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Suppose a wealth index is computed using information on a set of 14 assets that a household possesses. The index is generated using principal components, as the 14 individual asset variables are highly collinear. A OLS regression of education expenditures (in Rupees per household) on the wealth index yields a coefficient of 0.4.

It is harder to interpret coefficients principal components in terms of original regressors. But can I interpret wealth index as its own ie can I say - increase in wealth index by a score of 1 will increase education expenditure by 0.4?

Is there any alternative formulation of doing this exercise that may be preferred?

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  • $\begingroup$ Why would you want to use PCA on the 14 assets instead of adding up their values? $\endgroup$ – BKay Aug 26 at 13:31
  • $\begingroup$ Because they are highly collinear. $\endgroup$ – Elina Gilbert Aug 26 at 13:56
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It is harder to interpret coefficients principal components in terms of original regressors.

Indeed, since you are now dealing with components, you can't. That being said, if your original regressors are highly colinear and all capturing wealth-related phenomena, a possibility is that your first component is still somehow strongly related to wealth, and other components only controling for phenomena that are not related to wealth at all. Put differently, you could reduce the dimensionality of your problem to this unique first component, and maintain your interpretation.

But do not take what i am saying for granted: if you really want to know how abusive maintaining your economic interpretations would be, it's all a matter of visualizing/diagnozing the relative weights of your components, e.g. you need to check the eigen decomposition of your correlation matrix.

I recommend you to read Michael Friendly and Ernest Kwan (2009) "Where's Waldo? Visualizing Collinearity Diagnostics" The American Statistician, Vol. 63, No. 1 (Feb., 2009), pp. 56-65.

Also, Yoel Haitovsky (1966) "A Note on Regression on Principal Components" is very clear and concise.

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  • $\begingroup$ Any question @Elina ? $\endgroup$ – keepAlive Aug 26 at 21:37
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    $\begingroup$ Thank you for great explanation. Source is useful too $\endgroup$ – Elina Gilbert Aug 27 at 18:11

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