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To my best knowledge, in Ross's APT, it is assumed that the pricing model is the exact factor model.

Chamberlain (1983 ECTA) expanded it into the approximate factor model.

In the exact factor model, the covariance matrix of the error term is diagonal matrix with each variance on its diagonals, while in approximate factor model, the errors of the cross section can be "weakly correlated", which makes the covariance matrix non-diagonal matrix.

My questions are,

(1) Can we define "weakly correlated" in a rigorous statistical definition?

This paper(https://arxiv.org/pdf/1612.04990.pdf) defines it as "If we have an approximate factor structure, we expect a vanishing largest eigenvalue (of the covariance matrix)"

(2)Could you explain why the discussion of the strength of the correlation of the cross-section (off-diagonal of the covariance matrix) can be delivered with eigenvalues?

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  • $\begingroup$ (1) See Bai and Ng (2002, Econometrica). (2) In the strongly correlated case, the max eigenvalue is large and the min eigenvalue is close to zero. Find two eigenvalues of X'X for an nx2 matrix X, in two cases: One, the columns of X are independently generated from N(0,1); Two, the second column equals the first column. You will see it. $\endgroup$ – chan1142 Nov 16 '17 at 3:26

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