<blockquote>
  <p>It is harder to interpret coefficients principal components in terms of original regressors.</p>
</blockquote>

<p>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 <em>not related to wealth at all</em>. Put differently, you could <a href="https://en.m.wikipedia.org/wiki/Dimensionality_reduction" rel="nofollow noreferrer">reduce the dimensionality</a> of your problem to this unique first component, and maintain your interpretation.</p>

<p>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 <em>relative</em> weights of your components, e.g. you need to check the <a href="https://en.m.wikipedia.org/wiki/Eigendecomposition_of_a_matrix" rel="nofollow noreferrer">eigen decomposition</a> of your correlation matrix.</p>

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

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

CC BY-SA 4.0
2019-08-26T21:37:56.710Z
Any question @Elina ?
keepAlive
CC BY-SA 4.0
2019-08-27T18:11:39.617Z
Thank you for great explanation. Source is useful too
Elina Gilbert
CC BY-SA 4.0
2019-08-26T21:25:37.467Z
2019-08-26T21:35:49.640Z
2019-08-26T21:35:49.640Z
keepAlive
<p>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.</p>

<p>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?</p>

<p>Is there any alternative
 formulation of doing this exercise that may be preferred?  </p>

CC BY-SA 4.0
2019-08-26T13:31:11.810Z
Why would you want to use PCA on the 14 assets instead of adding up their values?
BKay
CC BY-SA 4.0
2019-08-26T13:56:25.920Z
Because they are highly collinear.
Elina Gilbert
<econometrics><regression>
Principal component analysis interpretation
Elina Gilbert
# Principal component analysis interpretation

(Asked by: username_0 on 2019-08-25)

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?

**Tags:** econometrics, regression
> **Comments:**
> **username_1**: Why would you want to use PCA on the 14 assets instead of adding up their values?  
> **username_0**: Because they are highly collinear.  

---

## 1 answers

### Answer by username_2 on 2019-08-26. Score: 2 | Accepted answer

> 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](https://en.m.wikipedia.org/wiki/Dimensionality_reduction) 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](https://en.m.wikipedia.org/wiki/Eigendecomposition_of_a_matrix) 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.

> **Comments:**
> **username_2**: Any question @Elina ?  
> **username_0**: Thank you for great explanation. Source is useful too  

---

