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Why in the Arbitrage Pricing Theory (APT), one of the assumptions is that the factors has to be orthonogal? what if not?

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    $\begingroup$ Do you have a citation for this? Campbell, Lo, and Mackinlay (on p. 220) doesn't have an orthogonality restriction on f in their introduction to APT. Nor, when they talk about portfolios of risk factors (on p. 223) do they require the portfolios to be orthogonal. $\endgroup$ – BKay Jan 26 '15 at 15:37
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If you want to describe excess returns in terms of exposure to common risk factors, you want the risk factors to be orthogonal. However, if you have $k$ factors with no perfect collinearity, you can always orthogonalize them and use those. You then call the orthogonalized factors the risk factors.

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  • $\begingroup$ what would happen if i use factors with no perfect collinearity? $\endgroup$ – jtomasrl Jan 26 '15 at 13:54
  • $\begingroup$ That was addressed here. "However, if you have k factors with no perfect collinearity, you can always orthogonalize them and use those." Did you mean something else? $\endgroup$ – jmbejara Jan 26 '15 at 15:37
  • $\begingroup$ I know i can always orthogonalize them, but what would happend if i dont orthogonalize them. If i use them just as they are. $\endgroup$ – jtomasrl Jan 26 '15 at 16:18
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    $\begingroup$ Then the linear equation (excess returns as a linear function of exposure to risk factors) doesn't really make sense the way you want it to. You want the coefficients in the factor model to represent the impact of marginally more exposure to one of the risk factors controlling for all the others. If they aren't orthogonal, then the coefficients don't have this interpretation. $\endgroup$ – jmbejara Jan 26 '15 at 20:56

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