on these topics from university. Can you help?

## Answer by nbbo2 (score 0)

https://quant.stackexchange.com/a/73431

On Page 4 "An exact factor model assumes in addition that Σu is a diagonal matrix. In contrast, an approximate factor model only assumes that Σu is a matrix with bounded L 1 or L 2 norm" for example, see Connor and Korajczyk (1993), Bai and Ng (2002), Fan et al. (2008), and the references therein." So the difference is: are the residual or idiosyncractic terms orthogonal or not.

In an exact factor model all the common influences on the stocks are accounted for by the known factors; what is left are random uncorrelated movements of each stock. In an approximate factor model the known factors take care of some of the common movements, but some stocks still tend to move together for other reasons, possibly because of omitted or unknown factors.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.